Returns

April 20, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains how returns of financial assets are computed and their interpretation in the world of finance.

Introduction

The main focus of any investment in financial markets is to make maximum profits within a coherent risk level. Returns in finance is a metric that inherently refers to the change in the value of any investment. Positive values of returns are interpreted as gains whereas negative values are interpreted as losses.

Returns are generally computed over standardized frequencies such as daily, monthly, yearly, etc. They can also be computed for specific time periods such as the holding period for ease of comparison and analysis.

Computation of returns

Consider an asset for a time period [t -1, t] with an initial price P_t-1 at time t-1 and final price P_t at time t (one period, two dates). Different forms of defining returns for the asset over period [t -1, t] are discussed below.

Arithmetic (percentage) returns

This is the simplest way for computation of returns.

The return over the period [t -1, t], denoted by R_t, is expressed as:

Arithmetic returns

Logarithmic returns

Logarithmic returns (or log returns) are also used commonly to express investment returns. The log return over the period [t-1, t], denoted by R_t is expressed as:

Logarithmic returns

Log returns provide the property of time-additivity to the returns which essentially means that the log returns over a given period can be simply added together to compute the total return over subperiods. This feature is particularly useful in statistical analysis and reduction of algorithmic complexity.

Logarithmic returns additivity

Log returns are also known as continuously compounded returns because the rate of log returns is equivalent to the continuously compounding interest rate for the asset at price P₀ and time period t.

img_SimTrade_compounded_returns

Link between arithmetic and logarithmic returns

The arithmetic return (R^ari) and the logarithmic return (R^log) are linked by the following formula:

Relation between arithmetic and logarithmic returns

Components of total returns

The total return on an investment is essentially composed of two components: the yield and the capital gain (or loss). The yield refers to the periodic income or cash-flows that may be received on the investment. For example, for an investment in stocks, the yield corresponds to the revenues of dividends while for bonds, it corresponds to interest payments.

On the other hand, capital gain (or loss) refers to the appreciation (or depreciation) in the price of the investment. Thus, the capital gain (or loss) for any asset is essentially the price change in the asset.

Total returns for a stock over the period [t -1, t], denoted by R_t, can hence be expressed as:

Total returns

Where
   P_t: Stock price at time t
   P_t-1: Stock price at time t-1
   D_t-1,t: Dividend obtained over the period [t -1, t]

Price changes and returns

Consider a stock with an initial price of 100€ at time t=0. Suppose the stock price drops to 50€ at time t=1. Thus, there is a change of -50% (minus sign representing the decrease in price) in the initial stock price.

Now for the stock price to reach back to its initial price (100€ in this case) at time t=2 from its price of 50€ at time t=1, it will require an increase of (100€-50€)/50€ = 100%. With arithmetic returns, the increase (+100%) has to be higher than the decrease (-50%).

Similarly, for a price drop of -25% in the initial stock price of 100€, we would require an increase of 33% in the next time period to reach back the initial stock price. Figure 1 illustrates this asymmetry between positive and negative arithmetic returns.

Figure 1. Evolution of price change as a measure of arithmetic returns.
img_SimTrade_price_change_evolution
Source: computation by the author.

If the return is defined as a logarithmic return, there is a symmetry between positive and negative logarithmic returns as illustrated in Figure 2.

Figure 2. Evolution of price change as a measure of logarithmic returns.
img_SimTrade_price_change_evolution
Source: computation by the author.

You can also download below the Excel file for computation of arithmetic returns and visualise the above price change evolution.

Internal rate of return (IRR)

Internal rate of return (IRR) is the rate at which a project undertaken by the firm break’s even. It is a financial metric used by financial analysts to compute the profitability from an investment and is calculated by equating the initial investment and the discounted value of the future cashflows i.e., making the net present value (NPV) equal to zero. The IRR is the sprecail value of the discount rate which makes the NPV equal to zero.

The IRR for a project can be computed as follows:

IRR formula

Where,
CF_t : Cashflow for time period t

The higher the IRR from a project, the more desirable it is to pursue with the project.

Ex ante and ex post returns

Ex ante and ex post are Latin expressions. Ex ante refers to “before an event” while ex post refers to “after the event”. In context of financial returns, the ex ante return corresponds to prediction or estimation of an asset’s potential future return and can be based on a financial model like the Capital Asset Pricing Model (CAPM). On the other hand, the ex post return corresponds to the actual return generated by an asset historically, and hence are lagging or backward-looking in nature. Ex post returns can be used to forecast ex ante returns for the upcoming period and together, both are used to make sound investment decisions.

About the author

The article was written in April 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Supply and Demand

April 15, 2022 by Diana Carolina SARMIENTO PACHON

Supply and Demand

In this article, Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the economic concept of supply and demand, which is key to understand the way markets work.

Supply and demand are the fundamental concepts that shape the way we make business and operate in the world. They construct both simple transactions such as the purchase of coffee or more compounded transactions such as the operations in the financial world. For this reason, it’s crucial to understand and uncover them deeper.

The basic concepts

Supply is referred to the amount available of a product that firms offer, whereas demand is the amount desired by consumers or households. When these quantities are equal, an equilibrium is reached and consequently a transaction takes place, leading to the well-known law of supply & demand which shapes the behavior of daily transactions and shifts in the economy. If price increases, then supply also increases; nonetheless, demand decreases as it’s more expensive for consumers to a buy good; on the contrary, if prices decline then supply also decline since producers would make less revenue whereas demand goes up as it is cheaper to buy. This dynamic takes place until the quantities of supply and demand are equal so that the optimum equilibrium is found.

Figure 1. Supply and demand.
img_Simtrade_risk_reduction_stocks
Source: computation by the author.

From another perspective, if demand escalates then price rises due to the high desirability of the good, meanwhile when demand drops it can create a surplus of supply which can drag the price down. Likewise, this scenario can be applied in financial markets e.g., in the case of a bullish sentiment in the market, there can be a positive speculation which creates a higher desirability for certain stock resulting in a decrease in price; nevertheless, when demand is low the price may drop because of a low or negative speculation on a specific stock.

Furthermore, the fundamental law of supply and demand can also explain the price movements seen in the financial markets. To illustrate, for a commodity such as coffee, if the surface of cultivation expands or if the harvest is good, it is very likely that the coffee price will sink as its supply will be abundant. Therefore, it is essential to consider the information about the market regularly as it can have a significant influence on the speculation of investors which will eventually define their demands and so the price of a stock. Consequently, it is very important to be able to determine how an announcement or any kind of information can affect the demand or even the supply of a stock, commodity, or financial instrument since this will define how markets will behave.

Special cases

However, it’s also important to mention that there are industries and situations in which the law of supply and demand does not apply. An instance of this is the luxury industry, in which the higher the product price set by firms, the higher the demand from consumers. This may be due to the value that costumers perceive by purchasing such items. Alternatively, oil is another example to be mentioned as its price has a low-price sensitivity which means that any change in its price won’t result in any significant demand changes, this could be due to the high necessity of oil in all industries which makes it crucial for daily operations.

Useful resources

Krugman, P. & Wells, R. (2012) Economics. 3rd edition. United States: Macmillan Learning.

Mankiw, G. (2016) The Market Forces of Supply and Demand (table of content) Principles of Economics. 8th edition. Boston: Cengage Learning.

Mankiw, G. (2016) The Market Forces of Supply and Demand (slides) Principles of Economics. 8th edition. Boston: Cengage Learning.

Deskara Supply and Demand: Law, Curves, and Examples

International Energy Agency (IEA) Supply and demand for oil

Sabiou M. Inoua and Vernon L. Smith The Classical Theory of Supply and Demand

About the author

The article was written in April 2022 by Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022)

Risk Aversion

April 14, 2022 by Diana Carolina SARMIENTO PACHON

Risk Aversion

In this article, Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the economic concept of risk aversion, which is key to understand the behavior of participants in financial markets.

Risk Aversion refers to the level of reluctance that an individual possesses towards risk. Specifically, it refers to the attitude of investors towards the risk underlying investments which will directly determine how portfolios are allocated or even how a stock may behave depending on market conditions. To elaborate, when market participants have higher risk aversion due to unfavorable market shocks e.g., natural disasters, bad news or scandals that affect a company or a security, this situation will cause a perception of higher risk leading to many selling, and thus decreasing prices. Therefore, risk aversion should be analyzed carefully.

Risk aversion and investor’s characteristics

It’s important to note that risk aversion can be highly variable over time as this notion changes along with investor profile, in other words with age, income, culture and other key factors, making it even more complex to evaluate than it appears in the traditional economics literature. To illustrate more accurately some of the factors that define an investor profile are:

Age

The older the person is, the more risk averse he or she is. On the contrary, younger individuals tend to be less risk averse which may be due to their high expectations and eagerness to attempt something new as well as the longer timeframe they have, whereas older people prefer safety and stability in their lives.

Income

Individuals with a smaller budget tend to have a higher risk aversion since they have fewer resources, and a loss would make a greater impact on them than a wealthy individual.

Past Losses

When an individual has already experienced some loss, she or he will be more wary of it since it’s now too costly to bear another loss; therefore, risk aversion will be significantly higher. An example of this is the post-crisis, as people have lost so much and this has had a negative impact on their lives, they tend to become more cautious of risk.

Investment Objective

For crucial events such as retirement or education, risk version tends to be higher as the individual cannot bear to risk for such a fundamental matter of his or her life.

Investment Horizon

Investors focused on short-term horizon tend to be more risk averse as they cannot take too much risk due to the short timeline.

Risk aversion and financial investments

Furthermore, risk aversion also takes into account more factors apart from those mentioned above, for this reason most of the time before creating the respective portfolio for an investor, financial advisors shape their client’s risk preferences in order to adjust the portfolio allocation to them. Many times, these can be conducted by questionnaires and tests that will accordingly assign a risk profile concluding with certain risk categories:

A Conservative profile refers to more risk averse individuals, the portfolios assigned for this type are mainly composed by both more secure & less volatile securities such as bonds, meanwhile stocks have a minimal participation.
A Moderate profile is attributed to more risk averse individuals who are willing to take more risk, however he or she does not want to step too much further. These portfolios are usually more diversified as they contain more types of securities in different percentages such as government & corporate bonds, and stocks.
An aggressive profile which is allocated to portfolios mainly composed in the highest percentage by the risky securities. For instance, the main securities could be stocks, specifically growth stocks or even crypto.

Due to all sensitive and private information used by financial institutions, financial regulatory entities are important to ensure the protection and transparency of information, thereby the Mifid (The Markets in Financial Instruments Directive) has been created in the European Union to fulfill such task through the use of rules and general standards.

Measure of risk of financial assets

Additionally, there are other mathematical metrics that can interfere in the risk profile, and depending on these the portfolio may be constructed:

Standard Deviation

It refers to the volatility of historical data, in other words how dispersed the data is over time which illustrates how risky the security may be. The higher the standard deviation, the higher the risk since this is suggesting that the stock is more variable and there is more uncertainty, thus a risk averse individual prefers a lower standard deviation.

Beta

It is linked with the systematic risk that comes with a stock, that is to say it illustrates the volatility compared to the market. Firstly, a beta equal to 1 indicates a volatility and movement equalizing the market, secondly a beta higher than 1 is referred to a security that is more volatile than the market, to illustrate B= 1.50 specifies 50% more volatility than the market. Thirdly, a beta less than 1 stipulates less volatility than the market. Therefore, the lower the beta the less risk exposure is found.

