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).

My experience at the Bank of Lebanon

December 14, 2021 by Jade RAKHA

My experience at the Bank of Lebanon

In this article, Jade RAKHA (ESSEC Business School, Bachelor of Business Administration, 2021) shares with us his experience at Banque Du Liban.

Banque Du Liban

Before the Central Bank of Lebanon was established in 1963, the Bank of Syria and Lebanon had the role of the central bank. For a period of 25 years, this commercial bank had the privilege of issuing the Lebanese currency.

All of the Central Bank’s expenses are paid by the government. However, 20% of its net profit is allocated to its general reserve, while the rest (80%) is for the state treasury. It is exempt from all duties and taxes and is not subject to the bankruptcy rules specified in the Trade Law applicable to banks. Nevertheless, the Central Bank is subject to the Trade Law in its relationship with third parties, it should conduct its operations and organize its accounts under commercial and banking rules.

Banque du Liban

The head office of the bank is divided into 19 directorates and 10 units. The Central Bank of Lebanon has 9 branches in different regions of Lebanon. The bodies established by the Central Bank are Supervisory Board, Special Investigation Commission, Higher Banking Authority, Capital Markets Authority, and MidClear.

The Central Bank has the power to issue regulations and to impose administrative penalties on all institutions under its control. It issues the appropriate regulatory texts to achieve its objectives. It has a special legal personality and administrative independence from the state. Its employees shall be appointed in accordance with the special law system specific to the institution and shall be subject to the Labor Law.

Moreover, the Central Bank of Lebanon enjoys financial independence, but with certain limits: all of its decisions should be approved by the Government’s commissioner, and the Governor shall submit balance sheets and operations reports yearly, to the Minister of Finance.

The Governor and his four deputies shall be appointed by a decree issued by the Council of Ministers upon the proposal of the Minister of Finance, every 6 years. They are obliged to work full-time throughout their tenure. The Governor has the widest powers to administer and operate the Central Bank of Lebanon. He is the legitimate representative of the Bank and signs all contracts and agreements. Concerning the Deputies of the Governor, they have no specific powers, but the Governor may appoint them tasks. In the event of the vacancy of the post of Governor, the First Governor shall assume office pending the appointment of an attractive governor. Such functions shall be assumed by the Second Deputy Governor in the absence of the First Deputy Governor.

The Central Council consists of seven members: The Governor as President, the four Deputy Governors, the Director General of the Ministry of Finance, and the Director General of the Ministry of Economy and Trade. It should be noted that the Director General of the Ministry of Finance and the Director General of the Ministry of Economy and Trade are not present as representatives of the Government but are present as themselves, which confirms the independence of the Central Bank of Lebanon in its decisions.

The Central Council has wide powers, since it determines monetary and credit policies, fixes the discount rate and interest rate of the bank’s loans, establishes regulations relating to bank operations, approves the balance sheet of the Central Bank’s expenses, and approves the establishment of banks and all private institutions to supervise the Central Bank of Lebanon.

As mentioned before, the Central Bank has the privilege of issuing the national currency. This privilege has been delegated to the Central Bank, which carries out this task exclusively under strict conditions and rules. The Money and Credit Law defines certain characteristics of the Lebanese currency but it has given the Central Bank the freedom to decide the size, texts, fees and other features of banknotes and coins.

To maintain the integrity of the banking system, the Central Bank of Lebanon takes all measures it deems appropriate to maintain the harmony between bank liquidity and the volume of credit in banks. Thus, it gives recommendations and uses the means that will ensure the conduct of sound banking work. This institution also acts as an adviser to the government, proposes measures it considers necessary for the economic sector and warns the government of operations it considers harmful.

My internship at Banque Du Liban

During this internship, I did not have any missions to do because it was not a practical internship, but more of a theoretical one: each day we had one speaker that came and presented the department in which he or she worked in so that we know the main tasks and roles the central bank has. I will not present all the departments of Banque Du Liban because that would take a lot of time even if it is very interesting. I’ll present some departments that are related to the financial sector and to the material that we tackle in the SimTrade course.

