What is Value-at-Risk (VaR) ?

VaR is a risk measure that estimates the maximum potential loss over a specified time horizon at a given confidence level.

What is Expected Shortfall (ES) ?

Expected Shortfall is the average loss expected if the Value-at-Risk threshold is breached.

How does the Mean Variance Framework work ?

The Mean Variance Framework evaluates investments by balancing expected returns against the standard deviation of returns.

Why are financial distributions said to have fatter tails than normal distributions ?

Financial distributions often exhibit more extreme outcomes than predicted by the normal distribution, leading to fatter tails.

What is the efficient frontier in portfolio management ?

The efficient frontier represents the set of optimal portfolios offering the highest expected return for a given level of risk.

How is VaR calculated using a normal distribution ?

VaR for a normal distribution can be calculated using the mean and standard deviation of the returns.

What is the significance of correlation in portfolio risk ?

Correlation between assets affects the overall risk and return of a portfolio, influencing diversification benefits.

What are coherent risk measures ?

Coherent risk measures satisfy properties like monotonicity, translation invariance, homogeneity, and subadditivity.

What is spectral risk measure ?

A spectral risk measure is a type of coherent risk measure where weights assigned to percentiles increase based on risk aversion.

Why might VaR not be sufficient for assessing risk ?

VaR does not account for the severity of losses beyond the VaR threshold, potentially underestimating extreme risks.

Measures of Financial Risk

Introduction

When portfolios are described by the mean and standard deviation of their returns, the assumption of normal distributions is natural. However, most financial variables have fatter tails than the normal distribution. In other words, extreme events are more likely to occur than the normal distribution would predict. This is relevant because risk managers are particularly interested in quantifying the probability of these adverse extreme events.
Value-at-risk (VaR) and expected shortfall are two risk measures focusing on adverse events. Expected shortfall, although less intuitive than VaR, has more desirable theoretical properties and is an example of a coherent risk measure.

The Mean Variance Framework

There is a trade-off between risk and return. The greater the risks taken, the greater the expected return that can be achieved. Here, the expected return term is used by statisticians to describe the average (or mean) return. It does not describe the return that is expected to happen.
One measure of risk is the standard deviation of returns.
For a risk-free asset, such as a Treasury instrument, the standard deviation of returns is zero with a fixed return over a time period. For example, if a Treasury instrument provides a one-year return equal to 3%, the expected return will be 3% with a standard deviation of 0.
Now consider the asset given in this table.

Probability Return

0.05 −20

0.25 0%

0.4 7%

0.25 15%

0.05 40%
The set of all investment opportunities can be characterized in a chart by plotting the various combinations of means and standard deviations. This figure shows the two assets:
1. An asset having a return with a mean of 2.00% and a standard deviation of zero
2. An asset having a return with a mean of 7.55% and a standard deviation of 10.9%

Probability	Return
0.05	−20
0.25	0%
0.4	7%
0.25	15%
0.05	40%

Combining Investments

Now consider two available investment assets –
1. Asset 1 with expected return 𝜇₁ and standard deviation of returns as 𝜎₁
2. Asset 2 with expected return 𝜇₂ and standard deviation of returns as 𝜎₂
Correlation between the two assets is 𝜌_1,2
Suppose an investor has a portfolio of these assets where the weights of the asset 1 and 2 are 𝑤₁ and 𝑤₂= 1 – 𝑤₁ respectively. The portfolio expected return (𝜇_𝑝 ) is therefore a weighted average of the expected return from the two investments:
$\mu_p=\mu_1 w_1+\mu_2 w_2$
The standard deviation of the portfolio’s return 𝜎_𝑝 is given by:
$\sigma_p=\sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\rho_{1,2}w_1w_2\sigma_1\sigma_2}$
These are plotted in Figure 2.
Suppose that 𝜇₁= 5% and 𝜇₂ = 8% ; 𝜎₁ = 12% and 𝜎₂ = 16% ; and 𝜌_1,2 = 0.25. This Table shows several possible outcomes for various weightings for asset 1 and asset 2.