Modern Portfolio Theory & Risk

Introduced by Harry Markowitz in 1950s, the Modern Portfolio Theory illustrates the optimum portfolio allocation that maximizes return given a specific level of risk, in which risk is measured by the standard deviation and the return by the average mean of the portfolio. This explanation also leads to the one- single period mean-variance theory which suggests various portfolio allocations depending on the trade-off between return and risk. However, there are more advance models which explain this scenario in a multiperiod by rebalancing or diversifying further.

Risk aversion and economic conditions

Risk aversion does not only shape the portfolio allocation and its diversification, but it also may have a significant impact on the market as a result of expectations. When there are booming economic times, individuals usually feel more confident and thus less risk averse as a consequence of positive expectations of future cash flows; however, when a recession is coming investors may shift to a more risk averse behavior making them feel afraid of the future which influences them to sell certain stocks and, in this way, making the price plump. Although it may be seen as a simple emotion that defines the fear of risk, it still impacts in a very large extent the financial market as it dictates the roles and strategies behind investing, and thereby it is crucial analyze it carefully.

▶ Youssef LOURAOUI Markowitz Modern Portfolio Theory

▶ Youssef LOURAOUI Implementing Markowitz asset allocation model

▶ Jayati WALIA Capital Asset Pricing Model (CAPM)

Useful resources

Díaz A and Esparcia C (2019) Assessing Risk Aversion From the Investor’s Point of View Frontiers in Psychology, 10:1490

Desjardins Online brokerage The Risk Aversion Coefficient

Coursera course Investment management

Crehana course Trading: How to invest in stocks (Trading: Como invertir en Bolsa)

About the author

The article was written in April 2022 by Diana Carolina SARMIENTO PACHON (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022)

A quick overview of the Bloomberg terminal…

April 11, 2022 by Louis DETALLE

A quick overview of the Bloomberg terminal…

In this article, Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains everything there is to know about the Bloomberg terminal which is a must-know in finance.

How to use the main functions of the Bloomberg terminal?

One may notice that the keyboard of the Bloomberg terminal is a little strange. Indeed, this keyboard called Starboard, and contains red, blue, green and yellow keys for specific functions in addition to your regular keys.

Functions are unique Bloomberg applications that provide analysis and information on securities,
sectors, regions and more. Each function is accessed by typing in its unique mnemonic (a short, memorable name) and then pressing the key located in the lowest-right sided area of the keyboard.

Let’s review together the different functions of the buttons:

The HELP button is perhaps the most useful button for those just starting out. If you have questions about anything on the terminal, simply press the button once and a Bloomberg specialist will be there to start a live chat with you to resolve your questions.

In order to benefit from the latest news, users can simply type NEWS and press enter to get the latest information on market trends, movements and other relevant news.

Those in the finance industry chat via Bloomberg Messaging, which is essentially equivalent to Facebook Messenger but on Bloomberg. It enables you to send a message to anyone on the device. This means that anyone in the industry can technically contact each other instantly. No need to ask for someone else’s number or find out the best way to get in touch.

Main users of the Bloomberg terminal

Traders, brokers, analysts, portfolio managers, investors and executives are the Terminal’s primary consumer base as they need to access the data provided by Bloomberg easily in order to do their job.

A subscription to the Bloomberg Terminal costs approximately $20,000 a year, but that does not stop its customers from renewing their subscriptions because of its usefulness.

Training webinars

First and foremost, the Bloomberg beginner should work on the document available on Bloomberg website, Getting started on the Bloomberg Terminal, which will give you the main information on the keys and their function.

The best next step to get used to the Bloomberg Terminal is to complete the certification
course: Bloomberg Market Concepts (BMC). BMC is an 8-hour e-learning course that will
provide a visual introduction to the financial markets and covers nearly 70 Terminal functions which is enough for whoever wants to start using Bloomberg.

Useful resources

Bloomberg’s website

Capital Markets (BMC) Certification’s website

About the author

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

Specific risk

April 1, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) explains the specific risk of financial assets, a key concept in asset pricing models and asset management in practice.

This article is structured as follows: we start with a reminder of portfolio theory and the central concept of risk in financial markets. We then introduce the concept of specific risk of an individual asset and especially its sources. We then detail the mathematical foundation of risk. We finish with an insight of the relationship between diversification and risk reduction with a practical example to test this concept.

Portfolio Theory and Risk

Markowitz (1952) and Sharpe (1964) created a framework for risk analysis based on their seminal contributions to portfolio theory and capital market theory. All rational profit-maximizing investors attempt to accumulate a diversified portfolio of risky assets and borrow or lend to achieve a risk level consistent with their risk preferences given a set of assumptions. They established that the key risk indicator for an individual asset in these circumstances is its correlation with the market portfolio (the beta).

The variance of returns of an individual asset can be decomposed as the sum of systematic risk and specific risk. Systematic risk refers to the proportion of the asset return variance that can be attributed to the variability of the whole market. Specific risk refers to the proportion of the asset return variance that is unconnected to the market and reflects the unique nature of the asset. Specific risk is often regarded as insignificant or irrelevant because it can be eliminated in a well-diversified portfolio.

Sources of specific risk

Specific risk can find its origin in business risk (in the assets side of the balance sheet) and financial risk (in the liabilities side of the balance sheet):

Business risk

Internal or external issues might jeopardize a business. Internal risk is directly proportional to a business’s operational efficiency. An internal risk would include management neglecting to patent a new product, so eroding the company’s competitive advantage.

Financial risk

This pertains to the capital structure of a business. To continue growing and meeting financial obligations, a business must maintain an ideal debt-to-equity ratio.

Mathematical foundations

Following the Capital Asset Pricing Model (CAPM), the return on asset i, denoted by R_i can be decomposed as

img_SimTrade_return_decomposition

Where:

R_i the return of asset i
E(R_i) the risk premium of asset i
β_i the measure of the risk of asset i
R_M the return of the market
E(R_M) the risk premium of the market
R_M – E(R_M) the market factor
ε_i represent the specific part of the return of asset i

The three components of the decomposition are the expected return, the market factor and an idiosyncratic component related to asset only. As the expected return is known over the period, there are only two sources of risk: systematic risk (related to the market factor) and specific risk (related to the idiosyncratic component).

The beta of the asset with the market is computed as:

Beta

Where:

σ_i,m : the covariance of the asset return with the market return
σ_m² : the variance of market return

The total risk of the asset measured by the variance of asset returns can be computed as:

Decomposition of total risk

Where:

β_i² * σ_m² = systematic risk
σ_εi² = specific risk

In this decomposition of the total variance, the first component corresponds to the systematic risk and the second component to the specific risk.

Decomposition of returns

We analyze the decomposition of returns on Apple stocks. Figure 1 gives for every month of 2021 the decomposition of Apple stock returns into three parts: expected return, market factor (systematic return) and an idiosyncratic component (specific return). We used historical price downloaded from the Bloomberg terminal for the period 1999-2022.

Figure 1. Decomposition of Apple stock returns:
expected return, systematic return and specific return.
Decomposition of asset returns Computation by the author (data: Bloomberg).

You can download below the Excel file which illustrates the decomposition of returns on Apple stocks.

Why should I be interested in this post?

Investors will be less influenced by single incidents if they possess a range of firm stocks across several industries, as well as other types of assets in a number of asset classes, such as bonds and stocks.

An investor who only bought telecommunication equities, for example, would be exposed to a high amount of unsystematic risk (also known as idiosyncratic risk). A concentrated portfolio can have an impact on its performance. This investor would spread out telecommunication-specific risks by adding uncorrelated positions to their portfolio, such as firms outside of the telecommunication market.

Useful resources

Academic research

Evans, J.L., Archer, S.H. 1968. Diversification and the Reduction of Dispersion: An Empirical Analysis. The Journal of Finance, 23(5): 761–767.

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.

Tole T.M. 1982. You can’t diversify without diversifying. The Journal of Portfolio Management. Jan 1982, 8 (2) 5-11.

About the author

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

Systematic risk

April 1, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents the systematic risk of financial assets, a key concept in asset pricing models and investment management theories more generally.

This article is structured as follows: we introduce the concept of systematic risk. We then explain the mathematical foundation of this concept. We present an economic understanding of market risk on recent events.

Portfolio Theory and Risk

Markowitz (1952) and Sharpe (1964) developed a framework on risk based on their significant work in portfolio theory and capital market theory. All rational profit-maximizing investors seek to possess a diversified portfolio of risky assets, and they borrow or lend to get to a risk level that is compatible with their risk preferences under a set of assumptions. They demonstrated that the key risk measure for an individual asset is its covariance with the market portfolio under these circumstances (the beta).

The fraction of an individual asset’s total variance attributable to the variability of the total market portfolio is referred to as systematic risk, which is assessed by the asset’s covariance with the market portfolio. Systematic risk can be decomposed into the following categories:

Interest rate risk

We are aware that central banks, such as the Federal Reserve, periodically adjust their policy rates in order to boost or decrease the rate of money in circulation in the economy. This has an effect on the interest rates in the economy. When the central bank reduces interest rates, the money supply expands, allowing companies to borrow more and expand, and when the policy rate is raised, the reverse occurs. Because this is cyclical in nature, it cannot be diversified.

Inflation risk

When inflation surpasses a predetermined level, the purchasing power of a particular quantity of money reduces. As a result of the fall in spending and consumption, overall market returns are reduced, resulting in a decline in investment.

Exchange Rate Risk

As the value of a currency reduces in comparison to other currencies, the value of the currency’s returns reduces as well. In such circumstances, all companies that conduct transactions in that currency lose money, and as a result, investors lose money as well.

Geopolitical Risks

When a country has significant geopolitical issues, the country’s companies are impacted. This can be mitigated by investing in multiple countries; but, if a country prohibits foreign investment and the domestic economy is threatened, the entire market of investable securities suffers losses.

Natural disasters

All companies in countries such as Japan that are prone to earthquakes and volcanic eruptions are at risk of such catastrophic calamities.

Following the Capital Asset Pricing Model (CAPM), the return on asset i, denoted by R_i can be decomposed as

img_SimTrade_return_decomposition

Where:

R_i the return of asset i
E(R_i) the risk premium of asset i
β_i the measure of the risk of asset i
R_M the return of the market
E(R_M) the risk premium of the market
R_M – E(R_M) the market factor
ε_i represent the specific part of the return of asset i

The beta of the asset with the market is computed as:

Beta

Where:

σ_i,m : the covariance of the asset return with the market return
σ_m² : the variance of market return

The total risk of the asset measured by the variance of asset returns can be computed as:

Decomposition of total risk

Where:

β_i² * σ_m² = systematic risk
σ_εi² = specific risk

In this decomposition of the total variance, the first component corresponds to the systematic risk and the second component to the specific risk.

Systematic risk analysis in recent times

The volatility chart depicts the evolution of implied volatility for the S&P 500 and US Treasury bonds – the VIX and MOVE indexes, respectively. Implied volatility is the price of future volatility in the option market. Historically, the two markets have been correlated during times of systemic risk, like as in 2008 (Figure 1).

Figure 1. Volatility trough time (VIX and MOVE index).

Sources: BlackRock Risk and Quantitative Analysis and BlackRock Investment Institute, with data from Bloomberg and Bank of America Merrill Lynch, October 2021 (BlackRock, 2021).

The VIX index has declined following a spike in September amid the equity market sell-off. It has begun to gradually revert to pre-Covid levels. The periodic, albeit brief, surges throughout the year underscore the underlying fear about what lies beyond the economic recovery and the possibility of a wide variety of outcomes. The MOVE index — a gauge of bond market volatility – has remained relatively stable in recent weeks, despite the rise in US Treasury yields to combat the important monetary policy to combat the effect of the pandemic. This could be a reflection of how central banks’ purchases of government bonds are assisting in containing interest rate volatility and so supporting risk assets (BlackRock, 2021).