Directorate of Financial Operations

The general mission of the Central Bank of Lebanon, as stipulated in Article 70 of the Monetary and Credit Law, is to preserve the cash to secure the basis for lasting economic and social growth, including:

Maintaining the integrity of the Lebanese currency
Maintaining economic stability
Maintaining the integrity of the banking system
Developing the monetary and financial market
Developing and organizing payment systems and tools
Developing and organizing money transfer operations including electronic transfers
Developing and organizing clearing and settlement of financial instruments, payment and trading bonds.

Thus, the Central Bank of Lebanon is responsible for monetary management in light of economic and political developments, securing liquidity for the banking and financial system, and it is the last resort to lending banks.

Most of these goals are under the responsibility of the Directorate of Financial Operations at the Central Bank of Lebanon, in charge of leading Macroeconomic policies. It is at the helm of the Economic Legislation, as well as the Fiscal and Monetary policies. Being in charge of Fiscal policy means that this department should prepare the general budget, collect taxes, give tax incentive… As for the Monetary policy, it should maintain the integrity of cash as well as low inflation rates.

The Directorate of Financial Operations determines interest at the level of the financial market. It can define either the size of cash or interest on cash. This department can also influence prices in the financial market by affecting the level of the money supply. If it wants to reduce the interest rate in the market, it increases the amount of cash (increase the money supply). Vice versa, if it aims to raise the price of money in the market, it reduces the amount of cash (reduction of the money supply).

The Special Investigation Commission (SIC)

The Special Investigation Commission is in charge of Financial Regulation and Supervision, more specifically fighting Money Laundering and Terrorism Financing. It is composed of: The Governor of the Central Bank of Lebanon as President, a member of the Council of Ministers, the Chairman of the Banking Supervision Committee or any of its members, the Appointed Judge of the Higher Banking Commission or an alternative judge.

The Financial Intelligence Unit (FIU) is a national central unit responsible for receiving, requesting, analyzing and providing the relevant authorities with financial and other information concerning possible cases of money laundering and terrorism financing.

According to the SIC, Money Laundering is defined as “any act or attempt to conceal a source of funds or assets resulting from a criminal act”. The goal of money laundering is turning junk money into clean money. The different stages are placement, layering, and integration. This process has a lot of consequences: it increases the rate of crimes, it leads to the instability of the economy, it weakens the integrity of financial institutions as well as the social and economic structure.

The definition of Terrorism Financing according to the SIC is “financial support, in any form, for terrorism and those who encourage, plan or participate in it”. It aims to raise and secure funds for terrorism. The stages of financing terrorism are the collection, transmission, and use.

The mission of the Special Investigation Commission is to receive notifications and requests for assistance, conduct investigations into suspected money laundering or terrorist financing offences, determine the seriousness of evidence, and take the appropriate decision.

The Foreign Exchange & International Operations Department

The Foreign Exchange & International Operations Department is divided into 5 different divisions:

The Transfer Division
The Documentary Credit Division
The Back Office
The Clearing Room
The Communication Division.

There are different types of traders, like Forex Traders, Money Market Trader, Capital Market Trader, Commodities Traders, Derivative Traders…

The Forex market is the simultaneous buying of one currency and selling another. During our stay at the Central Bank of Lebanon, we got the chance to stimulate this process with Mr. Patrick el Hajj. This market has more buyers and sellers and daily volume than any other market in the world and takes place in major financial institution across the globe (daily turnover is about $5 Trillion). All information on Foreign Exchange can be found on a system called Bloomberg.

Financial Assets Authority

The Financial Assets Authority consists of two main sections: The Recruitment Section and the Contributions Section. The Recruitment Section should prepare daily reports on the Bank’s foreign currency reserves, sources and uses and monitors interest rates and amounts paid, and follow-up financial markets and global interest and study their effects on the bank’s investments. The Contributions Section is responsible for studying the financial situation of companies in which the Bank of Lebanon contributes or in which it represents the Lebanese state through general assemblies and monitoring reports, and preparing any reports or studies related to contributions requested by the Governor or the Central Council that may reach the Investor Information Statement (Prospectus).

