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).

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).

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).

Implementing Black-Litterman asset allocation model

March 21, 2022 by Youssef LOURAOUI

In this article, Youssef Louraoui (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents an implementation of the Black-Litterman model, used to determine the expected return of a portfolio by integrating investor’s views regarding the performance of the underlying assets selected in the investment portfolio.

This article follows the following structure: first, we introduce the Black-Litterman model. We then present the mathematical foundations of this model. We conclude with an explanation of the methodology to build the spreadsheet with the results obtained. You will find in this post an Excel spreadsheet which implement the model.

Introduction

The Black-Litterman asset allocation model, established for the first time in the early 1990’s by Fischer Black and Robert Litterman, is a sophisticated strategy for dealing with unintuitive, highly concentrated, and input-sensitive portfolios. The most likely reason that more portfolio managers do not use the Markowitz model, which maximises return for a given degree of risk, is input sensitivity, a well-documented issue with mean-variance optimization.

The Black-Litterman Model employs a Bayesian technique to integrate an investor’s subjective views of expected returns on one or more assets with the market equilibrium vector (prior distribution) of expected returns to obtain a new, mixed estimate of expected returns. The new vector of returns (the posterior distribution) is a weighted complex average of the investor’s views and market equilibrium.

Mathematical foundation

The idea of the Black Litterman estimates is not to find the optimum portfolio weights as in the Markowitz optimization framework, but instead to find the expected return that would be used as an input to compute the optimum portfolio weights. This approach is referred as reversion portfolio optimization technique. The idea behind is that optimum weights are already observed in the market and captured in the market portfolio. We can approach the reasoning by maximizing the following utility function adjusted to the risk:

img_SimTrade_mathematical_foundation_Black_Litterman_6

w^T = transposed of portfolio weights
Π = Implied equilibrium excess return vector
A = price of risk or risk aversion factor
Σ = variance-covariance matrix

We take the partial derivative of U in terms of weights (w) and we derive the following expression:

img_SimTrade_mathematical_foundation_Black_Litterman_5

By setting the partial derivative equal to zero, we can maximize the utility function in term of weights. The proposed approach in the Black Litterman approach is that instead of seeking the optimal weights, which are incorporated in the market portfolio and thus computable via the market capitalization of the equities in the portfolio, we’ll isolate the Π (implied equilibrium excess return) to obtain the optimal expected returns for the portfolio:

We can deconstruct the Black-Litterman model as

img_SimTrade_mathematical_foundation_Black_Litterman_3

τ= scalar
P = Linking matrix
∑ = Variance-covariance matrix
Π= implied equilibrium excess return
A = Price of risk
w = weight vector
Ω = uncertainty of views

The first term of the formula is introduced in order to respect the constraint that the portfolio weights should be equal to one:

The second term of the formula is to compute a weighted average of the implied equilibrium excess return adjusted to the uncertainty of the returns by the view vector weighted with the uncertainty of views:

The final output E(R) is a vector of return n x 1 that represent the equilibrium returns of the market adjusted to investors views.

Implementation of the Black-Litterman asset allocation model in practice

To model a Black-Litterman portfolio allocation, we obtained 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.

The data is derived from the Bloomberg terminal. The first step is to calculate the logarithmic returns and excess returns on the selected assets (returns minus the risk-free rate). After calculating the logarithmic returns on each asset, we can estimate the capital asset pricing model’s returns (CAPM) expected returns. This information will be used to calculate the Black-Litterman expected returns on a comparative basis.

1. The first input for the model is the price of risk A, which represents the risk aversion of investor and is obtained by subtracting the expected return of the market the risk-free rate and divided by the variance of the market:

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_1

E(r_m)= expected market returns
r_f = risk-free rate
σ²_m = variance of market

2. We extract the respective market capitalization of each security to obtain their market weights in the portfolio. Given that our investable universe is made of five stocks, we can retrieve their respective market capitalization and compute the weights of each stock in relation to the sum of total market-capitalization in the portfolio.

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_2

Table 1 depicts the optimal weights obtained from their respective market capitalisation, coupled with the respective expected return and volatility.

Table 1. Asset characteristics of Apple, Amazon, Microsoft, Goldman Sachs, and Pfizer.

img_SimTrade_Black_Litterman_spreadsheet_2

Source: computation by the author.

3. We compute the variance-covariance matrix of logarithmic returns using the data analysis tool pack available in Excel (Table 2).

Table 2. Variance-covariance matrix of asset returns

img_SimTrade_Black_Litterman_spreadsheet_5

Source: computation by the author.

4. We compute the implied equilibrium excess return (Π) as the matrix calculation of the price of risk (A) times the matrix multiplication of the weights computed in step 4 times the variance-covariance matrix computed in step 3.

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_3

Π= implied equilibrium excess return
A = Price of risk
w = weight vector

5. The views are incorporated into the model. To achieve this, we provide three views to include into the model. If there are no views, the values will correspond to the market portfolio. The investment manager’s views for the expected return on certain of the portfolio’s assets regularly diverge from the Implied Equilibrium Return Vector (), which serves as the market-neutral starting point for the Black-Litterman model that quantifies the uncertainty associated with each view. The Black-Litterman Model can be used to depict such views in absolute or relative terms. As an illustration, let us suppose that the real and simulated portfolio will have the same views:

View 1: Apple will outperform Microsoft by .05 percent
View 2: Amazon will outperform Microsoft by .1 percent
View 3: Apple will outperform Amazon by .05 percent

To incorporate the vector Q of views, we create a link matrix P where the rows sum to zero. Figure 3 depicts the workings done in the spreadsheet.

Table 3. Views vector and Link Matrix (P)

img_SimTrade_Black_Litterman_spreadsheet_1

Source: computation by the author.