The regime map depicts the market risk environment in two dimensions by plotting market risk sentiment and the strength of asset correlations (Figure 2).

Figure 2. Regime map for market risk environment.

Source: BlackRock Risk and Quantitative Analysis and BlackRock Investment Institute, October 2021 (BlackRock, 2021).

Positive risk sentiment means that riskier assets, such as equities, are outperforming less risky ones. Negative risk sentiment means that higher-risk assets underperform lower-risk assets.

Due to the risk of fast changes in short-term asset correlations, investors may find it challenging to guarantee their portfolios are correctly positioned for the near future. When asset correlation is higher (as indicated by the right side of the regime map), diversification becomes more difficult and risk increases. When asset prices are less correlated (on the left side of the map), investors have greater diversification choices.

When both series – risk sentiment and asset correlation – are steady on the map, projecting risk and return becomes easier. However, when market conditions are unpredictable, forecasting risk and return becomes substantially more difficult. The map indicates that we are still in a low-correlation environment with a high-risk sentiment, which means that investors are rewarded for taking a risk (BlackRock, 2021). In essence, investors should use diversification to reduce the specific risk of their holding coupled with macroeconomic fundamental analysis to capture the global dynamics of the market and better understand the sources of risk.

Why should I be interested in this post?

Market risks fluctuate throughout time, sometimes gradually, but also in some circumstances dramatically. These adjustments typically have a significant impact on the right positioning of a variety of different types of investment portfolios. Investors must walk a fine line between taking enough risks to achieve their objectives and having the proper instruments in place to manage sharp reversals in risk sentiment.

Useful resources

Academic research

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

BlackRock, 2021. Market risk monitor

About the author

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

The Monte Carlo simulation method for VaR calculation

January 30, 2024March 24, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) explains the Monte Carlo simulation method for VaR calculation.

Introduction

Monte Carlo simulations are a broad class of computational algorithms that rely majorly on repeated random sampling to obtain numerical results. The underlying concept is to model the multiple possible outcomes of an uncertain event. It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models.

The Monte Carlo simulation method was invented by John von Neumann (Hungarian-American mathematician and computer scientist) and Stanislaw Ulam (Polish mathematician) during World War II to improve decision making under uncertain conditions. It is named after the popular gambling destination Monte Carlo, located in Monaco and home to many famous casinos. This is because the random outcomes in the Monte Carlo modeling technique can be compared to games like roulette, dice and slot machines. In his autobiography, ‘Adventures of a Mathematician’, Ulam mentions that the method was named in honor of his uncle, who was a gambler.

Calculating VaR using Monte Carlo simulations

The basic concept behind the Monte Carlo approach is to repeatedly run a large number of simulations of a random process for a variable of interest (such as asset returns in finance) covering a wide range of possible scenarios. These variables are drawn from pre-specified probability distributions that are assumed to be known, including the analytical function and its parameters. Thus, Monte Carlo simulations inherently try to recreate the distribution of the return of a position, from which VaR can be computed.

Consider the CAC40 index as our asset of interest for which we will compute the VaR using Monte Carlo simulations.

The first step in the simulation is choosing a stochastic model for the behavior of our random variable (the return on the CAC 40 index in our case).
A common model is the normal distribution; however, in this case, we can easily compute the VaR from the normal distribution itself. The Monte Carlo simulation approach is more relevant when the stochastic model is more complex or when the asset is more complex, leading to difficulties to compute the VaR. For example, if we assume that returns follow a GARCH process, the (unconditional) VaR has to be computed with the Monte Carlo simulation methods. Similarly, if we consider complex financial products like options, the VaR has to be computed with the Monte Carlo simulation methods.

In this post, we compare the Monte Carlo simulation method with the historical method and the variance-covariance method. Thus, we simulate returns for the CAC40 index using the GARCH (1,1) model.
Figure 1 and 2 illustrate the GARCH simulated daily returns and volatility for the CAC40 index.

Figure 1. Simulated GARCH daily returns for the CAC40 index.
img_SimTrade_CAC40_GARCH_ret
Source: computation by the author.

Figure 2. Simulated GARCH daily volatility for the CAC40 index.
img_SimTrade_CAC40_GARCH_vol
Source: computation by the author.

Next, we sort the distribution of simulated returns in ascending order (basically in order of worst to best returns observed over the period). We can now interpret the VaR for the CAC40 index in one-day time horizon based on a selected confidence level (probability).

For instance, if we select a confidence level of 99%, then our VaR estimate corresponds to the 1st percentile of the probability distribution of daily returns (the bottom 1% of returns). In other words, there are 99% chances that we will not obtain a loss greater than our VaR estimate (for the 99% confidence level). Similarly, VaR for a 95% confidence level corresponds to bottom 5% of the returns.

Figure 3 below represents the unconditional probability distribution of returns for the CAC40 index assuming a GARCH process for the returns.

Figure 3. Probability distribution of returns for the CAC40 index.
img_SimTrade_CAC40_MonteCarloVaR
Source: computation by the author.

From the above graph, we can interpret VaR for 99% confidence level as -3% i.e., there is a 99% probability that daily returns we obtain in future are greater than -3%. Similarly, VaR for 95% confidence level as -1.72% i.e., there is a 95% probability that daily returns we obtain in future are greater than -1.72%.

You can download below the Excel file for computation of VaR for CAC40 stock using Monte Carlo method involving GARCH(1,1) model for simulation of returns.

Advantages and limitations of Monte Carlo method for VaR

The Monte Carlo method is a very powerful approach to VAR due its flexibility. It can potentially account for a wide range of scenarios. The simulations also account for nonlinear exposures and complex pricing patterns. In principle, the simulations can be extended to longer time horizons, which is essential for risk measurement and to model more complex models of expected returns.

This approach, however, involves investments in intellectual and systems development. It also requires more computing power than simpler methods since the more is the number of simulations generated, the wider is the range of potential scenarios or outcomes modelled and hence, greater would be the potential accuracy of VaR estimate. In practical applications, VaR measures using Monte Carlo simulation often takes hours to run. Time requirements, however, are being reduced significantly by advances in computer software and faster valuation methods.

Useful resources

Jorion P. (2007) Value at Risk, Third Edition, Chapter 12 – Monte Carlo Methods, 321-326.

About the author

The article was written in March 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

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).

Implementing Markowitz asset allocation model

March 20, 2022 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) explains how to implement the Markowitz asset allocation model. This model is used to determine optimal asset portfolios based on the risk-return trade-off.

This article follows the following structure: first, we introduce the Markowitz 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 implements the Markowitz asset allocation model.

Introduction

Markowitz’s work is widely regarded as a pioneer work in financial economics and corporate finance due to its theoretical foundations and applicability in the financial sector. Harry Markowitz received the Nobel Prize in 1990 for his contributions to these disciplines, which he outlined in his 1952 article “Portfolio Selection” published in The Journal of Finance. His major work established the foundation for what is now commonly referred to as “Modern Portfolio Theory” (MPT).

To find the portfolio’s minimal variance, the Markowitz model uses a constrained optimization strategy. The goal of the Markowitz model is to take into account the expected return and volatility of the assets in the investable universe to provide an optimal weight vector that indicates the best allocation for a given level of expected return or the best allocation for a given level of volatility. The expected return, volatility (standard deviation of expected return), and the variance-covariance matrix to reflect the co-movement of each asset in the overall portfolio design are the major inputs for this portfolio allocation model for an n-asset portfolio. We’ll go over how to use this complex method to find the best portfolio weights in the next sections.

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.

Implementation of the Markowitz asset allocation model in practice

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 use the data analysis tool pack supplied in Excel to run a variance-covariance matrix for ease of computation (Table 2).

Table 2. Variance-covariance matrix of asset returns.
img_SimTrade_implementing_Markowitz_spreadsheet_4
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 10% desired expected return, we get the following results (Table 3).

Table 3. Asset weights for an optimal portfolio.
img_SimTrade_implementing_Markowitz_spreadsheet_2
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 (Figure 1).

Figure 1. Markowitz efficient portfolio frontier.
img_SimTrade_implementing_Markowitz_spreadsheet_3
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?

Modern Portfolio Theory (MPT) is at the heart of modern finance, shaping the modern investing landscape. MPT has become the cornerstone of current financial theory and practice. MPT has been around for nearly sixty years and shows no signs of slowing down. His theoretical contributions paved the way for more portfolio theories. This post helps you to grasp the theoretical model and its implementation.

Useful resources

Academic research

Petters, A. O., and Dong, X. 2016. An Introduction to Mathematical Finance and Applications. Springer Undergraduate Texts in Mathematics and Technology.

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).

Monte Carlo simulation method

March 17, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) explains the Monte Carlo simulation method and its applications in finance.

Introduction

The Monte Carlo method was invented by John von Neumann (Hungarian-American mathematician and computer scientist) and Stanislaw Ulam (Polish mathematician) during World War II to improve decision making under uncertain conditions. It is named after the popular gambling destination Monte Carlo, located in Monaco and home to many famous casinos. This is because the random outcomes in the Monte Carlo modeling technique can be compared to games like roulette, dice and slot machines. In his autobiography, ‘Adventures of a Mathematician’, Ulam mentions that the method was named in honor of his uncle, who was a gambler.

How Monte Carlo simulation works

The main idea is to repeatedly run a large number of simulations of a random process for a variable of interest (such as an asset price in finance) covering a wide range of possible situations. The outcomes of this variables are drawn from a pre-specified probability distribution that is assumed to be known, including the analytical function and its parameters. Thus, Monte Carlo simulations inherently try to recreate the entire distribution of asset prices.

Example: Apple stock

Consider the Apple stock as our asset of interest for which we will generate stock prices according to the Monte Carlo simulation method.

The first step in the simulation is choosing a stochastic model for the behavior of our random variable (the Apple stock price in our case). A commonly used model is the geometric Brownian motion (GBM) model. The model assumes that future asset price changes are uncorrelated over time and the probability distribution function of the future price is a log-normal distribution. The movements in price in GBM process can be expressed as:

img_SimTrade_GBM_process

with dS being the change in asset price in continuous time dt. dW is the Wiener process (W_t+1 – W_t is a random variable from the normal distribution N(0, 1)). σ represents the price volatility considering the unexpected changes that can result from external effects (σ is assumed to be constant over time). μdt together represents the deterministic return within the time interval with μ representing the growth rate of the asset price or the ‘drift’.

Integrating dS/S over a finite interval, we get :

img_SimTrade_simulated_asset_price

Where ε is a random number generated from a normal distribution N(0,1).

This equation thus gives us the evolution of the asset price from a simulated model from day t-1 to day t.

We can now generate a simulation path for 100 days using the above formula.

The figure below shows five simulations for the price of the Apple stock over 100 days with Δt = 1 day. The initial price for Apple stock (i.e, price at t=0) is $146.52.

Figure 1. Simulated Apple stock prices according to the Monte Carlo simulation method.
img_SimTrade_Apple_MonteCarloSim
Source: computation by author.

Thus, we can observe that the prices obtained by just these five simulations range from $100 to over $220.

You can download below the Excel file for generating Monte Carlo Simulations for Apple stock.

Applications in finance

The Monte Carlo simulation method is widely used in finance for valuation and risk analysis purposes.

One popular application is option pricing. For option contracts with complicated features (such as Asian options) or those with a combination of assets as their underlying, Monte Carlo simulations help generate multiple potential payoff scenarios for the option which are averaged out to determine the option price at the issuance date.

