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).

Option Spreads

January 7, 2022 by Akshit GUPTA

Option Spreads

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the different option spreads used to hedge a position in financial markets.

Introduction

In financial markets, hedging is implemented by investors to minimize the risk exposure 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 traders and investors 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.

Spreads are hedging strategies used in trading in which traders buy and sell multiple option contracts on the same underlying asset. In a spread strategy, the option type used to create a spread has to be consistent, either call options or put options. These are used frequently by traders to minimize their risk exposure on the positions in the underlying assets.

Bull Spread

In a bull spread, the investor buys a European call option on the underlying asset with strike price K₁ and sells a call option on the same underlying asset with strike price K₂ (with K₂ higher than K₁) with the same expiration date. The investor expects the price of the underlying asset to go up and is bullish about the stock. Bull spread is a directional strategy where the investor is moderately bullish about the underlying asset, she is investing in.

When an investor buys a call option, there is a limited downside risk (the loss of the premium) and an unlimited upside risk (gains). The bull spread reduces the potential downside risk on buying the call option, but also limits the potential profit by capping the upside. It is used as an effective hedge to limit the losses.

Market Scenario

When the price of underlying asset is expected to moderately move up, investors chose to execute a bull spread and the expiration date is chosen such that it occurs after the expected price movement. If the price decreases significantly by the expiration of the call options, the investor loses money by using a bull spread.

Example

In Figure 1 below, we represent the profit and loss function of a bull spread strategy using a long and a short call option. K₁ is the strike price of the long call i.e., €88 and K₂ is the strike price of the short call position i.e., €110. The premium of the long call is equal to €12.62, and the premium of the short call is equal to €1.16 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 bull spread.

Profit and loss (P&L) function of a bul spread

Source: computation by the author.

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

Bear Spread

In a bear spread, the investor expects the price of the underlying asset to moderately decline in the near future. In order to hedge against the downside, the investor buys a put option with strike price K₁ and sells another put option with strike price K₂, with K₁ lower than < K₂. Initially, this initial position leads to a cash outflow since the put option bought (with strike price K₁) has a higher premium than put option sold (with strike price K₂) as K₁ is lower than < K₂.

Market Scenario

When the price of underlying asset is expected to moderately move down, investors chose to execute a bear spread and the expiration date is chosen such that it occurs after the expected price movement. Bear spread is a directional strategy where the investor is moderately bearish about the stock he is investing in. If the price increases significantly by the expiration of the put options, the investor loses money by using a bear spread.

Example

In Figure 2 below, we represent the profit and loss function of a bear spread strategy using a long and a short put option. K₁ is equal to the strike price of the short put i.e., €90 and K₂ is equal to the strike price of the long put i.e., €105. The premium of the short put is equal to €0.86, and the premium long put is equal to €7.26 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 stock is 40% and the risk-free rate (r) is 1% (market data).

Figure 2. Profit and loss (P&L) function of a bear spread.

Profit and loss (P&L) function of a bear spread

Source: computation by the author.

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

Butterfly Spread

In a butterfly spread, the investor expects the price of the underlying asset to remain close to its current market price in the near future. Just as a bull and bear spread, a butterfly spread can be created using call options. In order to profit from the expected market scenario, the investor buys a call option with strike price K₁ and buys another call option with strike price K₃, where K₁ < K₃, and sells two call options at price K₂, where K₁ < K₂ < K₃. Initially, this initial position leads to a net cash outflow.

Market Scenario

When the price of underlying asset is expected to stay stable, investors chose to execute a butterfly spread and the expiration date is chosen such that the expected price movement occurs before the expiration date. Butterfly spread is a non-directional strategy where the investor expects the price to remain stable and close to the current market price. If the price movement is significant (either downward or upward) by the expiration of the call options, the investor loses money by using a butterfly spread.

Example

In Figure 3 below, we represent the profit and loss function of a butterfly spread strategy using call options. K₁ is equal to the strike price of the long call position i.e., €85 and K₂ is equal the strike price of the two short call positions i.e., €98 and K₃ is equal to the strike price of another long call position i.e., €111. The premium of the long call K₁ is equal to €15.332, the premium of the long call K₃ is equal to €0.993 and the premium of the short call K₂ is equal to €5.334 computed using the Black-Scholes-Merton model. The premium of the butterfly spread is then equal to €5.657 (= 15.332 + 0.993 -2*5.334), which corresponds to an outflow for the investor.

The time to maturity (T) is of 18 days (i.e., 0.071 years). At the time of valuation, the price of the (S₀) is €100, the volatility (σ) of stock is 40% and the risk-free rate (r) is 1% (market data).

Figure 3. Profit and loss (P&L) function of a butterfly spread.

Profit and loss (P&L) function of a butterfly spread

Source: computation by the author.

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

Note that bull, bear, and butterfly spreads can also be created from put options or a combination of call and put options.

▶ All posts about options

▶ Gupta A. Options

▶ Gupta A. The Black-Scholes-Merton model

▶ Gupta A. Option Greeks – Delta

▶ Gupta A. Hedging Strategies – Equities

Useful resources

Hull J.C. (2018) Options, Futures, and Other Derivatives, Tenth Edition, Chapter 12 – Trading strategies involving Options, 282-301.

About the author

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

The historical method for VaR calculation

January 30, 2024December 17, 2021 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents the historical method for VaR calculation.

Introduction

A key factor that forms the backbone for risk management is the measure of those potential losses that an institution is exposed to any investment. Various risk measures are used for this purpose and Value at Risk (VaR) is the most commonly used risk measure to quantify the level of risk and implement risk management.

VaR is typically defined as the maximum loss which should not be exceeded during a specific time period with a given probability level (or ‘confidence level’). VaR is used extensively to determine the level of risk exposure of an investment, portfolio or firm and calculate the extent of potential losses. Thus, VaR attempts to measure the risk of unexpected changes in prices (or return rates) within a given period. Mathematically, the VaR corresponds to the quantile of the distribution of returns.

The two key elements of VaR are a fixed period of time (say one or ten days) over which risk is assessed and a confidence level which is essentially the probability of the occurrence of loss-causing event (say 95% or 99%). There are various methods used to compute the VaR. In this post, we discuss in detail the historical method which is a popular way of estimating VaR.

Calculating VaR using the historical method

Historical VaR is a non-parametric method of VaR calculation. This methodology is based on the approach that the pattern of historical returns is indicative of the pattern of future returns.

The first step is to collect data on movements in market variables (such as equity prices, interest rates, commodity prices, etc.) over a long time period. Consider the daily price movements for CAC40 index within the past 2 years (512 trading days). We thus have 512 scenarios or cases that will act as our guide for future performance of the index i.e., the past 512 days will be representative of what will happen tomorrow.

For each day, we calculate the percentage change in price for the CAC40 index that defines our probability distribution for daily gains or losses. We can express the daily rate of returns for the index as:
img_historicalVaR_returns_formula

Where R_t represents the (arithmetic) return over the period [t-1, t] and P_t the price at time t (the closing price for daily data). Note that the logarithmic return is sometimes used (see my post on Returns).

Next, we sort the distribution of historical 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).