6. We compute omega to determine the degree of uncertainty associated with the views. While Black-Litterman paper used a value of tau equal to 0.25, an important number of academics went for calculating the tau equal to one. For the sake of simplifying the model, consider tau to be equal to one. This input is obtained by multiplying the Linking matrix by the variance-covariance matrix and transposing the Linking matrix (P).

img_SimTrade_Black_Litterman_formulas_for_spreadsheet_4

τ= scalar
P = Linking matrix
∑ = Variance-covariance matrix

7. We integrate all the values computed previously in the Black-Litterman model. Table 4 depicts the results obtained via the Black-Litterman allocation model.

Table 4. Results of the Black-Litterman allocation

img_SimTrade_Black_Litterman_spreadsheet_4

Source: computation by the author.

We can see that the results converge slightly to those from CAPM. Additionally, we can see that the views are reflected in the Black-Litterman expected returns. As a result, we can determine whether or not each view is satisfied. Indeed, Apple outperforms Amazon and Microsoft, while Amazon outperforms Microsoft.

You can download an Excel file to help you construct a portfolio via the Black-Litterman allocation model.

Why should I be interested in this post?

Modern Portfolio Theory is at the heart of modern finance, shaping the modern investing landscape. MPT has become the cornerstone of current financial theory and practice. MPT’s thesis is that winning the market is difficult and requires diversification and taking higher-than-average risks.

MPT has been around for nearly sixty years and shows no signs of slowing down. His theoretical contributions paved the way for more portfolio theory study. But Markowitz’s portfolio theory is sensitive to and depends on further ‘probabilistic’ expansion. This paper expanded on Markowitz’s previous work by incorporating investor views into the asset allocation process.

Useful resources

Academic research

Black, F. and Litterman, R. 1990. Asset Allocation: Combining Investors Views with Market Equilibrium. Goldman Sachs Fixed Income Research working paper

Black, F. and Litterman, R. 1991. Global Asset Allocation with Equities, Bonds, and Currencies. Goldman Sachs Fixed Income Research working paper

Black, F. and Litterman, R. 1992. Global Portfolio Optimization.Financial Analysts Journal, 28-43.

Idzorek, T.M. 2002. A step-by-step guide to Black-Litterman model. Incorporating user-specified confidence levels. Working Paper, 2-11.

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

About the author

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

Portfolio

November 26, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of portfolio, which is a basic element in asset management.

This article is structured as follows: we introduce the concept of portfolio. We give the basic modelling to define and characterize a portfolio. We then expose the different types of portfolios that investors can rely on to meet their financial goals.

Introduction

An investment portfolio is a collection of assets that an investor owns. These assets can be individual assets such as bonds and stocks or baskets of assets such as mutual funds or exchange-traded funds (ETFs). In a nutshell, this refers to any asset that has the potential to increase in value or generate income. When building a portfolio, investors usually consider the expected return and risk. A well-balanced portfolio includes a variety of investments.

Modelling of portfolios

Portfolio weights

At a point of time, a portfolio is fully defined by the weights (w) of the assets of the universe considered (N assets).

Portfolio weights

The sum of the portfolio weights adds up to one (or 100%):

Sum of the portfolio weights

The weight of a given asset i can be positive (for a long position in the asset), equal to zero (for a neutral position in the asset) or negative (for a short position in the asset):

Asset weight for a long position

Asset weight for a neutral position

Asset weight for a short position

Short selling is the process of selling a security without owning it. By definition, a short sell occurs when an investor borrows a stock, sells it, and then buys it later back to repay the lender.

The equally-weighted portfolio is defined as the portfolio with weights that are evenly distributed across the number of assets held:

Equally-weigthed portfolio

Portfolio return: the case of two assets

Over a given period of time, the returns on assets 1 and 2 are equal to r₁ and r₂. In the two-asset portfolio case, the portfolio return r_P is computed as

Return of a 2-asset portfolio

The expected return of the portfolio E(r_P) is computed as

Expected return of a 2-asset portfolio

The standard deviation of the portfolio return, σ(r_P) is computed as

Standard deviation of a 2-asset portfolio return

where:

σ₁ = standard deviation of asset 1
σ₂ = standard deviation of asset 2
σ_1,2 = covariance of assets 1 and 2
ρ_1,2 = correlation of assets 1 and 2

Investing in asset classes with low or no correlation to one another can help you increase portfolio diversification and reduce portfolio volatility. While diversification cannot guarantee a profit or eliminate the risk of investment loss, the ideal scenario is to have a mix of uncorrelated asset classes in order to reduce overall portfolio volatility and generate more consistent long-term returns. Correlation is depicted mathematically as the division of the covariance between the two assets by the individual standard deviation of the asset. Correlation is a more interpretable metric than covariance because it’s measurable within a defined rank. Correlation is measured between -1 and 1, with a high positive correlation showing that the assets move in tandem, while negative correlation depicts securities that have contrary price movements. The holy grail of investing is to invest in securities that offer a low correlation of the portfolio as a whole.

where:

σ_1,2 = covariance of assets 1 and 2
σ₁ = standard deviation of asset 1
σ₂ = standard deviation of asset 2

Correlation is a more interpretable metric than covariance because it’s measurable within a defined rank. Correlation is measured between -1 and 1, with high positive correlation showing that the assets move in tandem, while negative correlation depicts securities that have contrary price movements. The holy grail of investing is to invest in securities that offer a low correlation of the portfolio as a whole.

You can download an Excel file to help you construct a portfolio and compute the expected return and variance of a two-asset portfolio. Just introduce the inputs in the model and the calculations will be performed automatically. You can even draw the efficient frontier to plot the different combinations of portfolios that optimize the risk-return trade-off (to minimize the risk for a given level of expected return or to maximize the expected return for a given level of risk).