The Monte Carlo method is also used to assess potential risks by generating simulations of market variables affecting portfolios such as asset returns, interest rates, macroeconomic factors, etc. over different time periods. These simulations are then assessed as required for risk modelling and to compute risk metrics such as the value at Risk (VaR) of a position.

Other applications include personal finance planning and corporate project finance where simulations are generated to construct stochastic financial models for sensitivity analysis and net present value (NPV) projections.

Useful resources

Hull, J.(2008) Risk Management and Financial Institutions, Fifth Edition, Chapter 7 – Valuation and Scenario Analysis.

About the author

The article was written in March 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Excel functions for mortgage

February 5, 2024March 14, 2022 by Liangyao TANG

Excel functions for mortgage

In this article, Liangyao TANG (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022) explains the functions in Excel that are useful to study a mortgage. Mastery of Excel is an essential skill nowadays in financial analysis and modelling tasks. Proficiency in using Excel formulas can help analysts quickly process the data and build the models more concisely.

Mortgage

A mortgage is the type of loan used in real estate, vehicles, and other types of property purchasing activities. There are two parties in the mortgage contract: the borrower and the lender. The contract sets the terms and conditions about the principal amount, interest rate, interest type, payment period, maturity, and collaterals. The borrower is contracted to pay back the lender in a series of payments that contains part of the principal as well as the interests before the maturity date.

The mortgage is also subject to different terms according to the bank’s offers and macroeconomic cycle. There are two types of interest rates: the fixed-rate loan and the floating (variable) rate loan, in which the interest rate is a pre-determined rate (at the beginning of the period) and post-determined rate (at the end of the period).

Example of repayment schedule.

In this post, I will use the following example: a mortgage of $300,000 for property purchasing. The mortgage specifies a 5% fixed annual interest rate for 30 years, and the borrower should pay back the loan on a monthly basis. We can use Excel functions to calculate the periodic (monthly) payment and its two components, the principal repaid and the interests paid for a given period. The calculations are shown in the sample Excel file that you can download below.

PMT

The “PMT” (Payment) Excel function calculates the periodic mortgage payment.

The periodic repayment for a fixed-rate mortgage includes a portion of repayment to the principal and an interest payment. Since the mortgage has a given maturity date, the payment is calculated on a regular basis, for example, every month. All repayments are of equal amount throughout the loan period.

The mathematical formula for the periodic mortgage payment is:

Formula for the periodic mortgage payment

With the following notations:

PMT: the payment
P: the principal value
r: the interest rate
N: the total number of periods

The repayment schedule is a table which gives the periodic payment, and the principal repaid and the interests paid for a given period. It can be a large table. For example, the repayment schedule of a loan with 30 year maturity and monthly payment has 180 lines. In formal terms, the payment schedule would be:

Repayment schedule of a mortgage

The repayment schedule shows the payment amount of each period, and the remaining principal balance after each payment. The ‘P’ represents the principal amount at the beginning of the mortgage, and the remaining principal is subjective to an (1+r) times interests at each period. The remaining principal is the principal balance from last period minus the current payment. Therefore for period 1, the remaining balance is equal to P(1+r), which is the principal with one year of interest, minus the PMT value, which is the payment of the current period.

The syntax for the Excel function to calculate the periodic payment is: PMT(rate, nper, pv, [fv], [type]).

With the following notations:

PMT: the periodic payment of the loan
Nper: the total number of periods of the loan
PV : the principal (present value) of the loan
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0

The function is used explicitly in the case of a fixed interest rate to compute the (constant) periodic payment.

The PMT function will calculate the loan’s payment at a given level of interest rate, the number of periods, and the total value of the loan for principals at the beginning of the period (principal + interest).

When using the function, it is essential to always align the time unit of the interest rate and the unit of Nper. If the mortgage is compounding on a monthly basis, the number of periods should be the total number of months in the amortization, and the rate should be the monthly interest rate, which equals the annual rate divided by 12. . In the above example, the interest should be paid in a monthly basis, therefore the number of period (Nper) is equal to 12 month x 30 year = 360 periods. Since the annual interest rate is 5%, the monthly interest rate would equal to 5% divide by 12, which is 0.42% per month.

IPMT and PPMT

To supplement on the information about the monthly payment, we can also use the function IPMT and PPMT to calculate the principal repaid and the interest rate paid for a given period.

IPMT

IPMT is the Excel function that calculates the interest portion in each of the periodic payment.

The syntax of the Excel function to calculate the interest portion of the periodic payment is: IPMT(rate, per, nper, pv, [fv], [type]).

With the following notations:

IPMT: interest payment
rate: interest rate
per: current period number
nper: total number of periods
pv: present value
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0

The rate refers to the periodic interest rate, while the “nper” refers to the total number of payment periods, and the “per” refers to the period for which we want to calculate the interest.

PPMT

PPMT is the Excel function that calculates the principal portion of a periodic payment.

The syntax of the Excel function to calculate the principal portion of a periodic payment is: PPMT(rate, per, nper, pv, [fv], [type]).

With the following notations:

PPMT: principal payment
rate: interest rate
per: current period number
nper: total number of periods
pv: present value
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0

Those of the results should be consistent with the amortization schedule shown above. The principal repayment should equal to PMT per period minus the interest rate paid (IPMT).

RATE

Contrarily, if the user is given the periodic payment amount information and wants to find out about the interest rate used for the calculation, he/she can use the RATE function in Excel.

The syntax of the Excel function to calculate the rate is: RATE(nper, pmt, pv, [fv], [type], [guess]).

With the following notations:

RATE: the interest rate
nper: the total number of payment periods
pmt: the constant periodic payment
pv: the principal amount
[fv]: the future value of the loan (optional parameter). Default equal to 0
[type]: when payments are due (optional parameter). 0 = end of period, 1 = beginning of period. Default is 0
[guess]: your guess on the rate (optional parameter). Default is 10%

The RATE Excel function will automatically calculate the interest rate per period. The time unit of the interest rate is aligned with the compounding period; for example, if the mortgage is compounding on a monthly basis, the RATE function also returns a monthly interest rate.

Example with an Excel file

The use of the Excel functions PMT, IPMT, PPMT and RATE is illustrated in the Excel file that you can download below.

Useful resources

Forbes What is a mortgage

Rocket mortgage Types of mortgage

Ramsey How Do Student Loans Work?

Prof. Longin’s website Echéancier d’un crédit (mortgage calculator in French)

About the author

The article was written in March 2022 by Liangyao TANG (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2021-2022).

Eurobonds

March 7, 2022 by Akshit GUPTA

Eurobonds

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains Eurobonds traded in financial markets.

Introduction

In financial markets, bonds are debt securities used by issuers to raise capital from investors. In return investors get an interest payment on the principal invested over the life of the bond. The bonds can be issued by governments, municipalities, financial institutions, and companies. The duration of the bonds can cover different time periods.

Eurobonds are a special kind of bonds issued by companies or governments to raise capital from financial markets. These bonds are denominated in a currency different from the currency of the country where they originated. The Eurobonds help issuers to raise capital in a foreign currency and at a lower cost. Let’s take the example of an American company which would like to issue debt in euros to finance its operations in Europe. If it borrows in European markets, it will get a higher interest rate as it is less well known in the foreign markets that in the domestic market. With Eurobonds, the company can benefit from the same level of interest rates as for its domestic bonds, thereby lowering its cost of capital.

These instruments have a medium to long term maturity and are highly liquid in the market. They are traded over the counter (OTC) and the market for Eurobonds is made up of several financial institutions, issuers, investors, government bodies, and brokers. Many brokerages across the world provide trading platforms facilities to investors and borrowers for trading in different kinds of Eurobonds.

Characteristics of Eurobonds

Eurobonds are unsecured instruments and investors demand high yields on these instruments based on the credit ratings of the issuer. The issuer can issue Eurobonds in a foreign currency and a foreign land based on their capital needs. The name of a Eurobond carries the name of the currency in which they are dominated. For example, a French company willing to do business in the United States, can issue a Eurobond in the UK financial market denominated in US dollars which will be called as euro-dollar bond.

A Eurobond should not be confused with a foreign bond issued by an issuer in the foreign market denominated in the local currency of the investor. A Eurobond can be issued in a foreign country and can be denominated in a currency different from the local currency of the issuer. For example, a French company willing to invest in Japan can issue a Euro-yen bond in the US markets denominated in the local currency of Japan.

These bonds are traded electronically on different platforms and can have maturities ranging from 5 years to 30 years. The bonds can have fixed or floating interest rates with semi-annual or annual payments. These bonds have a relatively small face value making it attractive even to small investors.

Benefits of Eurobonds

Eurobonds can serve different benefits to issuers and investors.

Major advantage of Eurobonds for the issuers

Access to capital at lower rates – Companies can choose countries with lower interest rates to issue Eurobonds, thereby avoiding interest rate risks
Access to different bond maturities – As Eurobonds can have maturities ranging from 5 years to 30 years, companies can have a wide range of maturities to choose from depending on their requirements
Access to international markets – By issuing Eurobonds denominated in a different currency, companies can access different markets with more ease with a wide investor base.

Major advantage of Eurobonds for the investors

Access to international markets – By buying Eurobonds, investors can gain easy access to international markets thereby diversifying their fixed income portfolios.
Access to different bond maturities – As Eurobonds can have maturities ranging from 5 years to 30 years, borrowers can have a wide range of maturities to choose from depending on their investment profile.
High liquidity – As the market size for Eurobonds is very large, investors can enjoy higher liquidity and can exit their positions as per their needs.

Example

The figure below gives an example of Eurobonds issued by the Federal Republic of Nigeria.

Characteristics of the Eurobonds issuance.

Example of Eurobond issuance

Source: FMDQ.

▶ Akshit GUPTA Green bonds

▶ Jayati WALIA Fixed-income products

▶ Jayati WALIA Credit Risk

Useful resources

International Capital Market Association (ICMA) History of Eurobonds

About the author

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

Green bonds

March 7, 2022 by Akshit GUPTA

Green bonds

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains Green bonds traded in financial markets.

Introduction

A green bond is a fixed-income product that works like a conventional bond, except that the money invested in them is used exclusively to finance green projects that support environment preservation, sustainability and reduction of climate change (low-carbon economy). Green projects can include renewable energy such as solar and wind power, energy-efficient infrastructure, clean transportation and waste management and recycling.

In 2007, the European Investment Bank (EIB) issued the world’s first ever green bond under the name Climate Awareness Bond (CAB), which focused on renewable energy and energy efficiency projects. This was followed by the World Bank issuing its own green bonds, until 2012 when the first corporate green bond was issued. Since then the market for green bonds has grown tremendously creating all-time highs with every passing year. The greatest issuer of green bonds in 2020 was the French government with a combined issue size of nearly 13 billion USD.

Types of green bonds

Green bonds can be classified as the following: green “use of proceeds” bonds, green “use of proceeds revenue” bonds, green project bonds, and securitized green bonds.

Green “use of proceeds” bonds

The funds raised by these green bonds are invested in green projects but they are backed/secured by issuer’s assets. Hence, their ratings are the same as other debt instruments by the issuer. For instance, the Climate Awareness Bond issued by EIB is one such green bond.

Green “use of proceeds revenue” bonds

The funds raised are assigned to eligible green projects. However, bondholders have recourse to a specified revenue stream from the issuers which may or may not be related to the eligible green projects.

Green project bonds

Proceeds from green project bonds are used for specific projects, investors having a direct exposure to the green project itself.

