Modeling of the crude oil price

January 11, 2023 by Youssef LOURAOUI

Modeling of the crude oil price

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) models the market price of the crude oil.

This article is structured as follows: we introduce the crude oil market. Then, we present the mathematical foundations of Geometric Brownian Motion (GBM) model. We use this model to simulate the price of crude oil.

The crude oil market

The crude oil market represents the physical (cash or spot) and paper (futures) market where buyers and sellers acquire oil.

Nowadays, the global economy is heavily reliant on fossil fuels such as crude oil, and the desire for these resources frequently causes political upheaval due to the fact that a few nations possess the greatest reservoirs. The price and profitability of crude oil are significantly impacted by supply and demand, like in any sector. The top oil producers in the world are the United States, Saudi Arabia, and Russia. With a production rate of 18.87 million barrels per day, the United States leads the list. Saudi Arabia, which will produce 10.84 million barrels per day in 2022 and own 17% of the world’s proved petroleum reserves, will come in second. Over 85% of its export revenue and 50% of its GDP are derived from the oil and gas industry. In 2022, Russia produced 10.77 million barrels every day. West Siberia and the Urals-Volga area contain the majority of the nation’s reserves. 10% of the oil produced worldwide comes from Russia.

Throughout the late nineteenth and early twentieth centuries, the United States was one of the world’s largest oil producers, and U.S. corporations developed the technology to convert oil into usable goods such as gasoline. U.S. oil output declined significantly throughout the middle and latter decades of the 20th century, and the country began to import energy. Nonetheless, crude oil net imports in 2021 were at their second-lowest yearly level since 1985. Its principal supplier was the Organization of the Petroleum Exporting Countries (OPEC), created in 1960, which consisted of the world’s largest (by volume) holders of crude oil and natural gas reserves.

As a result, the OPEC nations wielded considerable economic power in regulating supply, and hence price, of oil in the late twentieth century. In the early twenty-first century, the advent of new technology—particularly hydro-fracturing, or fracking—created a second U.S. energy boom, significantly reducing OPEC’s prominence and influence.

Oil spot contracts and futures contracts are the two forms of oil contracts that investors can exchange. To the individual investor, oil can be a speculative asset, a portfolio diversifier, or a hedge for existing positions.

Spot contract

The spot contract price indicates the current market price for oil, while the futures contract price shows the price that buyers are ready to pay for oil on a delivery date established in the future.

Most commodity contracts bought and sold on the spot market take effect immediately: money is exchanged, and the purchaser accepts delivery of the commodities. In the case of oil, the desire for immediate delivery vs future delivery is limited, owing to the practicalities of delivering oil.

Futures contract

An oil futures contract is an agreement to buy or sell a specified number of barrels of oil at a predetermined price on a predetermined date. When futures are acquired, a deal is struck between buyer and seller and secured by a margin payment equal to a percentage of the contract’s entire value. The futures price is no guarantee that oil will be at that price on that date in the future market. It is just the price that oil buyers and sellers anticipate at the time. The exact price of oil on that date is determined by a variety of factors impacting the supply and demand. Futures contracts are more frequently employed by traders and investors because investors do not intend to take any delivery of commodities at all.

End-users of oil buy on the market to lock in a price; investors buy futures to speculate on what the price will be in the future, and they earn if they estimate correctly. They typically liquidate or roll over their futures assets before having to take delivery. There are two major oil contracts that are closely observed by oil market participants: 1) West Texas Intermediate (WTI) crude, which trades on the New York Mercantile Exchange, serves as the North American oil futures benchmark (NYMEX); 2) North Sea Brent Crude, which trades on the Intercontinental Exchange, is the benchmark throughout Europe, Africa, and the Middle East (ICE). While the two contracts move in tandem, WTI is more sensitive to American economic developments, while Brent is more sensitive to those in other countries.

Mathematical foundations of the Geometric Brownian Motion (GBM) model

The concept of Brownian motion is associated with the contribution of Robert Brown (1828). More formally, the first works of Brown were used by the French mathematician Louis Bachelier (1900) applied to asset price forecast, which prepared the ground of modern quantitative finance. Price fluctuations observed over a short period, according to Bachelier’s theory, are independent of the current price as well as the historical behaviour of price movements. He deduced that the random behaviour of prices can be represented by a normal distribution by combining his assumptions with the Central Limit Theorem. This resulted in the development of the Random Walk Hypothesis, also known as the Random Walk Theory in modern finance. A random walk is a statistical phenomenon in which stock prices fluctuate at random. We implement a quantitative framework in a spreadsheet based on the Geometric Brownian Motion (GBM) model. Mathematically, we can derive the price of crude oil via the following model:

where dS represents the price change in continuous time dt, dX the Wiener process representing the random part, and Μdt the deterministic part.

The probability distribution function of the future price is a log-normal distribution when the price dynamics is described with a geometric Brownian motion.

Modelling crude oil market prices

Market prices

We downloaded a time series for WTI from June 2017 to June 2022. We picked this timeframe to assess the behavior of crude oil during two main market events that impacted its price: Covid-19 pandemic and the war in Ukraine.

The two main parameters to compute in order to implement the model are the (historical) average return and the (historical) volatility. We eliminated outliers (the negative price of oil) to clean the dataset and obtain better results. The historical average return is 11.99% (annual return) and the historical volatility is 59.29%. Figure 1 helps to capture the behavior of the WTI price over the period from June 2017 to June 2022.

Figure 1. Crude oil (WTI) price.
img_SimTrade_WTI_price
Source: computation by the author (data: Refinitiv Eikon).

Market returns

Figure 2 represents the returns of crude oil (WTI) over the period. We can clearly see that the impact of the Covid-19 pandemic had important implications for the negative returns in during the period covering early 2020.

Figure 2. Crude oil (WTI) return.
img_SimTrade_WTI_return
Source: computation by the author (data: Refinitiv Eikon).

We compute the returns using the log returns approach.

img_SimTrade_log_return_WTI

where P_t represents the closing price at time t.

Figure 3 captures the distribution of the crude oil (WTI) daily returns in a histogram. As seen in the plot, the returns are skewed towards the negative tail of the distribution and show some peaks in the center of the distribution. When analyzed in conjunction, we can infer that the crude oil daily returns doesn’t follow the normal distribution.

Figure 3. Histogram of crude oil (WTI) daily returns. img_SimTrade_WTI_histogram Source: computation by the author (data: Refinitiv Eikon).

To have a better understanding of the crude oil behavior across the 1257 trading days retained for the period of analysis, it is interesting to run a statistical analysis of the four moments of the crude oil time series: the mean (average return), standard deviation (volatility), skewness (symmetry of the distribution), kurtosis (tail of the distribution). As captured by Table 1, crude oil performed positively over the period covered delivering a daily return equivalent to 0.05% (13.38% annualized return) for a daily degree of volatility equivalent to 3.74% (or 59.33% annualized). In terms of skewness, we can see that the distribution of crude oil return is highly negatively skewed, which implies that the negative tail of the distribution is longer than the right-hand tail (positive returns). Regarding the high positive kurtosis, we can conclude that the crude oil return distribution is more peaked with a narrow distribution around the center and show more tails than the normal distribution.

Table 1. Statistical moments of the crude oil (WTI) daily returns.
WTI statistical moment
Source: computation by the author (data: Refinitiv Eikon).

Application: simulation of future prices for the crude oil market

Understanding the evolution of the price of crude oil can be significant for pricing purposes. Some models (such as the Black-Scholes option pricing model) rely heavily on a price input and can be sensitive to this parameter. Therefore, accurate price estimation is at the core of important pricing models and thus having a good estimate of spot and future price can have a significant impact in the accuracy of the pricing implemented profitability of the trade.

We implement this framework and use a Monte Carlo simulation of 25 iterations to capture the different path that the WTI price can take over a period of 24 months. Figure 4 captures the result of the model. We plot the simulations in a 3D-graph to grasp the shape of the variations in each maturity. As seen from Figure 4, price peaked at the longer end of the maturity at a level near the 250$/bbl. Overall the shape is bumpy, with some local spikes achieved throughout the whole sample and across all the maturities (Figure 4).

Figure 4. Geometric Brownian Motion (GBM) simulations for WTI. WTI GBM simulation Source: computation by the author (Data: Refinitiv Eikon).

You can find below the Excel spreadsheet that complements the explanations about of this article.

Useful resources

Academic research

Bachelier, Louis (1900). Théorie de la Spéculation, Annales Scientifique de l’École Normale Supérieure, 3e série, tome 17, 21-86.

Bashiri Behmiri, Niaz and Pires Manso, José Ramos, Crude Oil Price Forecasting Techniques: A Comprehensive Review of Literature (June 6, 2013). SSRN Reseach Journal.

Brown, Robert (1828), “A brief account of microscopical observations made on the particles contained in the pollen of plants” in Philosophical Magazine 4:161-173.

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Equity market neutral strategy

January 11, 2023 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the equity market neutral strategy. The objective of the equity market neutral strategy is to benefit from both long and short positions while minimizing the exposure to the equity market fluctuations.

This article is structured as follow: we introduce the equity market neutral strategy. Then, we present a practical case study to grasp the overall methodology of this strategy. We conclude with a performance analysis of this strategy in comparison with a global benchmark (MSCI All World Index and the Credit Suisse Hedge Fund index).

Introduction

According to Credit Suisse (a financial institution publishing hedge fund indexes), an equity market neutral strategy can be defined as follows: “Equity Market Neutral funds take both long and short positions in stocks while minimizing exposure to the systematic risk of the market (i.e., a beta of zero is desired). Funds seek to exploit investment opportunities unique to a specific group of stocks, while maintaining a neutral exposure to broad groups of stocks defined for example by sector, industry, market capitalization, country, or region. There are a number of sub- sectors including statistical arbitrage, quantitative long/short, fundamental long/short and index arbitrage”. This strategy makes money by holding assets that are decorrelated from a specific benchmark. The strategy can potentially generate returns in falling markets.

Mathematical foundation for the beta

This strategy relies heavily on the beta, derived from the capital asset pricing model (CAPM). Under this framework, we can relate the expected return of a given asset and its risk:

Where :

E(r) represents the expected return of the asset
r_f the risk-free rate
β a measure of the risk of the asset
E(r_m) the expected return of the market
E(r_m) – r_f represents the market risk premium.

In this model, the beta (β) parameter is a key parameter and is defined as:

Where:

Cov(r, r_m) represents the covariance of the asset return with the market return
σ²(r_m) is the variance of market return.

The beta is a measure of how sensitive an asset is to market swings. This risk indicator aids investors in predicting the fluctuations of their asset in relation to the wider market. It compares the volatility of an asset to the systematic risk that exists in the market. The beta is a statistical term that denotes the slope of a line formed by a regression of data points comparing stock returns to market returns. It aids investors in understanding how the asset moves in relation to the market. According to Fama and French (2004), there are two ways to interpret the beta employed in the CAPM:

According to the CAPM formula, beta may be thought in mathematical terms as the slope of the regression of the asset return on the market return observed on different periods. Thus, beta quantifies the asset sensitivity to changes in the market return;
According to the beta formula, it may be understood as the risk that each dollar invested in an asset adds to the market portfolio. This is an economic explanation based on the observation that the market portfolio’s risk (measured by 〖σ(r_m)〗^2) is a weighted average of the covariance risks associated with the assets in the market portfolio, making beta a measure of the covariance risk associated with an asset in comparison to the variance of the market return.

Additionally, the CAPM makes a distinction between two forms of risk: systematic and specific risk. Systematic risk refers to the risk posed by all non-diversifiable elements such as monetary policy, political events, and natural disasters. By contrast, specific risk refers to the risk inherent in a particular asset and so is diversifiable. As a result, the CAPM solely captures systematic risk via the beta measure, with the market’s beta equal to one, lower-risk assets having a beta less than one, and higher-risk assets having a beta larger than one.

Application of an equity market neutral strategy

For the purposes of this example, let us assume that a portfolio manager wants to invest $100 million across a diverse equity portfolio while maintaining market-neutral exposure to market index changes. To create an equity market-neutral portfolio, we use five stocks from the US equity market: Apple, Amazon, Microsoft, Goldman Sachs, and Pfizer. Using monthly data from Bloomberg for the period from 1999 to 2022, we compute the returns of these stocks and their beta with the US equity index (S&P500). Using the solver function on Excel, we find the weights of the portfolio with the maximum expected return with a beta equal to zero.

Table 1 displays the target weights needed to build a portfolio with a neutral view on the equity market. As shown by the target allocation in Table 1, we can immediately see a substantial position of 186.7 million dollars on Pfizer while keeping a short position on the remaining equity positions of the portfolio totaling 86.7 million dollars in short positions. Given that the stocks on the short list have high beta values (more than one), this allocation makes sense. Pfizer is the only defensive stock and has a beta of 0.66 in relation to the S&P 500 index.

If the investment manager allocated capital in the following way, he would create an equity market neutral portfolio with a beta of zero:

Apple: -$4.6 million (-4.6% of the portfolio; a weighted-beta of -0.066)
Amazon: -$39.9 million (-39.9% of the portfolio; a weighted-beta of -0.592)
Microsoft: -$16.2 million (-16.2% of the portfolio; a weighted-beta of -0.192)
Goldman Sachs: -$26 million (-26% of the portfolio; a weighted-beta of -0.398)
Pfizer: $186.7 million (186.7% of the portfolio; a weighted-beta of 1.247)

Table 1. Target weights to achieve an equity market neutral portfolio.
Source: computation by the author (Data: Bloomberg)

You can find below the Excel spreadsheet that complements the explanations about the equity market neutral portfolio.

An extension of the equity market neutral strategy to other asset classes

A portfolio with a beta of zero, or zero systematic risk, is referred to as a zero-beta portfolio. A portfolio with a beta of zero would have an expected return equal to the risk-free rate. Given that its expected return is equal to the risk-free rate or is relatively low compared to portfolios with a higher beta. Such portfolio would have no correlation with market movements.

Since a zero-beta portfolio has no market exposure and would consequently underperform a diversified market portfolio, it is highly unlikely that investors will be interested in it during bull markets. During a bear market, it may garner some interest, but investors are likely to ask if investing in risk-free, short-term Treasuries is a better and less expensive alternative to a zero-beta portfolio.

For this example, we imagine the case of a portfolio manager wishing to invest 100M$ across a diversified portfolio, while holding a zero-beta portfolio with respect to a broad equity index benchmark. To recreate a diversified portfolio, we compiled a shortlist of trackers that would represent our investment universe. To maintain a balanced approach, we selected trackers that would represent the main asset classes: global stocks (VTI – Vanguard Total Stock Market ETF), bonds (IEF – iShares 7-10 Year Treasury Bond ETF and TLT – iShares 20+ Year Treasury Bond ETF), and commodities (DBC – Invesco DB Commodity Index Tracking Fund and GLD – SPDR Gold Shares).

To construct the zero-beta portfolio, we pulled a ten-year time series from Refinitiv Eikon and calculated the beta of each asset relative to the broad stock index benchmark (VTI tracker). The target weights to create a zero-beta portfolio are shown in Table 2. As captured by the target allocation in Table 2, we can clearly see an important weight for bonds of different maturities (56.7%), along with a 33.7% towards commodities and a small allocation towards global equity equivalent to 9.6% (because of the high beta value).

If the investment manager allocated capital in the following way, he would create a zero-beta portfolio with a beta of zero:

VTI: $9.69 million (9.69% of the portfolio; a weighted-beta of 0.097)
IEF: $18.99 million (18.99% of the portfolio; a weighted-beta of -0.029)
GLD: $18.12 million (18.12% of the portfolio; a weighted-beta of 0.005)
DBC: $15.5 million (15.50% of the portfolio; a weighted-beta of 0.070)
TLT: $37.7 million (37.7% of the portfolio; a weighted-beta of -0.143)

Table 2. Target weights to achieve a zero-beta portfolio.
Source: computation by the author. (Data: Reuters Eikon)

You can find below the Excel spreadsheet that complements the explanations about the zero beta portfolio.

Performance of the equity market neutral strategy

To capture the performance of the equity market neutral strategy, we use the Credit Suisse hedge fund strategy index. To establish a comparison between the performance of the global equity market and the equity market neutral strategy, we examine the rebased performance of the Credit Suisse managed futures index with respect to the MSCI All-World Index.