Since the historical VaR is estimated directly from data without estimating or assuming any other parameters, hence it is a non-parametric method.

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 top 1% of worst 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 top 5% of the worst returns.

Figure 1. Probability distribution of returns for the CAC40 index.
Historical method VaR
Source: computation by the author (data source: Bloomberg).

You can download below the Excel file for the VaR calculation with the historical method. The historical distribution is estimated with historical data from the CAC 40 index.

From the above graph, we can interpret VaR for 90% confidence level as -3.99% i.e., there is a 90% probability that daily returns we obtain in future are greater than -3.99%. Similarly, VaR for 99% confidence level as -5.60% i.e., there is a 99% probability that daily returns we obtain in future are greater than -5.60%.

Advantages and limitations of the historical method

The historical method is a simple and fast method to calculate VaR. For a portfolio, it eliminates the need to estimate the variance-covariance matrix and simplifies the computations especially in cases of portfolios with a large number of assets. This method is also intuitive. VaR corresponds to a large loss sustained over an historical period that is known. Hence users can go back in time and explain the circumstances behind the VaR measure.

On the other hand, the historical method has a few of drawbacks. The assumption is that the past represents the immediate future is highly unlikely in the real world. Also, if the horizon window omits important events (like stock market booms and crashes), the distribution will not be well represented. Its calculation is only as strong as the number of correct data points measured that fully represent changing market dynamics even capturing crisis events that may have occurred such as the Covid-19 crisis in 2020 or the financial crisis in 2008. In fact, even if the data does capture all possible historical dynamics, it may not be sufficient because market will never entirely replicate past movements. Finally, the method assumes that the distribution is stationary. In practice, there may be significant and predictable time variation in risk.

Useful resources

Jorion, P. (2007) Value at Risk , Third Edition, Chapter 10 – VaR Methods, 276-279.

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

About the author

The article was written in December 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

The variance-covariance method for VaR calculation

January 30, 2024December 17, 2021 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) presents the variance-covariance method for VaR calculation.

Introduction

The two key elements of VaR are a fixed period of time (say one or ten days) over which risk is assessed and a confidence level which is essentially the probability of the occurrence of loss-causing event (say 95% or 99%). There are various methods used to compute the VaR. In this post, we discuss in detail the variance-covariance method for computing value at risk which is a parametric method of VaR calculation.

Assumptions

The variance-covariance method uses the variances and covariances of assets for VaR calculation and is hence a parametric method as it depends on the parameters of the probability distribution of price changes or returns.

The variance-covariance method assumes that asset returns are normally distributed around the mean of the bell-shaped probability distribution. Assets may have tendency to move up and down together or against each other. This method assumes that the standard deviation of asset returns and the correlations between asset returns are constant over time.

VaR for single asset

VaR calculation for a single asset is straightforward. From the distribution of returns calculated from daily price series, the standard deviation (σ) under a certain time horizon is estimated. The daily VaR is simply a function of the standard deviation and the desired confidence level and can be expressed as:

img_VaR_single_asset

Where the parameter ɑ links the quantile of the normal distribution and the standard deviation: ɑ = 2.33 for p = 99% and ɑ = 1.645 for p = 90%.

In practice, the variance (and then the standard deviation) is estimated from historical data.
img_VaR_asset_variance

Where R_t is the return on period [t-1, t] and R the average return.

Figure 1. Normal distribution for VaR for the CAC40 index
Normal distribution VaR for the CAC40 index
Source: computation by the author (data source: Bloomberg).

You can download below the Excel file for the VaR calculation with the variance-covariance method. The two parameters of the normal distribution (the mean and standard deviation) are estimated with historical data from the CAC 40 index.

VaR for a portfolio of assets

Consider a portfolio P with N assets. The first step is to compute the variance-covariance matrix. The variance of returns for asset X can be expressed as:

Variance

To measure how assets vary with each other, we calculate the covariance. The covariance between returns of two assets X and Y can be expressed as:

Covariance

Where X_t and Y_t are returns for asset X and Y on period [t-1, t].

Next, we compute the correlation coefficients as:

img_correlation_coefficient

We calculation the standard deviation of portfolio P with the following formula:

img_VaR_std_dev_portfolio

img_VaR_std_dev_portfolio_2

Where w_i corresponds to portfolio weights of asset i.

Now we can estimate the VaR of our portfolio as:

img_portfolio_VaR

Where the parameter ɑ links the quantile of the normal distribution and the standard deviation: ɑ = 2.33 for p = 99% and ɑ = 1.65 for p = 95%.

Advantages and limitations of the variance-covariance method

Investors can estimate the probable loss value of their portfolios for different holding time periods and confidence levels. The variance–covariance approach helps us measure portfolio risk if returns are assumed to be distributed normally. However, the assumptions of return normality and constant covariances and correlations between assets in the portfolio may not hold true in real life.

Useful resources

Jorion P. (2007) Value at Risk, Third Edition, Chapter 10 – VaR Methods, 274-276.

About the author

The article was written in December 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Understanding Options and Options Trading Strategies

December 13, 2021 by Luis RAMIREZ

Understanding Options and Options Trading Strategies

In this article, Luis RAMIREZ (ESSEC Business School, Global BBA, 2019-2023) discusses the fundamentals behind options trading.

Financial derivatives

In order to understand and grasp the concept of options, knowledge of what is a derivative should be established. A financial instrument derivative is ultimately an instrument whose value derives from the value of an underlying asset (or multiple underlying assets). These underlying assets can of course be bonds, stocks, commodities, currencies, etc. Derivatives are widely common and used around the world; investment banks, commercial banks, and corporations (mainly multinational corporations) are all consistent users of derivatives. The purpose, or goal, behind derivatives is to manage risk, whether that be alleviating risk by hedging investments, or by taking on risk through speculative investments. To carry out this process, the investor must undertake one of the four types of derivatives. The four types are the following: options, forwards, futures, and swaps. In this article the focus will be solely placed on options.

What are options?

An option contract provides an investor the chance to either buy (for a call option) or sell (for a put option) the underlying asset, depending on what type of option they possess. Every option contract has an expiry date in which the investor can effectively exercise the option. A very important thing about options is that they provide investors the right, but not the obligation, to either buy or sell an asset (i.e., stock shares) at a price and at a date that have been agreed at the issuing of the cotnract.

Put options vs call options

Firstly, the main two different options are call and put options. Call options give investors (that bought the call option) the right to buy a stock at a certain price and at a certain date, and put options give investors (that bought the put option) the right to sell a stock at a certain price and at a certain date. The first step into acquiring options, either type, is paying a premium. This premium which is spent at the beginning of the process is the only loss that investors will face if the options are not exercised. However, the other side of the coin, options writers (sellers) are more exposed to risk as they are exposed to lose more than only the premium.

Sell-side vs buy-side

In an option contract, the price at which the asset is sold or bought is known as the strike price, or exercise price. When a call option has been bought, and the price of the share has
had a bullish trend and rises above the strike price, the investor can simply exercise his right to buy the share at the strike price, and then immediately sell it at the spot price, resulting in immediate profit. However, if the price of the share had a bearish trend and dropped below the strike price, the investor can decide not to exercise his right and will only lose the amount of premium paid in this case.