Securitized green bonds

These bonds are backed by a large group of green projects or assets.

Benefits of investing in green bonds for issuers

Lower cost of capital

Green bonds help environment focused companies to raise large amount of initial and working capital at lower costs to fund their ESG activities which require heavy initial investments. For example, companies can raise capital to fund a project focused towards generating renewable energy.

Brand value

Companies issuing green bonds enjoy an increase in the brand value and favourable reputation amongst the investors, as they are becoming more inclined towards sustainability.

Benefits of investing in green bonds for investors

Diversification

Over the years, the financial markets have seen an increased demand for green bonds amongst investors. Various factors have contributed to this increase including portfolio diversification, focus on socially responsible investments opportunities, fulfilment of ESG mandates of the financial institutions, etc.

Tax benefits

Investors can enjoy tax incentives on the investments made in green bonds. The interest incomes generated on these bonds are generally tax exempt or provide tax reductions to the investors. Thus, the issuers also benefit from lower interest rates due to the tax benefits.

Increase in liquidity

As the market size for green bonds is increasing, investors can enjoy higher liquidity and can exit their positions as per their needs.

Examples

The image below shows the listing of green bonds on Euronext.

Listing of green bonds on Euronext.

Listing of green bonds

Source: Euronext.

Useful resources

Corporate Finance Institute Eurobonds

ICMA History of Eurobonds

Euronext Listing of green bonds

About the author

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

The regulation of cryptocurrencies: what are we talking about?

March 5, 2022 by Hugo MEYER

The regulation of cryptocurrencies: what are we talking about?

In this article, Hugo MEYER (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2020-2021) presents the regulation of cryptocurrencies.

Introduction

The first cryptocurrency – Bitcoin – launched in 2008 by Satoshi Nakamoto had for ambition to “break the rules and change the world”.

Thirteen years later, cryptocurrency represents a 2$ trillion market, with an increasing institutional presence, from crypto hedge funds to large banks. Behind this bewildering evolution, public authorities lagged behind, slowly empowering with feverish regulation actions.

The lack of regulation in this burgeoning area has created an opening for boundless fraud and money laundering, forcing some countries to get to grips with the cryptocurrency’s pitfalls.

What are cryptocurrencies?

Cryptocurrencies are at the edge of revolutionizing the way we’ve been trading since thousands of years. By definition, a cryptocurrency is an encrypted, digital, and decentralized medium of exchange that allows two parties that could be located everywhere on the globe to transfer funds directly, without relying on any trusted third party.

Instead of being secured by public institutions and/or companies, these transfers are carried out on the Blockchain which is a “digital database or ledger containing information that can be simultaneously used and shared within a large decentralized, publicly accessible network”.

Example: To make it simple, let’s say that A wants to send money to B. This transaction is included in a ‘block’. This block is broadcasted to every member in the network, and then validated or not by them. Once validated, the block is added to the chain, triggering the money to move from A to B.

Figure 1. Process of transaction with the blockchain
Schema of the blockchain
Source: Institut des actuaires.

This distributed network provides an indelible and transparent record of transactions as the chain cannot be counterfeited. If someone tried to change any information contained in one block, the different parties of the network would not approve the transaction as they could check the whole history of the blockchain and compare it to the new one.

Thus, many cryptocurrencies such as Bitcoin, Ethereum and Monero rely on public blockchains to allow transactions in complete security and transparency.

“I do think Bitcoin is the first money that has the potential to do something like change the world” – Peter THIEL.

What is the regulation about?

By definition, regulation tally with the act of controlling something, or enacting an official rule. What does it imply for cryptocurrencies?

A cryptocurrency is entirely defined by its creator, that must foremost determine its characteristics. This creation process is divided into three steps:

Pick or create its blockchain platform
Choose a consensus algorithm
Design the blockchain architecture.

His creator defines the rules around it, while the ecosystem built accordingly to these rules regulate it and make it functional. Once the crypto is launched, it is impossible to modify its architecture and the rules. In this way, a cryptocurrency cannot be regulated, even by his founder. Thus, authorities do not have any grip with cryptocurrencies in themselves. They are auto-regulated by their initial algorithms, and nothing else.

Thereby, what are we talking about when dealing with the regulation of cryptocurrency?

Cryptocurrencies are mainly exchanged through platforms called “exchanges” such as Coinbase, Binance or eToroX. The first existing regulation framework is the accessibility to these platforms. For most of them, requirements like providing its identity are requested, following the Know Your Customer (KYC) compliance.

Secondly, the regulatory framework for these platforms depends on where they are based. Each country has a different approach of cryptocurrency, meaning that the regulation can be different in any of them.

For example, cryptocurrency exchanges are legal in the United States and fall under the regulatory scope of the Bank Secrecy Act (BSA). Therefore, exchanges service providers must register with FinCEN, implement an anti-money laundering (AML) and combating the financing of terrorism (CFT) program, maintain appropriate records, and submit reports to the authorities. It does not mean their trading activities are regulated.

These requirements permit exchanges to operate as licensed Money Service Businesses (MSBs), leading regulators to focus on anti-money laundering (AML) and due diligence measures, but not trading (and all the aspects of market manipulation).

Given the lack of significant regulatory oversight of actual trading activity, it is not surprising to see many cryptocurrency exchanges carry out questionable activities, such as offering leverage to their clients and wash trading, during times of market instability. But these are not the only problems raised by the lack of regulation.

Why should exchanges be more regulated?

The blockchain is a recent technology, understood by a few. As regulation always comes after innovation, the crypto market has been sidelined by public authorities for many years. The question of regulating it has recently appeared in response to the many downsides incurred to cryptos.

Customer protection

When investing in cryptocurrencies, the customer is lacking protection. An investor could be facing fake websites, hacking, and platform bankrupts without any legal recourse to recover his money. These situations could never happen in a traditional investment as it is institutionally regulated. To become more secure, exchanges must follow the example of itBit, an US-based exchange oversighted by the New York Department of Financial Services (DFS) and registered as a bank.

Illegal Financial flows & crime

Cryptocurrency can be used for illicit transactions and for laundering criminal proceeds that may or may not have started as cryptocurrency. These illicit transactions occur on the dark web, including the purchase/sale of illicit drugs and debit and credit card information. According to a study published in 2019 by Oxford Academics, 76$ billion of illegal activity per year involve bitcoin, which represents half of total Bitcoin transactions.

Cryptos can also be used for ransomware attacks, like the one that shut down the Colonial Pipeline in May 2021. This attack was one of many others high-profile instances of hackers seeking Bitcoin ransoms, that should tend to multiply in the upcoming years.

Price stability

Blockchain technology has increasingly become a speculative tool for investing and achieving high returns in the short term, leading to market volatility. These fast and unpredictable price changes also have a direct impact on the velocity, where more and more people hold their cryptocurrencies instead of selling or using them.

Plus, the volatility of cryptos prices may let the market suffer from illiquidity. The notion of liquidity for a financial asset refers to the ease with which an asset can be bought or sold (without a strong price impact, e.g., limit implicit transaction costs).

Tax evasion

One of the first problem that arise from tax evasion is taxation. Many countries have their own regulatory framework, either taxing cryptos as an asset (Israel), a financial asset (Bulgaria), or even a foreign currency (Switzerland). Once the taxation rule found out, authorities will tackle another problem: The investors resistance to report their gains.

Taking the example of USA, authorities ask filers on their income tax forms – like any form of income – whether they received or made any transactions with cryptocurrency. However, third-party reporting in the sector is scarce; making it even more difficult to attribute gains to one natural person.

Thus, how can regulation allow the crypto market to take over these pitfalls?

Worldwide market regulation

“Bitcoin is not unregulated. It is regulated by algorithm instead of being regulated by government bureaucracies” – Andreas Antonopoulos

Despite being a global phenomenon, every country does not hang up with the same type of regulation.

First, some countries have expanded their laws on money laundering, counterterrorism, and organized crimes to include cryptocurrency markets, and require banks and other financial institutions that facilitate such markets to conduct all the due diligence requirements imposed under such laws. For instance, Australia and Canada recently enacted laws to bring cryptocurrency transactions and institutions that facilitate them under the ambit of money laundering and counter-terrorist financing laws.

Some jurisdictions have gone even further and imposed restrictions on investments in cryptocurrencies. Some countries – Algeria, Bolivia, Morocco, Vietnam – explicitly ban any and all activities involving cryptocurrencies. Qatar and Bahrain consider that their citizens are forbidden from engaging in any kind of activities involving cryptocurrencies locally but allow them to do so outside their borders.

There are also countries that, while not banning their citizens from investing in cryptocurrencies, impose indirect restrictions by hindering transactions involving cryptocurrencies, such as China, Iran, or Thailand.

A limited number of countries regulate initial coin offerings (ICOs), which use cryptocurrencies as a mechanism to raise funds. Of the jurisdictions that address ICOs, some (mainly China, Macau, and Pakistan) ban them altogether, while most tend to focus on regulating them.

When it comes to taxation, the challenge appears to be how to categorize cryptocurrencies and the specific activities involving them. This matters primarily because whether gains are categorized as income or capital gains invariably determines the applicable tax bracket. For instance, in Israel, cryptos gains are taxed as assets, while there are subject to income tax in Spain and Argentina.

Advocates of digital currencies say that accepting cryptocurrencies is much more relevant than rejecting it. For instance, El Salvador became the 7th of September 2021 the first country in the world to make Bitcoin a legal tender. One day after, the “Regulation of the Bitcoin Law” entered into force, that establishes standards of conducts supervised by the Superintendency of the Financial System (SSF), the equivalent of the Securities Exchange Commission (SEC) in the United States or the Autorité des Marchés Financiers (AMF) in France. This regulation will bring much more protection to Bitcoin users, while setting up numerous programs in cybersecurity, anti-money laundering, and tax evasion.

Conclusion

As Bitcoin – and other cryptocurrencies – become more and more popular, regulation will have to step up altogether, despite asking extensive questions on its bounding by International Authorities.

Economic threat, exacerbated risks and investigation complications are all issues that can be counteracted by regulation laws on the crypto market. Central banks will play a major role in this governance, going along with their traditional missions such as ensuring price stability and a proper operating financial system.

Nevertheless, regulation may lead to underestimated consequences. As it goes on, crypto investment will progressively become “mainstream” and dismiss the first and most powerful investors. This trend might also push innovators to take a step back from it, thus decreasing the number of cryptocurrencies created and newly innovative blockchains.

Useful resources

Academic research

Sean, F. Jonathan, R K. Talis, P. 2019. Sex, Drugs, and Bitcoin: How much illegal activity is financed through cryptocurrencies?” The Review of Financial Studies. Vol. 32, p. 1798-1853.

Business Analysis

L, S. 2016. Who is Satoshi Nakamoto, The Economist explains.

Thiemann, A. 2021. Cryptocurrencies: An empirical View from a Tax Perspective, JRC Working Papers on Taxation and Structural Reforms. No 12/2021, European Commission, Joint Research Centre, Seville, JRC126109.

Global Legal Research Directorate. 2018. Regulation of Cryptocurrency Around the World. LL File No. 2018-016036 LRA-D-PUB-002438.

Ryan, H. 2021. U.S. Officials send mixed messages on crypto regulation. Here’s what it all means for investors. NextAdvisor.

American Overseas, 2021. Washington Monthly: Catching Bitcoin tax evaders.

Alexis, G. 2021. Crypto doesn’t have to enable tax cheats Bloomberg Opinion.

Douma, S. 2016. Bitcoin: The pros and cons of regulation. s1453297.

