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

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).

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).

Alpha

November 20, 2021 by Youssef LOURAOUI

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

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

Introduction

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

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

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

Figure 1. Alpha and the Security Market Line

Source: Computation by the author.

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

Academic Literature

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

Mathematical derivation of Jensen’s alpha

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

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

Why should I be interested in this post?

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

Useful resources

Academic research

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

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

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

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

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

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

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

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

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

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

Business analysis

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

About the author

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

Security Market Line (SML)

November 20, 2021 by Youssef LOURAOUI

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

This article is structured as follows: we first introduce the concept of Security Market Line (SML). We then present the mathematical foundations of the SML. We finish by presenting an investment strategy that can be implemented relying on the SML.

Security Market Line

The SML reflects the risk-return combinations accessible in the capital market at any given time for all risky assets. Investors would choose investments based on their risk appetites; some would only consider low-risk investments, while others would welcome high-risk investments. The SML is derived from the Capital Asset Pricing Model (CAPM), which describes the trade-off between risk and expected return for efficient portfolios.

The expected relationship between risk and return is depicted in Figure 1. It demonstrates that as perceived risk increases, investors’ required rates of return increase.

Figure 1. Security Market Line.

Source: Computation by the author.

Under the CAPM framework, all investors will choose a position on the capital market line by borrowing or lending at the risk-free rate, since this maximizes the return for a given level of risk. Whereas the CML indicates the rates of return of a specific portfolio, the SML represents the risk and return of the market at a given point in time and indicates the expected returns of individual assets. Also, while the measure of risk in the CML is the standard deviation of returns (total risk), the measure of risk in the SML is the systematic risk, or beta. Figure 2 depicts the SML line combined with four different assets. Asset A and B are above the SML line, which implies that they are overvalued. Asset C and D are below the SML which implies that they are undervalued. From Figure 2, we can implement an investment strategy by going long if the asset or portfolio lies under the SML and going short if the asset or portfolio is greater than the SML.

Figure 2. Security Market Line with a plot of different assets.

Source: Computation by the author.

Mathematical foundation

The SML plots an individual security’s expected rate of return against systematic, undiversifiable risk. The risk associated with an individual risky security is determined by the volatility of the security’s return, not by the market portfolio’s return. Individual risky securities bear a proportional share of the systematic risk. The only risk that an investor should be compensated for is systematic risk, which cannot be neutralized through diversification. This risk is quantified using the beta, which refers to a security’s sensitivity to market fluctuations. The slope of the SML is equal to the market risk premium and reflects the risk-reward trade-off at a particular point in time. We can define the line of the SML as:

Mathematically, we can deconstruct the SML as:

SML_formula

Where

E(R_i) represents the expected return of asset i
R_f is the risk-free interest rate
β_i measures the systematic risk of asset i
E(R_M) represents the expected return of the market
E[R_M – R_f] represents the market risk premium.

Beta and the market factor

William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966) independently developed the Capital Asset Pricing Model (CAPM). The CAPM was a significant evolutionary step forward in capital market equilibrium theory because it allowed investors to value assets correctly in terms of risk. The CAPM makes a distinction between two forms of risk: systematic and specific risk. Systematic risk refers to the risk posed by the market’s basic structure, its participants, and all non-diversifiable elements such as monetary policy, political events, and natural disasters. By contrast, specific risk refers to the risk inherent in a particular asset and so is diversifiable. As a result, the CAPM solely captures systematic risk via the beta measure, with the market’s beta equal to one, lower-risk assets having a beta less than one, and higher-risk assets having a beta larger than one.

In the late 1970s, the portfolio management industry sought to replicate the market portfolio return, but as financial research advanced and significant contributions were made, it enabled the development of additional factor characteristics to capture additional performance. This resulted in the development of what is now known as factor investing.

Estimation of the Security Market Line

You can download an Excel file with data to estimate the Security Market Line.

Why should I be interested in this post?

The security market line is frequently used by portfolio managers and investors to determine the suitability of an investment product for inclusion in a portfolio. The SML is useful for determining whether a security’s expected return is favourable in comparison to its level of risk. The SML is frequently used to compare two similar securities that offer approximately the same rate of return to determine which one has the lowest inherent market risk in relation to the expected rate of return. Additionally, the SML can be used to compare securities of comparable risk to determine which one offers the highest expected return for that level of risk.

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

Useful resources

Academic research

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

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

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

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

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

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

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

About the author

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

Capital Market Line (CML)

January 30, 2024November 20, 2021 by Youssef LOURAOUI

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

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

Capital Market Line

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

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

Risky assets

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

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

Risky assets and a risk-free asset

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

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

Source: Computation by the author.

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

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

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

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

Figure 3. Efficient frontier with different points.

Source: Computation by the author.

Mathematical representation

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

img_SimTrade_CML_equations_0

Where:

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

The expected return of the portfolio can be computed as:

img_SimTrade_CML_equations_1

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

img_SimTrade_CML_equations_2

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

img_SimTrade_CML_equations_3

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

Why should I be interested in this post?