Figure 1. Profit and loss (P&L) of a long position in a call option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

On the other hand, selling options differs. Selling options is commonly known as writing options. The way this works is that a writer receives the premium from a buyer, this is the maximum profit a writer can receive by selling call options. Normally, a call option writer is bearish, therefore he believes that the price of the stack will fall below that of the strike price. If indeed the share price falls below the strike price, the writer would profit the premium paid by the buyer, since the buyer would not exercise the option. However, if the share price surpassed the strike price, the writer would have to sell shares at the low strike price. The writer would then experience a loss, the size of the loss depends on how many shares and price the writer would have to use to cover the entire option contract. Clearly, the risk for call writers is much higher than the risk exposure call buyers when acquiring an option. To summarize, the call buyer can only lose the premium paid, and the call writer can face infinite risk because the price of a share can keep increasing.

Figure 2. Profit and loss (P&L) of a short position in a call option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

As for put options, put buyers usually believe the share price will decrease under the strike price. If this does eventually happen, the investor can simply exercise the put and sell at strike price, instead of a lower spot price. If the investor wants to go long, he can substitute the shares used in the option contract and buy them for a cheaper spot price after the put has been exercised. However, if the spot price is above the strike price, and the investor choses to not exercise the put, the loss will once again only be the cost of the premium.

Figure 3. Profit and loss (P&L) of a long position in a put option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

On the other hand, put writers think the share price will have a bullish trend throughout the duration of the option lifecycle. If the share price rises above strike price, the contract will expire, and the seller’s profit is the premium he received. If the share price decreases, and falls under the strike price, then the writer is obliged to buy shares at a strike price which higher than the spot price. This is when the risk is at the highest for a put writer, if the share price falls. Just like call writing, the loss can be hefty. Only that in the case of put writing, it happens if the share price tumbles down.

Figure 4. Profit and loss (P&L) of a short position in a put option
as a function of the price of the underlying asset at maturity
Profit and loss (P&L) as a function of the price of the underlying asset at maturity

Source: production by the author.

This can be shrunk down to knowing that call buyers can benefit from buying securities or assets at a lower price if the share price rises during the length of the option contract. Put buyers can benefit from selling assets at a higher strike price if the share price falls during the length of the option contract. As per writers, they receive a premium fee when writing options. However, it is not all positive points, option buyers need to pay the premium fee and discount this from their potential profit, and writers face an indefinite risk subject to the share price and quantity.

Figure 5. Market scenarios for buying and selling call and put options

Source: production by the author.

Option Trading Strategies

Four trading strategies have already been mentioned, selling or buying either puts or calls. However, there are several different option trading strategies and new ones are being produced frequently, anyhow the article will focus on five trading strategies that most, if not all, investors are familiar with.

Covered Call

This trading strategy consists in the writer selling call options against the stock that he already owns. It is ‘covered’ because it covers the writer when the buyer of the option exercises his right to buy the shares, due to the writer already owning them, meaning that the writer can deliver the shares. This strategy is often used as an income stream from premiums. This is an employable strategy for those who believe that the asset they own will only experience a small change in price. The covered call is considered a low-risk strategy, and if used appropriately with a reliable stock, it can be a source of income.

Figure 6. Profit and loss (P&L) of a covered call
as a function of the price of the underlying asset at maturity

Source: production by the author.

Married put

Like a covered call, in a married put the investor buys an asset and then buys a put option with the strike price being equal to the spot price. This is done to be protected against a decrease in the asset price. Of course, when buying an option, a premium must be paid, which is a downside for a married put strategy. However, the married put limits the loss an investor could incur in case of a price decrease. On the other hand, if the price increases, profit is unlimited. This strategy is often used for volatile stocks.

Figure 7. Profit and loss (P&L) of a married put
as a function of the price of the underlying asset at maturity

Source: production by the author.

Protective Collar

This strategy is done when an investor buys a put option where the strike price is lower than the spot price, as well as instantly writing a call option where the strike price is higher than the spot price, this must be done by the investor owning said asset. This strategy protects the investor from a decrease in price. If the share price increases, large profits will be capped, however large losses will be also capped. When performing a protective collar, the best possibility for an investor is that the share price rises to the call strike price.

Figure 8. Profit and loss (P&L) of a protective collar
as a function of the price of the underlying asset at maturity

Source: production by the author.

Bull Call Spread

In order to execute this strategy, an investor buys calls at the same time that he sells the equivalent order of calls, which have a higher strike price. Of course, both calls must be tied to the same asset. As seen on the name of this strategy, it is a strategy that an investor employs when he predicts a bullish trend. Just like the protective collar, it limits both, gains and losses.

Figure 9. Profit and loss (P&L) of a bull call spread
as a function of the price of the underlying asset at maturity

Source: production by the author.

Bear Put Spread

This strategy is like the Bull Call Spread, only that it is in terms of a put option. The investor buys put options while he sells put options at a lower strike price. This can be done when the investor foresees a bearish trend, just like its call counterpart, the Bear Put Spread limits losses and gains.

Figure 10. Profit and loss (P&L) of a bear put spread
as a function of the price of the underlying asset at maturity

Source: production by the author.

Importance of options on financial markets

As seen on the variety of option trading strategies, and the different factors that play into each strategy mentioned, and dozens of other out there to explore, this instrument is a very utilized tool for investors, and financial institutions. The ‘options within options’ are of a huge variety and so much could be done. Many people have strong feelings towards this derivative, whether it is a negative, or positive stance, it all depends on the profits it brings. There is a lot of work behind options, and just like any other investment, due diligence is a key aspect of the procedure.

Useful Resources

Hull J.C. (2015) Options, Futures, and Other Derivatives, Ninth Edition.

Prof. Longin’s website Pricer d’options standards sur actions – Calls et puts (in French)

About the author

Article written in December 2021 by Luis RAMIREZ (ESSEC Business School, Global BBA, 2019-2023).

Equity structured products

December 11, 2021 by Akshit GUPTA

Equity structured products

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) introduces equity structured products, which are complex financial products proposed to investors to benefit from market expectations.

Introduction

Structured products are pre-packaged product offerings which are designed as per the client’s risk-return profile. The returns on the investments in these products are based on the performance of the underlying assets. These underlying assets can include individual assets or indexes in various markets like equities, bonds and commodities, and derivatives on these underlying assets like futures, swaps, and options. The structured products are highly sophisticated products since they are tailor-made as per the client’s requirements and risk/return profile. These products have pre-defined features like maturity date, early – redemption mechanism, coupon payments (fixed or variable coupons), underlying asset, and the degree of capital protection. They can guarantee full or partial capital protection and a flexible degree of leverage as well.

Since these products follow a non-traditional investment strategy and can have different underlying assets, they remain in high demand in different market conditions, either bullish, bearish, stable, volatile, or uncertain. Structured products are normally issued by financial institutions and can either be traded on stock exchanges or over the counter (OTC).