w₁ w₂ μ_P σ_P

0.0 1.0 8.0% 16.0%

0.2 0.8 7.4% 13.6%

0.4 0.6 6.8% 11.8%

0.6 0.4 6.2% 10.8%

0.8 0.2 5.6% 10.9%

1.0 0.0 5.0% 12.0%

Risk-return combinations from the two asset portfolio
The previous result can be extended to the situation in which there are 𝑛 investments and 𝜇_𝑖 and 𝜎_𝑖 are respectively the return mean and standard deviation for the 𝑖^𝑡ℎ investment. When the weight given to the 𝑖^𝑡ℎ investment is 𝑤_𝑖, the mean and standard deviation of the portfolio return are
$\mu_p = \sum_{i=1}^n w_i \mu_i$
And
$\sigma_p = \sum_{i=1}^n \sum_{j=1}^n \rho_{ij} w_i w_j \sigma_i \sigma_j$
where 𝜌_𝑖,𝑗 is the correlation between the returns from investments 𝑖 and 𝑗.
Investors want high expected returns and low risks.
Of the six investments in the previous Figure, the upper four (corresponding to w_1= 0.0, 0.2, 0.4, and 0.6) form what is referred to as an efficient frontier. For each of these four investments, there is no other investment (or combination of investments) that has both a higher expected return and a lower standard deviation of return.
The investors would want to move as far into the upper-left region in the previous Figure. An investor can move further in the desired direction by incorporating additional investments. For example, a third investment can be combined with any combination of the first two investments to obtain new risk-return trade-offs. A fourth investment can be combined with any combination of the first three investments to obtain even more risk-return trade-offs, and so on.

w₁	w₂	μ_P	σ_P
0.0	1.0	8.0%	16.0%
0.2	0.8	7.4%	13.6%
0.4	0.6	6.8%	11.8%
0.6	0.4	6.2%	10.8%
0.8	0.2	5.6%	10.9%
1.0	0.0	5.0%	12.0%

The Efficient Frontier

When all possible combinations of all risky investments are considered, the efficient frontier of all risky investments is obtained. This is shown in the given Figure. All risk-return combinations in the shaded area below and to the right of the efficient frontier can be created. Those to the upper-left cannot be created.
Now let’s consider what happens when a risk-free investment earning a return of R_F is possible. This investment is represented by point F in this Figure. Consider a line drawn from F that is a tangent to the efficient frontier in the previous Figure. Suppose that M is the point of tangency. By allocating funds between the risk-free investment represented by point F and the risky portfolio represented by point M, risk-return combinations cane be obtained represented by any point on the line FM.
If it is assumed that borrowing is possible at the risk-free rate R_F , points can be obtained on the line FM that are beyond M.
Hence, when a risk-free investment is considered, the efficient frontier must be a straight line. There is a linear trade-off between expected return and standard deviation of return. The argument also shows that all investors should choose to invest in the same portfolio of risky assets, for example, the portfolio represented by point 𝑀. They should then reflect their risk appetite by borrowing or lending at the risk-free rate, 𝑅_𝐹 . Investors who are relatively risk-averse will choose points on the line 𝐹𝑀 that are close to 𝐹. Those who like taking risks will choose points close to 𝑀, or even points on the line 𝐹𝑀 that are beyond 𝑀.
The investment represented by point 𝑀 is referred to as the market portfolio. It is the portfolio consisting of all investments in the market with the proportional amount of any given investment in the portfolio being the same as the proportion of all available investments that it represents. To understand why this must be so, first consider the situation where investment 𝑋 is under-represented in portfolio 𝑀. Because all investors want to buy portfolio 𝑀, demand for investment 𝑋 will be less than supply, and its price will decline to a point where it is no longer under-represented in 𝑀. Similarly, consider the opposite situation where investment 𝑋 is over-represented in portfolio 𝑀. In this case, demand for investment 𝑋 would exceed supply, and its price would increase until the over-representation disappeared.
For the purpose of the previous analysis, the following assumptions were made –
1. It was assumed that all investors make the same assumptions about the mean and standard deviations of, and coefficients of correlation between, the returns from different investments.
2. It was also assumed that they care only about the mean and standard deviation of the returns from their portfolios, and that they can all borrow at the risk-free rate of interest.
These assumptions are, at best, only approximately true.
Efficient frontiers are closely related to the capital asset pricing model (CAPM). This is discussed further in the Foundations of Risk Management readings.

The Normal Distribution

The normal distribution is often assumed for the returns on financial variables. It is a convenient choice because all that is needed is the mean and standard deviation of the returns, and these can be calculated for a portfolio easily.
In practice, financial variables tend to have much fatter tails than normal distributions. This can be illustrated using data on the 𝑆&𝑃 500 Index. Over a 20-year period ending on December 14, 2017, the standard deviation of the daily change in the index was 1.214%.