Portfolio return: the case of N assets

Over a given period of time, the return on asset i is equal to r_i. The portfolio return can be computed as

Portfolio return

The expression of the portfolio return is then used to compute two important portfolio characteristics for investors: the expected performance measured by the average return and the risk measured by the standard deviation of returns.

The expected return of the portfolio is given by

Expected portfolio return

Because relying on multiple assets can get extremely computationally heavy, we can refer to the matrix form for more straightforward use. We basically compute the vector of weight with the vector of returns (NB: we have to pay attention to the dimension and to the properties of matrix algebra).

w = weight vector
r = returns vector

The standard deviation of returns of the portfolio is given by the following equivalent formulas:

Standard deviation of portfolio return

w_i = weight of asset i
w_j = weight of asset j
σ_i = standard deviation of asset i
σ_j = standard deviation of asset j
ρ_i,j = correlation of asset i,j

Standard deviation of portfolio return

where:

w_i² = squared weight of asset I
σ_i² = variance of asset i
w_i = weight of asset i
w_j = weight of asset j
σ_i = standard deviation of asset i
σ_j = standard deviation of asset j
ρ_i,j = correlation of asset i,j

We can use the matrix form for a more straightforward application due to the computational burden associated with relying on multiple assets. Essentially, we multiply the vector of weights with the variance-covariance matrix and the transposed weight vector (NB: we must pay attention to the dimension and to the properties of matrix algebra).

w = weight vector
∑ = variance-covariance matrix
w’ = transpose of weight vector

You can get an Excel file that will help you build a portfolio and calculate the expected return and variance of a three-asset portfolio. Simply enter the data into the model, and the calculations will be carried out automatically. You can even use the efficient frontier to plot the various portfolio combinations that best balance risk and reward (to minimize the risk for a given level of expected return or to maximize the expected return for a given level of risk).

Basic principles on portfolio construction

Diversify

Diversification, a core principle of Markowitz’s portfolio selection theory, is a risk-reduction strategy that entails allocating assets among a variety of financial instruments, sectors, and other asset classes (Markowitz, 1952). In more straightforward terms, it refers to the concept “don’t put all your eggs in one basket.” If the basket is dropped, all eggs are shattered; if many baskets are used, the likelihood of all eggs being destroyed is significantly decreased. Diversification may be accomplished by investments in a variety of companies, asset types (e.g., bonds, real estate, etc.), and/or commodities such as gold or oil.

Diversification seeks to enhance returns while minimizing risk by investing in a variety of assets that will react differently to the same event(s). Portfolio diversification methods should include not just diverse stocks inside and outside of the same industry, but also diverse asset classes, such as bonds and commodities. When there is an imperfect connection between assets (lower than one), the diversification effect occurs. It is a critical and successful risk mitigation method since risk mitigation may be accomplished without jeopardizing profits. As a result, any prudent investor who is cautious (or ‘risk averse’) will diversify to a certain extent.

Portfolio Asset Allocation

The term “asset allocation” refers to the proportion of stocks, bonds, and cash in a portfolio. Depending on your investing strategy, you’ll determine the percentage of each asset type in your portfolio to achieve your objectives. As markets fluctuate over time, your asset allocation is likely to go out of balance. For instance, if Tesla’s stock price increases, the percentage of your portfolio allocated to stocks will almost certainly increase as well.

Portfolio Rebalancing

Rebalancing is a term that refers to the act of purchasing and selling assets in order to restore your portfolio’s asset allocation to its original state and avoid disrupting your plan.

Reduce investment costs as much as possible

Commission fees and management costs are significant expenses for investors. This is especially important if you frequently purchase and sell stocks. Consider using a discount brokerage business to make your investment. Clients are charged much lesser fees by these firms. Also, when investing for the long run, it is advisable to avoid making judgments based on short-term market fluctuations. To put it another way, don’t sell your stocks just because they’ve taken a minor downturn in the near term.

Invest on a regular basis

It is critical to invest on a regular basis in order to strengthen your portfolio. This will not only build wealth over time, but it will also develop the habit of investing discipline.

Buying in the future

It’s possible that you have no idea how a new stock will perform when you buy it. To be on the safe side, avoid putting your entire position to a single investment. Start with a little investment in the stock. If the stock’s performance fulfils your expectations, you can gradually increase your investments until you’ve covered your entire position.

Types of portfolio

We detail below the different types of portfolios usually proposed by financial institutions that investors can rely on to meet their financial goals.

Aggressive Portfolio

As the name implies, an aggressive portfolio is one of the most frequent types of portfolio that takes a higher risk in the pursuit of higher returns. Stocks in an aggressive portfolio have a high beta, which means they present more price fluctuations compared to the market. It is critical to manage risk carefully in this type of portfolio. Keeping losses to a minimal and taking profits are crucial to success. It is suitable for a high-risk appetite investor.

Defensive Portfolio

A defensive portfolio is one that consists of stocks with a low beta. The stocks in this portfolio are largely immune to market swings. The goal of this type of portfolio is to reduce the risk of losing the principal. Fixed-income securities typically make up a major component of a defensive portfolio. It is suitable for a low-risk appetite investor.

Income Portfolio

Another typical portfolio type is one that focuses on investments that generate income from dividends (for stocks), interests (for bonds) or rents (for real estate). An income portfolio invests in companies that return a portion of their profits to shareholders, generating positive cash flow. It is critical to remember that the performance of stocks in an income portfolio is influenced by the current economic condition.

Speculative Portfolio

Among all portfolio types, a speculative portfolio has the biggest risk. Speculative investments could be made of different assets that possess inherently higher risks. Stocks from technology and health-care companies that are developing a breakthrough product, junk bonds, distressed investments among others might potentially be included in a speculative portfolio. When establishing a speculative portfolio, investors must exercise caution due to the high risk involved.