An equity structured products has mainly two components that include:

Fixed-Income product – A fixed-income security like a Treasury bond which fully or partially protects the capital of the investor.
Equity Instrument and Derivatives – An equity instrument (which can be a stock or an index option) which provides the additional pay-off of the product. The payoff of the equity instrument is linked to the performance of the underlying asset.

Underlying assets

The equity structured products can provide the investor an exposure to equity-linked products like an option contract on individual share, index, basket of shares, or indices.
The investor benefits from the performance of the underlying asset and is paid by means of regular coupons at specific observation days or a one-off payment at the end of the product life.

Apart from the traditional equities, the underlying asset for the structured products can also include indices like CAC 40, S&P 500, FTSE or any other. They can also be customized as per the investor’s need to include several different equities or indices.

Example of an equity linked structured product

For example, an investor wants to buy a structured product and invest EUR 1,000 for 3 years. She wants capital protection and at the same time, gain an exposure to the stocks of LVMH trading in the French equity markets.

A structurer can buy a 3-year zero-coupon French OAT (government bond) with a par value of EUR 1,000 at price of EUR 901. At maturity the bond will pay the principal amount of EUR 1,000.

For the remaining EUR 99, the structurer buys a call option on the shares of LVMH
trading at EUR 110. This provides the investor with a participation of 90% (i.e., 99/110) in the performance of the share of LVMH, the underlying asset.

Figure 1. Risk profile of a protective put position.

Source: computation by the author.

img_SimTrade_Options_Protective_Put

Pros and Cons of investing in equity structured products

Pros

Financial planning: Because of their defined maturity dates, structured products can be timed for costs like educational tuition fees and essential purchases and give investors peace of mind.
Risk hedging: Structured products generally offer some form of capital protection as a defensive barrier depending on an investor’s preferences. Thus, structured investments are available to minimize risk exposure.
Market access to diverse assets: Structured products allow investors to gain access to markets and asset classes that are not available through other securities.
Structured products can provide leveraged exposure to markets.

Cons

Market Risk: The return from investment can turn to zero or even negative in adverse market conditions
Liquidity Risk: For structured products, there is only one market maker for the investments and the issuers commit to making a competitive aftersales market in a place that is visible to the investor or their advisory
Counterparty Risk: Like most investments, structured products are subject to counterparty defaults. Issuer’s credit rating assessment and other information like credit default spreads, balance sheet strength etc. are essential

Useful resources

▶ Oesterreichische Nationalbank (2004), Financial Instruments Structured Products Handbook

▶ Akshit GUPTA History of Options markets

▶ Akshit GUPTA Option Trader – Job description

▶ Akshit GUPTA Options

▶ Akshit GUPTA Option Greeks – Delta

▶ Shengyu ZHENG Reverse convertibles

About the author

Article written in December 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Covered call

December 11, 2021 by Akshit GUPTA

Covered Call

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

Introduction

Covered call

The covered call strategy is a two-part strategy that essentially involves an investor writing a call option on an underlying security while simultaneously holding a long position in the same underlying. This action of buying an asset and writing calls on it at the same time is commonly referred as ‘buy write’. By writing a call option, the investor locks in the price of the underlying asset, thereby enjoying a short-term gain from the premium received.

Market scenario

The covered call is generally ideal if the investor has a neutral or slightly bullish outlook of the market wherein the potential future upside of the underlying asset owned by the investor is limited. This strategy is used by investors when they would prefer booking short-term profits on the assets than to keep holding it.

For instance, consider a ‘buy write’ situation where an investor buys shares of a stock (i.e., holds a long position in the stock) and simultaneously writes call options on them. The investor has a neutral view on the stock and doesn’t expect the price to rise much.
To book a short-term profit and also hedge any minor downsides in the stock price, the investor is writing call options on the stock at a strike price greater than or equal to the current price of the stock (i.e. out-of-the-money or at-the-money call options). The buyer of those call options would pay the investor a premium on those calls, whether or not the option is exercised. This is the covered call strategy in a nutshell.

Let us consider a covered call position with writing at-the money calls. One of following three scenarios may happen:

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

In this scenario, the market viewpoint of the investor holds correct and the profit from the strategy is the premium earned on the call options. In this case, the option holder does not exercise its call options, and the investor gets to keep the underlying stocks too.

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

In this scenario, since the price of the stock was already locked in through the call, the investor enjoys a short-term profit along with the premium. However, this also poses a risk in case the price of the stock rises substantially because the investor misses out on the opportunity.

Scenario 3: the stock price falls and calls expire out of the money

This is a negative scenario for the investor. There is limited protection from the downside through the premium earned on the call options. However, if the stock price falls below a certain break-even point, the losses for the investor can be considerable since there will be a fall in its underlying position.

Risk profile

In a covered call, the total cost of the investment is equal to the price of the underlying asset minus the premium earned by writing the call. However, the profit potential for the investment is limited and the maximum loss can be significantly high. The risk profile of the position is represented in Figure 1.

Figure 1. Risk profile of covered call position.

Source: computation by the author (based on the BSM model).

You can download below the Excel file for the computation of the Profit or Loss (P&L) function of the underlying position and covered call 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 short position in the call option (-Δ).

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

Figure 2. Delta of a covered call position.

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 investment is then equal to $14,400. She is neutral about the short-term prospects of the market. In order to gain from her market scenario, she decides to write an at-the-money call option at $144 on the Apple stock (lot size is 100) with a maturity of one month, using the covered call 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 call option is $3.87.

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

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

The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) minus the premium received on writing the calls ($387 = $3.87*100), which is equal to $14,013, i.e. $14,400 – $387.

As the stock price ($144) is equal to the strike price of the call options ($144), the value of the call options is equal to zero, and the investor earns a profit which is equal to the initial price of the call options (the premium), which is equal to $387.

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

The total cost of the initial investment is the cost of acquiring the Apple stocks ($14,400) minus the premium on writing the calls ($387 = $3.87*100), which is equal to $14,013, i.e. $14,400 – $387.

As the stock price has risen to $155, the call options are exercised by the option buyer, and the investor will have to sell the Apple stocks at the strike price of $144.

By executing the covered call strategy, the investor earns $387 (i.e. ($144-$144)*100 +$387) but misses the opportunity of earning higher profits by selling the stock at the current market price of $155.

Scenario 3: a decrease in the price of the underlying asset to $142

As the stock price is at $142, the call options are not exercised by the option buyer and the options expire worthless (out of the money).

As the buyer does not exercise the call options, the investor earns a profit which is equal to the price of the call options which is equal to $387. But his net profit decreases by the amount of the decrease in his position in the APPLE stocks which is equal to -$200 (i.e. ($142-$144)*100).

▶ All posts about Options

▶ Akshit GUPTA Options

▶ Akshit GUPTA Option Trader – Job description

▶ Akshit GUPTA The Black-Scholes-Merton model

▶ Akshit GUPTA Protective Put

▶ Akshit GUPTA Option Greeks – Delta

Useful Resources

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(1): 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 December 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Currency swaps

December 11, 2021 by Akshit GUPTA

Currency swaps

This article written by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) introduces the currency swaps used in financial markets.