Movement Actual results (%) Predicted by Normal Distribution (%)

> 1SD 21.79 31.73

> 2SD 5.01 4.55

> 3SD 1.65 0.27

> 4SD 0.60 0.01

> 5SD 0.32 0.00

> 6SD 0.12 0.00
Most financial variables are like the S&P 500 in that they have fatter tails than the normal distribution. Most portfolio returns that can be created also have fatter tails than the normal distribution. The situation is illustrated in this Figure. The actual distribution and the normal have the same standard deviation. However, the actual distribution is more peaked than the normal distribution and has fatter tails. This means that both small changes and large changes happen more often than the normal distribution would suggest, while intermediate changes happen less often. This is consistent with our 𝑆&𝑃 500 data.

Movement	Actual results (%)	Predicted by Normal Distribution (%)
> 1SD	21.79	31.73
> 2SD	5.01	4.55
> 3SD	1.65	0.27
> 4SD	0.60	0.01
> 5SD	0.32	0.00
> 6SD	0.12	0.00

VaR

Risk managers are concerned with the possibility of adverse events. The standard deviation of a distribution, although a useful measure in many situations, does not describe the tails of a probability distribution. VaR is an important risk measure that focuses on adverse events and their probability. The VaR for an investment opportunity is a function of two parameters:
1. The time horizon, and
2. The confidence level.
VaR is the loss that is not expected to be exceeded over the time horizon at the specified confidence level.
Suppose that the time horizon is ten days, and the confidence level is 99%. A 𝑉𝑎𝑅 of 𝑈𝑆𝐷 10 million would mean that we are 99% certain that the loss during the next ten days will be less than 𝑈𝑆𝐷 10 million. To put this another way, there is a probability of only 1% that the loss over the next ten days will be greater than 𝑈𝑆𝐷 10 million.
Regulators have historically used 𝑉𝑎𝑅 to determine the capital that banks must keep when internal models are used. For example, the rules introduced for market risk in 1996 based the required capital on the 𝑉𝑎𝑅 with a ten-day time horizon and a 99% confidence level. Meanwhile, Basel II based credit risk capital on the 𝑉𝑎𝑅 with a one-year time horizon and a 99.9% confidence level. In 2012, regulators announced their intention to change the market risk capital calculation process by replacing 𝑉𝑎𝑅 with estimated shortfall and lowering the confidence level to 97.5%.
The probability distribution of losses arising from assets can of following types –
1. Normal distribution
2. Uniform distribution
3. Discrete distribution
More sophisticated tools are required to calculate the 𝑉𝑎𝑅 when the probability distribution of losses are non-normal. These tools have been discussed in detail in upcoming chapters.

VaR Normal Distribution

Suppose the return from an investment over a specified time horizon has a normal distribution with mean 20 and standard deviation 30. This corresponds to a loss distribution with mean −20 and standard deviation 30. 𝑉𝑎𝑅 with 99% confidence interval can be calculated using the 𝑁𝑂𝑅𝑀.𝐼𝑁𝑉 function in Excel.
The first argument is the percentile of the distribution required (0.99), the second is the mean loss (−20), and the third is the standard deviation (30). 𝑁𝑂𝑅𝑀.𝐼𝑁𝑉(0.99,−20,30) gives 49.79. Thus, the 99th percentile of a normal loss distribution with a mean of −20 and a standard deviation of 30 is 49.79. This is the 𝑉𝑎𝑅 level when the confidence level is 99%.
This is also the loss level that has a 1% chance of being exceeded. More detailed way to calculate the 𝑉𝑎𝑅 will be discussed in upcoming chapters.

VaR Uniform Distribution

Assume the result of an investment with a uniform distribution where all outcomes between a profit of 30 and a loss of 20 are equally likely. In this case, the 𝑉𝑎𝑅 with a 99% confidence level can be calculated as –

VaR Discrete Distribution

Suppose that the outcomes from an investment are discrete. It is further assumed that three scenarios are possible, as shown in the table.
The final column shows the cumulative loss probability measured from the lowest to the highest. The first 88% of the cumulative loss distribution corresponds to a loss of 𝑈𝑆𝐷2 million. The part of the cumulative distribution between 88% and 98% corresponds to a loss of 𝑈𝑆𝐷 5 million. The part of the cumulative distribution between 98% and 100% corresponds to a loss of 𝑈𝑆𝐷 8 million.