The majority of the subjects mentioned in the session at the Central Bank of Lebanon were new to me, so I can clearly say that I gained a lot of knowledge from this experience and got access to information that can only be obtained from experts in the fields. I got the chance to get into the details of each department in the Central Bank, gaining knowledge of how it is divided, and how it works.

Personally, I gained a lot from this training. However, I admit I was expecting more work on the field than just theoretical sessions, and thus gaining more experience rather than knowledge. However, we had the spread of COVID-19 and the current economic situation in the country did not give us the chance to be able to visit the Central Bank and see how each department operates on a daily basis on the field.

Most speakers gave us great sessions and were able to convey their message properly. Nevertheless, if the schedule was a bit more appropriate, I believe we would have had a better experience.

Useful Resources

Banque du Liban

About the author

The article was written in December 2021 by Jade RAKHA (ESSEC Business School, Bachelor of Business Administration, 2021).

My experience at the startup BSD Investing

December 14, 2021 by Rohit SALUNKE

My experience at the startup BSD Investing

In this article, Rohit SALUNKE (ESSEC Business School, Grande Ecole Program – Master in Management, 2018-2021) shares his experience working in a startup and the evolution of his role and responsibilities…

About BSD Investing

BSD Investing is an independent research firm operating in the asset management industry. It primarily provides research and analysis on active vs passive fund performances for equity and debt funds present across the global financial markets.

The funds are domiciled in Europe (i.e., these are European funds investing in domestic and international markets) and span across 62 universes (i.e., global markets and investment styles).

Logo BDS Investing

The goal of the research is to provide BSD Investing clients with insights into the active vs passive performances and help them optimize the portfolio to get better risk adjusted returns.

The evolution of my missions

I started working at this startup in July 2019 in a small office in Saint-Lazare area in Paris, France. At the beginning, it was just me and two others, the founder, and her colleague. We started from scratch, trying to figure out the best data source to use, figuring out the process flow, the product and much more. After selecting Morningstar as our primary data provider, I began writing codes in Python to fetch the data, create our own portfolios and develop performance key performance indicators (KPIs) for those portfolios. Since we did not have any employee skilled in IT, I was the one who took charge of creating the entire IT architecture from data handling to reporting.

After working for about eight months, we hired our head of IT, and he took over the handling of the IT system and made it much more efficient. That’s when I got the chance to devote my entire focus in developing quantitative models and simulations for the performances of our portfolios.

I started off with researching various technical indicators that gave insights into the market performances and how active funds fared in comparison to passive funds. A major portion of my time was devoted to simulating portfolio performances using various indicator signals. The signals were basically an indication to increase or decrease the active allocation to the portfolio.

Apart from this, I helped a lot with creating marketing materials, conducting market research, interviewing portfolio managers to understand the asset management industry and their needs.

In addition, I work very closely with the IT head to implement our models and key indicators onto the website.

Active and Passive funds

The asset management industry is broadly divided into two management styles: the active style and the passive style. Both styles follow a benchmark that could be an index (e.g., CAC 40), or a combination of indices, or a new portfolio that represent the entire universe of financial instruments in the category that the manager wants to invest in (e.g., Emerging countries large cap ESG funds).

Passive style

The goal of the passive fund manager is to create a portfolio that tracks closely the chosen benchmark. But, since a benchmark consists of a large number of stocks, investing in all of them is not very feasible or cost effective. Therefore, the passive manager creates a portfolio using a smaller number of instruments that aims to replicate the returns from the benchmark.

A good metric to measure the performance of a passive fund is tracking error. It is the divergence between the fund returns and those of the benchmark. A low tracking error means that the fund is tracking the benchmark closely and thus is performing well. Since, a passive fund (also known as ETF or index fund) manager does not aim to outperform the benchmark, but just to simply replicate its returns. Therefore, passive funds charge low fees.

Active style

An active manager on the other hand aims to beat the funds benchmark through stock picking, sector rotation and/or other methods. He or she thus takes a larger risk than a passive fund manager and needs a lot more research, expertise, and management. Therefore, an active fund generally charges more fees than a passive fund. For example, among the France large cap funds, an average passive fund charges around 0.25% of fees, whereas an active manager’s fee may range from 1-2%, in some cases more than 4%.