Introduction

In financial markets, currency swaps are a derivative contract in which two counterparties exchange a stream of interest payments and principal amount in one currency with a stream of interest payments and principal amount in another currency. The life of the swap is for a pre-defined number of years. The interest payments are based on a pre-determined principal amount and can include the exchange of:

A fixed interest rate for a fixed interest rate
A fixed interest rate for a floating interest rate
A floating interest rate for a floating interest rate

Another way of understanding currency swaps can be that a counterparty A borrows funds from another counterparty B in a currency different from its domestic currency and lends funds in their domestic currency to the counterparty B. The principal amount is specified in each of the two currencies and is exchanged at the beginning and the maturity of the swap contract. Currency swaps differ from interest rate swaps as the principal amount is exchanged between the counterparties for currency swaps. The principal amounts set in the beginning of the exchange are usually equivalent to the exchange rate at that given time (the spot rate).

However, the exchange rate for the principal amounts at the end of the swap are decided between the counterparties at the time of entering the contract. Usually, it is equivalent to the initial exchange rate of the agreement.

Cross currency swaps can be used by different counterparties to reduce their exposure to exchange rate fluctuations and to benefit from lower interest rates to finance transactions in a foreign currency. These swaps also provide arbitrage opportunities between interest rates in different markets to the counterparties.

Types of currency swap contracts

Currency swap contracts can be classified into three types based on the interest rates that are to be exchanged on the contract.

Fixed for fixed currency swaps

In a fixed for fixed currency swap, the interest rates are exchanged between the counterparties based on a pre-determined fixed interest rates in both currencies.

For example, two counterparties, say Apple & LVMH, decides to enter a fixed for fixed currency swap. Apple wants to expand its operations in Europe and needs to borrow €87 million whereas LVMH wants to fund an acquisition it did in the US and requires $100 million. The companies resort to debt financing to fund their operations and takes a loan in their domestic currencies (due to cheaper borrowing rates in their respective countries). Apple takes a loan in USD for a fixed interest rate of 2% per annum, and LVMH takes a domestic loan in EUR for a fixed interest rate of 1.6% per annum.

Both the parties enter into a currency swap wherein Apple decides to pay $100 million to LVMH in exchange for €87 million ($1 = €0.87). On the principal amounts, Apple pays 1.6% in euros in interest rate to LVMH, and LVMH pays 2% in dollars to Apple. This is an illustration of a fixed for fixed currency swap.

Fixed for floating currency swaps

In a fixed for floating currency swap, a counterparty receives the interest payment based a fixed interest rate and pays the interest rates based on a floating interest rate. The rates are pre-determined at the time of entering the agreement.

If we take the case for fixed for floating currency swaps in the above example, LVMH pays at a fixed interest rate of 2% per annum and receives at a floating interest rate which is indexed to the 6-month Euribor.

Floating for floating currency swaps

In a floating for floating currency swap, a counterparty receives and pays the interest payment based floating interest rates that are pre-determined at the time of entering the agreement. The floating interest rates are usually indexed to the LIBOR rates.

If we take the case of floating for floating currency swaps in the above example, LVMH pays a floating interest rate indexed to the 6-month USD Libor and receives a rate based on the 6-month Euribor.

Interest rates on a currency swap

Currency swaps can be used in different market situations based on the needs of different counterparties. The floating for floating currency swap is considered as a basic swap and is most commonly used in financial markets. The interest rates for a floating for floating swaps are usually determined based on the LIBOR rates +/- spreads. The spreads are based on the dynamics of demand and supply for a currency swap. Higher spreads can imply higher demand for a particular currency swap.

The spreads also include the credit risk of a counterparty. The credit risk implies the possibility of a default on payments by a counterparty specified in the currency swap agreement.

Example – Fixed for fixed currency swap

For example, two counterparties, say Apple and LVMH, decides to enter a fixed for fixed currency swap. Apple wants to expand their operations in Europe and needs to borrow €87 million whereas LVMH wants to fund an acquisition they did in USA and requires $100 million. The companies resort to debt financing to fund their operations and takes a loan in their domestic currencies (due to cheaper borrowing rates in their respective countries).

Apple takes a loan in USD for a fixed interest rate of 2% and LVMH takes a domestic loan in EUR for a fixed interest rate of 1.6%. Both the parties enter into a 5-year currency swap on 1st November 2021 wherein Apple decides to pay $1 million to LVMH in exchange for €0.87 million ($1 = €0.87). As interest payments, Apple pays 1.6% per annum fixed rate to LVMH and received 2% per annum fixed rate semi-annually. The table below shows the pricing of currency swap.

Figure 1. Pricing of currency swap
mgsimtrade_Currencyswaps_Leg 1 .
imgsimtrade_Currencyswaps_Leg 2
Source: computation by the author.

▶ Alexandre VERLET Understanding financial derivatives: swaps

▶ Alexandre VERLET Understanding financial derivatives: forwards

▶ Alexandre VERLET Understanding financial derivatives: options

▶ Akshit GUPTA Options

Useful Resources

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

About the author

Article written in December 2021 by Akshit GUPTA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Standard deviation

February 1, 2025December 9, 2021 by Jayati WALIA

Standard deviation

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022) presents an overview of standard deviation and its use in financial markets.

Definition

The standard deviation is a measure that indicates how much data scatter around the mean. The idea is to measure how an observation deviates from the mean on average./p>

Mathematical formulae

The first step to compute the standard deviation is to compute the mean. Considering a variable X, the arithmetic mean of a data set with N observations, X₁, X₂ … X_N, is computed as:

img_arithmetic_mean

In the data set analysis, we also consider the dispersion or variability of data values around the central tendency or the mean. The variance of a data set is a measure of dispersion of data set values from the (estimated) mean and can be expressed as:

variance

Note that in the above formula we divide by N-1 because the mean is not known but estimated (usual case in finance). If the mean is known with certainty (when dealing the whole population not a sample), then we divide by N.

A problem with variance, however, is the difficulty of interpreting it due to its squared unit of measurement. This issue is resolved by using the standard deviation, which has the same measurement unit as the observations of the data set (such as percentage, dollar, etc.). The standard deviation is computed as the square root of variance:

standard deviation

A low value standard deviation indicates that the data set values tend to be closer to the mean of the set and thus lower dispersion, while a high standard deviation indicates that the values are spread out over a wider range indication higher dispersion.

Measure of volatility

For financial investments, the X variable in the above formulas would correspond to the return on the investment computed on a given period of time. We usually consider the trade-off between risk and reward. In this context, the reward corresponds to the expected return measured by the mean, and the risk corresponds to the standard deviation of returns.

In financial markets, the standard deviation of asset returns is used as a statistical measure of the risk associated with price fluctuations of any particular security or asset (such as stocks, bonds, etc.) or the risk of a portfolio of assets (such as mutual funds, index mutual funds or ETFs, etc.).

Investors always consider a mathematical basis to make investment decisions known as mean-variance optimization which enables them to make a meaningful comparison between the expected return and risk associated with any security. In other words, investors expect higher future returns on an investment on average if that investment holds a relatively higher level of risk or uncertainty. Standard deviation thus provides a quantified estimate of the risk or volatility of future returns.