The equity market neutral strategy generated an annualized return of -0.18% with an annualized volatility of 7.5%, resulting in a Sharpe ratio of -0.053. During the same time period, the Credit Suisse Hedge Fund index had an annualized return of 4.34 percent with an annualized volatility of 5.64 percent, resulting in a Sharpe ratio of 0.174. With a neutral market beta exposure of 0.04, the results are consistent with the theory that this approach does not carry the equity risk premium. This aspect justifies the underperformance.

Figure 1 gives the performance of the equity market neutral funds (Credit Suisse Equity Market Neutral Index) compared to the hedge funds (Credit Suisse Hedge Fund index) and the world equity funds (MSCI All-World Index) for the period from July 2002 to April 2021.

Figure 1. Performance of the equity market neutral strategy.

Source: computation by the author (Data: Bloomberg)

You can find below the Excel spreadsheet that complements the explanations about the Credit Suisse equity market neutral strategy.

Why should I be interested in this post?

Understanding the performance and risk of the equity market neutral strategy might assist investors in incorporating this hedge fund strategy into their portfolio allocation.

Useful resources

Academic research

Pedersen, L. H., 2015. Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined. Princeton University Press.

Business Analysis

Credit Suisse Hedge fund strategy

Credit Suisse Hedge fund performance

Credit Suisse Equity market neutral strategy

Credit Suisse Equity market neutral performance benchmark

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fixed-income arbitrage strategy

January 10, 2023 by Youssef LOURAOUI

Fixed-income arbitrage strategy

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the fixed-income arbitrage strategy which is a well-known strategy used by hedge funds. The objective of the fixed-income arbitrage strategy is to benefit from trends or disequilibrium in the prices of fixed-income securities using systematic and discretionary trading strategies.

This article is structured as follow: we introduce the fixed-income arbitrage strategy principle. Then, we present a practical case study to grasp the overall methodology of this strategy. We also present a performance analysis of this strategy and compare it a benchmark representing all hedge fund strategies (Credit Suisse Hedge Fund index) and a benchmark for the global equity market (MSCI All World Index).

Introduction

According to Credit Suisse (a financial institution publishing hedge fund indexes), a fixed-income arbitrage strategy can be defined as follows: “Fixed-income arbitrage funds attempt to generate profits by exploiting inefficiencies and price anomalies between related fixed-income securities. Funds limit volatility by hedging out exposure to the market and interest rate risk. Strategies include leveraging long and short positions in similar fixed-income securities that are related either mathematically or economically. The sector includes credit yield curve relative value trading involving interest rate swaps, government securities and futures, volatility trading involving options, and mortgage-backed securities arbitrage (the mortgage-backed market is primarily US-based and over-the-counter)”.

Types of arbitrage

Fixed-income arbitrage makes money based on two main underlying concepts:

Pure arbitrage

Identical instruments should have identical price (this is the law of one price). This could be the case, for instance, of two futures contracts traded on two different exchanges. This mispricing could be used by going long the undervalued contract and short the overvalued contract. This strategy uses to work in the days before the rise of electronic trading. Now, pure arbitrage is much less obvious as information is accessible instantly and algorithmic trading wipe out this kind of market anomalies.

Relative value arbitrage

Similar instruments should have a similar price. The fundamental rationale of this type of arbitrage is the notion of reversion to the long-term mean (or normal relative valuations).

Factors that influence fixed-income arbitrage strategies

We list below the sources of market inefficiencies that fixed-income arbitrage funds can exploit.

Market segmentation

Segmentation is of concern for fixed-income arbitrageurs. In financial institutions, the fixed-income desk is split into different traders looking at specific parts of the yield curve. In this instance, some will focus on very short, dated bonds, others while concentrate in the middle part of the yield curve (2-5y) while other while be looking at the long-end of the yield curve (10-30y).

Regulation

Regulation has an implication in the kind of fixed-income securities a fund can hold in their books. Some legislations regulate actively to have specific exposure to high yield securities (junk bonds) since their probability of default is much more important. The diminished popularity linked to the tight regulation can make the valuation of those bonds more attractive than owning investment grade bonds.

Liquidity

Liquidity is also an important concern for this type of strategy. The more liquid the market, the easier it is to trade and execute the strategy (vice versa).

Volatility

Large market movements in the market can have implications to the profitability of this kind of strategy.

Instrument complexity

Instrument complexity can also be a reason to have fixed-income securities. The events of 2008 are a clear example of how banks and regulators didn’t manage to price correctly the complex instruments sold in the market which were highly risky.

Application of a fixed-income arbitrage

Fixed-income arbitrage strategy makes money by focusing on the liquidity and volatility factors generating risk premia. The strategy can potentially generate returns in both rising and falling markets. However, understanding the yield curve structure of interest rates and detecting the relative valuation differential between fixed-income securities is the key concern since this is what makes this strategy profitable (or not!).

We present below a case study related tot eh behavior of the yield curves in the European fixed-income markets inn the mid 1990’s

The European yield curve differential during in the mid 1990’s

The case showed in this example is the relative-value trade between Germany and Italian yields during the period before the adoption of the Euro as a common currency (at the end of the 1990s). The yield curve should reflect the future path of interest rates. The Maastricht treaty (signed on 7th February 1992) obliged most EU member states to adopt the Euro if certain monetary and budgetary conditions were met. This would imply that the future path of interest rates for Germany and Italy should converge towards the same values. However, the differential in terms of interest rates at that point was nearly 350 bps from 5-year maturity onwards (3.5% spread) as shown in Figure 1.

Figure 1. German and Italian yield curve in January 1995.

Source: Motson (2022) (Data: Bloomberg).

A fixed-income arbitrageur could have profited by entering in an interest rate swap where the investor receives 5y-5y forward Italian rates and pays 5y-5y German rates. If the Euro is introduced, then the spread between the two yield curves for the 5-10y part should converge to zero. As captured in Figure 2, the rates converged towards the same value in 1998, where the spread between the two rates converged to zero.

Figure 2. Payoff of the fixed-income arbitrage strategy.

Source: Motson (2022) (Data: Bloomberg).

Performance of the fixed-income arbitrage strategy

Overall, the performance of the fixed-income arbitrage between 1994-2020 were smaller on scale, with occasional large drawdowns (Asian crisis 1998, Great Financial Crisis of 2008, Covid-19 pandemic 2020). This strategy is skewed towards small positive returns but with important tail-risk (heavy losses) according to Credit Suisse (2022). To capture the performance of the fixed-income arbitrage strategy, we use the Credit Suisse hedge fund strategy index. To establish a comparison between the performance of the global equity market and the fixed-income arbitrage strategy, we examine the rebased performance of the Credit Suisse index with respect to the MSCI All-World Index.

Over a period from 2002 to 2022, the fixed-income arbitrage strategy index managed to generate an annualized return of 3.81% with an annualized volatility of 5.84%, leading to a Sharpe ratio of 0.129. Over the same period, the Credit Suisse Hedge Fund index Index managed to generate an annualized return of 5.04% with an annualized volatility of 5.64%, leading to a Sharpe ratio of 0.197. The results are in line with the idea of global diversification and decorrelation of returns derived from the global macro strategy from global equity returns. Overall, the Credit Suisse fixed-income arbitrage strategy index performed better than the MSCI All World Index, leading to a higher Sharpe ratio (0.129 vs 0.08).

Figure 3 gives the performance of the fixed-income arbitrage funds (Credit Suisse Fixed-income Arbitrage Index) compared to the hedge funds (Credit Suisse Hedge Fund index) and the world equity funds (MSCI All-World Index) for the period from July 2002 to April 2021.

Figure 3. Performance of the fixed-income arbitrage strategy.
Global macro performance
Source: computation by the author (Data: Bloomberg).

You can find below the Excel spreadsheet that complements the explanations about the fixed-income arbitrage strategy.

Why should I be interested in this post?

The fixed-income arbitrage strategy aims to profit from market dislocations in the fixed-income market. This can be implemented, for instance, by investing in inexpensive fixed-income securities that the fund manager predicts that it will increase in value, while simultaneously shorting overvalued fixed-income securities to mitigate losses. Understanding the profits and risks associated with such a strategy may aid investors in adopting this hedge fund strategy into their portfolio allocation.

Useful resources

Academic research

Pedersen, L. H., 2015. Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined. Princeton University Press.

Motson, N. 2022. Hedge fund elective. Bayes (formerly Cass) Business School.

Business Analysis

Credit Suisse Hedge fund strategy

Credit Suisse Hedge fund performance

Credit Suisse Fixed-income arbitrage strategy

Credit Suisse Fixed-income arbitrage performance benchmark

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Global macro strategy

January 9, 2023 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the global macro equity strategy, one of the most widely known strategies in the hedge fund industry. The goal of the global macro strategy is to look for trends or disequilibrium in equity, bonds, currency or alternative assets based on broad economic data using a top-down approach.

This article is structured as follow: we introduce the global macro strategy principle. Then, we present a famous case study to grasp the overall methodology of this strategy. We conclude with a performance analysis of this strategy in comparison with a global benchmark (MSCI All World Index and the Credit Suisse Hedge Fund index).

Introduction

According to Credit Suisse, a global macro strategy can be defined as follows: “Global Macro funds focus on identifying extreme price valuations and leverage is often applied on the anticipated price movements in equity, currency, interest rate and commodity markets. Managers typically employ a top-down global approach to concentrate on forecasting how political trends and global macroeconomic events affect the valuation of financial instruments. Profits are made by correctly anticipating price movements in global markets and having the flexibility to use a broad investment mandate, with the ability to hold positions in practically any market with any instrument. These approaches may be systematic trend following models, or discretionary.”

This strategy can generate returns in both rising and falling markets. However, asset screening is of key concern, and the ability of the fund manager to capture the global macro picture that is driving all asset classes is what makes this strategy profitable (or not!).

The greatest trade in history

The greatest trade in history (before Michael Burry becomes famous for anticipating the Global financial crisis of 2008 linked to the US housing market) took place during the 1990’s when the UK was intending to join the Exchange Rate Mechanism (ERM) founded in 1979. This foreign exchange (FX) system involved eight countries with the intention to move towards a single currency (the Euro). The currencies of the countries involved would be adjustably pegged with a determined band in which they can fluctuate with respect to the Deutsche Mark (DEM), the currency of Germany considered as the reference of the ERM.

Later in 1992, the pace at which the countries adhering to the ERM mechanism were evolving at different rate of growth. The German government was in an intensive spending following the reunification of Berlin, with important stimulus from the German Central Bank to print more money. However, the German government was very keen on controlling inflation to satisfactory level, which was achieved by increasing interest rates in order to curb the inflationary pressure in the German economy.

In the United Kingdom (UK), another macroeconomic picture was taking place: there was a high unemployment coupled with already relatively high interest rates compared to other European economies. The Bank of England was put in a very tight spot because they were facing two main market scenarios:

To increase interest rates, which would worsen the economy and drive the UK into a recession
To devalue the British Pound (GBP) by defending actively in the FX market, which would cause the UK to leave the ERM mechanism.

The Bank of England decided to go with the second option by defending the British Pound in the FX market by actively buying pounds. However, this strategy would not be sustainable over time. Soros (and other investors) had seen this disequilibrium and shorted British Pound and bought Deutsche Mark. The situation got completely off control for the Bank of England that in September 1992, they decided to increase interest rates, which were already at 10% to more than 15% to calm the selling pressure. Eventually, the following day, the Bank of England announced the exit of the UK from the ERM mechanism and put a hold on the increase of interest rate to the 12% until the economic conditions get better. Figure 1 gives the evolution of the exchange rate between the British Pound (GBP) and the Deutsche Mark (DEM) over the period 1991-1992.

Figure 1. Evolution of the GBP-DEM (British Pound / Deutsche Mark FX rate).
Global macro performance
Source: Bloomberg.

It was reported that Soros amassed a position of $10 billion and gained a whopping $1 billion for this trade. This event put Soros in the scene as the “man who broke the Bank of England”. The good note about this market event is that the UK economy emerged much healthier than the European countries, with UK exports becoming much more competitive as a result of the pound devaluation, which led the Bank of England to cut rates cut down to the 5-6% level the years following the event, which ultimately helped the UK economy to get better.

Performance of the global macro strategy

Overall, the performance of the global macro funds between 1994-2020 was steady, with occasional large drawdowns (Asian crisis 1998, Dot-com bubble 2000’s, Great Financial Crisis of 2008, Covid-19 pandemic 2020). On a side note, the returns seem smaller and less volatile since 2000 onwards (Credit Suisse, 2022).

To capture the performance of the global macro strategy, we use the Credit Suisse hedge fund strategy index. To establish a comparison between the performance of the global equity market and the global macro hedge fund strategy, we examine the rebased performance of the Credit Suisse index with respect to the MSCI All-World Index. Over a period from 2002 to 2022, the global macro strategy index managed to generate an annualized return of 7.85% with an annualized volatility of 5.77%, leading to a Sharpe ratio of 0.33. Over the same period, the MSCI All World Index managed to generate an annualized return of 6.00% with an annualized volatility of 15.71%, leading to a Sharpe ratio of 0.08. The low correlation of the long-short equity strategy with the MSCI All World Index is equal to -0.02, which is close to zero. The results are in line with the idea of global diversification and decorrelation of returns derived from the global macro strategy from global equity returns. Overall, the Credit Suisse hedge fund strategy index performed better worse than the MSCI All World Index, leading to a higher Sharpe ratio (0.33 vs 0.08).

Figure 2 gives the performance of the global macro funds (Credit Suisse Global Macro Index) compared to the hedge funds (Credit Suisse Hedge Fund index) and the world equity funds (MSCI All-World Index) for the period from July 2002 to April 2021.

Figure 2. Performance of the global macro strategy.

Source: computation by the author (data: Bloomberg).

You can find below the Excel spreadsheet that complements the explanations about the global macro hedge fund strategy.

Why should I be interested in this post?

Global macro funds seek to profit from market dislocations across different asset classes. reduce negative risk while increasing market upside. They might, for example, invest in inexpensive assets that the fund managers believe will rise in price while simultaneously shorting overvalued assets to cut losses. Other strategies used by global macro funds to lessen market volatility can include leverage and derivatives. Understanding the profits and risks of such a strategy might assist investors in incorporating this hedge fund strategy into their portfolio allocation.

Useful resources

Academic research

Pedersen, L. H., 2015. Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined. Princeton University Press.

Business Analysis

Credit Suisse Hedge fund strategy

Credit Suisse Hedge fund performance

Credit Suisse Global macro strategy

Credit Suisse Global macro performance benchmark

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Interest rate term structure and yield curve calibration

January 9, 2023 by Youssef LOURAOUI

Interest rate term structure and yield curve calibration

In this article, Youssef LOURAOUI (Bayes Business School,, MSc. Energy, Trade & Finance, 2021-2022) presents the usage of a widely used model for building the yield curve, namely the Nelson-Seigel-Svensson model for interest rate calibration.

This article is structured as follows: we introduce the concept of the yield curve. Next, we present the mathematical foundations of the Nelson-Siegel-Svensson model. Finally, we illustrate the model with practical examples.

Introduction

Fine-tuning the term structure of interest rates is the cornerstone of a well-functioning financial market. For this reason, the testing of various term-structure estimation and forecasting models is an important topic in finance that has received considerable attention for several decades (Lorenčič, 2016).

The yield curve is a graphical representation of the term structure of interest rates (i.e. the relationship between the yield and the corresponding maturity of zero-coupon bonds issued by governments). The term structure of interest rates contains information on the yields of zero-coupon bonds of different maturities at a certain date (Lorenčič, 2016). The construction of the term structure is not a simple task due to the scarcity of zero-coupon bonds in the market, which are the basic elements to estimate the term structure. The majority of bonds traded in the market carry coupons (regular paiement of interests). The yields to maturity of coupon bonds with different maturities or coupons are not immediately comparable. Therefore, a method of measuring the term structure of interest rates is needed: zero-coupon interest rates (i.e. yields on bonds that do not pay coupons) should be estimated from the prices of coupon bonds of different maturities using interpolation methods, such as polynomial splines (e.g. cubic splines) and parsimonious functions (e.g. Nelson-Siegel).