Hybrid Portfolio

A hybrid portfolio is one that includes passive investments and offers a lot of flexibility. The cornerstone of a hybrid portfolio is typically made up of blue-chip stocks and high-grade corporate or government bonds. A hybrid portfolio provides diversity across many asset classes while also providing stability by combining stocks and bonds in a predetermined proportion.

Socially Responsible Portfolio

A socially responsible portfolio is based on environmental, social, and governance (ESG) criteria. It allows investors to make money while also doing good for society. Socially responsible or ESG portfolios can be structured for any level of risk or investment aim and can be built for growth or asset preservation. The important thing is that they prefer stocks and bonds that aim to reduce or eliminate environmental impact or promote diversity and equality.

Why should I be interested in this post?

Useful resources

Academic research

Mangram, M.E., 2013. A simplified perspective of the Markowitz Portfolio Theory. Global Journal of Business Research, 7(1): 59-70.

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

Business analysis

Edelweiss, 2021.What is a portfolio?

Forbes, 2021.Investing basics: What is a portfolio?

JP Morgan Asset Management, 2021.Glossary of investment terms: Portfolio

About the author

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

Black-Litterman Model

November 21, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Black-Litterman model, used to determine optimal asset allocation in a portfolio. The Black-Litterman model takes the Markowitz model one step further: it incorporates an investor’s own views in determining asset allocations.

This article is structured as follows: we introduce the Black-Litterman model. We then present the mathematical foundations of the model to understand how the method is derived. We finish with an example to illustrate how we can implement a Black-Litterman asset allocation in practice.

Introduction

The Black-Litterman asset allocation model, developed by Fischer Black and Robert Litterman in the early 1990’s, is a complex method for dealing with unintuitive, highly concentrated, input-sensitive portfolios produced by the Markowitz model. The most likely reason why more portfolio managers do not employ the Markowitz paradigm, in which return is maximized for a given level of risk, is input sensitivity, which is a well-documented problem with mean-variance optimization.

The Black-Litterman model employs a Bayesian technique to integrate an investor’s subjective views on expected returns for one or more assets with the market equilibrium expected returns (prior distribution) of expected returns to get a new, mixed estimate of expected returns. The new vector of expected returns (the posterior distribution) is a complex, weighted average of the investor’s views and the market equilibrium.

The purpose of the Black-Litterman model is to develop stable, mean-variance efficient portfolios based on an investor’s unique insights that overcome the problem of input sensitivity. According to Lee (2000), the Black-Litterman Model “essentially mitigates” the problem of estimating error maximization (Michaud, 1989) by dispersing errors throughout the vector of expected returns.

The vector of expected returns is the most crucial input in mean-variance optimization; yet, Best and Grauer (1991) demonstrate that this input can be very sensitive in the final result. Black and Litterman (1992) and He and Litterman (1999) investigate various potential projections of expected returns in their search for a fair starting point: historical returns, equal “mean” returns for all assets, and risk-adjusted equal mean returns. They demonstrate that these alternate forecasts result in extreme portfolios, which have significant long and short positions concentrated in a small number of assets.

Mathematical foundation of Black-Litterman model

It is important to introduce the Black-Litterman formula and provide a brief description of each of its elements. In the formula below, the integer k is used to represent the number of views and the integer n to express the number of assets in the investment set (NB: the superscript ’ indicates the transpose and -1 indicates the inverse).

Where:

E[R] = New (posterior) vector of combined expected return (n x 1 column vector)
τ = Scalar
Σ = Covariance matrix of returns (n x n matrix)
P = Identifies the assets involved in the views (k x n matrix or 1 x n row vector in the special case of 1 view)
Ω = Diagonal covariance matrix of error terms in expressed views representing the level of confidence in each view (k x k matrix)
П = Vector of implied equilibrium expected returns (n x 1 column vector)
Q = Vector of views (k x 1 column vector)

Traditionally, personal views are used for prior distribution. Then observed data is used to generate a posterior distribution. The Black-Litterman Model assumes implied returns as the prior distribution and personal views alter it. The basic procedure to find the Black-Litterman model is: 1) Find implied returns 2) Formulate investor views 3) Determine what the expected returns are 4) Find the asset allocation for the optimal portfolio.

Black-Litterman asset allocation in practice

An investment manager’s views for the expected return of some of the assets in a portfolio are frequently different from the the Implied Equilibrium Return Vector (Π), which represents the market-neutral starting point for the Black-Litterman model. representing the uncertainty in each view. Such views can be represented in absolute or relative terms using the Black-Litterman Model. Below are three examples of views stated in the Black and Litterman model (1990).

View 1: Merck (MRK) will generate an absolute return of 10% (Confidence of View = 50%).
View 2: Johnson & Johnson (JNJ) will outperform Procter & Gamble (PG) by 3% (Confidence of View = 65%).
View 3: GE (GE) will beat GM (gm), Wal-Mart (WMT), and Exxon (XOM) by 1.5 percent (Confidence of View = 30%).

An absolute view is exemplified by View 1. It instructs the Black-Litterman model to set Merck’s return at 10%.

Views 2 and 3 are relative views. Relative views are more accurate representations of how investment managers feel about certain assets. According to View 2, Johnson & Johnson’s return will be on average 3 percentage points higher than Procter & Gamble’s. To determine if this will have a good or negative impact on Johnson & Johnson in comparison to Procter & Gamble, their respective Implied Equilibrium returns must be evaluated. In general (and in the absence of constraints and other views), the model will tilt the portfolio towards the outperforming asset if the view exceeds the difference between the two Implied Equilibrium returns, as shown in View 2.