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

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

Useful resources

Academic research

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

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

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

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

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

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

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

About the author

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

Active Investing

November 20, 2021 by Youssef LOURAOUI

In this article, Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) elaborates on the concept of active investing, which is a core investment strategy that relies heavily on market timing and stock picking as the two main drivers of financial performance.

This article is structured as follows: we introduce the concept of active investing in asset management. Next, we present an overview of the academic literature regarding active investing. We finish by presenting some basic principles on active investing.

Introduction

Active investing is an approach for going beyond matching a benchmark’s performance and instead aiming to outperform it. Alpha may be calculated using the CAPM framework, by comparing the fund manager’s expected return with the expected market return (Jensen, 1968). The search for alpha is done through two very different types of investment approaches: stock picking and market timing.

Stock picking

Stock picking is a method used by active managers to select assets based on a variety of variables such as their intrinsic value, the growth rate of dividends, and so on. Active managers use the fundamental analysis approach, which is based on the dissection of economic and financial data that may impact the asset price in the market.

Market timing

Market timing is a trading approach that involves entering and exiting the market at the right time. In other words, when rising outlooks are expected, investors will enter the market, and when downward outlooks are expected, investors will exit. For instance, technical analysis, which examines price and volume of transactions over time to forecast short-term future evolution, and fundamental analysis, which examines the macroeconomic and microeconomic data to forecast future asset prices, are the two techniques on which active managers base their decisions.

Review of academic literature on active investing

As fund managers tried strategies to beat the market, financial literature delved deeper into the mechanism to achieve this purpose. Jensen’s groundbreaking work in the early ’70s gave rise to the concept of alpha in the tracking of a fund’s performance to distinguish between the fund’s manager’s ability to generate abnormal returns and the part of the returns due to luck (Jensen, 1968).

Jensen develops a risk-adjusted measure of portfolio performance that quantifies the contribution of a manager’s forecasting ability to the fund’s returns. He used the measure to quantify the predictive ability of 115 mutual fund managers from 1945 to 1964—that is, their ability to produce returns above those expected given the risk level of each portfolio.

Not only does the evidence on mutual fund performance indicate that these 115 funds on average were unable to forecast security prices accurately enough to outperform a buy-and-hold strategy, but there is also very little evidence that any individual fund performed significantly better than what we would expect from mutual random chance. Additionally, it is critical to highlight that these conclusions hold even when fund returns are measured net of management expenses (that is assume their bookkeeping, research, and other expenses except brokerage commissions were obtained free). Thus, on average, the funds did not appear to be profitable enough in their trading activity to cover even their brokerage expenses.

Core principles of active investing

First principle: market efficiency varies between asset classes.

Investment information is not always readily available in all markets. For less efficient asset classes, an “active” management strategy offers a larger possibility to outperform the market, whereas a “passive” investment strategy may be more appropriate for highly efficient asset classes. In other words, there are compelling advantages for incorporating both active and passive techniques into an overall portfolio.

For example, Wall Street analysts cover a huge portion of US large size shares, making it harder to locate cheap companies. For this highly efficient asset class, a passive investment strategy may be more cost-effective in some cases. On the other side, emerging market equities are sometimes under-researched and difficult to appraise, providing an active manager with additional opportunities to identify mispriced companies. The critical point here is to notice the distinctions and then make the appropriate decisions.

Second principle: market efficiency varies across asset classes.

Within practically every asset class, active and passive management strategies can alternate as winners periodically. Even the most efficient asset classes can occasionally benefit from active management over passive. The reason is substantially distinct from the one stated in Principle One. Principle Two is related to the “Grossman-Stiglitz Paradox”: If markets are fully efficient, there is no reason to investigate them; yet markets can only be perfectly efficient for as long as they are regularly investigated. When investors run out of patience researching stocks in a highly efficient market, passive investment becomes appealing, reopening the door to opportunities for active research. This can result in an annual cycle of active/passive trends.

In some investing environments, active strategies have tended to benefit investors more, while passive strategies have tended to outperform in others. For instance, active managers may outperform more frequently than passive managers when the market is turbulent, or the economy is deteriorating. On the other way, when certain securities within the market move in lockstep or when stock valuations are more consistent, passive strategies may be preferable. Investors may gain from combining passive and active strategies in a way that exploits these insights, depending on the opportunity in various areas of the capital markets. Market conditions, on the other hand, vary constantly, and it frequently takes an intelligent eye to determine when and how much to skew toward passive rather than active investments (Morgan Stanley, 2021).

It’s worth noting that attaining consistently successful active management has historically been more challenging in some asset classes and segments of the market, such as large US company stocks. As a result, it may make sense to be more passive in certain areas and more active in asset classes and segments of the market where active investing has historically been more rewarding, such as overseas stocks in emerging markets and smaller U.S. corporations (Morgan Stanley, 2021).

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 broadening your knowledge of finance.

Useful resources

Academic research

Grossman, S., Stiglitz, J., 1980. On the impossibility of Informationally efficient markets. The American Economic Review, 70(3), 393-408.