In the context of financial securities, the higher the standard deviation, the greater is the dispersion between each return and the mean, which indicates a wider price range and hence greater volatility. Similarly, the lower the standard deviation, the lesser is the dispersion between each return and the mean, which indicates a narrower price range and hence lower volatility for the security.

Example: Apple Stock

To illustrate the concept of volatility in financial markets, we use a data set of Apple stock prices. At each date, we compute the volatility as the standard deviation of daily stock returns over a rolling window corresponding to the past calendar month (about 22 trading days). This daily volatility is then annualized and expressed as a percentage.

Figure 1. Stock price and volatility of Apple stock.

price and volatility for Apple stock
Source: computation by the author (data source: Bloomberg).

You can download below the Excel file for the calculation of the volatility of stock returns. The data used are for Apple for the period 2020-2021.

Useful resources

Wikipedia Standard Deviation

About the author

The article was written in November 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

Logistic Regression

December 9, 2021 by Jayati WALIA

In this article, Jayati WALIA (ESSEC Business School, Grande Ecole – Master in Management, 2019-2022) presents an overview of logistic regression and its application in finance.

Introduction

Logistic regression is a predictive analysis regression method that is used in classification to determine whether an output that is categorical, belongs to a particular class or category. Mathematically, this means that the dependent variable in regression is dichotomous or binary i.e., it can take the values 0 or 1. Logistic regression is used to describe data and explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.

For instance, consider a weather forecasting situation. If we wish to predict the likelihood of whether it will rain or not on a particular day, linear regression is not going to be of use in this scenario because our outcome or value of dependent variable is unbounded. On the other hand, a binary logistic regression model will provide with a classified outcome (1: it will rain; 0: it will not rain).

Logistic regression analysis is valuable for predicting the likelihood of an event. It helps determine the probabilities between any two classes. In essence, logistic regression helps solve probability and classification problems.

Logistic Function

Logistic regression model uses the sigmoid function to map the output of a linear equation between 0 and 1. The sigmoid function is an S-shaped curve and can be expressed as:

sigmoid function

Figure 1. Sigmoid function curve.

img_sigmoid_function_curve

Source: computation by the author.

For logistic regression, we initially model the relationship between the dependent and independent variables as a linear equation as follows:

linear equation for logistic regression

wherein Y is the dependent variable (i.e., the variable we want to predict) and X is the explanatory variables (i.e., the variables we use to predict the dependent variable). β₀, β₁, β₂… β_N are regression coefficients that are generally estimated using the maximum likelihood estimation method.

This equation is mapped to the sigmoid function to squeeze the value of the outcome (Y) from a large scale to within the range 0 – 1. We get our logistic regression equation as:

logistic regression equation

The dependent variable Y is assumed to follow a Bernoulli distribution with parameter p defined as p = Probability(Y = 1). Thus, the main use-case of a logistic model is that with given observations of the variables (X₁,X₂ …, X_N) we estimate the probability p that the outcome Y is equal to 1.

Note that the logistic regression model is sensitive to outliers and the number of explanatory variables should be less than the total observations to avoid overfitting. The logistic regression model is generally combined with artificial neural networks to make it more suitable to assess complex relationships. In practice, it is performed using programming languages like Python and R which possess powerful libraries (packages) to evaluate the models.

Applications

Logistic regression is a relatively simple and efficient method for binary classification problems. It is a classification model that achieves very good performance with linearly separable classes or categories and is extensively employed in various industries such as medicine, gaming, hospitality, retail, etc.

In finance, the logistic regression model is commonly used to model the credit risk of individuals and small and medium enterprises. For companies, this model is used to predict their bankruptcy probability. Such a method is called credit scoring. To construct a logistic regression model for credit scoring of corporate firms, the independent variables are usually financial ratios computed with the information contained in financial statements: EBIT margin, return on equity (RoE), debt to equity (D/E), liquidity ratio, EBIT/Total Assets, etc. Further predictive statistical metrics like p-value and correlation test for multicollinearity can be used to narrow down to the variables with most contribution to the model.

Useful resources

Wikipedia Maximum Likelihood Estimation

Towards Data Science Logistic Regression

About the author

The article was written in November 2021 by Jayati WALIA (ESSEC Business School, Grande Ecole Program – Master in Management, 2019-2022).

How to compute the net present value of an investment in Excel

December 7, 2021 by Maite CARNICERO MARTINEZ

How to compute the net present value of an investment in Excel

In this article, Maite CARNICERO MARTINEZ (ESSEC Business School, Global Bachelor of Business Administration, 2021-2022, exchange student from the University of Salamanca) explains how to compute the net present value of an investment in Excel.

When the time comes that one must choose what project to embark on, there are several measures to compare the available options, such as the internal rate of return, the payback method and the net present value (also known as the “discounted cash flow” method or DCF). In this article, I will focus on the last one of these tools, which is the preferred by most financial analysts.

A project is a temporary, unique and progressive endeavor to produce a tangible or intangible result, for instance, a new product or a competitive advantage. It normally entails the execution of some tasks over a period of time, conditioned to limitations related to cost, quality or performance. During its implementation, an initial investment and a series of cash flows are to be generated at different times. Some examples of projects are: developing a new service, building a factory, and implementing a new process.

The Net Present Value (NPV) compares the present value of the future cash flows with the investment made at the beginning. The computation of the present value uses a the required rate of return. It takes into account the time value of money, translating future cash flows into today’s value, since the buying power of money today is greater that the buying power of the same amount in the future.

The NPV is the basis of the discounted cash flow model (DCF) which allows investors to compare the initial cash flow of expenditure against the present value of future cash flows. It could be used to evaluate whether an important investment is worthwhile, but also in mergers and acquisitions and to compare companies, like Warren Buffet does, because once we have calculated the different NPVs we will know which investment has the biggest gain.

To sum up, the NPV allows us to do evaluate investments from a financial point of view and select the best one.

Modelling of an investment

How can we calculate it?

The mathematical formula for the NPV is given by:

CF_t = cash flows of each period (from t=0 to t=T)
T = number of periods
r = discount rate or interested rate required of the investment. It is the rate of return that the investors expect on their investment

For a classical project, the first cash flow, CF₀, is negative and corresponds to the initial cost of the project and the following cash flows, CF_t for t=1 to t=T, are assumed to be positive. The NPV can be rewritten as

This formula clearly shows that the NPV compares the first cash flow on the one hand, and the present value of future cash flows on the other hand. As the initial cash flow is negative and the present value of future cash flows is positive, the sign of the NPV depends on relative weight of these two components.

Investment decision

The NPV can be used as a criterion for the investment decision.

If the NPV is positive, the investment should be made as it creates value.
If the NPV is zero, the investment should be made or not.
If the NPV is negative, the investment should not be made as it destroys value.

Advantages

The NPV of an investment is easy to calculate, specially nowadays with financial calculators and spreadsheets like Excel.
The NPV measures the effect of the investment on the firm’s value.
The NPV It takes into account the maturity of each cash flow.