As explained in an interesting paper that I read (Lorenčič, 2016), the prediction of the term structure of interest rates is a basic requirement for managing investment portfolios, valuing financial assets and their derivatives, calculating risk measures, valuing capital goods, managing pension funds, formulating economic policy, making decisions about household finances, and managing fixed income assets . The pricing of fixed income securities such as swaps, bonds and mortgage-backed securities depends on the yield curve. When considered together, the yields of non-defaulting government bonds with different characteristics reveal information about forward rates, which are potentially predictive of real economic activity and are therefore of interest to policy makers, market participants and economists. For instance, forward rates are often used in pricing models and can indicate market expectations of future inflation rates and currency appreciation/depreciation rates. Understanding the relationship between interest rates and the maturity of securities is a prerequisite for developing and testing the financial theory of monetary and financial economics. The accurate adjustment of the term structure of interest rates is the backbone of a well-functioning financial market, which is why the refinement of yield curve modelling and forecasting methods is an important topic in finance that has received considerable attention for several decades (Lorenčič, 2016).

The most commonly used models for estimating the zero-coupon curve are the Nelson-Siegel and cubic spline models. For example, the central banks of Belgium, Finland, France, Germany, Italy, Norway, Spain and Switzerland use the Nelson-Siegel model or a type of its improved extension to fit and forecast yield curves (BIS, 2005). The European Central Bank uses the Sonderlind-Svensson model, an extension of the Nelson-Siegel model, to estimate yield curves in the euro area (Coroneo, Nyholm & Vidova-Koleva, 2011).

Mathematical foundation of the Nelson-Siegel-Svensson model

In this article, we will deal with the Nelson-Siegel extended model, also known as the Nelson-Siegel-Svensson model. These models are relatively efficient in capturing the general shapes of the yield curve, which explains why they are widely used by central banks and market practitioners.

Mathematically, the formula of Nelson-Siegel-Svensson is given by:

img_SimTrade_NSS_equation

where

τ = time to maturity of a bond (in years)
β₀ = parameter to capture for the level factor
β₁= parameter to capture the slope factor
β₂ = parameter to capture the curvature factor
β₃ = parameter to capture the magnitude of the second hump
λ₁ and λ₂ = parameters to capture the rate of exponential decay
exp = the mathematical exponential function

The parameters β₀, β₁, β₂, β₃, λ₁ and λ₂ can be calculated with the Excel add-in “Solver” by minimizing the sum of squared residuals between the dirty price (market value, present value) of the bonds and the model price of the bonds. The dirty price is a sum of the clean price, retrieved from Bloomberg, and accrued interest. Financial research propose that the Svensson model should be favored over the Nelson-Siegel model because the yield curve slopes down at the very long end, necessitating the second curvature component of the Svensson model to represent a second hump at longer maturities (Wahlstrøm, Paraschiv, and Schürle, 2022).

Application of the yield curve structure

In financial markets, yield curve structure is of the utmost importance, and it is an essential market indicator for central banks. During my last internship at the Central Bank of Morocco, I worked in the middle office, which is responsible for evaluating risk exposures and profits and losses on the positions taken by the bank on a 27.4 billion euro foreign reserve investment portfolio. Volatility evaluated by the standard deviation, mathematically defined as the deviation of a random variable (asset prices or returns in my example) from its expected value, is one of the primary risk exposure measurements. The standard deviation reveals the degree to which the present return deviates from the expected return. When analyzing the risk of an investment, it is one of the most used indicators employed by investors. Among other important exposures metrics, there is the VaR (Value at Risk) with a 99% confidence level and a 95% confidence level for 1-day and 30-day positions. In other words, the VaR is a metric used to calculate the maximum loss that a portfolio may sustain with a certain degree of confidence and time horizon.

Every day, the Head of the Middle Office arranges a general meeting in which he discusses a global debriefing of the most significant overnight financial news and a debriefing of the middle office desk for “watch out” assets that may present an investment opportunity. Consequently, the team is tasked with adhering to the investment decisions that define the firm, as it neither operates as an investment bank nor as a hedge fund in terms of risk and leverage. As the central bank is tasked with the unique responsibility of safeguarding the national reserve and determining the optimal mix of low-risk assets to invest in, it seeks a good asset strategy (AAA bonds from European countries coupled with American treasury bonds). The investment mechanism is comprised of the segmentation of the entire portfolio into three principal tranches, each with its own features. The first tranche (also known as the security tranche) is determined by calculating the national need for a currency that must be kept safe in order to establish exchange market stability (mostly based on short-term positions in low-risk profile assets) (Liquid and high rated bonds). The second tranche is based on a buy-and-hold strategy and a market approach. The first entails taking a long position on riskier assets than the first tranche until maturity, with no sales during the asset’s lifetime (riskier bonds and gold). The second strategy is based on the purchase and sale of liquid assets with the expectation of better returns.

Participants in the market are accustomed to categorizing the debt of eurozone nations. Germany and the Netherlands, for instance, are regarded as “core” nations, and their debt as safe-haven assets (Figure 1). Due to the stability of their yield spreads, France, Belgium, Austria, Ireland, and Finland are “semi-core” nations (Figure 1). Due to their higher bond yields and more volatile spreads, Spain, Portugal, Italy, and Greece are called “peripheral” (BNP Paribas, 2019) (Figure 2). The 10-year gap represents the difference between a country’s 10-year bond yield and the yield on the German benchmark bond. It is a sign of risk. Therefore, the greater the spread, the greater the risk. Figure 3 represents the yield curve for the Moroccan bond market.

Figure 1. Yield curves for core countries (Germany, Netherlands) and semi-core (France, Austria) of the euro zone.

Source: computation by the author.

Figure 2. Yield curves for peripheral countries of the euro zone
(Spain, Italy, Greece and Portugal).
Yield curves for semi-core countries of the euro zone
Source: computation by the author.

Figure 3. Yield curve for Morocco.

Source: computation by the author.

This example provides a tool comparable to the one utilized by central banks to measure the change in the yield curve. It is an intuitive and simplified model created in an Excel spreadsheet that facilitates comprehension of the investment process. Indeed, it is capable of continuously refreshing the data by importing the most recent quotations (in this case, retrieved from investing.com, a reputable data source).

One observation can be made about the calibration limits of the Nelson-Seigel-Svensson model. In this sense, when the interest rate curve is in negative levels (as in the case of the structure of the Japanese curve), the NSS model does not manage to model negative values, obtaining a result with substantial deviations from spot rates. This can be interpreted as a failure of the NSS calibration approach to model a negative interest rate curve.

In conclusion, the NSS model is considered as one of the most used and preferred models by central banks to obtain the short- and long-term interest rate structure. Nevertheless, this model does not allow to model the structure of the curve for negative interest rates.

Excel file for the calibration model of the yield curve

You can download an Excel file with data to calibrate the yield curve for different countries. This spreadsheet has a special macro to extract the latest data pulled from investing.com website, a reliable source for time-series data.

Why should I be interested in this post?

Predicting the term structure of interest rates is essential for managing investment portfolios, valuing financial assets and their derivatives, calculating risk measures, valuing capital goods, managing pension funds, formulating economic policy, deciding on household finances, and managing fixed income assets. The yield curve affects the pricing of fixed income assets such as swaps, bonds, and mortgage-backed securities. Understanding the yield curve and its utility for the markets can aid in comprehending this parameter’s broader implications for the economy as a whole.

Useful resources

Academic research

Lorenčič, E., 2016. Testing the Performance of Cubic Splines and Nelson-Siegel Model for Estimating the Zero-coupon Yield Curve. NGOE, 62(2), 42-50.

Wahlstrøm, Paraschiv, and Schürle, 2022. A Comparative Analysis of Parsimonious Yield Curve Models with Focus on the Nelson-Siegel, Svensson and Bliss Versions. Springer Link, Computational Economics, 59, 967–1004.

Business Analysis

BNP Paribas (2019) Peripheral Debt Offers Selective Opportunities

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School,, MSc. Energy, Trade & Finance, 2021-2022).

Minimum Volatility Portfolio

January 9, 2023 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of Minimum Volatility Portfolio, which is derived from Modern Portfolio Theory (MPT) and also in practice to build investment funds.

This article is structured as follows: we introduce the concept of Minimum Volatility Portfolio. Next, we present some interesting academic findings, and we finish by presenting a theoretical example to support the explanations given in this article.

Introduction

The minimum volatility portfolio represents a portfolio of assets with the lowest possible risk for an investor and is located on the far-left side of the efficient frontier. Note that the minimum volatility portfolio is also called the minimum variance portfolio or more precisely the global minimum volatility portfolio (to distinguish it from other optimal portfolios obtained for higher risk levels).

Modern Portfolio Theory’s fundamental notion had significant implications for portfolio construction and asset allocation techniques. In the late 1970s, the portfolio management business attempted to capture the market portfolio return. However, as financial research progressed and some substantial contributions were made, new factor characteristics emerged to capture extra performance. The financial literature has long encouraged taking on more risk to earn a higher return. However, this is a common misconception among investors. While extremely volatile stocks can produce spectacular gains, academic research has repeatedly proved that low-volatility companies provide greater risk-adjusted returns over time. This occurrence is known as the “low volatility anomaly,” and it is for this reason that many long-term investors include low volatility factor strategies in their portfolios. This strategy is consistent with Henry Markowitz’s renowned 1952 article, in which he embraces the merits of asset diversification to form a portfolio with the maximum risk-adjusted return.

Academic Literature

Markowitz is widely regarded as a pioneer in financial economics and finance due to the theoretical implications and practical applications of his work in financial markets. Markowitz received the Nobel Prize in 1990 for his contributions to these fields, which he outlined in his 1952 Journal of Finance article titled “Portfolio Selection.” His seminal work paved the way for what is now commonly known as “Modern Portfolio Theory” (MPT).

In 1952, Harry Markowitz created modern portfolio theory with his work. Overall, the risk component of MPT may be evaluated using multiple mathematical formulations and managed through the notion of diversification, which requires building a portfolio of assets that exhibits the lowest level of risk for a given level of expected return (or equivalently a portfolio of assets that exhibits the highest level of expected return for a given level of risk). Such portfolios are called efficient portfolios. In order to construct optimal portfolios, the theory makes a number of fundamental assumptions regarding the asset selection behavior of individuals. These are the assumptions (Markowitz, 1952):

The only two elements that influence an investor’s decision are the expected rate of return and the variance. (In other words, investors use Markowitz’s two-parameter model to make decisions.) .
Investors are risk averse. (That is, when faced with two investments with the same expected return but two different risks, investors will favor the one with the lower risk.)
All investors strive to maximize expected return at a given level of risk.
All investors have the same expectations regarding the expected return, variance, and covariances for all hazardous assets. This assumption is known as the homogenous expectations assumption.
All investors have a one-period investment horizon.

Only in theory does the minimum volatility portfolio (MVP) exist. In practice, the MVP can only be estimated retrospectively (ex post) for a particular sample size and return frequency. This means that several minimum volatility portfolios exist, each with the goal of minimizing and reducing future volatility (ex ante). The majority of minimum volatility portfolios have large average exposures to low volatility and low beta stocks (Robeco, 2010).

Example

To illustrate the concept of the minimum volatility portfolio, we consider an investment universe composed of three assets with the following characteristics (expected return, volatility and correlation):

Asset 1: Expected return of 10% and volatility of 10%
Asset 2: Expected return of 15% and volatility of 20%
Asset 3: Expected return of 22% and volatility of 35%

Correlation between Asset 1 and Asset 2: 0.30
Correlation between Asset 1 and Asset 3: 0.80
Correlation between Asset 2 and Asset 3: 0.50

The first step to achieve the minimum variance portfolio is to construct the portfolio efficient frontier. This curve represents all the portfolios that are optimal in the mean-variance sense. After solving the optimization program, we obtain the weights of the optimal portfolios. Figure 1 plots the efficient frontier obtained from this example. As captured by the plot, we can see that the minimum variance portfolio in this three-asset universe is basically concentrated on one holding (100% on Asset 1). In this instance, an investor who wishes to minimize portfolio risk would allocate 100% on Asset 1 since it has the lowest volatility out of the three assets retained in this analysis. The investor would earn an expected return of 10% for a volatility of 10% annualized (Figure 1).

Figure 1. Minimum Volatility Portfolio (MVP) and the Efficient Frontier.

Source: computation by the author.

Excel file to build the Minimum Volatility Portfolio

You can download below an Excel file in order to build the Minimum Volatility portfolio.

Why should I be interested in this post?

Portfolio management’s objective is to optimize the returns on the entire portfolio, not just on one or two stocks. By monitoring and maintaining your investment portfolio, you can accumulate a sizable capital to fulfil a variety of financial objectives, including retirement planning. This article helps to understand the grounding fundamentals behind portfolio construction and investing.

Useful resources

Academic research

Lintner, John. 1965a. Security Prices, Risk, and Maximal Gains from Diversification. Journal of Finance, 20, 587-616.

Lintner, John. 1965b. The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.Review of Economics and Statistics 47, 13-37.

Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7, 77-91.

Sharpe, William F. 1963. A Simplified Model for Portfolio Analysis. Management Science, 19, 425-442.

Sharpe, William F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19, 425-442.

Business analysis

Robeco, 2010 Ten things you should know about minimum volatility investing.

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Moments d’une distribution statistique

April 8, 2024January 7, 2023 by Shengyu ZHENG

Moments d’une distribution statistique

Dans cet article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) présente les quatre premiers moments d’une distribution statistique : la moyenne, la variance, la skewness et la kurtosis.

Variable aléatoire

Une variable aléatoire est une variable dont la valeur est déterminée d’après la réalisation d’un événement aléatoire. Plus précisément, la variable (X) est une fonction mesurable depuis un ensemble de résultats (Ω) à un espace mesurable (E).

X : Ω → E

X est une variable aléatoire réelle à condition que l’espace mesurable (E) soit, ou fasse partie de, l’ensemble des nombres réels (ℝ).

Je présente un exemple avec la rentabilité d’un investissement dans l’action Apple. La figure 1 ci-dessous représente la série temporelle de la rentabilité journalière de l’action Apple sur la période allant de novembre 2017 à novembre 2022.

Figure 1. Série temporelle de rentabilités de l’action Apple.
Série de rentabilité
Source : calcul par l’auteur (données : Yahoo Finance).

Figure 2. Histogramme des rentabilités de l’action Apple.
Histogramme de rentabilité
Source : calcul par l’auteur (données : Yahoo Finance).

Moments d’une distribution statistique

Le moment d’ordre r ∈ ℕ est un indicateur de la dispersion de la variable aléatoire X. Le moment ordinaire d’ordre r est défini, s’il existe, par la formule suivante :

m_r = 𝔼 (X^r)

Nous avons aussi le moment centré d’ordre r défini, s’il existe, par la formule suivante :

c_r = 𝔼([X-𝔼(X)]^r)

Moment d’ordre un : la moyenne

Définition

La moyenne ou l’espérance mathématique d’une variable aléatoire est la valeur attendue en moyenne si la même expérience aléatoire est répétée un grand nombre de fois. Elle correspond à une moyenne pondérée par probabilité des valeurs que peut prendre cette variable, et elle est donc connue comme la moyenne théorique ou la vraie moyenne.

Si une variable X prend une infinité de valeurs x₁, x₂,… avec les probabilités p₁, p₂,…, l’espérance de X est définie comme :

Μ = m₁= 𝔼(X) = ∑^∞_i=1p_ix_i

L’espérance existe à condition que cette somme soit absolument convergente.

Estimation statistique

La moyenne empirique est un estimateur de l’espérance. Cet estimateur est sans biais, convergent (selon la loi des grands nombres), et distribué normalement (selon le théorème centrale limite).

A partir d’un échantillon de variables aléatoire réelles indépendantes et identiquement distribuées (X₁,…,X_n), la moyenne empirique est donc :

X̄ = (∑ⁿ_i=1x_i)/n

Pour une loi normale centrée réduite (μ = 0 et σ = 1), la moyenne est égale à zéro.

Moment d’ordre deux : la variance

Définition

La variance (moment d’ordre deux) est une mesure de la dispersion des valeurs par rapport à sa moyenne.

Var(X) = σ ² = 𝔼[(X-μ)²]

Elle exprime l’espérance du carré de l’écart à la moyenne théorique. Elle est donc toujours positive.

Pour une loi normale centrée réduite (μ = 0 et σ = 1), la variance est égale à un.