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

Forbes, 2021. Active or Passive investing? Two principles provide the answer

JP Morgan Asset Management, 2021. Investing

Morgan Stanley, 2021. Active vs Passive management

About the author

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

The IRR, XIRR and MIRR functions in Excel

June 4, 2024November 19, 2021 by Léopoldine FOUQUES

The IRR, XIRR and MIRR functions in Excel

In this article, Léopoldine FOUQUES (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021) presents the IRR function in Excel to compute the internal rate of return of a series of cash flows.

About Excel

Excel is by far the most used financial modeling tool across the world to build models and perform analysis. Knowing which Excel function to use can help employees in the financial sector (financial analysts, fund managers, risk managers, traders, etc.) to work faster and build a more powerful model.

The internal rate of return (IRR)

Definition

The computation of the internal rate of return (IRR) is based on the net present value (NPV) of an investment. In financial modelling, an investment is represented by a series of cash flows: CF₀, CF₁, CF₂, …, CF_T. For a classic investment, the first cash flow, CF₀, is negative (outflow) and the future cash flows, CF₁, CF₂, …, CF_T are positive (inflows).

The net present value (NPV) of an investment is computed according to the following formula:

where r is the discount rate that takes into account the risk of the project.

The IRR corresponds to the value of the discount rate for which the NPV is equal to 0:

The IRR is the solution of a non-linear equation:

Use in finance

One of the most important functions is the Internal Rate of Return (IRR) function, as it’s an easy function to compare an investment’s return, based on a series of cash flows.

The function is very useful in financial modeling. Indeed, it’s frequently used to compare scenarios before deciding about a project. An example is when a company is presented with two opportunities: one is investing in a new factory and the second is expanding its existing factory.

By using IRR, we can estimate the IRR for each scenario and verify which one is higher than the average cost of capital of the business (the Weighted Average Cost of Capital or WACC) is a calculation of a firm’s cost of capital in which each category is proportionally weighted).

The Excel functions to compute the IRR

Building a math-based calculation is time-consuming and complicated, so Excel offers three functions for the calculation of the internal rate of return: IRR, MIRR, and XIRR.

The IRR function

The IRR function uses one required argument and one optional:

The values: they represent the series of cash flows, including net income value and investments.
The guessed number for the expected internal rate of return. If omitted, the function will default to 0.1 (= 10%).

You can download the Excel file below in which I illustrate the use of the IRR function in Excel based on a simple example.

Note that the IRR corresponds to a period rate. Monthly cash flows lead to a monthly IRR, quarterly cash flows lead to a quarterly IRR; and annual cash flows lead to an annual IRR. As, in practice, the standard is to work annual rates, monthly and quarterly IRR have to annualized.

Note that the use of the IRR function assumes that the period between each cash flow is the same (equal-size payment periods), for example one year.

From the IRR function to the XIRR function

If the period between each cash flow is not the same, the IRR function should not be used. It is the case with monthly cash flows as the months of the year may contain 28, 29, 30 or 31 days.

In this case, the XIRR function comes into play to calculate a correct internal rate of return, taking into consideration the periods of different sizes.

The XIRR function has three arguments:

The values
The dates for cash outflows and inflows.
The guessed number for the expected internal rate of return (optional argument).

You can download the Excel file below in which I illustrate the use of the IRR and XIRR functions in Excel based on a simple example.

From the IRR function to the MIRR function (Modified Internal Rate of Return)

The MIRR function is quite the same as the IRR function, except that it takes into consideration both the cost of borrowing the initial investment funds (discount rate) and reinvestment rates for future cash flows.

In contrast to IRR, MIRR assumes that cash flows from a project are reinvested at the firm’s cost of capital (rate of return on a portfolio company’s existing securities).

To compute the MIRR, the Excel function uses the following parameters:

The values
The guessed number for the expected internal rate of return (optional argument).
The financial rate: the finance rate of interest paid
The reinvest rate: the interest rate earned from the reinvested profit

Where FV represents the Future Value of positive cash flows at the cost of capital for the company, PV represents the Present Value of negative cash flows at the financing cost of the company, and T represents the number of periods.

You can download the Excel file below in which I illustrate the use of the IRR and MIRR functions in Excel based on a simple example.

Some of the accountants say that the MIRR function is less valid than the other because not all the flows are reinvested fully. Although we can use a less important interest rate to compensate the partial investment; but we think the best approach will be the inclusion of the three calculations (IRR, XIRR, and MIRR).

Limits of the IRR

The non-linear equation for obtaining the IRR may have one solution, several solutions or no solution according to the sequence of cash flows. These represent limits of the IRR as an investor would like one value when estimating its investment.

Another limit of the IRR as a decision criterion for investing is that the result is not in agreement with the decision criterion based on NPV, which represents the value created by the investment.

You can download the Excel file below in which I provide an example to illustrate the limit of the IRR when selecting investment when two projects are available.

Useful resources

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

About the author

The article was written in November 2021 by Léopoldine FOUQUES (ESSEC Business School, Global Bachelor of Business Administration, 2017-2021).

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