Disadvantages

In order to compute the NPV, the discount rate has to be specified and it is a difficult issue.
The calculations are based on assumptions and estimations and the reality can differ from them.
Misestimations can be found in the initial investment, on the discount rate and on the projected returns of the project.
The NPV formula presumes that the cash flows are immediately reinvested at the same rate as the discount rate.
It presumes that the negative cash flows are financed with resources whose cost is also the discount rate.

How to compute the NPV on Excel?

Example

Excel is an extended tool in the financial world, also to calculate the NPV. Let’s take an example to illustrate how we can use it: we are offered a project in which we have to invest 42,000 euros and we will receive 8,400 euros the first year, 9,000 the second, 10,300 the third, 11,700 the fourth and 13,000 the last year.

Assuming that the discount rate is 6% per year, what will be the NPV?

Hand-made computation

We can do a hand-made computation of the NPV:

We find a NPV of €1,564.43. As the NPV of the investment is positive, we will take the project.

Computation with Excel

We can also use Excel to compute the NPV:

Useful resources

longin.fr website Cours Gestion financière (in French).

Mazars Excel IRR Function And Other Ways To Calculate IRR In Excel

Economipedia NPV definition (in Spanish)

HBR NPV use and calculation

HBR NPV limitations

MyManagementGuide Project definition

About the author

The article was written in December 2021 by Maite CARNICERO MARTINEZ (ESSEC Business School, Global Bachelor of Business Administration, 2021-2022, exchange student from the University of Salamanca).

Systematic risk and specific risk

November 30, 2021 by Youssef LOURAOUI

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

This article is structured as follows: we introduce the concept of systematic and specific risk. We then explain the mathematical foundation of this concept. We finish with an insight that sheds light on the relationship between diversification and risk reduction.

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. In the article systematic risk, we develop the economic sources of systematic risk: interest rate risk, inflation risk, exchange rate risk, geopolitical risk, and natural risk.

Additionally, due to the asset’s unique characteristics, an individual asset exhibits variance that is unrelated to the market portfolio (the asset’s non-market variance). Specific risk is the term for non-market variance, and it is often seen as minor because it can be eliminated in a large diversified portfolio. In the article specific risk, we develop the economic sources of specific risk: business risk and financial risk.

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 expected return of asset i
β_i the measure of the risk of asset i
R_M the return of the market
E(R_M) the expected return of the market
R_M – E(R_M) the market factor
ε_i 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

Total risk can be deconstructed into two main blocks:

Total risk formula

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.

Effect of diversification on portfolio risk

Diversification’s objective is to reduce the portfolio’s standard deviation. This assumes an imperfect correlation between securities. Ideally, as investors add securities, the portfolio’s average covariance decreases. How many securities must be included to create a portfolio that is completely diversified? To determine the answer, investors must observe what happens as the portfolio’s sample size increases by adding securities with some positive correlation. Figure 1 illustrates the effect of diversification on portfolio risk, more precisely on total risk and its two components (systematic risk and specific risk).

Figure 1. Effect of diversification on portfolio risk

Source: Computations from the author.

The critical point is that by adding stocks that are not perfectly correlated with those already held, investors can reduce the portfolio’s overall standard deviation, which will eventually equal that of the market portfolio. At that point, investors eliminated all specific risk but retained market or systematic risk. There is no way to completely eliminate the volatility and uncertainty associated with macroeconomic factors that affect all risky assets. Additionally, investors can reduce systematic risk by diversifying globally rather than just within the United States, as some systematic risk factors in the United States market (for example, US monetary policy) are not perfectly correlated with systematic risk variables in other countries such as Germany and Japan. As a result, global diversification eventually reduces risk to a global systematic risk level.

You can download below two Excel files which illustrate the effect of diversification on portfolio risk.

The first Excel file deals with the case of independent assets with the same profile (risk and expected return).

Figure 2 depicts the risk reduction of total risk in as we increase the number of assets in the portfolio. We manage to reduce half of the overall portfolio volatility by adding five assets to the portfolio. However, the decrease becomes more and more marginal as we add more assets.

Figure 2. Risk reduction of the portfolio. img_SimTrade_systematic_specific_risk_1 Source: Computations from the author.

Figure 3 depicts the overall risk reduction of a portfolio. The benefit of diversification are more evident when we add the first 5 assets in the portfolio. As depicted in Figure 2, the diversification starts to fade at a certain point as we keep adding more assets in the portfolio. It can be seen in this figure how the specific risk is considerably reduced as we add more assets because of the effect of diversification. Systematic risk (market risk) is more constant and doesn’t change drastically as we diversify the portfolio. Overall, we can clearly see that diversification helps decrease the total risk of a portfolio considerably.

Figure 3. Risk decomposition of the portfolio. img_SimTrade_systematic_specific_risk_2 Source: Computations from the author.

The second Excel file deals with the case of dependent assets with the different characteristics (expected return, volatility, and market beta).

Academic research

A series of studies examined the average standard deviation for a variety of portfolios of randomly chosen stocks with varying sample sizes. Evans and Archer (1968) and Tole (1982) calculated the standard deviation for portfolios up to a maximum of twenty stocks. The results indicated that the majority of the benefits of diversification were obtained relatively quickly, with approximately 90% of the maximum benefit of diversification being obtained from portfolios of 12 to 18 stocks. Figure 1 illustrates this effect graphically.

This finding has been modified in two subsequent studies. Statman (1987) examined the trade-off between diversification benefits and the additional transaction costs associated with portfolio expansion. He concluded that a portfolio that is sufficiently diversified should contain at least 30–40 stocks. Campbell, Lettau, Malkiel, and Xu (2001) demonstrated that as the idiosyncratic component of an individual stock’s total risk (specific risk) has increased in recent years, it now requires a portfolio to contain more stocks to achieve the same level of diversification. For example, they demonstrated that the level of diversification possible in the 1960s with only 20 stocks would require approximately 50 stocks by the late 1990s (Reilly and Brown, 2012).

Figure 4. Effect of diversification on portfolio risk Source: Computation from the author.

You can download below the Excel file which illustrates the effect of diversification on portfolio risk with real assets (Apple, Microsoft, Amazon, etc.). The effect of diversification on the total risk of the portfolio is already significant with the addition of few stocks.

We can appreciate the decomposition of total risk in the below figure with real asset. We can appreciate how asset with low beta had the lowest systematic out of the sample analyzed (i.e. Pfizer). For the whole sample, specific risk is a major concern which makes the major component of risk of each stock. This can be mitigated by holding a well-diversified portfolio that can mitigate this component of risk. Figure 5 depicts the decomposition of total risk for assets (Apple, Microsoft, Amazon, Goldman Sachs and Pfizer).

Figure 5. Decomposition of total risk Source: Computation from the author.

You can download below the Excel file which deconstructs the risk of assets (Apple, Microsoft, Amazon, Goldman Sachs, and Pfizer).

Why should I be interested in this post?

If you’re an investor, understanding the source of risk is essential in order to build balanced portfolios that can withstand market corrections and downturns.

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

Useful resources

Academic research

Campbell, J.Y., Lettau, M., Malkiel, B.G. and Xu, Y. 2001. Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk. The Journal of Finance, 56: 1-43.

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.

Reilly, R.K., Brown C.K. 2012. Investment Analysis & Portfolio Management, Tenth Edition. 239-245.

