After completing this reading you should be able to:

- Describe the mean-variance framework and the efficient frontier.
- Explain the limitations of the mean-variance framework with respect to assumptions about return distributions.
- Define the VaR measure of risk, describe assumptions about return distributions and holding period, and explain the limitations of VaR.
- Define the properties of a coherent risk measure and explain the meaning of each property.
- Explain why VaR is not a coherent risk measure.
- Explain and calculate Expected Shortfall (ES), and compare and contrast VaR and ES.
- Describe spectral risk measures, and explain how VaR and ES are special cases of spectral risk measures.
- Describe how the results of scenario analysis can be interpreted as coherent risk measures.

## The Mean-Variance Framework and the Efficient Frontier

The mean-variance framework uses the expected mean and standard deviation to measure the financial risk of portfolios. Under this framework, it’s necessary to assume that returns follow a specified distribution, usually the normal distribution.

The normal distribution is particularly common because it concentrates most of the data around the mean return. 66.7% of returns occur within plus or minus one of the standard deviation of the mean. A whopping 95% of the returns occur within plus or minus two standard deviations of the mean.

Investors are normally concerned with downside risk and are therefore interested in probabilities that lie to the left of the expected mean.

**The efficient frontier** represents the set of optimal portfolios that offers the highest expected return for a defined level of risk or the lowest risk for a given level of expected return. This concept can be represented on a graph pitting the expected return (Y-axis) against the standard deviation (X-axis).

For every point on the efficient frontier, there is at least one portfolio that can be constructed from all available investments that has the expected risk and return corresponding to that point. Portfolios that do not lie on the efficient frontier are suboptimal: those that lie below the line do not provide enough return for the level of risk, and those that lie on the right of the line have a higher level of risk for the defined rate of return.

The choice between optimal portfolios \(A\), \(B\), and \(D\) above will depend on an individual investor’s appetite for risk. A very risk-averse investor will choose portfolio \(A\) because it offers an optimal return at the lowest risk, whereas an investor with room for more risk might pick \(D\) because it offers the highest optimal return at the highest risk.

## Limitations of the mean-variance framework with respect to assumptions about the return distributions

The mean-variance framework is unreliable when the assumption of normality is not met. If the return distribution is not symmetric, the standard deviation is not a reliable or relevant measure of probabilities at the extreme ends of the distribution.

The normality assumption is only strictly appropriate in the presence of a zero-skew (symmetric) distribution. If the distribution leans to the left or to the right – something that often happens with financial returns – the mean-variance framework churns out misleading estimates of risk.

## Value at Risk (VaR) as a Measure of Risk

VaR can be defined as the maximum amount of loss, under normal business conditions, that can be incurred with a given confidence interval. It can also be viewed as the worst possible loss under normal conditions over a specified period. Suppose an analyst calculates the monthly VaR as $100 million at 95% confidence: What does this imply?

This simply means that under normal conditions, in 95% of the months, we expect the fund to make a profit or lose no more than $100 million. Put differently, the probability of losing $100 million or more in any given month is 5%.

### Limitations of VaR

- It does not describe the
**worst possible**loss. Indeed, as seen from the example above, we would expect the $100 million loss mark to be breached 5 times out of a hundred for a 95% confidence level. - VaR does not describe the losses in the left tail. It indicates the probability of a value occurring but stops short of describing the distribution of losses in the left tail.
- Two arbitrary parameters are used in its calculation – the confidence level and the holding period. The confidence level indicates the probability of obtaining a value greater than or equal to VaR. The holding period is the time span during which we expect the loss to be incurred, say, a week, month, day, or year. VaR increases at an increasing rate as the confidence level increases. VaR also increases with increases in the holding period.
- VaR estimates are subject to both model risk and implementation risk. Model risks arise from incorrect assumptions while implementation risk is the risk of errors from the implementation process.

## Properties of a Coherent Risk measure

A risk measure summarizes the entire distribution of dollar returns \(X\) by one number, \(\rho \left( X \right) \). There are four desirable properties every risk measure should possess. These are:

**Monotonicity:**If \({ X }_{ 1 }\le { X }_{ 2 },\rho \left( { X }_{ 1 } \right) \ge \rho \left( { X }_{ 2 } \right) \)Interpretation: If a portfolio has systematically lower values than another, in each state of the world, it must have greater risk.

**Subadditivity:**\(\rho \left( { X }_{ 1 }+{ X }_{ 2 } \right) \le \rho \left( { X }_{ 1 } \right) +\rho \left( { X }_{ 2 } \right) \)Interpretation: When two portfolios are combined, their total risk should be less than (or equal to) the sum of their individual risks. Merging of portfolios ought to reduce risk.

**Homogeneity:**\(\rho \left( kX \right) =k\rho \left( X \right) \)Interpretation: Increasing the size of a portfolio by a factor \(k\) should result in a proportionate scale in its risk measure.

**Translation invariance:**\( \rho \left( X+h \right) =\rho \left( X \right) -h \)Interpretation: Adding cash \(h\) to a portfolio should reduce its risk by \(h\). Like \(X\), \(h\) is measured in dollars.