View 3 shows that the number of outperforming assets does not have to equal the number of failing assets, and that the labels “outperforming” and “underperforming” are relative terms. Views that include several assets with a variety of Implied Equilibrium returns are less intuitive, generalizing more challenges. In the absence of constraints and other views, the view’s assets are divided into two mini-portfolios: a long and a short portfolio. The relative weighting of each nominally outperforming asset is proportional to that asset’s market capitalization divided by the sum of the market capitalization of the other nominally outperforming assets of that particular view. Similarly, the relative weighting of each nominally underperforming asset is proportional to that asset’s market capitalization divided by the sum of the market capitalizations of the other nominally underperforming assets. The difference between the net long and net short positions is zero. The real outperforming asset(s) from the expressed view may not be the mini-portfolio that receives the good view. In general, the model will overweight the “outperforming” assets if the view is greater than the weighted average Implied Equilibrium return differential.

Why should I be interested in this post?

Modern Portfolio Theory (MPT) is at the heart of modern finance and its core foundations are structuring the modern investing panorama. MPT has established itself as the foundation for modern financial theory and practice. MPT’s premise is that beating the market is difficult, and those that do it by diversifying their portfolios appropriately and accepting higher-than-average investment risks.

MPT has been around for almost sixty years, and its popularity is unlikely to wane anytime soon. Its theoretical contributions have laid the groundwork for more theoretical research in the field of portfolio theory. Markowitz’s portfolio theory, however, is vulnerable to and dependent on continuing ‘probabilistic’ development and expansion. This article shed light on an enhancement of the initial Markowitz work by going a step further: to incorporate the views of the investors in the asset allocation process.

Useful resources

Academic research

Best, M.J., and Grauer, R.R. 1991. On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results.The Review of Financial Studies, 315-342.

Black, F. and Litterman, R. 1990. Asset Allocation: Combining Investors Views with Market Equilibrium. Goldman Sachs Fixed Income Research working paper

Black, F. and Litterman, R. 1991. Global Asset Allocation with Equities, Bonds, and Currencies. Goldman Sachs Fixed Income Research working paper

Black, F. and Litterman, R. 1992. Global Portfolio Optimization.Financial Analysts Journal, 28-43.

He, G. and Litterman, R. 1999. The Intuition Behind Black-Litterman Model Portfolios. Goldman Sachs Investment Management Research, working paper.

Idzorek, T.M. 2002. A step-by-step guide to Black-Litterman model. Incorporating user-specified confidence levels. Working Paper, 2-11.

Lee, W., 2000, Advanced theory and methodology of tactical asset allocation. Fabozzi and Associates Publications.

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

Michaud, R.O. 1989. The Markowitz Optimization Enigma: Is Optimized Optimal?. Financial Analysts Journal, 31-42.

Mossin, J. 1966. Equilibrium in a Capital Asset Market. Econometrica, 34(4): 768-783.

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.

About the author

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

Passive Investing

November 21, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of passive investing.

This article will offer a concise summary of the academic literature on passive investment. After that, we’ll discuss the fundamental principles of passive investment. The article will finish by establishing a link between passive strategies and the Efficient Market Hypothesis.

Review of academic literature on passive investing

We can retrace the foundations of passive investing to the theory of portfolio construction developed by Harry Markowitz. For his theoretical implications, Markowitz’s work is widely regarded as a pioneer in financial economics and corporate finance. For his contributions to these disciplines, which he developed in his thesis “Portfolio Selection” published in The Journal of Finance in 1952, Markowitz received the Nobel Prize in economics in 1990. His ground-breaking work set the foundation for what is now known as ‘Modern Portfolio Theory’ (MPT).

William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966) separately developed the Capital Asset Pricing Model (CAPM). The CAPM was a huge evolutionary step forward in capital market equilibrium theory because it enabled investors to appropriately value assets in terms of their risk. The asset management industry intended to capture the market portfolio return in the late 1970s, defined as a hypothetical collection of investments that contains every kind of asset available in the investment universe, with each asset weighted in proportion to its overall market participation. A market portfolio’s expected return is the same as the market’s overall expected return. But as financial research evolved and some substantial contributions were made, new factor characteristics emerged to capture some additional performance.

Core principles of passive investing

Positive outlook: The core element of passive investing is that investors can expect the stock market to rise over the long run. A portfolio that mimics the market will appreciate in lockstep with it.

Low cost: A passive strategy has low transaction costs (commissions and market impact) due to its steady approach and absence of frequent trading. While management fees required by funds are unavoidable, most exchange traded funds (ETFs) – the vehicle of choice for passive investors – charge much below 1%.

Diversification: Passive strategies automatically provide investors with a cost-effective method of diversification. This is because index funds diversify their risk by investing in a diverse range of securities from their target benchmarks.

Reduced risk: Diversification almost usually results in lower risk. Investors can also diversify their holdings more within sectors and asset classes by investing in more specialized index funds.

Passive investing and Efficient Market Hypothesis

The Efficient Market Hypothesis (EMH) asserts that markets are efficient, meaning that all information is incorporated into market prices (Fama, 1970). The passive investing strategy is built on the concept of “buy-and-hold,” or keeping an investment position for a lengthy period without worrying about market timing. This latter technique is frequently implemented through the purchase of exchange-traded funds (ETF) that aim to closely match a given benchmark to produce a performance that is comparable to the underlying index or benchmark. The index might be broad-based, such as the S&P500 index in the US equity market for instance, or more specialized, such as an index that monitors a specific sector or geographical zone.

A study from Bloomberg on index funds suggests that passive investments lead 11.6 trillion $ in the US domestic equity-fund market. Passive investing accounts for approximately 54% of the market, owing largely to the growth of funds tracking the S&P 500, the total US stock market, and other broad US indexes. Large-cap stocks in the United States are widely recognized as the world’s most efficient equity market, contributing to passive investing’s dominance. The $6.2 trillion in passive assets represents less than a sixth of the US stock market, which currently has a market capitalization of approximately $40.4 trillion (Bloomberg, 2021).