About the author

The article was written in March 2022 by Hugo MEYER (ESSEC Business School, Master in Strategy & Management of International Business (SMIB), 2020-2021).

Black-Scholes-Merton option pricing model

February 4, 2024March 3, 2022 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) explains the Black-Scholes-Merton model to price options.

The Black-Scholes-Merton model (or the BSM model) is the world’s most popular option pricing model. Developed in the beginning of the 1970s, this model introduced to the world, a mathematical way of pricing options. Its success was essentially a starting point for new forms of financial derivatives in the knowledge that they could be priced accurately using the ideas and analyses pioneered by Black, Scholes and Merton and it set the foundation for the flourishing of modern quantitative finance. Myron Scholes and Robert Merton were awarded the Nobel Prize for their work on option pricing in 1997. Unfortunately, Fischer Black had died several years earlier but would certainly have been included in the prize had he been alive, and he was also listed as a contributor by Scholes and Merton.

Today, the Black-Scholes-Merton formula is widely used by traders in investment banks to price and hedge option contracts. Options are used by investors to hedge their portfolios to manage their risks.

Assumptions of the BSM Model

As any model, the BSM model relies on a set of assumptions:

The model considers European options, which we can only be exercised at their expiration date.
The price of the underlying asset follows a geometric Brownian motion (corresponding to log-normal distribution for the price at a given point in time).
The risk-free rate remains constant over time until the expiration date.
The volatility of the underlying asset price remains constant over time until the expiration date.
There are no dividend payments on the underlying asset.
There are no transaction costs on the underlying asset.
There are no arbitrage opportunities.

The BSM equation

The value of an option is a function of the price of the underlying stock and its statistical behavior over the life of the option.

A commonly used model is Geometric Brownian Motion (GBM). GBM assumes that future asset price differences are uncorrelated over time and the probability distribution function of the future prices is a log-normal distribution (or equivalently the probability distribution function of the future returns is a normal distribution). The price movements in a GBM process can be expressed as:

GBM equation

with dS being the change in the underlying asset price in continuous time dt and dX the random variable from the normal distribution (N(0, 1) or Wiener process). σ is the volatility of the underlying asset price (it is assumed to be constant). μdt represents the deterministic return within the time interval with μ representing the growth rate of asset price or the ‘drift’.

Therefore, option price is determined by these parameters that describe the process followed by the asset price over a period of time. The Black-Scholes-Merton equation governs the price evolution of European stock options in financial markets. It is a linear parabolic partial differential equation (PDE) and is expressed as:

BSM model equation

Where V is the value of the option (as a function of two variables: the price of the underlying asset S and time t), r is the risk-free interest rate (think of it as the interest rate which you would receive from a government debt or similar debt securities) and σ is the volatility of the log returns of the underlying security (say stocks).

The key idea behind the equation is to hedge the option and limit exposure to market risk posed by the asset. This is achieved by a strategy known as ‘delta hedging’ and it involves replicating the option through an equivalent portfolio with positions in the underlying asset and a risk-free asset in the right way so as to eliminate risk.

Thus, from the BSM equation we can derive the BSM formulae that describe the price of call and put options over their life time.

The BSM formulae

Note that the type of option we are valuing (call or put), the strike price and the maturity date do not appear in the above BSM equation. These elements only appear in the ‘final condition’ i.e., the option value at maturity, called the payoff function.

For a call option, the payoff C is given by:

C_T = max⁡(S_T – K; 0)

For a put option, the payoff is given by:

P_T = max⁡(K – S_T; 0)

The BSM formula is a solution to the BSM equation, given the boundary conditions (given by the payoff equations above). It calculates the price at time t for both a call and a put option.

The value for a call option at time t is given by:

Call option value equation

The value for a put option at time t is given by:

Put option value equation

where

With the notations:
S_t: Price of the underlying asset at time t
t: Current date
T: Expiry date of the option
K: Strike price of the option
r: Risk-free interest rate
σ: Volatility (the standard deviation of the return on the underlying asset)
N(.): Cumulative distribution function for a normal (Gaussian) distribution. It is the probability that a random variable is less or equal to its input (i.e. d₁ and d₂) for a normal distribution. Thus, 0 ≤ N(.) ≤ 1

Figure 1 gives the graphical representation of the value of a call option at time t as a function of the price of the underlying asset at time t as given by the BSM formula. The strike price for the call option is 50€ with a maturity of 0.25 years and volatility of 50% in the underlying.

Figure 1. Call option value

Source: computation by author.

Figure 2 gives the graphical representation of the value of a put option at time t as a function of the price of the underlying asset at time t as given by the BSM formula. The strike price for the put option is 50€ with a maturity of 0.25 years and volatility of 50% in the underlying.

Figure 2. Put option value
Source: computation by author.

You can download below the Excel file for option pricing with the BSM Model.

Some Criticisms and Limitations

American options

The Black-Scholes-Merton model was initially developed for European options. This is a limitation of the equation for American options which can be exercised at any time before the expiry date. The BSM model would then not accurately determine the option value (an important case when the underlying asset pays a discrete dividend).

Stocks paying dividends

Also, in reality, most stocks pay dividends, and no dividends was an assumption in the initial BSM model, which analysts now eliminated by accommodating the dividend yield in the formula if required.

Constant volatility

Another limitation is the use of constant volatility. Volatility is the measure of risk based on the standard deviation of the return on the underlying asset. In reality the value of an asset will change randomly, not with a specific constant pattern regarding the way it can change.

Finally, the assumption of no transaction cost neglects the liquidity risk in the market since transaction costs are clearly incurred in the real world and there exists a bid-offer spread on most underlying assets. For the most heavily traded stocks, this cost may be low but for others it may lead to an inaccuracy.

Useful resources

Black F. and M. Scholes (1973) The Pricing of Options and Corporate Liabilities The Journal of Political Economy 81, 637-654.

Merton R.C. (1973) Theory of Rational Option Pricing Bell Journal of Economics 4, 141–183.

About the author

The article was written in March 2022 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Quick review of the most famous investments frauds ever…

February 20, 2022 by Louis DETALLE

Quick review of the most famous investments frauds ever…

In this article, Louis DETALLE (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) explains what a tax specialist works on, on a daily basis.

The most famous amongst all frauds ever: Charles Ponzi

Born in 1882, this Italian man has built himself quite a reputation in the fraud industry due to his invention: the Ponzi Pyramid. That consisted of a financial business plan that promised a 50% interest rate within 45 days to the investors that picked his solution. You may wonder how it was possible to reach such rates, well it was possible because Charles Ponzi reimbursed the old investors their initial investment plus the interests with the money collected from newer investors and so on… In fact, Ponzi developed an idea that he encountered in the “Banca Zarossi” in Montreal, that relied on a similar principle that made it impossible to reimburse all the clients if they came altogether asking for their savings.

At the end of his fraud, in 1919, Charles Ponzi had managed to convince nearly 40 000 investors to commit to his business plan for $15 million which account for several dozens of current billion dollars. To this day, Ponzi is still considered the father of financial fraud and several others drew from his example.

The Great Bernard Madoff

Bernard Madoff was a New York hedge fund manager who promised the most experienced investors his hedge fund would provide a 7% annual growth whatever the economic conjuncture. His fund relied on the same principle as the Ponzi system that Bernard Madoff hid successfully thanks to his fame in the finance sector. In fact, his renown made all this fraud possible and explains how institutions such as HSBC, Santander, BNP Paribas or Nomura got played. In 2008, when the trick was no longer viable, a 65 billion dollar fraud was unveiled…

The unviable mechanism behind this type of fraud: the Ponzi pyramid

A Ponzi pyramid is a fraudulent financial scheme that enables its creator to offer investors unusually high rates for very limited risk. The offer may seduce lots of investors which will only see their money back if newer investors contribute later. The scam is named after its inventor, Charles Ponzi, has been repeated several times.

However, it must be stated that a Ponzi scheme cannot last… Indeed, let’s consider the following example: Investor A invests 10 euros. The fraud promises to pay back twice as much two months later. Two months later, the company approaches new investors with the same promise. Investors B & C invest 10 euros. Their money is used to pay back the 20 euros promised to Customer A and so on…

In the example, for each round of new investors which corresponds to the maturity of the round of investments, the fraud must convince twice as many investors to invest as during the last row, in order to multiply the funds by 2 (so that the previous row of investors be reimbursed). In the following example, after 20 rounds of investors, the fraud will have to gather 10 485 760 € in order to reimburse the 19^th round of investors. As you can see, the scheme had already exceeded its viable size due to an exponential growth which can only cause the loss of the last round of investors and the dreadful financial consequences that comes with it.

About the author

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

Stress Testing used by Financial Institutions

January 16, 2022 by Jayati WALIA

Stress Testing used by Financial Institutions

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) introduces the concept of Stress testing used by financial institutions to estimate the impact of extraordinary market conditions characterized by a high level of volatility like stock market crashes.

Introduction

Asset price movements in financial markets are based on several local or global factors which can include economic developments, risk aversion, asset-specific financial information amongst others. These movements may lead to adverse situations which can cause unpredicted losses to financial institutions. Since the financial crisis of 2008, the need for resilience of financial institutions against market shocks has been exemplified, and regulators around the world have implemented strict measures to ensure financial stability and stress testing has become an imperative part of those measures.

Stress testing techniques were applied in the 1990s by most large international banks. In 1996, the need for stress testing by financial institutions was highlighted by the Basel Committee on Banking Supervision (BCBS) in its regulation recommendations (Basel Capital Accord). Following the 2008 financial crisis, focus on stress testing to ensure adequate capital requirements was further enhanced under the Dodd-Frank Wall Street reform Act (2010) in the United States.

Financial institutions use stress testing as a tool to assess the susceptibility of their portfolios to potential adverse market conditions and protect the capital thus ensuring stability. Institutions create extreme scenarios based on historical, hypothetical, or simulated macro-economic and financial information to measure the potential losses on their investments. These scenarios can incorporate single market variable (such as asset prices or interest rates) or a group of risk factors (such as asset correlations and volatilities).

Thus, stress tests are done using statistical models to simulate returns based on portfolio behavior under exceptional circumstances that help in gauging the asset quality and different risks including market risk, credit risk and liquidity risk. By using the results of the stress tests, the institutions evaluate the quality of their processes and implement further controls or measures required to strengthen them. They can also be prepared to use different hedging strategies to mitigate the potential losses in case of an adverse event.

Types of Stress testing

Stress testing can be based on different sets of information incorporated in the tests. These sets of information can be of two types: historical stress testing and hypothetical stress testing.

Historical stress testing

In this approach, market risk factors are analyzed using historical information to run the stress tests which can include incorporating information from previous crisis episodes in order to measure potential losses the portfolio may incur in case a similar situation reoccurs. For example, the downfall in S&P500 (approximately 30% during February 2020-March 2020) due to the Covid pandemic could be used to gauge future downsides if any such event occurs again. A drawback of this approach is that historical returns alone may not provide sufficient information about the likelihood of abnormal but plausible market events.

The extreme value theory can be used for calculation of VaR especially for stress testing. considers the distribution of extreme returns instead of all returns i.e., extreme price movements observed during usual periods (which correspond to the normal functioning of markets) and during highly volatile periods (which correspond to financial crises). Thus, these extreme values cover almost all market conditions ranging from the usual environments to periods of financial crises which are the focus of stress testing.