Estimation statistique

A partir d’un échantillon (X₁,…,X_n), nous pouvons estimer la variance théorique à l’aide de la variance empirique :

S² = (∑ⁿ_i=1(x_i – X̄)²)/n

Cependant, cet estimateur est biaisé, parce que 𝔼(S²) = (n-1)/(n) σ². Nous avons donc un estimateur non-biaisé Š² = (∑ⁿ_i=1(x_i – X̄)²)/(n-1)

Application en finance

La variance correspond à la volatilité d’un actif financier. Une variance élevée indique une dispersion plus importante, et ce n’est pas favorable du regard des investisseurs rationnels qui présentent de l’aversion au risque. Ce concept est un paramètre clef dans la théorie moderne du portefeuille de Markowitz.

Moment d’ordre trois : la skewness

Définition

La skewness (coefficient d’asymétrie en bon français) est le moment d’ordre trois, défini comme ci-dessous :

γ₁ = 𝔼[((X-μ)/σ)³]

La skewness mesure l’asymétrie de la distribution d’une variable aléatoire. On distingue trois types de distributions selon que la distribution est asymétrique à gauche, symétrique, ou asymétrique à droite. Un coefficient d’asymétrie négatif indique une asymétrie à gauche de la distribution, dont la queue gauche est plus importante que la queue droite. Un coefficient d’asymétrie nul indique une symétrie, les deux queues de la distribution étant aussi importante l’une que l’autre. Enfin, un coefficient d’asymétrie positif indique une asymétrie à droite de la distribution, dont la queue droite est plus importante que la queue gauche.

Pour une loi normale, la skewness est égale à zéro car cette loi est symétrique par rapport à la moyenne.

Moment d’ordre quatre : la kurtosis

Définition

La kurtosis (coefficient d’acuité en bon français) est le moment d’ordre quatre, défini par :

β₂ = 𝔼[((X-μ)/σ)⁴]

Il décrit l’acuité d’une distribution. Un coefficient d’acuité élevé indique que la distribution est plutôt pointue en sa moyenne, et a des queues de distribution plus épaisses (fatter tails en anglais).

Le coefficient d’une loi normale est de 3, autrement dit, une distribution mésokurtique. Au-delà de ce seuil, une distribution est appelée leptokurtique. Les distributions présentes au marché financier sont principalement leptokurtique, impliquant que les valeurs anormales et extrêmes sont plus fréquentes que celles d’une distribution gaussienne. Au contraire, un coefficient d’acuité de moins de 3 indique une distribution platykurtique, dont les queues sont plus légères.

Pour une loi normale, la kurtosis est égale à trois.

Exemple : distribution des rentabilités d’un investissement dans l’action Apple

Nous donnons maintenant un exemple en finance en étudiant la distribution des rentabilités de l’action Apple. Dans les données récupérées de Yahoo! Finance pour la période allant de novembre 2017 à novembre 2022, on se sert de la colonne du cours de clôture pour calculer les rentabilités journalières. Nous utilisons des fonctions Excel afin de calculer les quatre premiers moments de la distribution empirique des rentabilités de l’action Apple comme indiqué dans la table ci-dessous.

Moments de l’action Apple

Pour une distribution normale standard (centrée réduite), la moyenne est de zero, la variance est de 1, le skewness est de zéro, et le kurtosis est de 3. À comparaison avec une distribution normale, la distribution de rentabilité de l’action Apple a une moyenne légèrement positive. Cela signifie qu’à long terme, la rentabilité de l’investissement dans cet actif est positive. Son skewness est négatif, indiquant l’asymétrie vers la gauche (les valeurs négatives). Son kurtosis est supérieur de 3, ce qui indique que les extrémités sont plus épaisses que la distribution normale.

Fichier Excel pour calculer les moments

Vous pouvez télécharger le ficher Excel d’analyse des moments de l’action Apple en suivant le lien ci-dessous :

Autres article sur le blog SimTrade

▶ Shengyu ZHENG Catégories de mesures de risques

▶ Shengyu ZHENG Mesures de risques

Ressources

Articles académiques

Robert C. Merton (1980) On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics, 8:4, 323-361.

Données

Yahoo! Finance Données de marché pour l’action Apple

A propos de l’auteur

Cet article a été écrit en janvier 2023 par Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Arbitrage Pricing Theory (APT)

April 18, 2025January 2, 2023 by Youssef LOURAOUI

Arbitrage Pricing Theory (APT)

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the concept of arbitrage portfolio, a pillar concept in asset pricing theory.

This article is structured as follows: we present an introduction for the notion of arbitrage portfolio in the context of asset pricing, we present the assumptions and the mathematical foundation of the model and we then illustrate a practical example to complement this post.

Introduction

Arbitrage pricing theory (APT) is a method of explaining asset or portfolio returns that differs from the capital asset pricing model (CAPM). It was created in the 1970s by economist Stephen Ross. Because of its simpler assumptions, arbitrage pricing theory has risen in favor over the years. However, arbitrage pricing theory is far more difficult to apply in practice since it requires a large amount of data and complicated statistical analysis.The following points should be kept in mind when understanding this model:

Arbitrage is the technique of buying and selling the same item at two different prices at the same time for a risk-free profit.
Arbitrage pricing theory (APT) in financial economics assumes that market inefficiencies emerge from time to time but are prevented from occurring by the efforts of arbitrageurs who discover and instantly remove such opportunities as they appear.
APT is formalized through the use of a multi-factor formula that relates the linear relationship between the expected return on an asset and numerous macroeconomic variables.

The concept that mispriced assets can generate short-term, risk-free profit opportunities is inherent in the arbitrage pricing theory. APT varies from the more traditional CAPM in that it employs only one factor. The APT, like the CAPM, assumes that a factor model can accurately characterize the relationship between risk and return.

Assumptions of the APT model

Arbitrage pricing theory, unlike the capital asset pricing model, does not require that investors have efficient portfolios. However, the theory is guided by three underlying assumptions:

Systematic factors explain asset returns.
Diversification allows investors to create a portfolio of assets that eliminates specific risk.
There are no arbitrage opportunities among well-diversified investments. If arbitrage opportunities exist, they will be taken advantage of by investors.

To have a better grasp on the asset pricing theory behind this model, we can recall in the following part the foundation of the CAPM as a complementary explanation for this article.

Capital Asset Pricing Model (CAPM)

William Sharpe, John Lintner, and Jan Mossin separately developed a key capital market theory based on Markowitz’s work: the Capital Asset Pricing Model (CAPM). The CAPM was a huge evolutionary step forward in capital market equilibrium theory, since it enabled investors to appropriately value assets in terms of systematic risk, defined as the market risk which cannot be neutralized by the effect of diversification. In their derivation of the CAPM, Sharpe, Mossin and Litner made significant contributions to the concepts of the Efficient Frontier and Capital Market Line. The seminal contributions of Sharpe, Litner and Mossin would later earn them the Nobel Prize in Economics in 1990.

The CAPM is based on a set of market structure and investor hypotheses:

There are no intermediaries
There are no limits (short selling is possible)
Supply and demand are in balance
There are no transaction costs
An investor’s portfolio value is maximized by maximizing the mean associated with projected returns while reducing risk variance
Investors have simultaneous access to information in order to implement their investment plans
Investors are seen as “rational” and “risk averse”.

Under this framework, the expected return of a given asset is related to its risk measured by the beta and the market risk premium:

CAPM risk beta relation

Where :

E(r_i) represents the expected return of asset i
r_f the risk-free rate
β_i the measure of the risk of asset i
E(r_m) the expected return of the market
E(r_m)- r_f the market risk premium.

In this model, the beta (β) parameter is a key parameter and is defined as:

CAPM beta formula

Where:

Cov(r_i, r_m) represents the covariance of the return of asset i with the return of the market
σ²(r_m) is the variance of the return of the market.

According to the CAPM formula, beta may be thought in mathematical terms as the slope of the regression between the asset return and the market return. Thus, beta quantifies the asset sensitivity to changes in the market return.
According to the beta formula, it may be understood as the risk that each dollar invested in an asset adds to the market portfolio. This is an economic explanation based on the observation that the market portfolio’s risk (measured by σ²(r_m)) is a weighted average of the covariance risks associated with the assets in the market portfolio, making beta a measure of the covariance risk associated with an asset in comparison to the variance of the market return.

Mathematical foundations

The APT can be described formally by the following equation

APT expected return formula

Where

E(r_p) represents the expected return of portfolio p
r_f the risk-free rate
β_k the sensitivity of the return on portfolio p to the k^th factor (f_k)
λ_k the risk premium for the k^th factor (f_k)
K the number of risk factors

Richard Roll and Stephen Ross found out that the APT can be sensible to the following factors:

Expectations on inflation
Industrial production (GDP)
Risk premiums
Term structure of interest rates

Furthermore, the researchers claim that an asset will have varied sensitivity to the elements indicated above, even if it has the same market factor as described by the CAPM.

Application

For this specific example, we want to understand the asset price behavior of two equity indexes (Nasdaq for the US and Nikkei for Japan) and assess their sensitivity to different macroeconomic factors. We extract a time series for Nasdaq equity index prices, Nikkei equity index prices, USD/CHY FX spot rate and US term structure of interest rate (10y-2y yield spread) from the FRED Economics website, a reliable source for macroeconomic data for the last two decades.

The first factor, which is the USD/CHY (US Dollar/Chinese Renminbi Yuan) exchange rate, is retained as the primary factor to explain portfolio return. Given China’s position as a major economic player and one of the most important markets for the US and Japanese corporations, analyzing the sensitivity of US and Japanese equity returns to changes in the USD/CHY Fx spot rate can help in understanding the market dynamics underlying the US and Japanese equity performance. For instance, Texas Instrument, which operates in the sector of electronics and semiconductors, and Nike both have significant ties to the Chinese market, with an overall exposure representing approximately 55% and 18%, respectively (Barrons, 2022). In the example of Japan, in 2017 the Japanese government invested 117 billion dollars in direct investment in northern China, one of the largest foreign investments in China. Similarly, large Japanese listed businesses get approximately 18% of their international revenues from the Chinese market (The Economist, 2019).

The second factor, which is the 10y-2y yield spread, is linked to the shape of the yield curve. A yield curve that is inverted indicates that long-term interest rates are lower than short-term interest rates. The yield on an inverted yield curve decreases as the maturity date approaches. The inverted yield curve, also known as a negative yield curve, has historically been a reliable indicator of a recession. Analysts frequently condense yield curve signals to the difference between two maturities. According to the paper of Yu et al. (2017), there is a significant link between the effects of varying degrees of yield slope with the performance of US equities between 2006 and 2012. Regardless of market capitalization, the impact of the higher yield slope on stock prices was positive.

The APT applied to this example can be described formally by the following equation:

APT expected return formula example

Where

E(r_p) represents the expected return of portfolio p
r_f the risk-free rate
β_{p, Chinese FX} the sensitivity of the return on portfolio p to the USD/CHY FX spot rate
β_{p, US spread} the sensitivity of the return on portfolio p to the US term structure
λ_{Chinese FX} the risk premium for the FX risk factor
λ_{US spread} the risk premium for the interest rate risk factor

We run a first regression of the Nikkei Japanese equity index returns onto the macroeconomic variables retained in this analysis. We can highlight the following: Both factors are not statistically significant at a 10% significance level, indicating that the factors have poor predictive power in explaining Nikkei 225 returns over the last two decades. The model has a low R2, equivalent to 0.48%, which indicates that only 0.48% of the behavior of Nikkei performance can be attributed to the change in USD/CHY FX spot rate and US term structure of the yield curve (Table 1).

Table 1. Nikkei 225 equity index regression output.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 1 and 2 captures the linear relationship between the USD/CHY FX spot rate and the US term structure with respect to the Nikkei equity index.

Figure 1. Relationship between the USD/CHY FX spot rate with respect to the Nikkei 225 equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 2. Relationship between the US term structure with respect to the Nikkei 225 equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

We conduct a second regression of the Nasdaq US equity index returns on the retained macroeconomic variables. We may emphasize the following: Both factors are not statistically significant at a 10% significance level, indicating that they have a limited ability to predict Nasdaq returns during the past two decades. The model has a low R2 of 4.45%, indicating that only 4.45% of the performance of the Nasdaq can be attributable to the change in the USD/CHY FX spot rate and the US term structure of the yield curve (Table 2).

Table 2. Nasdaq equity index regression output.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 3 and 4 captures the linear relationship between the USD/CHY FX spot rate and the US term structure with respect to the Nasdaq equity index.

Figure 3. Relationship between the USD/CHY FX spot rate with respect to the Nasdaq equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Figure 4. Relationship between the US term structure with respect to the Nasdaq equity index.
Time-series regression
Source: computation by the author (Data: FRED Economics)

Applying APT

We can create a portfolio with similar factor sensitivities as the Arbitrage Portfolio by combining the first two index portfolios (with a Nasdaq Index weight of 40% and a Nikkei Index weight of 60%). This is referred to as the Constructed Index Portfolio. The Arbitrage portfolio will have a full weighting on US equity index (100% Nasdaq equity index). The Constructed Index Portfolio has the same systematic factor betas as the Arbitrage Portfolio, but has a higher expected return (Table 3).

Table 3. Index, constructed and Arbitrage portfolio return and sensitivity table. img_SimTrade_portfolio_sensitivity
Source: computation by the author (Data: FRED Economics)

As a result, the Arbitrage portfolio is overvalued. We will then buy shares of the Constructed Index Portfolio and use the profits to sell shares of the Arbitrage Portfolio. Because every investor would sell an overvalued portfolio and purchase an undervalued portfolio, any arbitrage profit would be wiped out.

Excel file for the APT application

You can find below the Excel spreadsheet that complements the example above.

Why should I be interested in this post?

In the CAPM, the factor is the market factor representing the global uncertainty of the market. In the late 1970s, the portfolio management industry aimed to capture the market portfolio return, but as financial research advanced and certain significant contributions were made, this gave rise to other factor characteristics to capture some additional performance. Analyzing the historical contributions that underpins factor investing is fundamental in order to have a better understanding of the subject.

Useful resources

Academic research

Lintner, J. (1965) The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. The Review of Economics and Statistics 47(1): 13-37.

Lintner, J. (1965) Security Prices, Risk and Maximal Gains from Diversification. The Journal of Finance 20(4): 587-615.

Roll, Richard & Ross, Stephen. (1995). The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning. Financial Analysts Journal 51, 122-131.

Ross, S. (1976) The arbitrage theory of capital asset pricing Journal of Economic Theory 13(3), 341-360.

Sharpe, W.F. (1963) A Simplified Model for Portfolio Analysis. Management Science 9(2): 277-293.

Sharpe, W.F. (1964) Capital Asset Prices: A theory of Market Equilibrium under Conditions of Risk. The Journal of Finance 19(3): 425-442.

Yu, G., P. Fuller, D. Didia (2013) The Impact of Yield Slope on Stock Performance Southwestern Economic Review 40(1): 1-10.

Business Analysis

Barrons (2022) Apple, Nike, and 6 Other Companies With Big Exposure to China.

The Economist (2019) Japan Inc has thrived in China of late.

Time series

FRED Economics (2022) Chinese Yuan Renminbi to U.S. Dollar Spot Exchange Rate (DEXCHUS).

FRED Economics (2022) 10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity (T10Y2Y).

FRED Economics (2022) NASDAQ Composite Index (NASDAQCOM).

FRED Economics (2022) Nikkei Stock Average, Nikkei 225 (NIKKEI225).

About the author

The article was written in January 2023 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Interest Rate Swaps

December 29, 2022 by Akshit GUPTA

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the derivative contract of interest rate swaps used to hedge interest rate risk in financial markets.

Introduction

In financial markets, interest rate swaps are derivative contracts used by two counterparties to exchange a stream of future interest payments with another for a pre-defined number of years. The interest payments are based on a pre-determined notional principal amount and usually include the exchange of a fixed interest rate for a floating interest rate (or sometimes the exchange of a floating interest rate for another floating interest rate).

While hedging does not necessarily eliminate the entire risk for any investment, it does limit or offset any potential losses that the investor can incur.

Forward rate agreements (FRA)

To understand interest rate swaps, we first need to understand forward rate agreements in financial markets.

In an FRA, two counterparties agree to an exchange of cashflows in the future based on two different interest rates, one of which is a fixed rate and the other is a floating rate. The interest rate payments are based on a pre-determined notional amount and maturity period. This derivative contract has a single settlement date. LIBOR (London Interbank Offered Rate) is frequently used as the floating rate index to determine the floating interest rate in the swap.