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.

Statman, M. 1987. How Many Stocks Make a Diversified Portfolio?. The Journal of Financial and Quantitative Analysis, 22(3), 353–363.

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 November 2021 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).

Portfolio

November 26, 2021 by Youssef LOURAOUI

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

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

Introduction

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

Modelling of portfolios

Portfolio weights

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

Portfolio weights

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

Sum of the portfolio weights

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

Asset weight for a long position

Asset weight for a neutral position

Asset weight for a short position

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

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

Equally-weigthed portfolio

Portfolio return: the case of two assets

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

Return of a 2-asset portfolio

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

Expected return of a 2-asset portfolio

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

Standard deviation of a 2-asset portfolio return

where:

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

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

where:

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

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

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

Portfolio return: the case of N assets

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

Portfolio return

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

The expected return of the portfolio is given by

Expected portfolio return

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

w = weight vector
r = returns vector

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

Standard deviation of portfolio return

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

Standard deviation of portfolio return

where:

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

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

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

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

Basic principles on portfolio construction

Diversify

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

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

Portfolio Asset Allocation

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

Portfolio Rebalancing

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

Reduce investment costs as much as possible

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

Invest on a regular basis

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

Buying in the future

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

Types of portfolio

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

Aggressive Portfolio

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

Defensive Portfolio

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

Income Portfolio

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

Speculative Portfolio

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

Hybrid Portfolio

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

Socially Responsible Portfolio

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

Why should I be interested in this post?

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

Useful resources

Academic research

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

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

Business analysis

Edelweiss, 2021.What is a portfolio?

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

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

About the author

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

Black-Litterman Model

November 21, 2021 by Youssef LOURAOUI

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

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

Introduction

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

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

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

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

Mathematical foundation of Black-Litterman model

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

Where:

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

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

Black-Litterman asset allocation in practice

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

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

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

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

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

Why should I be interested in this post?

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

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

Useful resources

Academic research

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

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

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

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

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

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

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

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

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

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

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

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

About the author

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

Passive Investing

November 21, 2021 by Youssef LOURAOUI

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

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

Review of academic literature on passive investing

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

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

Core principles of passive investing

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

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

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

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

Passive investing and Efficient Market Hypothesis

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

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

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

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

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

Estimation of the S&P500 return

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

Why should I be interested in this post?

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

Useful resources

Academic research

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

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

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

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

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

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

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

Business analysis

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

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

About the author

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

Beta

November 20, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) explains the concept of beta, one of the most fundamental concepts in the financial industry, which is heavily used in asset management to assess the risk of assets and portfolios.

This article is structured as follows: we introduce the concept of beta in asset management. Next, we present the mathematical foundations of the concept. We finish with an interpretation of beta values for risk analysis.

Introduction

The (market) beta represents the sensitivity of an individual asset or a portfolio to the fluctuations of the market. This risk measure helps investors to predict the movements of their assets according to the movements of the market overall. It measures the asset risk in comparison with the systematic risk inherent to the market.

In practice, the beta for a portfolio (fund) in respect to the market M represented by a predefined index (the S&P 500 index for example) indicates the fund’s sensitivity to the index. Essentially, the fund’s beta to the index attempts to capture the amount of money made (or lost) when the index increases (or decreases) by a specified amount.

Graphically, the beta represents the slope of the straight line through a regression of data points between the asset return in comparison to the market return for different time periods. It is a traditional risk measure used in the asset management industry. To give a more insightful explanation, a regression analysis has been performed using data for the Apple stock (APPL) and the S&P500 index to see how the stock behaves in relation to the market fluctuations (monthly data for the period July 2018 – June 2020). Figure 1 depicts the regression between Apple stock and the S&P500 index (excess) returns. The estimated beta is between zero and one (beta = 0.3508), which indicates that the stock price fluctuates less than the market index.

Figure 1. Linear regression of the Apple stock return on the S&P500 index return.

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

Mathematical derivation of Beta

Use of beta

William Sharpe, John Lintner, and Jan Mossin separately developed key capital markets theory as a result of Markowitz’s previous works: the Capital Asset Pricing Model (CAPM). The CAPM was a huge evolutionary step forward in capital market equilibrium theory since it enabled investors to appropriately value assets in terms of systematic risk, defined as the market risk which cannot be neutralized by the effect of diversification.

The CAPM expresses the expected return of an asset a function of the risk-free rate, the beta of the asset, and the expected return of the market. The main result of the CAPM is a simple mathematical formula that links the expected return of an asset to these different components. For an asset i, it is given by:

CAPM risk beta relation

Where:

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

The risk premium for asset i is equal to β_i(E(r_m)- r_f), that is the beta of asset i, β_i, multiplied by the risk premium for the market, E(r_m)- r_f.

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

CAPM beta formula

Where:

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

Excel file to compute the beta

You can download below an Excel file with data for Apple stock returns and the S&P500 index returns (used as a representation of the market). This Excel file computes the beta of apple with the S&P500 index.

Interpretation of the beta

Beta helps investors to explain how the asset moves compared to the market. More specifically, we can consider the following cases for beta values:

β = 1 indicates a fluctuation between the asset and its benchmark, thus the asset tends to move at a similar rate than the market fluctuations. A passive ETF replicating an index will present a beta close to 1 with its associated index.
0 < β < 1 indicates that the asset moves at a slower rate than market fluctuations. Defensive stocks, stocks that deliver consistent returns without regarding the market state like P&G or Coca Cola in the US, tend to have a beta with the market lower than 1.
β > 1 indicates a more aggressive effect of amplification between the asset price movements with the market movements. Call options tend to have higher betas than their underlying asset.
β = 0 indicates that the asset or portfolio is uncorrelated to the market. Govies, or sovereign debt bonds, tend to have a beta-neutral exposure to the market.
β < 0 indicates an inverse effect of market fluctuation impact in the asset volatility. In this sense, the asset would behave inversely in terms of volatility compared to the market movements. Put options and Gold typically tend to have negative betas.

Why should I be interested in this post?

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

Useful resources

Academic research

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

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

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

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

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

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

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

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

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

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

Business analysis

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

Man Institute, 2021. How to calculate the Beta of a portfolio to a factor

Nasdaq, 2021. Beta

About the author

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

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Stress Testing used by Financial Institutions

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Types of Stress testing

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Hypothetical stress testing

EBA Regulations

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Protective Put

Introduction

Buying a protective put

Market scenario

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

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

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

Risk profile

Example

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

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

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

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Straddle and Strangle

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Straddle

Market Scenario

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Strangle

Market Scenario

Example

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Books

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Analysis of synergy-based theories for M&A

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Introduction

M&A’s main theories

Management synergy

Operating synergy

Diversification and strategic synergy

Financial synergy

Undervaluation theory

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Option Spreads

Introduction

Bull Spread

Market Scenario

Example

Bear Spread

Market Scenario

Example

Butterfly Spread

Market Scenario

Example

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Introduction

Calculating VaR using the historical method

Advantages and limitations of the historical method

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Introduction

Assumptions

VaR for single asset

VaR for a portfolio of assets

Advantages and limitations of the variance-covariance method