Hypothetical stress testing

In this method, hypothetical scenarios are constructed in order to measure the vulnerability of portfolios to different risk factors. Simulation techniques are implemented to anticipate scenarios that may incur extreme losses for the portfolios. For example, institutions may run a stress test to determine the impact of a decline of 3% in the GDP (Gross Domestic Product) of a country on their fixed income portfolio based in that country. However, a drawback of this approach is estimating the likelihood of the generated hypothetical scenario since there is no evidence to back the possibility of it ever happening.

EBA Regulations

In order to ensure the disciplined functioning and stability of the financial system in the EU, the European Banking Authority (EBA) facilitates the EU-wide stress tests in cooperation with European Central Bank (ECB), the European Systemic Risk Board (ESRB), the European Commission (EC) and the Competent Authorities (CAs) from all relevant national jurisdictions. These stress tests are conducted every 2 years and include the largest banks supervised directly by the ECB. The scenarios, key assumptions and guidelines implemented in the stress tests are jointly developed by EBA, ESRB, ECB and the European Commission and the individual and aggregated results are published by the EBA.

The purpose of this EU-wide stress testing is to assess how well banks are able to cope with potentially adverse economic and financial shocks. The stress test results help to identify banks’ vulnerabilities and address them through informed supervisory decisions.

Useful resources

Wikipedia: Stress testing

EBA Guidelines: EU-wide stress testing

Longin F. (2000) From VaR to stress testing : the extreme value approach” Journal of Banking and Finance N°24, pp 1097-1130.

▶ Walia J. Quantitative Risk Management

▶ Walia J. Value at Risk

▶ Walia J. The historical method for VaR calculation

▶ Walia J. The variance-covariance method for VaR calculation

About the author

Article written in January 2022 by Jayati Walia (ESSEC Business School, Master in Management, 2019-2022).

Protective Put

January 12, 2022 by Akshit GUPTA

Protective Put

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the concept of protective put using option contracts.

Introduction

Hedging is a strategy implemented by investors to reduce the risk in an existing investment. In financial markets, hedging is an effective tool used by investors to minimize the risk exposure and change the risk profile for any investment in securities. While hedging does not necessarily eliminate the entire risk for any investment, it does limit the potential losses that the investor can incur.

Option contracts are commonly used by market participants (traders, investors, asset managers, etc.) as hedging mechanisms due to their great flexibility (in terms of expiration date, moneyness, liquidity, etc.) and availability. Positions in options are used to offset the risk exposure in the underlying security, another option contract or in any other derivative contract. There are various popular strategies that can be implemented through option contracts to minimize risk and maximize returns, one of which is a protective put.

Buying a protective put

A put option gives the buyer of the option, the right but not the obligation, to sell a security at a predefined date and price.

A protective put also called as a synthetic long option, is a hedging strategy that limits the downside of an investment. In a protective put, the investor buys a put option on the stock he/she holds in its portfolio. The protective put option acts as a price floor since the investor can sell the security at the strike price of the put option if the price of the underlying asset moves below the strike price. Thus, the investor caps its losses in case the underlying asset price moves downwards. The investor has to pay an option premium to buy the put option.

The maximum payoff potential from using this strategy is unlimited and the potential downside/losses is limited to the strike price of the put option.

Market scenario

A put option is generally bought to safeguard the investment when the investor is bullish about the market in the long run but fears a temporary fall in the prices of the asset in the short term.

For example, an investor owns the shares of Apple and is bullish about the stock in the long run. However, the earnings report for Apple is due to be released by the end of the month. The earnings report can have a positive or a negative impact on the prices of the Apple stock. In this situation, the protective put saves the investor from a steep decline in the prices of the Apple stock if the report is unfavorable.

Let us consider a protective position with buying at-the money puts. One of following three scenarios may happen:

Scenario 1: the stock price does not change, and the puts expire at the money.

In this scenario, the market viewpoint of the investor does not hold correct and the loss from the strategy is the premium paid on buying the put options. In this case, the option holder does not exercise its put options, and the investor gets to keep the underlying stocks.

Scenario 2: the stock price rises, and the puts expire in the money.

In this scenario, since the price of the stock was locked in through the put option, the investor enjoys a short-term unrealized profit on the underlying position. However, the put option will not be exercised by the investor and it will expire worthless. The investor will lose the premium paid on buying the puts.

Scenario 3: the stock price falls, and the puts expire out of the money.

In this scenario, since the price of the stock was locked in through the put option, the investor will execute the option and sell the stocks at the strike price. There is protection from the losses since the investor holds the put option.

Risk profile

In a protective put, the total cost of the investment is equal to the price of the underlying asset plus the put price. However, the profit potential for the investment is unlimited and the maximum losses are capped to the put option price. The risk profile of the position is represented in Figure 1.

Figure 1. Profit or Loss (P&L) function of the underlying position and protective put position.

Protective put

Source: computation by the author.

You can download below the Excel file for the computation of the Profit or Loss (P&L) function of the underlying position and protective put position.

The delta of the position is equal to the sum of the delta of the long position in the underlying asset (+1) and the long position in the put option (Δ). The delta of a long put option is negative which implies that a fall in the asset price will result in an increase in the put price and vice versa. However, the delta of a protective put strategy is positive. This implies that in a protective put strategy, the value of the position tends to rise when the underlying asset price increases and falls when the underlying asset prices decreases.

Figure 2 represents the delta of the protective put position as a function of the price of the underlying asset. The delta of the put option is computed with the Black-Scholes-Merton model (BSM model).

Figure 2. Delta of a protective put position.
Delta Protective put
Source: computation by the author (based on the BSM model).

You can download below the Excel file for the computation of the delta of a protective put position.

Example

An investor holds 100 shares of Apple bought at the current price of $144 each. The total initial investment is equal to $14,400. He is skeptical about the effect of the upcoming earnings report of Apple by the end of the current month. In order to avoid losses from a possible downside in the price of the Apple stock, he decides to purchase at-the-money put options on the Apple stock (lot size is 100) with a maturity of one month, using the protective put strategy.

We use the following market data: the current price of Appel stock is $144, the implied volatility of Apple stock is 22.79% and the risk-free interest rate is equal to 1.59%.

Based on the Black-Scholes-Merton model, the price of the put option $3.68.

Let us consider three scenarios at the time of maturity of the put option:

Scenario 1: stability of the price of the underlying asset at $144

The market value of the investment $14,400. The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) plus the cost of buying the put options ($368 = $3.68*100), which is equal to $14,768, (i.e. ($14,400 + $368)).

As the stock price is stable at $144, the investor will not execute the put option and the option will expire worthless.

By not executing the put option, the investor incurs a loss which is equal to the price of the put option which is $368.

Scenario 2: an increase in the price of the underlying asset to $155

The market value of the investment $15,500. The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) plus the cost of buying the put options ($368 = $3.68*100), which is equal to $14,768, (i.e. ($14,400 + $368)).

As the stock price is at $155, the investor will not execute the put option and hold on the underlying stock.

By not executing the put option, the investor incurs a loss which is equal to the price of the put option which is $368.

Scenario 3: a decrease in the price of the underlying asset to $140

The market value of the investment $14,000. The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) plus the cost of buying the put options ($368 = $3.68*100), which is equal to $14,768, (i.e. ($14,400 + $368)).

As the stock price has decreased to $140, the investor will execute the put option and sell the Apple stocks at $144. By executing the put option, the investor will protect himself from incurring a loss of $400 (i.e.($144-$140)*100) due to a decrease in the Apple stock prices.

▶ All posts about Options

▶ Akshit GUPTA Options

▶ Akshit GUPTA The Black-Scholes-Merton model

▶ Akshit GUPTA Option Greeks – Delta

▶ Akshit GUPTA Covered call

▶ Akshit GUPTA Option Trader – Job description

Useful Resources

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Trading strategies involving Options, 276-295.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4(1): 141–183.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

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

Straddle and strangle strategy

January 12, 2022 by Akshit GUPTA

Straddle and Strangle

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the strategies of straddle and strangle based on options.

Introduction

In financial markets, hedging is implemented by investors to minimize the risk exposure and maximize the returns for any investment in securities. While hedging does not necessarily eliminate the entire risk for an investment, it does limit or offset any potential losses that the investor can incur.

Option contracts are commonly used by investors / traders as hedging mechanisms due to their great flexibility (in terms of expiration date, moneyness, liquidity, etc.) and availability. Positions in options are used to offset the risk exposure in the underlying security, another option contract or in any other derivative contract. Option strategies can be directional or non-directional.

Directional strategy is when the investor has a specific viewpoint about the movement of an asset price and aims to earn profit if the viewpoint holds true. For instance, if an investor has a bullish viewpoint about an asset and speculates that its price will rise, she/he can buy a call option on the asset, and this can be referred as a directional trade with a bullish bias. Similarly, if an investor has a bearish viewpoint about an asset and speculates that its price will fall, she/he can buy a put option on the asset, and this can be referred as a directional trade with a bearish bias.

On the other hand, non-directional strategies can be used by investors when they anticipate a major market movement and want to gain profit irrespective of whether the asset price rises or falls, i.e., their payoff is independent of the direction of the price movement of the asset but instead depends on the magnitude of the price movement. There are various popular non-directional strategies that can be implemented through a combination of option contracts to minimize risk and maximize returns. In this post, we are interested in straddle and strangle.

Straddle

In a straddle, the investor buys a European call and a European put option, both at the same expiration date and at the same strike price. This strategy works in a similar manner like a strangle (see below). However, the potential losses are a bit higher than incurred in a strangle if the stock price remains near the central value at expiration date.

A long straddle is when the investor buys the call and put options, whereas a short straddle is when the investor sells the call and put options. Thus, whether a straddle is long or short depends on whether the options are long or short.

Market Scenario

When the price of underlying is expected to move up or down sharply, investors chose to go for a long straddle and the expiration date is chosen such that it occurs after the expected price movement. Scenarios when a long straddle might be used can include budget or company earnings declaration, war announcements, election results, policy changes etc.
Conversely, a short straddle can be implemented when investors do not expect a significant movement in the asset prices.

Example

In Figure 1 below, we represent the profit and loss function of a straddle strategy using a long call and a long put option. K₁ is the strike price of the long call i.e., €98 and K₂ is the strike price of the long put position i.e., €98. The premium of the long call is equal to €5.33, and the premium of the long put is equal to €3.26 computed using the Black-Scholes-Merton model. The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the underlying asset (S₀) is €100, the volatility (σ) of the underlying asset is 40% and the risk-free rate (r) is 1% (market data).

Figure 1. Profit and loss (P&L) function of a straddle position.

Source: computation by the author.

You can download below the Excel file for the computation of the straddle value using the Black-Scholes-Merton model.

Strangle

In a strangle, the investor buys a European call and a European put option, both at the same expiration date but different strike prices. To benefit from this strategy, the price of the underlying asset must move further away from the central value in either direction i.e., increase or decrease. If the stock prices stay at a level closer to the central value, the investor will incur losses.

Like a straddle, a long strangle is when the investor buys the call and put options, whereas a short strangle is when the investor sells (issues) the call and put options. The only difference is the strike price, as in a strangle, the call option has a higher strike price than the price of the underlying asset, while the put option has a lower strike price than the price of the underlying asset.

Strangles are generally cheaper than straddles because investors require relatively less price movement in the asset to ‘break even’.

Market Scenario

The long strangle strategy can be used when the trader expects that the underlying asset is likely to experience significant volatility in the near term. It is a limited risk and unlimited profit strategy because the maximum loss is limited to the net option premiums while the profits depend on the underlying price movements.

Similarly, short strangle can be implemented when the investor holds a neutral market view and expects very little volatility in the underlying asset price in the near term. It is a limited profit and unlimited risk strategy since the payoff is limited to the premiums received for the options, while the risk can amount to a great loss if the underlying price moves significantly.