The payoff of the contract is as shown in the formula below:

(LIBOR – Fixed Interest Rate) * Notional amount * Number of days / 100

Interest rate swaps (IRS)

An interest rate swap is a hedging mechanism wherein a pre-defined series of forward rate agreements to buy or sell the floating interest rate at the same fixed interest rate.

In an interest rate swap, the position taken by the receiver of the fixed interest rate is called “long receiver swaps” and the position taken by the payer of the fixed interest rate is called “long payer swaps”.

How does an interest rate swap work?

Interest rate swaps can be used in different market situations based on a counterparty’s prediction about future interest rates.

For example, when a firm paying a fixed rate of interest on an existing loan believes that the interest rate will decrease in the future, it may enter an interest rate swap agreement in which it pays a floating rate and receives a fixed rate to benefit from its expectation about the path of future interest rates. Conversely, if the firm paying a floating interest rate on an existing loan believes that the interest rate will increase in the future, it may enter an interest rate swap in which it pays a fixed rate and receives a floating rate to benefit from its expectation about the path of future interest rates.

Example

Let’s consider a 4-year swap between two counterparties A and B on January 1, 2021. In this swap, counterparty A agrees to pay a fixed interest rate of 3.60% per annum to counterparty B every six months on an agreed notional amount of €10 million. Counterparty B agrees to pay a floating interest rate based on the 6-month LIBOR rate, currently at 2.60%, to Counterparty A on the same notional amount. Here, the position taken by Counterparty A is called long payer swap and the position taken by Counterparty B is called the long receiver swap. The projected cashflow receipt to Counterparty A based on the assumed LIBOR rates is shown in the below table:

Table 1. Cash flows for an interest rate swap.

Source: computation by the author

In the above example, a total of eight payments (two per year) are made on the interest rate swap. The fixed rate payment is fixed at €180,000 per observation date whereas the floating payment rate depend on the prevailing LIBOR rate at the observation date. The net receipt for the Counterparty A is equal to €77,500 at the end of 5 years. Note that in an interest rate swap the notional amount of €10 million is not exchanged between the counterparties since it has no financial value to either of the counterparties and that is why it is called the “notional amount”.

Note that when the two counterparties enter the swap, the fixed rate is set such that the swap value is equal to zero.

Excel file for interest rate swaps

You can download below the Excel file for the computation of the cash flows for an interest rate swap.

▶ Jayati WALIA Derivative Markets

▶ Akshit GUPTA Forward Contracts

▶ Akshit GUPTA Options

Useful resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 7 – Swaps, 180-211.

www.longin.fr Pricer of interest swaps

About the author

Article written in December 2022 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Asset allocation techniques

December 19, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the concept of asset allocation, a pillar concept in portfolio management.

This article is structured as follows: we introduce the notion of asset allocation, and we use a practical example to illustrate this notion.

Introduction

An investment portfolio is a collection of assets that are owned by an investor. Individual assets, such as bonds and stocks, as well as asset baskets, such as mutual funds or exchange-traded funds, can be employed. When constructing a portfolio, investors often consider both the projected return and risk. A well-balanced portfolio includes a wide range of investments to benefit from diversification.

The asset allocation is one of the processes in the portfolio construction process. At this point, the investor (or fund manager) must divide the available capital into a number of assets that meet the criteria in terms of risk and return trade-off, while adhering to the investment policy, which specifies the amount of exposure an investor can have and the amount of risk the fund manager can hold in his or her portfolio.

The next phase in the process is to evaluate the risk and return characteristics of the various assets. The analyst develops economic and market expectations that can be used to develop a recommended asset allocation for the customer. The distribution of equities, fixed-income securities, and cash; sub asset classes, such as corporate and government bonds; and regional weightings within asset classes are all decisions that must be taken in the portfolio’s asset allocation. Real estate, commodities, hedge funds, and private equity are examples of alternative assets. Economists and market strategists may set the top-down view on economic conditions and broad market movements. The returns on various asset classes are likely to be altered by economic conditions; for example, equities may do well when economic growth has been surprisingly robust whereas bonds may do poorly if inflation soars. These situations will be forecasted by economists and strategists.

The top-down approach

A top-down approach begins with assessment of macroeconomic factors. The investor examines markets and sectors based on the existing and projected economic climate in order to invest in those that are predicted to perform well. Finally, funding is evaluated for specific companies within these categories.

The bottom up approach

A bottom-up approach focuses on company-specific variables such as management quality and business potential rather than economic cycles or industry analysis. It is less concerned with broad economic trends than top-down analysis is, and instead focuses on company particular.

Types of asset allocations

Arnott and Fabozzi (1992) divide asset allocation into three types: 1) policy asset allocation; 2) dynamic asset allocation; and 3) tactical asset allocation.

Policy asset allocation

The policy asset allocation decision is a long-term asset allocation decision in which the investor aims to assess a suitable long-term “normal” asset mix that represents an optimal mixture of controlled risk and enhanced return. The strategies that offer the best prospects of achieving strong long-term returns are inherently risky. The strategies that offer the greatest safety tend to offer very moderate return opportunities. The balancing of these opposing goals is known as policy asset allocation. The asset mix (i.e., the allocation among asset classes) is mechanistically altered in response to changing market conditions in dynamic asset allocation. Once the policy asset allocation has been established, the investor can focus on the possibility of active deviations from the regular asset mix established by policy. Assume the long-run asset mix is established to be 60% equities and 40% bonds. A variation from this mix under certain situations may be tolerated. A decision to diverge from this mix is generally referred to as tactical asset allocation if it is based on rigorous objective measurements of value. Tactical asset allocation does not consist of a single, well-defined strategy.

Dynamic asset allocation

The term “dynamic asset allocation” can refer to both long-term policy decisions and intermediate-term efforts to strategically position the portfolio to benefit from big market swings, as well as aggressive tactical strategies. As an investor’s risk expectations and tolerance for risk fluctuate, the normal or policy asset allocation may change. It is vital to understand what aspect of the asset allocation decision is being discussed and in what context the words “asset allocation” are being used when delving into asset allocation difficulties.

Tactical asset allocation

Tactical asset allocation broadly refers to active strategies that seek to enhance performance by opportunistically adjusting the asset mix of a port- folio in response to the changing patterns of reward available in the capi- tal markets. Notably, tactical asset allocation tends to refer to disciplined techniques for evaluating anticipated rates of return on various asset classes and constructing an asset allocation response intended to capture larger rewards.

Asset allocation application: an example

For this example, lets suppose the fictitious following scenario with real data involved:

Mr. Dubois recently sold his local home construction company in the south of France to a multinational homebuilder with a nationwide reach. He accepted a job as regional manager for that national homebuilder after selling his company. He is now thinking about the financial future for himself and his family. He is looking forward to his new job, where he enjoys his new role and where he will earn enough money to meet his family’s short- and medium-term liquidity demands. He feels strongly that he should not invest the profits of the sale of his company in real estate because his income currently rely on the state of the real estate market. He speaks with a financial adviser at his bank about how to invest his money so that he can retire comfortably in 20 years.

The initial portfolio objective they created seek a nominal return goal of 7% with a Sharpe ratio of at least 1 (for this example, we consider the risk-free rate to be equal to zero). The bank’s asset management division gives Mr Dubois and his adviser with the following data (Figure 1) on market expectations.

Figure 1. Risk, return and correlation estimates on market expectation.
Time-series regression
Source: computation by the author (Data: Refinitiv Eikon).

In order to replicate a global asset allocation approach, we shortlisted a number of trackers that would represent our investment universe. To keep a well-balanced approach, we took trackers that would represent the main asset classes: global equities (VTI – Vanguard Total Stock Market ETF), bonds (IEF – iShares 7-10 Year Treasury Bond ETF and TLT – iShares 20+ Year Treasury Bond ETF) and commodities (DBC – Invesco DB Commodity Index Tracking Fund and GLD – SPDR Gold Shares). To create the optimal asset allocation, we extracted the equivalent of a ten-year timeframe from Refinitiv Eikon to capture the overall performance of the portfolio in the long run. As captured in Figure 1, the global equities was the best performing asset class during the period covered (13.02% annualised return), followed by long term bond (4.78% annualised return) and by gold (4.65% annualised return).

Figure 2. Asset class performance (rebased to 100).
Time-series regression
Source: computation by the author (Data: Refinitiv Eikon).

After analyzing the historical return on the assets retained, as well as their volatility and covariance (and correlation), we can apply Mean-Variance portfolio optimization to determine the optimal portfolio. The optimal asset allocation would be the end outcome of the optimization procedure. The optimal portfolio, according to Markowitz’ seminal study on portfolio construction, will seek to create the best risk-return trade-off for an investor. After performing the calculations, we notice that investing 42.15% in the VTI fund, 30.69% in the IEF fund, 24.88% in the TLT fund, and 2.28% in the GLD fund yields the best asset allocation. As reflected in this asset allocation, the investor intends to invest his assets in a mix of equities (about 43%) and bonds (approximately 55%), with a marginal position (around 3%) in gold, which is widely employed in portfolio management as an asset diversifier due to its correlation with other asset classes. As captured by this asset allocation, we can clearly see the defensive nature of this portfolio, which relies significantly on the bond part of the allocation to operate as a hedge while relying on the equities part as the main driver of returns.

As shown in Figure 3, the optimal asset allocation has a better Sharpe ratio (1.27 vs 0.62) and is captured farther along the efficient frontier line than a naive equally-weighted allocation . The only portfolio with the needed characteristics is the optimal one, as the investor’s goal was to attain a 7% projected return with a minimum Sharpe ratio of 1.

Figure 3. Optimal asset allocation and the Efficient Frontier plot.
Time-series regression
Source: computation by the author (Data: Refinitiv Eikon).

Will this allocation, however, continue to perform well in the future? The market’s reliance on future expectations, return, volatility, and correlation predictions, as well as the market regime, will ultimately determine how much the performance predicted by this study will really change in the future.

Excel file for asset allocation

You can find below the Excel spreadsheet that complements the example above.

Why should I be interested in this post?

The purpose of portfolio management is to maximize (expected) returns on the entire portfolio, not just on one or two stocks for a given level of risk. By monitoring and maintaining your investment portfolio, you can build a substantial amount of wealth for a variety of financial goals, such as retirement planning. This post facilitates comprehension of the fundamentals underlying portfolio construction and investing.

Useful resources

Academic research

Arnott, R. D., and F. J. Fabozzi. 1992. The many dimensions of the asset allocation decision. In Active asset allocation, edited by R. Arnott and F. J. Fabozzi. Chicago: Probus Publishing.

Fabozzi, F.J., 2009. Institutional Investment Management: Equity and Bond Portfolio Strategies and Applications. I (4-6). John Wiley and Sons Edition.

Pamela, D. and Fabozzi, F., 2010. The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management. John Wiley and Sons Edition.

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Quantitative equity investing

December 19, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of quantitative equity investing, a type of investment approach in the equity trading space.

This article follows the following structure: we introduce the quantitative equity investing. We present a review of the major types of quantitative equity strategies and we finish with a conclusion.

Introduction

Quantitative equity investing refers to funds that uses model-driven decision making when trading in the equity space. Quantitative analysts program their trading rules into computer systems and use algorithmic trading, which is overseen by humans.

Quantitative investing has several advantages and disadvantages over discretionary trading. The disadvantages are that the trading rule cannot be as personalized to each unique case and cannot be dependent on “soft” information such human judgment. These disadvantages may be lessened as processing power and complexity improve. For example, quantitative models may use textual analysis to examine transcripts of a firm’s conference calls with equity analysts, determining whether certain phrases are commonly used or performing more advanced analysis.

The advantages of quantitative investing include the fact that it may be applied to a diverse group of stocks, resulting in great diversification. When a quantitative analyst builds an advanced investment model, it can be applied to thousands of stocks all around the world at the same time. Second, the quantitative modeling rigor may be able to overcome many of the behavioral biases that commonly impact human judgment, including those that produce trading opportunities in the first place. Third, using past data, the quant’s trading principles can be backtested (Pedersen, 2015).

Types of quantitative equity strategies

There are three types of quantitative equity strategies: fundamental quantitative investing, statistical arbitrage, and high-frequency trading (HFT). These three types of quantitative investing differ in various ways, including their conceptual base, turnover, capacity, how trades are determined, and their ability to be backtested.

Fundamental quantitative investing

Fundamental quantitative investing, like discretionary trading, tries to use fundamental analysis in a systematic manner. Fundamental quantitative investing is thus founded on economic and financial theory, as well as statistical data analysis. Given that prices and fundamentals only fluctuate gradually, fundamental quantitative investing typically has a turnover of days to months and a high capacity (meaning that a large amount of money can be invested in the strategy), owing to extensive diversification.

Statistical arbitrage

Statistical arbitrage aims to capitalize on price differences between closely linked stocks. As a result, it is founded on a grasp of arbitrage relations and statistics, and its turnover is often faster than that of fundamental quants. Statistical arbitrage has a lower capacity due to faster trading (and possibly fewer stocks having arbitrage spreads).

High Frequency Trading (HFT)

HFT is based on statistics, information processing, and engineering, as the success of an HFT is determined in part by the speed with which they can trade. HFTs focus on having superfast computers and computer programs, as well as co-locating their computers at exchanges, actually trying to get their computer as close to the exchange server as possible, using fast cables, and so on. HFTs have the fastest trading turnover and, as a result, the lowest capacity.

The three types of quants also differ in how they make trades: Fundamental quants typically make their deals ex ante, statistical arbitrage traders make their trades gradually, and high-frequency traders let the market make their transactions. A fundamental quantitative model, for example, identifies high-expected-return stocks and then buys them, almost always having their orders filled; a statistical arbitrage model seeks to buy a mispriced stock but may terminate the trading scheme before completion if prices have moved adversely; and, finally, an HFT model may submit limit orders to both buy and sell to several exchanges, allowing the market to determine which ones are hit. Because of this trading structure, fundamental quant investing can be simulated with some reliability via a backtest; statistical arbitrage backtests rely heavily on assumptions on execution times, transaction costs, and fill rates; and HFT strategies are frequently difficult to simulate reliably, so HFTs must rely on experiments.

Table 1. Quantitative equity investing main categories and characteristics.

Source: Source: Pedersen, 2015.

Conclusion

Quants run their models on hundreds, if not thousands, of stocks. Because diversification eliminates most idiosyncratic risk, firm-specific shocks tend to wash out at the portfolio level, and any single position is too tiny to make a major impact in performance.

An equity market neutral portfolio eliminates total stock market risk by being equally long and short. Some quants attempt to establish market neutrality by ensuring that the long side’s dollar exposure equals the dollar worth of all short bets. This technique, however, is only effective if the longs and shorts are both equally risky. As a result, quants attempt to balance market beta on both the long and short sides. Some quants attempt to be both dollar and beta neutral.

Why should I be interested in this post?

It may provide an opportunity for investors to diversify their global portfolios. Including hedge funds in a portfolio can help investors obtain absolute returns that are uncorrelated with typical bond/equity returns.

For practitioners, learning how to incorporate hedge funds into a standard portfolio and understanding the risks associated with hedge fund investing can be beneficial.

Understanding if hedge funds are truly providing “excess returns” and deconstructing the sources of return can be beneficial to academics. Another challenge is determining whether there is any “performance persistence” in hedge fund returns.

Getting a job at a hedge fund might be a profitable career path for students. Understanding the market, the players, the strategies, and the industry’s current trends can help you gain a job as a hedge fund analyst or simply enhance your knowledge of another asset class.

Useful resources

Academic research

Pedersen, L. H., 2015. Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined. Chapter 9 : 133 – 164. Princeton University Press.

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Optimal portfolio

December 19, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the concept of optimal portfolio, which is central in portfolio management.

This article is structured as follows: we first define the notion of an optimal portfolio (in the mean-variance framework) and we then illustrate the concept of optimal portfolio with an example.

Introduction

An investor’s investment portfolio is a collection of assets that he or she possesses. Individual assets such as bonds and equities can be used, as can asset baskets such as mutual funds or exchange-traded funds (ETFs). When constructing a portfolio, investors typically evaluate the expected return as well as the risk. A well-balanced portfolio contains a diverse variety of investments.

An optimal portfolio is a collection of assets that maximizes the trade-off between expected return and risk: the portfolio with the highest expected return for a given level of risk, or the portfolio with the lowest risk for a given level of expected return.