Figure 1 depicts the historical monthly returns of the S&P500 highlighting the contraction periods in orange. It is considered as a key benchmark that is heavily tracked by passive instruments like Exchange Traded Funds and Mutual Funds. In a two-decade timeframe analysis, the S&P managed to offer an annualised 5.56% return on average coupled with a 15.16% volatility.

Figure 1. S&P500 historical returns (Jan 2000 – November 2021).

Source: Computation by the author (data source: Thomson Reuters).

Estimation of the S&P500 return

You can download an Excel file with data for the S&P500 index returns (used as a representation of the market).

Why should I be interested in this post?

If you are a business school or university undergraduate or graduate student, this content will help you in grasping the concept of passive investing, which is in practice key to investors, and which has attracted a lot of attention in academia.

Useful resources

Academic research

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

Mangram, M.E., 2013. A simplified perspective of the Markowitz Portfolio Theory. Global Journal of Business Research, 7(1): 59-70.

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

Mossin, J. 1966. Equilibrium in a Capital Asset Market.Econometrica, 34(4): 768-783.

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.

Business analysis

JP Morgan Asset Management, 2021.Glossary of investment terms: Passive Investing

Bloomberg, 2021. Passive likely overtakes active by 2026, earlier if bear market

About the author

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

Alpha

November 20, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of alpha, one of the fundamental parameters for portfolio performance measure.

This article is structured as follows: we introduce the concept of alpha in asset management. Next, we present some interesting academic findings on the alpha. We finish by presenting the mathematical foundations of the concept.

Introduction

The alpha (also called Jensen’s alpha) is defined as the additional return delivered by the fund manager on the overall performance of the portfolio compared to the market performance (Jensen, 1968). A key issue in finance (and particularly in portfolio management) has been evaluating the performance of portfolio managers. The term ‘performance’ encompasses at least two independent dimensions (Sharpe, 1967): 1) The portfolio manager’s ability to boost portfolio returns by successful forecasting of future security prices; and 2) The portfolio manager’s ability to minimize (via “efficient” diversification) the amount of “insurable risk” borne by portfolio holders.

The primary hurdle to evaluating a portfolio’s performance in these two categories has been a lack of a solid grasp of the nature and assessment of “risk”. Risk aversion appears to predominate in the capital markets, and as long as investors accurately perceive the “riskiness” of various assets, this indicates that “risky” assets must on average give higher returns than less “risky” assets. Thus, when evaluating portfolios’ performance, the implications of varying degrees of risk on their returns must be considered (Sharpe, 1967).

One way of representing the performance is by linking the performance of a portfolio to the security market line (SML). Figure 1 depicts the relation between the portfolio performance in relation to the security market line. As illustrated in Figure 1 below, Fund A has a negative alpha as it is located under the SML, implying a negative performance of the fund manager compared to the market. Fund B has a positive alpha as it is located above the SML, implying a positive performance of the fund manager compared to the market.

Figure 1. Alpha and the Security Market Line

Source: Computation by the author.

You can download below an Excel file with data to compute Jensen’s alpha for fund performance analysis.

Academic Literature

Jensen develops a risk-adjusted measure of portfolio performance that quantifies the contribution of a manager’s forecasting ability to the fund’s returns. In the first empirical study to assess the outperformance of fund managers, Jensen aimed at quantifying the predictive ability of 115 mutual fund managers from 1945 to 1964. He looked at their ability to produce returns above the expected return given the risk level of each portfolio. Not only does the evidence on mutual fund performance indicate that these 115 funds on average were unable to forecast security prices accurately enough to outperform a buy-and-hold strategy, but there is also very little evidence that any individual fund performed significantly better than what we would expect from mutual random chance. Additionally, it is critical to highlight that these conclusions hold even when fund returns are measured net of management expenses (that is assume their bookkeeping, research, and other expenses except brokerage commissions were obtained free). Thus, on average, the funds did not appear to be profitable enough in their trading activity to cover even their brokerage expenses.

Mathematical derivation of Jensen’s alpha

The portfolio performance metric given below is derived directly from the theoretical results of Sharpe (1964), Lintner (1965a), and Treynor (1965) capital asset pricing models. All three models assume that (1) all investors are risk-averse and single-period expected utility maximizers, (2) all investors have identical decision horizons and homogeneous expectations about investment opportunities, (3) all investors can choose between portfolios solely based on expected returns and variance of returns, (4) all transaction costs and taxes are zero, and (5) all assets are infinitely fungible. With the extra assumption of an equilibrium capital market, each of the three models produces the following equation for the expected one-period return defined by (Jensen, 1968):

E(r): the expected return of the fund
r_f: the risk-free rate
E(r_m): the expected return of the market
β(E(r_m) – r_f): the systematic risk of the portfolio
α: the alpha of the portfolio (Jensen’s alpha)

Why should I be interested in this post?

If you are a business school or university student, this post will help you to understand the fundamentals of investment.

Useful resources

Academic research

Fama, Eugene F. 1965. The Behavior of Stock Market Prices.Journal of Business 37, 34-105.

Fama, Eugene F. 1967. Risk, Return, and General Equilibrium in a Stable Paretian Market. Chicago, IL: University of Chicago.Unpublished manuscript.

Fama, Eugene F. 1968. Risk, Return, and Equilibrium: Some Clarifying Comments. Journal of Finance, 23, 29-40.

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.

Sharpe, William F. 1966. Mutual Fund Performance. Journal of Business39, Part 2: 119-138.

Treynor, Jack L. 1965. How to Rate Management of Investment Funds.Harvard Business Review 18, 63-75.

Business analysis

JP Morgan Asset Management, 2021.Glossary of investment terms: Alpha

About the author

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

Capital Market Line (CML)

January 30, 2024November 20, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the Capital Market Line (CML), a key concept in asset pricing derived from the Capital Asset Pricing Model (CAPM).