Loss (million) Probability (%) Cumulative Probability Range (%)

2 88 0 to 88

5 10 88 to 98

8 2 98 to 100
With a 99% confidence level (C.I.), the 𝑉𝑎𝑅 is 𝑈𝑆𝐷 8 million as 99% falls into the 98% to 100% cumulative probability range, and the loss for this range is 𝑈𝑆𝐷 8 million. At 98% C.I., there is some ambiguity. It could be argued that the 98% to 100% range is preferred and the 𝑉𝑎𝑅 is 𝑈𝑆𝐷 8 million, or that it is the 88% to 98% range and the 𝑉𝑎𝑅 is 𝑈𝑆𝐷 5 million. One approach here is to set the 𝑉𝑎𝑅 equal to the average of the two answers, or 𝑈𝑆𝐷 6.5 million.

Loss (million)	Probability (%)	Cumulative Probability Range (%)
2	88	0 to 88
5	10	88 to 98
8	2	98 to 100

Expected Shortfall

One problem with 𝑉𝑎𝑅 is that it does not say anything about how bad losses might be when they exceed the 𝑉𝑎𝑅 level. Suppose that the 𝑉𝑎𝑅 with a 99% confidence level is 𝑈𝑆𝐷 20 million. Two situations can be considered-
1. If the loss is greater than 𝑈𝑆𝐷 20 million (a 1% chance), it is inconceivable that it will be greater than 𝑈𝑆𝐷 30 million.
2. If the loss is greater than 𝑈𝑆𝐷 20 million (a 1% chance), it is most likely to be 𝑈𝑆𝐷 100 million.
The 𝑉𝑎𝑅 for both situations is 𝑈𝑆𝐷 20 million, but the second situation is clearly riskier than the first. The second situation could arise when a trader has sold a credit default swap as part of a portfolio. This is a contract that provides a payoff if a particular counterparty defaults. The default might have a small probability, say 0.9%, but lead to a loss of 𝑈𝑆𝐷 100 million. 𝑉𝑎𝑅 does not distinguish between the two situations because it sets the risk measure equal to a particular percentile of the loss distribution and takes no account of possible losses beyond the 𝑉𝑎𝑅 level.
Expected shortfall is a risk measure that does take account of expected losses beyond the VaR level. The Expected Shortfall (ES) is the expected loss after the VaR threshold has been breached. It can be calculated as the probability weighted average of tail losses after the corresponding VaR level.
In the example given previously, the expected shortfall with a 99% confidence level would be much greater in the second situation than in the first situation.
When losses are normally distributed with mean 𝜇 and standard deviation 𝜎 the expected shortfall is –
$\mu + \sigma \frac{e^{U^2/2}}{(1-X)\sqrt{2\pi}}$
where, 𝑋 is the confidence level, 𝑈 is the point in the standard normal distribution that has a probability 𝑋% of being exceeded.

Coherent Risk Measures

A coherent risk measure satisfies the following conditions –
- Monotonicity: If (regardless of what happens) a portfolio always produces a worse result than another portfolio, it should have a higher risk measure.
- Translation Invariance: If an amount of cash 𝐾 is added to a portfolio, its risk measure should decrease by 𝐾.
- Homogeneity: Changing the size of a portfolio by multiplying the amounts of all the components by 𝜆 results in the risk measure being multiplied by 𝜆.
- Subadditivity: For any two portfolios, A and B, the risk measure for the portfolio formed by merging A and B should be no greater than the sum of the risk measures for portfolios A and B.
Artzner et al. showed that expected shortfall has all four properties, while 𝑉𝑎𝑅 has only the first three properties. There are circumstances when 𝑉𝑎𝑅 does not satisfy the fourth diversification property.
A risk measure can be defined by assigning weights to the percentiles of the loss distribution. Expected shortfall is by no means the only coherent risk measure than can be constructed. For example, a spectral risk measure is a type of coherent risk measure where the weights assigned to percentiles increase in a way that reflects risk aversion. One way that has been suggested to achieve this is to make weights proportional to:
$e^{-\frac{(1-P)}{\gamma}}$ where, 𝑃 is the percentile (expressed in decimal form), and 𝛾is a constant reflecting the user’s degree of risk aversion. The lower the value of 𝛾, the greater is the degree of risk aversion.