Active managers are alpha seeking. Alpha is the excess return that an active manager generates compared to its benchmark. There are multiple ways to calculate alpha. One such way is using the CAPM model. We predict the expected return of the portfolio using the CAPM model. Subtracting this return from the active managers portfolio gives the alpha. A passive funds alpha is supposed to be zero.

Fund of funds managers create a portfolio of active and/or passive funds to meet their return and risk objectives.

Best style?

In the asset management industry, there is an ongoing debate about which management style is better and are the extra fees charged by the active managers really worth it?

In the US, for example, the active funds have performed very poorly as compared to the ETFs. Whereas, in Europe the performance was mixed and in Japan, the active funds performed better. However, these are the results over the entire period of 10 years. There have been many periods when the active funds outperformed.

Taking the recent example, after the Covid-19 pandemic, the markets went haywire. Since then, in most of the universes active funds have outperformed the passive funds. Therefore, higher returns can be achieved by understanding the markets and allocating the portfolio to the right management style at the right time.

My key learnings

Working in a startup is always challenging and the job comes with heavy responsibilities.

And although working in a startup sounds very interesting, most of the work during the very beginning is quite tedious when it comes to data handling. I spent a few months just understanding the data, checking for errors from the source, figuring out ways to deal with data errors and so on.

Once, I started working on the quantitative models and the simulations, I felt that my work has just begun. During this time, I learnt a big lesson regarding building quantitative models. I build very sophisticated models including machine learning models such as neural networks, gradient boosted trees and so on. However, despite the good results, I had to use simpler logistic models because selling overly sophisticated models would become very difficult.

People in the asset management industry need to know what the real meaning of the data is. And giving recommendations using a black box model does not make it very easy to understand the functioning of the model.

Working on the various indicators, trying to understand their correlations with active and passive fund performances gave me good insights about them. For instance, one good variable that works the best for me is dispersion. This is the standard deviation of returns among funds or stocks. During periods of high dispersion, I observed that active funds generally outperformed the passive funds. I saw a similar result during periods of high volatility. An explanation to this is that a high dispersion could signify a period of high inefficiency in the market, which the active managers could take advantage of. When markets are highly efficient, it makes sense to invest in ETFs, and reduce your costs. Whereas, during periods of high inefficiency, a good active manager could be worth the higher fees that he/she charges. As described above, since March 2019, the active managers have generally outperformed the passive funds across many universes. And this period is also marked with high dispersion and volatility.

In addition, we found that bear periods were more conducive to active outperformance, while bull periods were not. This can be understood since the volatility and dispersion is generally high during bear periods. However, periods after March 2019 were an anomaly to this, since although the markets are in a bull run, there is high dispersion and volatility, and the active funds are outperforming the ETFs.

Knowledge and skills required

For this job I had to have strong data skills, coding skills as well as sound knowledge about finance.
In addition, since I had little to no guidance in my role, I had to come up with my own tasks, define the product and its objects and then learn the essential skills to build it.

Therefore, there was a lot of market research, visits to stackoverflow, reading research papers, cold mailing portfolio managers and so on. Thus, project management and communication skills are essential.

Hard Skills

Python
SQL
HTML
MorningstarDirect
Capital Markets
Portfolio Management, Optimization …
Risk Management
Market Research

Soft Skills

Communication
Project Management
Leadership
Entrepreneurial Thinking
Ability to handle pressure
Dedication to your project and display of ownership

Useful resources

BSD Investing

Morningstar

About the author

The article was written in December 2021 by Rohit SALUNKE (ESSEC Business School, Grande Ecole Program – Master in Management, 2018-2021).

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

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 covered call.

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).

Climate change’s impact on the financial sector

December 7, 2021 by Maite CARNICERO MARTINEZ

Climate change’s impact on the financial sector

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 climate change has affected the financial sector.