This article is structured as follows: we first introduce the concept. We then illustrate how to estimate the capital market line (CML). We finish by presenting the mathematical foundations of the CML.

Capital Market Line

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

Let us consider two cases: 1) when investors have access to risky assets only; 2) when investors have access to risky assets and a risk-free asset (earning a constant interest rate, 2% for example below).

Risky assets

In the case of risky assets only, the efficient frontier (the set of optimal portfolios) is represented below in Figure 1.

Figure 1. Efficient frontier with risky assets only.
img_Simtrade_CML_graph_1
Source: Computation by the author.

Risky assets and a risk-free asset

In the case of risky assets and a risk-free asset, the efficient frontier (the set of optimal portfolios) is represented below in Figure 2. In this case, the efficient frontier is a straight line called the Capital Market Line (CML).

Figure 2. Efficient frontier with risky assets and a risk-free asset.

Source: Computation by the author.

The CML joins the risk-free asset and the tangency portfolio, which is the intersection with the efficient frontier with risky assets only. We can reasonably conclude from Figure 2 that, to increase expected return, an investor has to increase the amount of risk he or she takes to attain returns higher than the risk-free interest rate. As a result, the Sharpe ratio of the market portfolio equals the slope of the CML. If the Sharpe ratio is more than the CML, an investment strategy can be implemented, such as buying assets if the Sharpe ratio is greater than the CML and selling assets if the Sharpe ratio is less than the CML (Drake and Fabozzi, 2011).

Investors who allocate their money between a riskless asset and the risky market portfolio M can expect a return equal to the risk-free rate plus compensation for the number of risk units σ_P) they accept. This result is in line with the underlying notion of all investment theory: investors perform two services in the capital markets for which they might expect to be compensated. First, they enable someone else to utilize their money in exchange for a risk-free interest rate. Second, they face the risk of not receiving the promised returns in exchange for their invested capital. The term E(r_M)- R_f) / σ_M refers to the investor’s expected risk premium per unit of risk, which is also known as the expected compensation per unit of risk taken.

Figure 3 represents the Capital Market Line which connect the risk-free asset to the efficient frontier line. The straight line in Figure 3 represents a combination of a risky portfolio and a riskless asset. Any combination of the risk-free asset and Portfolio A is similarly outperformed by some combination of the risk-free asset and Portfolio B. Continue drawing a line from R_f to the efficient frontier with increasing slopes until you reach Portfolio M’s point of tangency. All other possible portfolio combinations that investors could build are outperformed by the collection of portfolio possibilities along Line R_f-M, which is the CML. The CML, in this sense, represents a new efficient frontier that combines the Markowitz efficient frontier of risky assets with the ability to invest in risk-free securities. The CML’s slope is (E(r_M)- R_f) / σ(_M), which is the highest risk premium compensation that investors can expect for each unit of risk they take on (Reilly and Brown, 2012) (Figure 3).

If we fully invest our cash on the risk-free rate, we would be exactly on the y axis with an expected return of 2%. Each time we move along the curve that connects the risk-free rate to the optimum market portfolio, we allocate less weight to the risk-free rate, and we overweight more on riskier assets (Point A). Points M represents the optimal risky portfolio in the efficient frontier line, which minimizes the overall portfolio variance. It would have a weighting of 45% in stock A and a 55% in stock B, which would offer a 26.23% annualized return for a 17.27% annualized volatility. Point B represents a portfolio composition that is based on a leveraged position of 140% on the optimal risky portfolio and a short position on the risk-free asset of -40% (Figure 3).

Figure 3. Efficient frontier with different points.

Source: Computation by the author.

Mathematical representation

We can define the CML as the line that is tangent to the efficient frontier which connects the risk-free asset with the market portfolio:

img_SimTrade_CML_equations_0

Where:

σ_P: the volatility of portfolio P
R_f: the risk-free interest rate
E(R_M): the expected return of the market M
σ_M: the volatility of the market M
E[R_M– R_f]: the market risk premium.

The expected return of the portfolio can be computed as:

img_SimTrade_CML_equations_1

The Sharpe Ratio is shown in parenthesis, and it compares the performance of an investment, such as a security or portfolio, to the performance of a risk-free asset after adjusting for risk. It is calculated by dividing the difference between the investment returns and the risk-free return by the standard deviation of the investment returns. It denotes the additional amount of return that an investor receives for each unit of risk increase (Sharpe, 1963). We can define it mathematically as:

img_SimTrade_CML_equations_2

We can identify the following relationship between the slope of the CML and the Sharpe ratio of the market portfolio, defined mathematically as follows:

img_SimTrade_CML_equations_3

A simple strategy for stock selection is to buy assets with Sharpe ratios that are higher than the CML and sell those with Sharpe ratios that are lower. Indeed, the efficient market hypothesis implies that beating the market is impossible. As a result, all portfolios should have a Sharpe ratio that is lower than or equal to the market. As a result, if a portfolio (or asset) has a higher Sharpe ratio than the market, this portfolio (or asset) has a higher return per unit of risk (i.e. volatility), which contradicts the efficient market hypothesis. The alpha is the abnormal excess return over the market return at a given level of risk.

Why should I be interested in this post?

Sharpe ratio is a popular tool for assessing portfolio risk/return in finance. The Sharpe ratio informs the investor precisely which portfolio has the best performance among the available options. This simplifies the investor’s decision-making process. The higher the ratio, the greater the return for each unit of risk.

If you are a business school or university undergraduate or graduate student, this content will help you in broadening your knowledge of finance.

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.

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

Mossin, J. 1966. Equilibrium in a Capital Asset Market. Econometrica, 34(4): 768-783.

Reilly, R. K., Brown C. K., 2012. Investment Analysis & Portfolio Management, Tenth Edition.

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.