## Why VaR is Not a Coherent Risk Measure

Value at risk is not a coherent risk measure because it fails the subadditivity test. Here’s an illustration:

Suppose we want to calculate the VaR of a portfolio at 95% confidence over the next year of two zero-coupon bonds ( A and B) scheduled to mature in one year. Assume that:

- The current yield on each of the two bonds is 0;
- The bonds have different issuers;
- Each bond has a probability of 4% of defaulting over the next year;
- The event of default in either bond is independent of the other; and
- The recovery rate upon default is 30%.

Given these conditions, the 95% VaR for holding either of the bonds is 0 because the probability of default is less than 5%. Now, what’s the probability ‘P’ that at least one bond defaults?

$$ P=0.04\times 0.96+0.96\times 0.04+0.04\times 0.04=7.84% $$

The probability of at least one default is 7.84%, which exceeds 5%.

So if we held a portfolio that consisted of 50% \(A\) and 50% \(B\), then the \(95\% \quad VaR =0.7 \times 0.5+0 \times 0.5=35\%\). This clearly violates the subadditivity principle, and VaR is therefore not a coherent risk measure.

## Expected Shortfall (Conditional VaR)

The expected shortfall, also known as the conditional VaR, is the average of losses defined by the probability. In other words, it is expected loss given that the portfolio return already lies below the pre-specified worst-case quantile return (e.g. \({ 5 }^{ th }\) percentile).

Suppose the 5% VaR for a fund is -25%. Therefore, 5% of the time, the fund earns a return that’s less than -25%. The expected shortfall gives as the expected value of all returns falling at or below the 5 percentile return. As such, ES is a larger loss than the VaR. However, unlike the VaR, ES satisfies the subadditivity property.

The ES is considered a better risk measure than VaR because:

- It meets all the desirable qualities of a coherent risk measure, including subadditivity
- It has less restrictive assumptions regarding risk/return decision rules
- Unlike the VaR, ES gives an estimate of the magnitude of a loss for unfavorable events

## Spectral Risk Measures

A spectral risk measure is a risk measure given as a weighted average of return quantiles from the loss distribution. It’s given as a weighted average of outcomes where bad outcomes typically have more weight. A spectral risk measure is always a coherent risk measure.

The Expected shortfall is a special case of risk spectrum measurement where the weighting function is set to \(\left[ { 1 }/{ \left( 1-confidence\quad level \right) } \right] \) for tail losses, with all the other quantiles having a weight of zero.

VaR is also a special case of risk spectrum measurement but one that places no weight on tail losses.

## Interpreting the Results of Scenario analysis as Coherent Risk Measures

Following a scenario analysis, the results can be interpreted as coherent risk measures. To do this, loss outcomes are assigned probabilities. The losses can be viewed as representative of tail distributions of the relevant distribution function. The arithmetic average of the losses gives the expected shortfall for the distribution. Since the ES is a coherent risk measure, the results of scenario analysis are also coherent.

## Questions

### Question 1

You have been given the following 30 ordered percentage returns of an asset:

\(\left[ -18,-16,-14,-12,-10,-9,-7,-7,-6,-6,-6,-5,-5,-4,-4,-4,-2,-1,0,0,2,3,6,12,12,13,15,15,18,28,29 \right] \)

The VaR and expected shortfall, at 90% confidence level, are closest to:

$$

\begin{array}{|c|c|c|}

\hline

{} & VaR & Expected \quad shortfall \\ \hline

A. & 14 & 17 \\ \hline

B. & 14 & 13 \\ \hline

C. & 12 & 16 \\ \hline

D. & 12 & 24 \\ \hline

\end{array}

$$

The correct answer is **B**

To locate the full 10% tail “outside” the VaR quantile, we take the third worst return, which is -14. However, recall that VaR need not be represented as a negative.

The ES is the arithmetic average of losses at and beyond 90% Thus,

$$ ES=\frac { 18+16+14 }{ 3 } =17 $$

### Question 2

Ann Conway, FRM, has spent the last several months trying to develop a new risk measure to appraise a set of defaultable zero-coupon bonds owned by her employer. Prior to its use, her supervisor has asked her to demonstrate that it’s a coherent risk measure. The results are listed below:

Given:

- \(X\) and \(y\) are state-contingent payoffs of two different bond portfolios
- \(P\left( x \right) \) and \(P\left( y \right) \) are risk measures for portfolio \(x\) and portfolio \(y\), respectively.
- \(K\) and \(l\) are arbitrary constants, with \(k>0\)

Which of the following equations shows that Conway’s risk measure is not coherent?

- \(P\left( kx \right) =kP\left( x \right) \)
- \(P\left( x \right) +P\left( y \right) \ge P\left( x+y \right) \)
- \(P\left( x \right) \le P\left( y \right) \quad if\quad x\le y\)
- \(P\left( x+l \right) =P\left( x \right) -l\)

The correct answer is **C**.

**C**, as represented above, shows that the risk measure does not satisfy the monotonicity property. Monotonicity requires that \(P\left( x \right) \ge P\left( y \right) \quad if\quad x\le y\). (If a portfolio has systematically lower values than another, in each state of the world, it must have greater risk.)

**A** demonstrates that the measure satisfies the homogeneity property.

**B** demonstrates that the measure satisfies the subadditivity property

**D** demonstrates that the measure satisfies the translation invariance property