For the last two decades, the environmental awareness has progressively increased. People are worried about the consequences of the greenhouse effect and the gas emissions and environmental policies play an important role in governments’ agendas. At first, the focus was mainly on the economic activity’s repercussion on the environment and developing less polluting practices. But many actors have acknowledged the major repercussions of climate change on countries, firms, the financial sector and the global economy. Therefore, the management of the risks associated to climate change is crucial.

To illustrate this point, take for instance PG&E, California’s principal power supplier. After the California wildfires in 2018, the firm had to declare bankruptcy due to the damage that its infrastructure had suffered and the amount of obligations that it had to pay because of being responsible of the ignition. Nevertheless, the drought and the extremely high temperatures were crucial for the start, expansion and severity of the fires. Also, the insurance sector is one of the most exposed to the physical risks of the climate change.

There are some regions and sectors specially exposed to these risks, but in a globalized world, they will affect, more or less, all the countries and sectors and will challenge the financial sector. Many agreements have been signed to approach the situation, such as the Kyoto Protocol, the GRI Sustainability Reporting Standards in 1997 (the Global Reporting Initiative, GRI, is a ONG created by Coalition for Environmentally Responsible Economies, CERES; and the United Nations Environment Programme, UNEP), the 2030 Agenda in 2015 or the Green Pact in 2019.

Financial markets’ adaptation

Financial markets have to be adapted, like the 2008’s crisis proved, with measures to ensure the transparency of the environmental risks and the criteria applied. That is the consequence of the “greenwashing” (a marketing method that consists in the use of the ecological argument to communicate with the public to portray the business as an eco-responsible one, when that is in fact not true. Such practice is deceptive and can be likened to false advertising) a phenomenon that appeared due to the increasing demand of “green” financial products, which are financial initiatives, processes, products or services either designed to protect the environment or to manage how the environment impacts finance and the investment. The problem is the lack of clarity when we have to determine if a financial product is sustainable. Sustainable financial products are the ones that take into account social and environmental factors when investing. That can be evaluated using the ESG (Environmental, Social and Governance) criteria.

Despite the risks inherent to climate change, it is important to note that it also comes along with many opportunities related to finding solutions and alternatives to the problem, like the renewables energies, whose value has increased that much, that some are even talking about the formation of a “green bubble”. Also, there is an increasing demand of green bonds (fixed-income assets destined to the financing of projects that meet environmental criteria, sustainable activities or projects to minimize the climate change), green loans and deposits, and energetic efficiency plans.

Some important sustainable financial products:

Assets:

Sustainable loans and sustainability linked loans
Sustainable credits: green credits, social credits
Green mortgages
Agribusiness

Investment funds:

Green
Social
Mixed

Liabilities:

Green and social bond principles
Transition bonds
ODS-linked bonds
Sustainability linked bonds

Risk associated to climate change on the financial sector

But, how to measure the risk of climate change? First of all, we have to rank the exposures depending on criteria like the geographical area, business model, gas emissions or vulnerability towards physical changes. A good starting point is the TCFD (Task Force on Climate-Related Financial Disclosures) recommendations on what are the most exposed sectors. Then, using different scenarios, we have to measure the risk. Nevertheless, there is little evidence that the market has put a price on each type of climatic risk, and even when they are included on the risk measurement of the asset, they are not completely incorporated or they are subject to a very high uncertainty, resulting in a very high volatility.

To sum up, climate change and its raising awareness have completely altered the financial markets outlook. A new concern about the risk of the climate change has been brought up, but also new areas of investment that look very promising, even with the possibility of the development of a new bubble. Therefore, I think that those risks need to be taken into account and its importance is only starting to grow. In the future, we will see how climate change plays a great role in the financial markets.

Useful resources

Mazars Managing climate change risks

Task Force on Climate-related Financial Disclosures (TFCD) Climate change is a financial risk to the global economy

The Guardian The evolution of greenwashing

Reuters PG&E charged with manslaughter for sparking California wildfire

NPR PG&E’s equipment started the fire

Investopedia ESG criteria

Novethic What is greenwashing?

Satander Sustainable finance

“The Green Qualifications Workbook” by the Chartered Banker Institute.

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).

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

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).

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