To obtain the optimal portfolio, Markowitz sought to optimize the following dual program:

The first optimization seeks to maximize expected return with respect to a specific level of risk, subject to the short-selling constraint (weights of the portfolio should be equal to one).

img_SimTrade_implementing_Markowitz_2

The second optimization seeks to minimize the variance of the portfolio with respect to a specific level of expected return, subject to the short-selling constraint (weights of the portfolio should be equal to one).

img_SimTrade_implementing_Markowitz

Mathematical foundations

The investment universe is composed of N assets characterized by their expected returns μ and variance-covariance matrix V. For a given level of expected return μ_P, the Markowitz model gives the composition of the optimal portfolio. The vector of weights of the optimal portfolio is given by the following formula:

With the following notations:

w_P = vector of asset weights of the portfolio
μ_P = desired level of expected return
e = identity vector
μ = vector of expected returns
V = variance-covariance matrix of returns
V^-1 = inverse of the variance-covariance matrix
t = transpose operation for vectors and matrices

A, B and C are intermediate parameters computed below:

img_SimTrade_implementing_Markowitz_2

The variance of the optimal portfolio is computed as follows

img_SimTrade_implementing_Markowitz_3

To calculate the standard deviation of the optimal portfolio, we take the square root of the variance.

Optimal portfolio application: the case of two assets

To create the optimal portfolio, we first obtain monthly historical data for the last two years from Bloomberg for two stocks that will comprise our portfolio: Apple and CMS Energy Corporation. Apple is in the technology area, but CMS Energy Corporation is an American company that is entirely in the energy sector. Apple’s historical return for the two years covered was 41.86%, with a volatility of 35.11%. Meanwhile, CMS Energy Corporation’s historical return was 13.95% with a far lower volatility of 15.16%.

According to their risk and return profiles, Apple is an aggressive stock pick in our example, but CMS Energy is a much more defensive stock that would serve as a hedge in our example. The correlation between the two stocks is 0.19, indicating that they move in the same direction. In this example, we will consider the market portfolio, defined as a theoretical portfolio that reflects the return of the whole investment universe, which is captured by the wide US equities index (S&P500 index).

As captured in Figure 1, CMS Energy suffered less severe losses than Apple. When compared to the red bars, the blue bars are far more volatile and sharp in terms of the size of the move in both directions.

Figure 1. Apple and CMS Energy Corporation return breakdown.
Time-series regression
Source: computation by the author (Data: Bloomberg)

After analyzing the historical return on both stocks, as well as their volatility and covariance (and correlation), we can use Mean-Variance portfolio optimization to find the optimal portfolio. According to Markowitz’ foundational study on portfolio construction, the optimal portfolio will attempt to achieve the best risk-return trade-off for an investor. After doing the computations, we discover that the optimal portfolio is composed of 45% Apple stock and 55% CMS Energy corporation stock. This portfolio would return 26.51% with a volatility of 19.23%. As captured in Figure 2, the optimal portfolio is higher on the efficient frontier line and has a higher Sharpe ratio (1.27 vs 1.23 for the theoretical market portfolio).

Figure 2. Optimal portfolio.
Optimal portfolio plot 2 asset
Source: computation by the author (Data: Bloomberg)

You can find below the Excel spreadsheet that complements the example above.

Optimal portfolio application: the general case

We generated a large time series to obtain useful results by downloading the equivalent of 23 years of market data from a data provider (in this example, Bloomberg). We generate the variance-covariance matrix after obtaining all necessary statistical data, which includes the expected return and volatility indicated by the standard deviation of the returns for each stock during the provided period. Table 1 depicts the expected return and volatility for each stock retained in this analysis.

Table 1. Asset characteristics of Apple, Amazon, Microsoft, Goldman Sachs, and Pfizer.
img_SimTrade_implementing_Markowitz_spreadsheet_1
Source: computation by the author.

We can start the optimization task by setting a desirable expected return after computing the expected return, volatility, and the variance-covariance matrix of expected return. With the data that is fed into the appropriate cells, the model will complete the optimization task. For a 20% desired expected return, we get the following results (Table 2).

Table 2. Asset weights for an optimal portfolio.
Optimal portfolio case 1
Source: computation by the author.

To demonstrate the effect of diversification in the reduction of volatility, we can form a Markowitz efficient frontier by tilting the desired anticipated return with their relative volatility in a graph. The Markowitz efficient frontier is depicted in Figure 1 for various levels of expected return. We highlighted the portfolio with 20% expected return with its respective volatility in the plot (Figure 3).

Figure 3. Optimal portfolio plot for 5 asset case.
Optimal portfolio case 1
Source: computation by the author.

You can download the Excel file below to use the Markowitz portfolio allocation model.

Why should I be interested in this post?

The purpose of portfolio management is to maximize the (expected) returns on the entire portfolio, not just on one or two stocks. By monitoring and maintaining your investment portfolio, you can build a substantial amount of wealth for a variety of financial goals such as retirement planning. This post facilitates comprehension of the fundamentals underlying portfolio construction and investing.

Useful resources

Academic research

Pamela, D. and Fabozzi, F., 2010. The Basics of Finance: An Introduction to Financial Markets, Business Finance, and Portfolio Management. John Wiley and Sons Edition.

Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Long-short equity strategy

December 17, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the long-short equity strategy, one of pioneer strategies in the hedge fund industry. The goal of the long-short equity investment strategy is to buy undervalued stocks and sell short overvalued ones.

This article is structured as follow: we introduce the long-short strategy principle. Then, we present a practical case study to grasp the overall methodology of this strategy. We conclude with a performance analysis of this strategy in comparison with a global benchmark (MSCI All World Index).

Introduction

According to Credit Suisse, a long-short strategy can be defined as follows: “Long-short equity funds invest on both long and short sides of equity markets, generally focusing on diversifying or hedging across particular sectors, regions, or market capitalizations. Managers have the flexibility to shift from value to growth; small to medium to large capitalization stocks; and net long to net short. Managers can also trade equity futures and options as well as equity related securities and debt or build portfolios that are more concentrated than traditional long-only equity funds.”

This strategy has the particularity of potentially generate returns in both rising and falling markets. However, stock selection is key concern, and the stock picking ability of the fund manager is what makes this strategy profitable (or not!). The trade-off of this approach is to reduce market risk but exchange it for specific risk. Another key characteristic of this type of strategy is that overall, funds relying on long-short are net long in their trading exposure (long bias).

Equity strategies

In the equity universe, we can separate long-short equity strategies into discretionary long-short equity, dedicated short bias, and quantitative.

Discretionary long-short

Discretionary long-short equity managers typically decide whether to buy or sell stocks based on a basic review of the value of each firm, which includes evaluating its growth prospects and comparing its profitability to its valuation. By visiting managers and firms, these fund managers also evaluate the management of the company. Additionally, they investigate the accounting figures to judge their accuracy and predict future cash flows. Equity long-short managers typically predict on particular companies, but they can also express opinions on entire industries.

Value investors, a subset of equity managers, concentrate on acquiring undervalued companies and holding these stocks for the long run. A good illustration of a value investor is Warren Buffett. Since companies only become inexpensive when other investors stop investing in them, putting this trading approach into practice frequently entails being a contrarian (buy assets after a price decrease). Because of this, cheap stocks are frequently out of favour or purchased while others are in a panic. Traders claim that deviating from the standard is more difficult than it seems.

Dedicated short bias

Like equity long-short managers, dedicated short bias is a trading technique that focuses on identifying companies to sell short. Making a prediction that the share price will decline is known as short selling. Similar to how purchasing stock entails profiting if the price increases, holding a short position entail profiting if the price decreases. Dedicated short-bias managers search for companies that are declining. Since dedicated short-bias managers are working against the prevailing uptrend in markets since stocks rise more frequently than they fall (this is known as the equity risk premium), they make up a very small proportion of hedge funds.

Most hedge funds in general, as well as almost all equity long-short hedge funds and dedicated short-bias hedge funds, engage in discretionary trading, which refers to the trader’s ability to decide whether to buy or sell based on his or her judgement and an evaluation of the market based on past performance, various types of information, intuition, and other factors.

Quantitative

The quantitative investment might be seen as an alternative to this traditional style of trading. Quants create systems that methodically carry out the stated definitions of their trading rules. They use complex processing of ideas that are difficult to analyse using non-quantitative methods to gain a slight advantage on each of the numerous tiny, diversified trades. To accomplish this, they combine a wealth of data with tools and insights from a variety of fields, including economics, finance, statistics, mathematics, computer science, and engineering, to identify relationships that market participants may not have immediately fully incorporated in the price. Quantitative traders use computer systems that use these relationships to generate trading signals, optimise portfolios considering trading expenses, and execute trades using automated systems that send hundreds of orders every few seconds. In other words, data is fed into computers that execute various programmes under the supervision of humans to conduct trading (Pedersen, 2015).

Example of a long-short equity strategy

The purpose of employing a long-short strategy is to profit in both bullish and bearish markets. To measure the profitability of this strategy, we implemented a long-short strategy from the beginning of January 2022 to June 2022. In this time range, we are long Exxon Mobile stock and short Tesla. The data are extracted from the Bloomberg terminal. The strategy of going long Exxon Mobile and short Tesla is purely educational. This strategy’s basic idea is to profit from rising oil prices (leading to a price increase for Exxon Mobile) and rising interest rates (leading to a price decrease for Tesla). Over the same period, the S&P 500 index has dropped 23%, while the Nasdaq Composite has lost more than 30%. The Nasdaq Composite is dominated by rapidly developing technology companies that are especially vulnerable to rising interest rates.

Overall, the market’s net exposure is zero because we are 100% long Exxon Mobile and 100% short Tesla stock. This strategy succeeded to earn significant returns in both the long and short legs of the trade over a six-month timeframe. It yielded a 99.5 percent return, with a 36.8 percent gain in the value of the Exxon Mobile shares and a 62.8 percent return on the short Tesla position. Figure 1 shows the overall performance of each equity across time.

Figure 1. Long-short equity strategy performance over time
Time-series regression
Source: computation by the author (Data: Bloomberg)

You can find below the Excel spreadsheet that complements the example above.

Performance of the long-short equity strategy

To capture the performance of the long-short equity strategy, we use the Credit Suisse hedge fund strategy index. To establish a comparison between the performance of the global equity market and the long-short hedge fund strategy, we examine the rebased performance of the Credit Suisse index with respect to the MSCI All-World Index. Over a period from 2002 to 2022, the long-short equity strategy index managed to generate an annualised return of 5.96% with an annualised volatility of 7.33%, leading to a Sharpe ratio of 0.18. Over the same period, the MSCI All World Index managed to generate an annualised return of 6.00% with an annualised volatility of 15.71%, leading to a Sharpe ratio of 0.11. The low correlation of the long-short equity strategy with the MSCI All World Index is equal to 0.09, which is closed to zero. Overall, the Credit Suisse hedge fund strategy index performed somewhat slightly worse than the MSCI All World Index, but presented a much lower volatility leading to a higher Sharpe ratio (0.18 vs 0.11).

Figure 2. Performance of the long-short equity strategy compared to the MSCI All-World Index across time.
Time-series regression
Source: computation by the author (Data: Bloomberg)

You can find below the Excel spreadsheet that complements the explanations about the Credit Suisse hedge fund strategy index.

Why should I be interested in this post?

Long-short funds seek to reduce negative risk while increasing market upside. They might, for example, invest in inexpensive stocks that the fund managers believe will rise in price while simultaneously shorting overvalued stocks to cut losses. Other strategies used by long-short funds to lessen market volatility include leverage and derivatives. Understanding the profits and risks of such a strategy might assist investors in incorporating this hedge fund strategy into their portfolio allocation.

Useful resources

Academic research

Pedersen, L. H., 2015. Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined. Princeton University Press.

Business Analysis

BlackRock Long-short strategy

BlackRock Investment Outlook

Credit Suisse Hedge fund strategy

Credit Suisse Hedge fund performance

Credit Suisse Long-short strategy

Credit Suisse Long-short performance benchmark

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth two-step regression method: the case of K risk factors

December 12, 2022 by Youssef LOURAOUI

Fama-MacBeth two-step regression method: the case of K risk factors

This article is structured as follows: we introduce the Fama-MacBeth two-step regression method. Then, we present the mathematical foundation that underpins their approach for K risk factors. We provide an illustration for the 3-factor mode developed by Fama and French (1993).

Introduction

Risk factors are frequently employed to explain asset returns in asset pricing theories. These risk factors may be macroeconomic (such as consumer inflation or unemployment) or microeconomic (such as firm size or various accounting and financial metrics of the firms). The Fama-MacBeth two-step regression approach found a practical way for measuring how correctly these risk factors explain asset or portfolio returns. The aim of the model is to determine the risk premium associated with the exposure to these risk factors.

The first step is to regress the return of every asset against one or more risk factors using a time-series approach. We obtain the return exposure to each factor called the “betas” or the “factor exposures” or the “factor loadings”.

The second step is to regress the returns of all assets against the asset betas obtained in Step 1 using a cross-section approach. We obtain the risk premium for each factor. Then, Fama and MacBeth assess the expected premium over time for a unit exposure to each risk factor by averaging these coefficients once for each element.

Mathematical foundations

We describe below the mathematical foundations for the Fama-MacBeth regression method for a K-factor application. In the analysis, we investigated the Fame-French three factor model in order to understand their significance as a fundamental driver of returns for investors under the Fama-MacBeth framework.

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed every day over the time period [0, T].
The risk factors denoted by F_k for k equal from 1 to K.

Step 1: time-series analysis of returns

For each asset i from 1 to N, we estimate the following linear regression model:

Fama-French time-series regression

From this model, we obtain the β_{i, Fk} which is the beta associated with the k^th risk factor.

Step 2: cross-sectional analysis of returns

For each period t from 1 to T, we estimate the following linear regression model:

Fama-French cross-sectional regression

Application: the Fama-French 3-Factor model

The Fama-French 3-factor model is an illustration of Fama-MacBeth two-step regression method in the case of K risk factors (K=3). The three factors are the market (MKT) factor, the small minus big (SMB) factor, and the high minus low (HML) factor. The SMB factor measures the difference in expected returns between small and big firms (in terms of market capitalization). The HML factor measures the difference in expected returns between value stocks and growth stock.

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed every day over the time period [0, T].
The risk factors denoted by F_MKT associated to the MKT risk factor, F_SMB associated to the MKT risk factor which measures the difference in expected returns between small and big firms (in terms of market capitalization) and F_HML associated to 𝐻𝑀𝐿 (“High Minus Low”) which measures the difference in expected returns between value stocks and growth stock

Step 1: time-series regression

img_SimTrade_Fama_French_time_series_regression

Step 2: cross-sectional regression

img_SimTrade_Fama_French_cross_sectional_regression

Figure 1 represents for a given period the cross-sectional regression of the return of all individual assets with respect to their estimated individual beta for the MKT factor.

Figure 1. Cross-sectional regression for the market factor.
Cross-section regression for the MKT factor Source: computation by the author.

Figure 2 represents for a given period the cross-sectional regression of the return of all individual assets with respect to their estimated individual beta for the SMB factor.

Figure 2. Cross-sectional regression for the SMB factor.
Cross-section regression for the SMB factor Source: computation by the author.

Figure 3 represents for a given period the cross-sectional regression of the return of all individual assets with respect to their estimated individual beta for the SMB factor.

Figure 3. Cross-sectional regression for the HML factor.
Cross-section regression for the HML factor Source: computation by the author.

Empirical study of the Fama-MacBeth regression

Fama-MacBeth seminal paper (1973) was based on an analysis of the market factor by assessing constructed portfolios of similar betas ranked by increasing values. This approach helped to overcome the shortcoming regarding the stability of the beta and correct for conditional heteroscedasticity derived from the computation of the betas for individual stocks. They performed a second time the cross-sectional regression of monthly portfolio returns based on equity betas to account for the dynamic nature of stock returns, which help to compute a robust standard error and assess if there is any heteroscedasticity in the regression. The conclusion of the seminal paper suggests that the beta is “dead”, in the sense that it cannot explain returns on its own (Fama and MacBeth, 1973).

Empirical study: Stock approach for a K-factor model

We collected a sample of 440 significant firms’ end-of-day stock prices in the US economy from January 3, 2012 to December 31, 2021. We calculated daily returns for each stock as well as the factor used in this analysis. We chose the S&P500 index to represent the market since it is an important worldwide stock benchmark that captures the US equities market.