Example

In Figure 2 below, we represent the profit and loss function of a strangle strategy using a long call and a long put option. K₁ is the strike price of the long call i.e., €98 and K₂ is the strike price of the long put position i.e., €108. The premium of the long call is equal to €5.33, and the premium of the long put is equal to €9.47 computed using the Black-Scholes-Merton model. The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the underlying asset (S₀) is €100, the volatility (σ) of the underlying asset is 40% and the risk-free rate (r) is 1% (market data).

Figure 2. Profit and loss (P&L) function of a strangle position.

Source: computation by the author..

You can download below the Excel file for the computation of the strangle value using the Black-Scholes-Merton model.

▶ All posts about Options

▶ Akshit GUPTA Options

▶ Akshit GUPTA The Black-Scholes-Merton model

▶ Akshit GUPTA Option Spreads

▶ Akshit GUPTA Option Trader – Job description

Useful resources

Academic research articles

Black F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities” The Journal of Political Economy, 81, 637-654.

Merton R.C. (1973) “Theory of Rational Option Pricing” Bell Journal of Economics, 4, 141–183.

Books

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition, Chapter 10 – Trading strategies involving Options, 276-295.

Wilmott P. (2007) Paul Wilmott Introduces Quantitative Finance, Second Edition, Chapter 8 – The Black Scholes Formula and The Greeks, 182-184.

About the author

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

Analysis of synergy-based theories for M&A

January 11, 2022 by Suyue MA

Analysis of synergy-based theories for M&A

In this article, Suyue MA (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021) analyzes the synergy-based theories for M&As.

This article is structured as follows: I will first share with my professional experience. I will introduce the concepts of M&A and a brief analysis of past M&A market activity. We then expose the different theories based on synergies emphasized by companies in M&A deals.

About myself

I have been interested in finance ever since I started my study at ESSEC Business School in 2017. By acknowledging more about finance, during my 2nd year of study, I decided to build up my career in corporate finance, focusing on the primary market. By sending around 400 resumes to different companies and banks, I finally worked in the field of M&A. Until now, I have finished four internships in the field of corporate finance, private equity, capital-raising advisory, and mergers and acquisitions (M&A).

In this article, I would like to share with you about some very important M&A theories based on synergies that most of companies decided to execute as effective corporate strategies.

Introduction

M&As are defined as consolidation of companies, and it refers to corporate finance, corporate strategy, and corporate management, dealing with selling, buying, or combination of different firms, which can create resources, financing, and business development to a firm to grow its business without the need of creating a new business entity. Normally, a merger occurs between companies that have related interests with a similar company size or market cap. In addition, a merger is commonly understood as a fusion of two companies, which the bigger and better company will remain its name and status while the other one will disappear and not exist as a unique business entity. Nevertheless, acquisition means that a company is going to pay a certain price (in cash or stocks) to buyout or acquire the target company’s part of or full of stock right, achieving the controlling right or assets of the company that is being acquired, but the legal person’s status will remain.

To put in a nutshell, based on the historical M&A transactions, the primary objectives behind a merger or acquisition are to create long-term shareholder value, achieve larger market share, and improve the company’s efficiency. However, obviously, there are also a great number of M&A activities failed to reach such goals or even ruined companies. According to the collated research and a recent Harvard Business Review report in 2021, the M&A’s failure rate sites between 70% to 90%, which is an extremely high figure even though the report takes all rage of business, culture factors, and objectives factors into considerations. Thus, it remains doubtful whether a M&A transaction can help company’s development and create shareholder’s value.

Nowadays, companies use M&A for various reasons because companies are always facing the issues of dealing with global competition, market globalization, and constant technology innovation. It is now a fact that M&A has become the most popular corporate strategy around the world. We may ask why the management and shareholder boards are using merger and acquisition to promote the company’s advancement and shareholders’ return instead of other strategies, such as doing investments and innovations. According to the aforementioned report, some finance professionals believe that such transactions create short-cut for companies’ growth and market share, since the companies do not need to start a business sector over again, in which the risk of running a successful business is high and the cost of capital is high as well. On the contrary if both buy-side and sell-side companies can find synergies that benefit each other, ideally, they will gain more revenues due to the positive reaction, and therefore create value for their shareholders. Thus, here I will dig deeper in the following theories and synergies to better understand the aim and purpose of M&A.

Figure 1. Number and value of merger and acquisition deals worldwide from 1985 to 2020.

Number and value of merger and acquisition deals worldwide from 1985 to 2020

Source: Institute of Mergers, Acquisitions and Alliances (IMAA)

Figure 2. Number and value of merger and acquisition deals in the United States from 1985 to 2020.

Number and value of merger and acquisition deals in the United States from 1985 to 2020

Source: Institute of Mergers, Acquisitions and Alliances (IMAA)

The figures above are about the number and value of M&A transactions in both U.S. and worldwide in the last two and half decades (1985-2020). The reason why I choose these geographic locations is because the global M&A transactions’ number and value can provide us the activity level of the market; secondly, the U.S. market has the most active level from all time, and therefore, by viewing such figures, it can provide us a very clear overview of the market. According to both figures above, both M&A’s value and transactions are increasing stably except three serious drops in year of 2000, 2008, and 2020. The first drop is because of 2000’s financial crisis that happened in most of developed countries; the second drop happened right after the U.S. subprime crisis, and the last drop just happened from years of 2019 to 2021, in which the whole world was shut down because of COVID-19 virus. A great number of big companies went bankrupt and most of financial institutions had to stop their operations. What is more, we can find that after each recession, the value and number of M&A transactions rebounded rapid to the average level and kept increasing the volume within the following years. As I mentioned previously, although M&As have a super rate of failure, the success rate of successful company’s transactions must surpass the risks involved. Consequently, it is not difficult to explain why companies are keeping entering M&A transactions.

M&A’s main theories

The history of mergers and acquisitions exists for more than a hundred years, and financial professionals and scholars came forward with a great number of merger and acquisition theories. Most of these theories are based on the motives and benefits of merger and acquisition, and several major models have been developed. The following part is a brief introduction of these theories.

Efficiency theory assumes that both the acquiring company and the target company are interested in maximizing shareholder value, that the merger is a value-adding investment for both the acquiring company and the target company; the total benefits of the merger (the sum of the values of target and acquiring companies after vs before the deal) are positive. Efficiency theories are powerful in explaining the motivation of mergers, but the exact motivation of mergers in terms of synergies and efficiency improvements requires further examination and analysis and is beyond the scope of this dissertation. The different sources of efficiency theory based on value addition can be divided into the following areas: management synergy, operating synergy, diversification and strategic synergy, financial synergy, and undervaluation theory.

Management synergy

Since there are differences between the management capabilities of any two firms, the merger and acquisition activity may enable the more efficient management capabilities to diffuse in the new post-acquisition firm, bringing about efficiency improvements. For example, a relatively efficient firm may improve the management and operations of the acquired firm by acquiring a relatively inefficient firm to improve efficiency, thus increasing the value of the acquired firm; or a firm with relatively poor management efficiency may acquire a firm with higher management efficiency to improve its own efficiency, thus acquiring the organizational capital unique to the acquired firm.

Operating synergy

Operating synergies assume that there are economies of scale and economies of scope, which are cost advantages reaped by companies when production becomes efficient, in an industry, and that through merger and acquisition, companies can improve their original operating efficiency. In this theory, merger and acquisition can create great value.

The scale of the enterprise before the merger is far from the economies of scale, and the enterprise entity (consortium) formed after the merger can minimize the cost or maximize the profit in production, personnel, equipment, management, and sales. On the other hand, through vertical mergers, enterprises at different stages of development in the industry can be combined to reduce transaction costs and obtain effective synergies. Economies of scope mean that companies can use their existing product manufacturing and sales experience to produce related add-on products at a lower cost. For example, in the automotive industry, additional production of small cars and various vans would benefit from the existing automotive technology and manufacturing experience.

Diversification and strategic synergy

Companies can diversify their operations through M&A activities, which can diversify risks and stabilize revenue streams and provide employees with greater security and advancement opportunities; ensure continuity of the corporate team and organization; secure the company’s reputation. For strategic synergy, the company can acquire new management skills and organizational costs through M&A to increase the ability to enter new growth areas or overcome new competitive threats.

Financial synergy

One source of financial synergy is the lower cost of internal and external financing. For example, companies with high internal cash flow and low investment opportunities should have excess cash flow, while companies with lower internal capital production capacity and significant investment opportunities should require additional financing. Therefore, merger of these two firms may have the advantage of lower internal capital costs. On the other hand, the combined firm’s ability to leverage debt is greater than the sum of ability of the two firms before the merger, which provides a tax saving advantage.

Undervaluation theory

This theory suggests that the most direct basis for M&A comes from the difference in the value of the target company as judged by different investors and market players, since there is no purely efficient stock market in the world, it is possible that market value of the target company is lower than its true or potential value for some reason. The main reasons for undervaluation are: first, the inability of the target company’s management to realize the full potential of the company. The second reason could be insider information, because the M&A firm has information about the true value of the target company that is not known to the outside world. Thirdly, the Q-ratio. This is the ratio of the market value of the firm’s securities over the replacement cost of its assets. When inflation persists, as the Q ratio falls below one, it is cheaper to acquire an existing firm than to build a new one.

Useful resources

Institute of Mergers, Acquisitions and Alliances (IMAA) M&A Statistics.

Christensen, C.M., Alton, R., Rising, C., Waldeck, A., (March 2011) The Big Idea: The New M&A Playbook Harvard Business Review (89):48-57.

Dineros-De Guzman, C., (May 2019) Creating value through M&A PWC.

About the author

The article was written in January 2022 by Suyue MA (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021).

Introduction

Computation of returns

Arithmetic (percentage) returns

Logarithmic returns

Link between arithmetic and logarithmic returns

Components of total returns

Price changes and returns

Internal rate of return (IRR)

Ex ante and ex post returns

Related posts on the SimTrade blog

About the author

Supply and Demand

The basic concepts

Special cases

Useful resources

About the author

Risk Aversion

Risk aversion and investor’s characteristics

Age

Income

Past Losses

Investment Objective

Investment Horizon

Risk aversion and financial investments

Measure of risk of financial assets

Standard Deviation

Beta

Modern Portfolio Theory & Risk

Risk aversion and economic conditions

Related posts

Useful resources

About the author

A quick overview of the Bloomberg terminal…

How to use the main functions of the Bloomberg terminal?

Main users of the Bloomberg terminal

Training webinars

Related posts on the SimTrade blog

Useful resources

About the author

Portfolio Theory and Risk

Sources of specific risk

Business risk

Financial risk

Mathematical foundations

Decomposition of returns

Why should I be interested in this post?

Related posts on the SimTrade blog

Useful resources

Academic research

About the author

Portfolio Theory and Risk

Interest rate risk

Inflation risk

Exchange Rate Risk

Geopolitical Risks

Natural disasters

Systematic risk analysis in recent times

Why should I be interested in this post?

Related posts on the SimTrade blog

Useful resources

Academic research

Business analysis

About the author

Introduction

Calculating VaR using Monte Carlo simulations

Advantages and limitations of Monte Carlo method for VaR

Related posts on the SimTrade blog

Useful resources

About the author

Introduction

Mathematical foundation

Implementation of the Black-Litterman asset allocation model in practice

Why should I be interested in this post?

Related posts on the SimTrade blog

Useful resources

Academic research

About the author

Introduction

Mathematical foundations

Implementation of the Markowitz asset allocation model in practice