About the author

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

My experience as a portfolio manager in a central bank

November 29, 2020 by Youssef LOURAOUI

My experience as a portfolio manager in a central bank

During my studies at ESSEC Business School, I had the chance to attend the SimTrade course. This course helped me to secure an internship as a risk manager at Bank Al-Maghrib (the central bank of Morocco) as I was asked during my interviews technical questions about financial markets that were covered during the course.

In this article, Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2020) shares his experience as an intern in the risk management department (middle office) at the Central Bank of Morocco (Bank Al-Maghrib) in 2020.

Bank Al-Maghrib

The central bank of Morocco was founded in 1959 after Morocco proclaimed its independence. It is a 100% state-owned bank that regulates the markets and the economy by implementing monetary and economic policies to ensure the welfare in terms of the parity of prices and the control of inflation. Inflation is a major economic indicator that possesses strategic importance and is part of the major focus for the central bank.

Bank Al-Maghrib

I describe below my experience at Bank Al-Maghrib.

My internship at Bank Al-Maghrib

I was affected at the middle office department, which is in charge of measuring risk exposures and profits and losses on the positions taken by the bank on an investment portfolio of 27,4 billion euros of foreign reserve. One of the key risk exposure metrics is volatility measured by the standard deviation statistically defined as the dispersion of a random variable (asset prices or returns in my case) from its expected value. The standard deviation indicates how much the current return is deviating from its expected historical returns. It is one of the most widely used metrics for investors when analyzing the risk of an investment. Among other key exposures metric, there is what it is called the VaR (Value at Risk) at 99% and a 95% confidence level for 1-day and 30-day positions. In other words, the VaR is a metric used to compute how much loss can the portfolio incur at a % degree of confidence for a given time horizon.

Every day, the Head of the Middle Office organizes a general meeting where he talks about global debriefing of the main financial news that happened overnight and debriefing the middle office desk for the “watch out” assets that could have a potential investment opportunity. Accordingly, the team has also the task of staying in line with the investment decision that characterizes the organization, as it does not operate as an investment banking corporation nor a hedge fund in the risk and leverage used. As the central bank has the special task of keeping safe the national reserve and searching for a good mix to invest in a low risk asset (AAA bonds from European countries coupled with American treasury bonds).

My task aimed to get a hand on the investment mechanism in the middle office of the bank. The investment mechanism consists of the division of the overall portfolio into three main tranches where each one has its characteristics. The first tranche (called also the security tranche) is calculated by analyzing the national need for a currency that needs to be kept safe to establish welfare on the exchange market (based mainly on short term position in low-risk profile asset (Liquid and high rated bonds). The second tranche is based on buy and hold and a market strategy. The first one consists of taking a long position on more risky assets than the first tranche till maturity, there is no selling during the lifetime of the asset (riskier bonds and gold). The second strategy is based on buying and selling liquid assets for an expectation of yielding higher returns.

During my time at the middle office desk, I’ve managed to develop a tool to represent the investment mechanism used for asset allocation. The tool, developed in an Excel spreadsheet, is an intuitive and simplified model that enables the understanding of the investment mechanism. Indeed, it is capable of continuously refreshing the data by importing the most recent quotations (from data providers like Bloomberg or Reuters as the two main financial data providers) to allow for an update of the different exposures and thus allow to respect the proportions of portfolio allocations. It has also a dynamic risk management tool to effectively compute draw-downs (a peak-to-trough decline during a specific period for an investment) and stressed conditions, as I experienced how the markets reacted to the novel Covid-19 pandemic with one of the most historic market movements in a long time.

Some of the key learning outcomes:

The introduction to data analysis by manipulating large datasets

Portfolio optimization based on the Markowitz efficient frontier

Dynamic portfolio allocation based on the fundamentals of the modern portfolio theory

The theory of efficient markets to understand how the markets evolve and move in a different direction as a reaction to events.

Front office, middle office and back office

My internship was also a good opportunity to discover the different departments of the bank: the front office, the middle office, and the back office:

The front office directly deals with the individual or corporate clients of the bank. Salespeople propose adequate products and solutions to the clients (they are in front of them!). Traders intervene in the financial markets on behalf of the clients or for the bank itself (proprietary trading). To answer the demand of clients, financial engineers and quants also develop new products and the associated mathematical models to price them. One of the main trends that are emerging in the front office is the automatization with the help of AI and algorithmic trading that is taken some room in the trading desks. At this time the bank didn’t implement any technology based on high-frequency trading, but it is taking the financial industry by surprise and it goes a long way back, nearly decades ago since the first usage of algorithmic trading.

The middle office situated between the front office and the back office (somewhere in the middle!) deals with the risk management of the bank. Risk managers control the traders’ positions (respect of constraints such as value-at-risk limits and stress tests) and compute the profits and losses (P&L) on traders’ positions daily.

The back-office deals with the conformity and the security check of every trade to ensure a proper settlement.

Note that the frontiers between the front, middle, and back-office may change from one bank to another. And last but not the least, the IT people are also supporting all three departments to make the whole system work. In other words, they are in charge of the maintenance of the technical infrastructure that the bank uses daily to operate fluently, as all the departments are dependent on internal software to intermediate and operate in the market or to communicate between each department of the bank or with another organization. The IT desk has great importance in offering a flawless experience for the employees when using the internal electronic infrastructure. There is the backbone of the bank skeleton.

All in all, the SimTrade module served me well as I managed to gain quickly the necessary knowledge and bridge the gap that I had to be in the best position to achieve the missions I’ve been affected. I especially used the content of Period 2 of the SimTrade certificate, which deals with market information. The concepts of trading and investing were also obviously useful for the development of my portfolio management tools.

Useful resources

Bank of Morocco

About the author

The article was written in November 2020 by Youssef LOURAOUI (ESSEC Business School, Global Bachelor of Business Administration, 2020).

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