Time-series regression

To assess the multi-factor regression, we used the Fama-MacBeth 3-factor model as the main factors assessed in this analysis. We regress the average returns for each stock against their factor betas. The first regression is statistically tested. This time-series regression is run on a subperiod of the whole period from January 03, 2012, to December 31, 2018. We use a t-statistic to explain the regression’s behavior. Because the p-value is in the rejection zone (less than the significance level of 5%), we can conclude that the factors can first explain an investor’s returns. However, as we will see later in the article, when we account for a second regression as proposed by Fama and MacBeth, the factors retained in this analysis are not capable of explaining the return on asset returns on its own. The stock approach produces statistically significant results in time-series regression at 10%, 5%, and even 1% significance levels. As shown in Table 1, the p-value is in the rejection range, indicating that the factors are statistically significant.

Table 1. Time-series regression t-statistic result.
Cross-section regression Source: computation by the author.

Cross-sectional regression

Over a second period from January 04, 2019, to December 31, 2021, we compute the dynamic regression of returns at each data point in time with respect to the betas computed in Step 1.

That being said, when the results are examined using cross-section regression, they are not statistically significant, as indicated by the p-value in Table 2. We are unable to reject the null hypothesis. The premium investors are evaluating cannot be explained solely by the factors assessed. This suggests that factors retained in the analysis fail to adequately explain the behavior of asset returns. These results are consistent with the Fama-MacBeth article (1973).

Table 2. Cross-section regression t-statistic result.
Source: computation by the author.

Excel file

You can find below the Excel spreadsheet that complements the explanations covered in this article.

Why should I be interested in this post?

Fama-MacBeth made a significant contribution to the field of econometrics. Their findings cleared the way for asset pricing theory to gain traction in academic literature. The Capital Asset Pricing Model (CAPM) is far too simplistic for a real-world scenario since the market factor is not the only source that drives returns; asset return is generated from a range of factors, each of which influences the overall return. This framework helps in capturing other sources of return.

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth regression method: the stock approach vs the portfolio approach

December 12, 2022 by Youssef LOURAOUI

Fama-MacBeth regression method: the stock approach vs the portfolio approach

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Fama-MacBeth regression method used to test asset pricing models and addresses the difference when applying the regression method on individual stocks or portfolios composed of stocks with similar betas.

This article is structured as follow: we introduce the Fama-MacBeth testing method. Then, we present the mathematical foundation that underpins their approach. We conduct an empirical analysis on both the stock and the portfolio approach. We conclude with a discussion on econometric issues.

Introduction

As a reminder, the Fama-MacBeth regression method is composed on two steps: step 1 with a time-series regression and step 2 with a cross-section regression.

The first step is to regress the return of every stock against one or more risk factors using a time-series regression. We obtain the return exposure to each factor called the “betas” or the “factor exposures” or the “factor loadings”.

The second step is to regress the returns of all stocks against the asset betas obtained in the first step using a cross-section regression for different periods. We obtain the risk premium for each factor used to test the asset pricing model.

The implementation of this method can be done with individual stocks or with portfolios of stocks as proposed by Fama and MacBeth (1973). Their argument is the better stability of the beta when considering portfolios. In this article we illustrate the difference with the two implementations.

Fama and MacBeth (1973) implemented with individual stocks

We downloaded a sample of daily prices of stocks composing the S&P500 index over the period from January 03, 2012, to December 31, 2021 (we selected the stocks present from the beginning to the end of the period reducing our sample from 500 to 440 stocks). We computed daily returns for each stock and for the market factor retained in this study. To represent the market, we chose the S&P500 index, an important global stock benchmark capturing the US equity market.

The procedure to derive the Fama-MacBeth regression using the stock approach can be achieved as follow:

Step 1: time-series regression

We compute the beta of the stocks with respect to the market factor for the period covered (time-series regression). We estimate the beta of each stock related to the S&P500 index. The beta is computed as the slope of the linear regression of the stock return on the market return (S&P500 index return). This time-series regression is run on a subperiod of the whole period from January 03, 2012, to December 31, 2018.

Step 2: cross-sectional regression

Over a second period from January 04, 2019, to December 31, 2021, we compute the dynamic regression of returns at each data point in time with respect to the betas computed in Step 1.

With this procedure, we obtain a risk premium that would represent the relationship between the stock returns at each data point in time with their respective beta for the sample analyzed.

Test the statistical significance of the results obtained from the regression

Results in the time-series regression using the stock approach are statistically significant. As shown in Table 1, the p-value is in the rejection area, which implies that the factor that the market factor can be considered as a driver of return.

Table 1. Time-series regression t-statistic result.
img_SimTrade_Fama_MacBeth_cross_sectional_regression_stat_result Source: computation by the author.

However, when analyzed in the cross-section regression, the results are not statistically significant anymore. As shown in Table 2, the p-value is not in the rejection area. We cannot reject the null hypothesis (H0: non significance of the market factor). Market factor alone cannot explain the premium investors are considering.
This means that the market factor fails to explain properly the behavior of asset returns, which undermines the validity of the CAPM framework. These results are in line with the Fama-MacBeth paper (1973).

Table 2. Cross-section regression t-statistic result.
img_SimTrade_Fama_MacBeth_cross_sectional_regression_stat_result Source: computation by the author.

You can find below the Excel spreadsheet that complements the explanations covered in this part of the article (implementation of the Fama and MacBeth (1973) method with individual stocks).

Fama and MacBeth (1973) implemented with portfolios of stocks

The procedure to derive the Fama-MacBeth regression using the portfolio approach can be achieved as follow:

Step 1: time-series regression

We first compute the beta of the stocks with respect to the market factor for the period covered (time-series regression). We estimate the beta of each stock related to the S&P500 index. The beta is computed as the slope of the linear regression of the stock return on the market return (S&P500 index return). This time-series regression is run on a subperiod of the whole period from January 03, 2012, to December 31, 2015. We build twenty portfolios based on stock betas ranked in ascending order. The betas of the portfolios are then estimated again on a subperiod from January 04, 2016, to December 31, 2018.

It is challenging to maintain beta stability over time. Fama-MacBeth aimed to remedy this shortcoming through its novel technique. However, some issues must be addressed. When betas are calculated using a monthly time series, the statistical noise in the time series is significantly reduced in comparison to shorter time frames (i.e., daily observation). When portfolio betas are constructed, the coefficient becomes considerably more stable than when individual betas are evaluated. This is due to the diversification impact that a portfolio can produce, which considerably reduces the amount of specific risk.

Step 2: cross-sectional regression

Over a second period from January 03, 2019, to December 31, 2021, we compute the dynamic regression of portfolio returns at each data point in time with respect to the betas computed in Step 1.

With this procedure, we obtain a risk premium that would represent the relationship between the portfolio returns at each data point in time with their respective beta for the sample analyzed.

Test the statistical significance of the results obtained from the regression

Results in the cross-section regression using the portfolio approach are not statistically significant. As captured in Table 3, the p-value is not in the rejection area, which implies that the factor is statistically insignificant and that the market factor cannot be considered as a driver of return.

Table 3. Cross-section regression with portfolio approach t-statistic result.
img_SimTrade_Fama_MacBeth_Portfolio_cross_sectional_regression_stat_result Source: computation by the author.

You can find below the Excel spreadsheet that complements the explanations covered in this part of the article (implementation of the Fama and MacBeth (1973) method with portfolios of stocks).

Econometric issues

Errors in data measurement

Because regression uses a sample instead of the entire population, a certain margin of error must be accounted for since the authors derive estimated betas for the sample.

Asset return heteroscedasticity

In econometrics, heteroscedasticity is an important concern since it results in unequal residual variance. This indicates that a time series exhibiting some heteroscedasticity has a non-constant variance, which renders forecasting ineffective because the time series will not revert to its long-run mean.

Asset return autocorrelation

Standard errors in Fama-MacBeth regressions are solely corrected for cross-sectional correlation. This method does not fix the standard errors for time-series autocorrelation. This is typically not a concern for stock trading, as daily and weekly holding periods have modest time-series autocorrelation, whereas autocorrelation is larger over long horizons. This suggests that Fama-MacBeth regression may not be applicable in many corporate finance contexts where project holding durations are typically lengthy.

Why should I be interested in this post?

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Fama-MacBeth regression method: Analysis of the market factor

December 12, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Fama-MacBeth two-step regression method used to test asset pricing models. The seminal paper by Fama and MacBeth (1973) was based on an investigation of the market factor by evaluating portfolios of stocks with similar betas. In this article I will elaborate on the methodology and assess the statistical significance of the market factor as a fundamental driver of return.

This article is structured as follow: we introduce the Fama-MacBeth testing method used in asset pricing. Then, we present the mathematical foundation that underpins their approach. I then apply the Fama-MacBeth to recent US stock market data. Finally, I expose the limitations of their approach and conclude to discuss the generalization of the original study to other risk factors.

Introduction

The two-step regression method proposed by Fama-MacBeth was originally used in asset pricing to test the Capital Asset Pricing Model (CAPM). In this model, there is only one risk factor determining the variability of returns: the market factor.

The first step is to regress the return of every asset against the risk factor using a time-series approach. We obtain the return exposure to the factor called the “beta” or the “factor exposure” or the “factor loading”.

The second step is to regress the returns of all assets against the asset betas obtained in Step 1 during a given time period using a cross-section approach. We obtain the risk premium associated with the market factor. Then, Fama and MacBeth (1973) assess the expected premium over time for a unit exposure to the risk factor by averaging these coefficients once for each element.

Mathematical foundations

We describe below the mathematical foundations for the Fama-MacBeth two-step regression method.

Step 1: time-series analysis of returns

The model considers the following inputs:

The return of N assets denoted by R_i for asset i observed over the time period [0, T].
The risk factor denoted by F for the market factor impacting the asset returns.

For each asset i (for i varying from 1 to N) we estimate the following time-series linear regression model:

Fama MacBeth time-series regression

From this model, we obtain the following coefficients: α_i and β_i which are specific to asset i.

Figure 1 represents for a given asset (Apple stocks) the regression of its return with respect to the S&P500 index return (representing the market factor in the CAPM). The slope of the regression line corresponds to the beta of the regression equation.

Figure 1. Step 1: time-series regression for a given asset (Apple stock and the S&P500 index).
Source: computation by the author.

Step 2: cross-sectional analysis of returns

For each period t (from t equal 1 to T), we estimate the following cross-section linear regression model:

Fama MacBeth cross-section regression

Figure 2 plots for a given period the cross-sectional returns and betas for a given point in time.

Figure 2 represents for a given period the regression of the return of all individual assets with respect to their estimated individual market beta.

Figure 2. Step 2: cross-section regression for a given time-period.

Source: computation by the author.

We average the gamma obtained for each data point. This is the way the Fama-MacBeth method is used to test asset pricing models.

Empirical study of the Fama-MacBeth regression

The seminal paper by Fama and MacBeth (1973) was based on an analysis of the market factor by assessing constructed portfolios of similar betas ranked by increasing values. This approach helped to overcome the shortcoming regarding the stability of the beta and correct for conditional heteroscedasticity derived from the computation of the betas for individual stocks. They performed a second time the cross-sectional regression of monthly portfolio returns based on equity betas to account for the dynamic nature of stock returns, which help to compute a robust standard error and assess if there is any heteroscedasticity in the regression. The conclusion of the seminal paper suggests that the beta is “dead”, in the sense that it cannot explain returns on its own (Fama and MacBeth, 1973).

Empirical study: Stock approach

We downloaded a sample of end-of-month stock prices of large firms in the US economy over the period from March 31, 2016, to March 31, 2022. We computed monthly returns. To represent the market, we chose the S&P500 index.

We then applied the Fama-MacBeth two-step regression method to test the market factor (CAPM).

Figure 3 depicts the computation of average returns and the betas and stock in the analysis.

Figure 3. Computation of average returns and betas of the stocks.
Source: computation by the author.

Figure 4 represents the first step of the Fama-MacBeth regression. We regress the average returns for each stock with their respective betas.

Figure 4. Step 1 of the regression: Time-series analysis of returns
Source: computation by the author.

The initial regression is statistically evaluated. To describe the behavior of the regression, we employ a t-statistic. Since the p-value is in the rejection area (less than the significance limit of 5 percent), we can deduce that the market factor can at first explain the returns of an investor. However, as we are going deal in the later in the article, when we account for a second regression as formulated by Fama and MacBeth (1973), the market factor is not capable of explaining on its own the return of asset returns.

Figure 5 represents Step 2 of the Fama-MacBeth regression, where we perform for a given data point a regression of all individual stock returns with their respective estimated market beta.

Figure 5. Step 2: cross-sectional analysis of return.
Source: computation by the author.

Figure 6 represents the hypothesis testing for the cross-sectional regression. From the results obtained, we can clearly see that the p-value is not in the rejection area (at a 5% significance level), hence we cannot reject the null hypothesis. This means that the market factor fails to explain properly the behavior of asset returns, which undermines the validity of the CAPM framework. These results are in line with Fama-MacBeth (1973).

Figure 6. Hypothesis testing of the cross-sectional regression.
Source: computation by the author.

Excel file for the Fama-MacBeth two-step regression method

You can find below the Excel spreadsheet that complements the explanations covered in this article to implement the Fama-MacBeth two-step regression method.

Limitations of the Fama-McBeth approach

Selection of the market index

For the CAPM to be valid, we need to determine if the market portfolio is in the Markowitz efficient curve. According to Roll (1977), the market portfolio is not observable because it cannot capture all the asset classes (human capital, art, and real estate among others). He then believes that the returns cannot be captured effectively and hence makes the market portfolio, not a reliable factor in determining its efficiency.

Furthermore, the coefficients estimated in the time-series regressions are sensitive to the market index chosen for the study. These shortcomings must be taken into account when assessing CAPM studies.

Stability of the coefficients

The beta of individual assets are not stable over time. Fama and MacBeth attempted to address this shortcoming by implementing an innovative approach.

When betas are computed using a monthly time-series, the statistical noise of the time series is considerably reduced as opposed to shorter time frames (i.e., daily observation).

Using portfolio betas makes the coefficient much more stable than using individual betas. This is due to the diversification effect that a portfolio can achieve, reducing considerably the amount of specific risk.

Conclusion

Why should I be interested in this post?

Useful resources

Academic research

Brooks, C., 2019. Introductory Econometrics for Finance (4th ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108524872

Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

Roll R., 1977. A critique of the Asset Pricing Theory’s test, Part I: On Past and Potential Testability of the Theory. Journal of Financial Economics, 1, 129-176.

American Finance Association & Journal of Finance (2008) Masters of Finance: Eugene Fama (YouTube video)

Business Analysis

NEDL. 2022. Fama-MacBeth regression explained: calculating risk premia (Excel). [online] Available at: [Accessed 29 May 2022].

About the author

The article was written in December 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Forex exchange markets

December 2, 2022 by Nakul PANJABI

Forex exchange markets

In this article, Nakul PANJABI (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2024) explains how the foreign exchange markets work.

Forex Market

Forex trading can be simply defined as exchange of a unit of one currency for a certain unit of another currency. It is the act of buying one currency while simultaneously selling another.

Foreign exchange markets (or Forex) are markets where currencies of different countries are traded. Forex market is a decentralised market in which all trades take place online in an over the counter (OTC) format. By trading volume, the forex market is the largest financial market in the world with a daily turnover of 6.6 trillion dollars in 2019. At present, it is worth 2,409 quadrillion dollars. Major currencies traded are USD, EUR, GBP, JPY, and CHF.

Players

The main players in the market are Central Banks, Commercial banks, Brokers, Traders, Exporters and Importers, Immigrants, Investors and Tourists.

Central banks

Central banks are the most important players in the Forex Markets. They have the monopoly in the supply of currencies and therefore, tremendous influence on the prices. Central Banks’ policies tend to protect aggressive fluctuations in the Forex Markets against the domestic currency.

Commercial banks

The second most important players of the Forex market are the Commercial Banks. By quoting, on a daily basis, the foreign exchange rates for buying and selling they “Make the Market”. They also function as Clearing Houses for the Market.

Brokers

Another important group is that of Brokers. Brokers do not participate in the market but acts as a link between Sellers and Buyers for a commission.

Types of Transactions in Forex Markets

Some of the transactions possible in the Forex Markets are as follows:

Spot transaction

As spot transaction uses the spot rate and the goods (currencies) are exchanges over a two-day period.

Forward transaction

A forward transaction is a future transaction where the currencies are exchanged after 90 days of the deal a fixed exchange rate on a defined date. The exchange rate used is called the Forward rate.

Future transaction

Futures are standardized Forward contracts. They are traded on Exchanges and are settled daily. The parties enter a contract with the exchange rather than with each other.

Swap transaction

The Swap transactions involve a simultaneous Borrowing and Lending of two different currencies between two investors. One investor borrows the currency and lends another currency to the second investor. The obligation to repay the currencies is used as collateral, and the amount is repaid at forward rate.

Option transaction

The Forex Option gives an investor the right, but not the obligation to exchange currencies at an agreed rate and on a pre-defined date.

Peculiarities of Forex Markets

Trading of Forex is not much different from trading of any other asset such as stocks or bonds. However, it might not be as intuitive as trading of stocks or bonds because of its peculiarities. Some peculiarities of the Forex market are as follows:

Going long and short simultaneously

Since the goods traded in the market are currencies themselves, a trade in the Forex market can be considered both long and short position. Buying dollars for euros can be profitable in cases of both dollar appreciation and euro depreciation.

High liquidity and 24-hour market

As mentioned above, the Forex market has the largest daily trading volume. This large volume of trading implies the highly liquid feature of Forex Assets. Moreover, Forex market is open 24 hours 5 days a week for retail traders. This is due to the fact that Forex is exchanged electronically over the world and anyone with an internet connection can exchange currencies in any Forex market of the world. In fact for Central banks and related organisations can trade over the weekends as well. This can cause a change in the price of currencies when the market opens to retail traders again after a gap of 2 days. This risk is known as Gapping risk.

High leverage and high volatility

Extremely high leverage is a common feature of Forex trades. Using high leverage can result in multiple fold returns in favourable conditions. However, because of high trading volume, Forex is very volatile and can go in either upward or downward spiral in a very short time. Since every position in the Forex market is a short and long position, the exposure from one currency to another is very high.

Hedging

Hedging is one of the main reasons for a lot of companies and corporates to enter into a Forex Market. Forex hedging is a strategy to reduce or eliminate risk arising from negative movement in the Exchange rate of a particular currency. If a French wine seller is about to receive 1 million USD for his wine sales then he can enter into a Forex futures contract to receive 900,000 EUR for that 1 million USD. If, at the date of payment, the rate of 1 million USD is 800,000 EUR the French wine seller will still get 900,000 EUR because he hedged his forex risk. However, in doing so, he also gave up any gain on any positive movement in the EUR-USD exchange rate.

Useful resources

Academic resources

Solnik B. (1996) International Investments Addison-Wesley.

Business resources

DailyFX / IG The History of Forex

DailyFX / IG Benefits of forex trading

DailyFX / IG Foreign Exchange Market: Nature, Structure, Types of Transactions

About the author

The article was written in December 2022 by Nakul PANJABI (ESSEC Business School, Grande Ecole Program – Master in Management, 2021-2024).

Exchange-traded funds and Tracking Error

November 16, 2022 by Micha FISCHER

Exchange-traded funds and Tracking Error

In this article, Micha FISHER (University of Mannheim, MSc. Management, 2021-2023) explains the concept of Tracking Error in the context of exchange traded funds (ETF).

This article will offer a short introduction to the concept of exchange-traded funds, will then describe several reasons for the existence of tracking errors and finish with a concise example on how tracking error can be calculated.

Exchange-traded funds

An exchange-traded fund is conceptionally very close to classical mutual funds, with the key difference being, that ETFs are traded on a stock exchange during the trading day. Most ETFs are so-called index funds and thus they try to replicate an existing index like the S&P 500 or the CAC 40. This sort of passive investing is aimed at following or tracking the underlying index as closely as possible. However, actively managed ETFs with the aim of outperforming the market do exist as well and typically come with higher management fees. There are several types of ETFs covering equity index funds, commodities or currencies with classical equity index funds being the most prominent.

The total volume of global ETF portfolios has increased substantially over the last two decades. At the beginning of the century total asset volume was in the low triple digit billions measured in USD. According to research by the Wall Street Journal total assets in ETF investments surpassed nine trillion USD in 2021.

The continuing attractiveness of exchange-traded index funds can be explained with the very low management fees, the clarity of the product objective, and the high liquidity of the investment vehicle. However, although especially the market leaders like BlackRock, the Vanguard Group or State Street offer products that come extremely close to mirroring their underlying index, exchange-traded funds do not perfectly track the evolution of the underlying index. This phenomenon is known as tracking error and will be discussed in detail below.

Theoretical measure of the Tracking Error

Simply speaking, the tracking error of an ETF is the difference in the returns of the underlying index (I for index) and the returns of the ETF itself (E for ETF). For a specific period, it is computed by taking the standard deviation of the differences between the two time-series.

Formula for tracking error

Theoretically, it is possible to fully replicate an index in a portfolio and thus reach a tracking error of zero. However, there are several reasons why this is not achievable in practice.

Origins of the Tracking Error

The most important and obvious reason is that the Net Asset Value (NAV) of index funds is necessarily lower than the NAV of its underlying index. An index itself has no liabilities, as it is strictly speaking an instrument of measurement. On the other hand, even a passively managed index fund comes with expenses to pay for infrastructure, personnel, and marketing. These liabilities decrease the Net Asset Value of the fund. In general, a higher tracking error could indicate that the fund is not working efficiently compared to products of competitors with the same underlying index.

Another origin of tracking error can be found in specific sector ETFs and more niche markets with not enough liquidity. When the trading volume of a stock is very low, buying / selling the stock would increase / decrease the price (price impact). In this case an ETF could buy more liquid stocks with the aim to mirror the value development of the illiquid stock, which in turn could lead to a higher tracking error.

Another source of tracking error that occurs more severely in dividend-focused ETFs is the so-called cash drag. High dividend payments that are not instantly reinvested drag down the fund performance in contrast to the underlying index.

Of course, transaction fees of the marketplaces can reduce the fund performance as well. This is especially true if large rebalancing efforts are necessary due to a change of the index composition.

Lastly, there are also ways to reduce the effects described above. Funds can engage in security lending to earn additional money. In this case, the fund lends individual assets within the portfolio to other investors (mostly short sellers) for an agreed period in return for lending fees and possible interest. It should be noted, that while this might reduce tracking error, it also exposes the fund to additional counterparty risk.

Tracking Error: An Example

The sheet posted below shows a simple example of how the tracking error can be computed. To not include hundreds of individual shares, the example transformed the top ten positions within the Nasdaq-100 index into an artificial “Nasdaq-10” index. Although the data for the 23rd of September is accurate, the future data is of course randomly simulated.

By using the individual weights of the index components and their corresponding weights, the index returns for the next three months can be computed.

Figure 1: Three-months simulation of “Nasdaq-10” index.
Three-months simulation of Nasdaq-10 index
Source: computation by the author.

At this point our made-up ETF is introduced with an initial investment of 100 million USD. This ETF fully replicates the Nasdaq-10 index by holding shares in the same proportion as the index. In this example only the management and marketing fees are incorporated. Security lending, index changes and transaction fees and dividends are omitted. Also, all the portfolio shares are highly liquid and allow for full replication. The fund works with small expenses for personnel of only ten thousand USD per month. Additionally, once per quarter, a marketing campaign costs additionally fifty thousand USD.

Figure 2: Computation of ETF return and tracking error.
Computation of ETF-return and Tracking Error
Source: computation by the author.

Calculating the net asset value (NAV) gives us the monthly returns of the fund which in turn allows us to calculate the three-month standard deviation of the tracking difference. Additionally, the Total Expense Ratio can be calculated as the percentage of expenses per year divided by the total asset value of the fund.

This example gives us a Total Expense Ratio of nearly 0.3 percent per annum which is within the competitive area of real passive funds. Vanguard is able to replicate the FTSE All-World index with 0.2 percent. However, the calculated tracking error is obviously smaller than most real tracking errors with only 0.0002, as only management fees were considered. Exemplary, Vanguards FTSE All-World ETF had an historical tracking error of 0.042 in 2021, due to the reasons mentioned in the section above.

Excel file for computing the tracking error of an ETF

You can also download below the Excel file for the computation of the tracking error of an ETF.

Why should I be interested in this post?

ETFs in all forms are one of the major developments in the area of portfolio management over the last two decades. They are also a very interesting option for private investments.

Although they are conceptually very simple it is important to understand the finer metrics that vary between different service providers as even small differences can have a large impact over a longer investment period.

Useful resources

Academic articles

Roll R. (1992) A Mean/Variance Analysis of Tracking Error, The Journal of Portfolio Management, 18 (4) 13-22.

Business

ET Money What is Tracking Error in Index Funds and How it Impacts Investors?

About the author

The article was written in November 2022 by Micha FISHER (University of Mannheim, MSc. Management, 2021-2023).

Approaches to investment

November 14, 2022 by Henri VANDECASTEELE

Approaches to investment

In this article, Henri VANDECASTEELE (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the two main approaches to investment: fundamental analysis and technical analysis.

Fundamental analysis

Fundamental analysis (FA) is a way of determining the fundamental value of a securities by looking at linked economic and financial elements. Fundamental analysts look at everything that might impact the value of a security, from macroeconomic issues like the state of the economy and industry circumstances to microeconomic elements like management performance. All stock analysis attempts to evaluate if a security’s value in the larger market is right. Fundamental research is often conducted from a macro to micro viewpoint in order to find assets that the market has not valued appropriately. To get at a fair market valuation for the stock, analysts often look at the overall status of the economy, then the strength of the specific industry, before focusing on individual business performance.

Fundamental analysis evaluates the value of a stock or any other form of investment using publicly available data. An investor, for example, might undertake fundamental research on a bond’s value by looking at economic variables like interest rates and the overall status of the economy, then reviewing information about the bond issuer, such as probable changes in its credit rating.

The aim is to arrive at a figure that can be compared to the present price of an asset to determine whether it is undervalued or overpriced.

Fundamental analysis is based on both qualitative and quantitative publicly available historical and current data. This includes company statements, historical stock market data, company press releases, financial year statements, investor presentations, information found on internet fora, media articles, and broker/analyst reports.

Technical analysis

Technical analysis (TA) is a trading discipline that analyzes statistical trends acquired from trading activity, such as price movement and volume, to evaluate investments and uncover trading opportunities.

Technical analysis, as opposed to fundamental analysis, focuses on the examination of price and volume. Fundamental analysis aims to estimate a security’s worth based on business performance such as sales and earnings. Technical analysis methods are used to examine how variations in price, volume, and implied volatility are affected by supply and demand for a security. Any security with past trading data can benefit from technical analysis. This includes stocks, futures, commodities, bonds, currencies and other securities. In fact, technical analysis is much more common in commodities and forex markets where traders focus on short-term price fluctuations.

Technical analysis is commonly used to generate short-term trading signals from various charting tools, but it also helps to improve the assessment of securities strengths or weaknesses compared to one of the broader markets or sectors increase. This information helps analysts improve their overall rating estimates.

Technical analysis is performed on quantitative data only that recent and historical, but publicly available. It leverages mainly market information, namely daily transaction volumes, stock price, spread, volatility, … and performs trend analyses.

Link with market efficiency

When linking both approaches to investment to the market efficiency theory, we can state that fundamental analysis assumes that financial markets are not efficient in the semi-strong sense, whereas technical analysis assumes that financial markets are not efficient in the weak sense. But the trading activity of both fundamental analysts and technical analysts make the markets more efficient.

Useful resources

SimTrade course Market information

About the author

The article was written in November 2022 by Henri VANDECASTEELE (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022).

Understand the mechanism of inflation in a few minutes?

November 1, 2022 by Louis DETALLE

Understand the mechanism of inflation in a few minutes?

In this article, Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains everything you have to know about inflation.

What is inflation and how can it make us poorer?

In a liberal economy, the prices of goods and services consumed vary over time. In France, for example, when the price of wheat rises, the price of wheat flour rises and so the price of a loaf of bread may also rises as a consequence of the rise in the price of the raw materials used for its production… This small example is only designed to make the evolution of prices concrete for one good only. It helps us understand what happens when the increase in price happens not only for a loaf of bread, but for all the goods of an economy.

Inflation is when prices rise overall, not just the prices of a few goods and services. When this is the case, over time, each unit of money buys fewer and fewer products. In other words, inflation gradually erodes the value of money (purchasing power).

If we take the example of a loaf of bread which costs €1 in year X, while the price of the 20g of wheat flour contained in a loaf is 20 cents. In year X+1, if the 20g of wheat flour now costs 22 cents, i.e., a 10% increase over one year, the price of the loaf of bread will have to reflect this increase, otherwise the baker will be the only one to suffer the increase in the price of his raw material. The price of a loaf of bread will then be €1.02.

We can see that here, with one euro, i.e., the same amount of the same currency, from one year to the next, it is not possible for us to buy a loaf of bread because it costs €1.02 and not €1 anymore.

This is a very schematic way of understanding the mechanism of inflation and how it destroys the purchasing power of consumers in an economy.

How is the inflation computed and what does a x% inflation mean?

In France, Insee (Institut national de la statistique et des études économiques in French) is responsible for calculating inflation. It obtains it by comparing the price of a basket of goods and services each month. The content of this basket is updated once a year to reflect household consumption patterns as closely as possible. In detail, the statistics office uses the distribution of consumer expenditure by item as assessed in the national accounts, and then weights each product in proportion to its weight in household consumption expenditure.

What is important to understand is that Insee calculates the price of an overall household expenditure basket and evaluates the variation of its price over time.

When inflation is announced at X%, this means that the overall value spent in the year by a household will increase by X%.

However, if the price of goods increases but wages remain the same, then purchasing power deteriorates, and this is why low-income households are the most affected by the rise in the price of everyday goods. Indeed, low-income households can’t easily cope with a 10% increase in price of their daily products, whereas the middle & upper classes can better deal with such a situation.

What can we do to reduce inflation?

It is the regulators who control inflation through major macroeconomic levers. It is therefore central banks and governments that can act and they do so in various ways (as an example, we use the context of the War in Ukraine in 2022):

They raise interest rates: when inflation is too high, central banks raise interest rates to slow down the economy and bring inflation down. This is what the European Central Bank (ECB) has just done because of the economic consequences of the War in Ukraine. The economic sanctions have seen the price of energy commodities soar, which has pushed up inflation.

Blocking certain prices: This is what the French government is still doing on energy prices. Thus, in France, the increase in gas and electricity tariffs will be limited to 15% for households, compared to a freeze on gas prices and an increase limited to 4% for electricity in 2022. Without this “tariff shield”, the French would have had to endure an increase of 120%.

Distribute one-off aid: These measures are often considered too costly and can involve an increase in salaries.

Bear in mind that “miracle” methods do not exist, otherwise inflation would never be a subject discussed in the media. However, these three methods are the most used by governments and central banks but only time will tell us whether they succeed.

Figure 1. Inflation in France.

Source: Insee / Les Echos.

Useful resources

Inflation rates across the World

Insee’s forecast of the French inflation rate

About the author

The article was written in October 2022 by Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023).

Modeling of the crude oil price

The crude oil market

Spot contract

Futures contract

Mathematical foundations of the Geometric Brownian Motion (GBM) model

Modelling crude oil market prices

Market prices

Market returns

Application: simulation of future prices for the crude oil market

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Useful resources

Academic research

About the author

Introduction

Mathematical foundation for the beta

Application of an equity market neutral strategy

An extension of the equity market neutral strategy to other asset classes

Performance of the equity market neutral strategy

Why should I be interested in this post?

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Financial techniques

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Business Analysis

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Fixed-income arbitrage strategy

Introduction

Types of arbitrage

Pure arbitrage

Relative value arbitrage

Factors that influence fixed-income arbitrage strategies

Market segmentation

Regulation

Liquidity

Volatility

Instrument complexity

Application of a fixed-income arbitrage

The European yield curve differential during in the mid 1990’s

Performance of the fixed-income arbitrage strategy

Why should I be interested in this post?

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The greatest trade in history

Performance of the global macro strategy

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Interest rate term structure and yield curve calibration

Introduction

Mathematical foundation of the Nelson-Siegel-Svensson model

Application of the yield curve structure

Excel file for the calibration model of the yield curve

Why should I be interested in this post?

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Hedge funds

Financial techniques

Other

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Academic research

Business Analysis

About the author

Introduction

Academic Literature

Example

Excel file to build the Minimum Volatility Portfolio

Why should I be interested in this post?

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Academic research

Business analysis