Chapter 4 - Measures Of Investment Risk

Question 1

Variance of return

Answer 1

It is defined as
Integral from -inf to +inf of (μ-x)^2 * f(x) dx
Where μ is the mean return at the end of the chosen period and f(x) is the probability density function of the return.

Question 2

Advantages of variance of return over all other measures (2)

Answer 2

1. It is mathematically tractable
2. It leads to elegant solutions for optimal portfolios, withing the context of mean-variance portfolio theory.

This ease of use should not be lightly disregarded and to justify using a more complicated measure it would have to be shown that it was more theoretically correct and that it leads to significantly different choices than the use of variance.

Question 3

Why is variance of return not the most suitable measure of investment risk

Answer 3

Most mathematical investment theories of investment risk use variance of return as a measure of risk.
However, it is not obvious that variance necessarily corresponds to investors’ perception of risk.

Question 4

Semi-variance of return

Answer 4

It is defined as
Integral from -inf to μ of (μ-x)^2 * f(x) dx
where μ is the mean return at the end of the chosen period

Question 5

Argument against variance / Advantage of downside semi-variance

Answer 5

It’s not uncertainty in general that investors dislike, but the uncertainty of poor returns that causes concern. Downside semi-variance focuses on this aspect of risk.

The main argument against the use of the variance as a measure of risk is that most investors do not dislike uncertainty of returns as such; raher they dislike the possibility of low returns. One measure that seeks to quantify this view is downside semi-variance.

Question 6

Disadvantages of semi-variance (4)

Answer 6

1. It is not easy to handle mathematically. (it can become mathematically intractable, even for relatively straightforward distributions of returns)
2. It takes no account of variability above the mean. ( which may make this measure unsuitable for comparing assets)
3. If returns on assets are symmetrically distributed, semi-variance is proportional to variance. (so it gives no extra information)
4. It measures downside risk relative to the mean rather than another benchmark that might be more relevant to the investor.

Question 7

Shortfall probabilities

Answer 7

A shortfall probability measures the probability of returns falling below a certain level. For continuous variable, the risk measure is given by:
Integral from -inf to L of f(x) dx
where L is a chosen benchmark level.
The benchmark level can be expressed as the return on a benchmark fund if this is more appropriate than an absolute level.

Question 8

Value at Risk (VaR)

Answer 8

Value at risk generalises the likelihood of underperforming by providing a statistical measure of downside risk.
It is defined as
Continuous: VaR(X) = -t where P(X < t) = p
Discrete: VaR(X) = -t where t = max{x:P(X < x) <= p}
VaR represents the maximum potential loss on a portfolio over a given future time period with a given degree of confidence (1-p).

So for example a 99% one day VaR is the maximum loss on a portfolio over a one day period with 99% confidence, ie there is a 1% probability of a greater loss.

Question 9

What is the potential problem with using VaR?

Answer 9

VaR is based on assumptions that may not be immediately apparent. It is often calculated assuming that investment returns are normally distributed.

Question 10

What types of portfolios may exhibit non-normal distributions?

Answer 10

Portfolios exposed to 
1. Credit risk
2. Systematic bias
3. Derivatives 
may exhibit non-normal distributionsy.

Question 11

Discuss the usefulness of VaR when portfolio returns exhibit non-normal distributions.

Answer 11

The usefulness of VaR in these situations depends on modelling skewed or fat-tailed distributions of returns, either in the form of statistical distributions or via Monte Carlo simulations. However the further one gets out into the “tails” of the distributions, the more lacking the data and hence the more arbitrary the choice of the underlying probability becomes.

Question 12

Expected shortfall

Answer 12

The risk measure can be expressed as the expected shortfall below a certain level.
It is given by
Continuous: E[max(L-X,0)] = integral from -inf to L of (L-x) f(x) dx
Discrete: E[max(L-X,0)] = Σ where x < L of (L-x) P(X=x)
where L is the chosen benchmark level.

Question 13

Advantages of shortfall measures / use of shortfall measures in general

Answer 13

Shortfall measures are useful for monitoring a fund’s exposure to risk because the expected underperformance relative to a benchmark is a concept that is apparently easy to understand. As with semi-variance however, no attention is paid to the distribution of outperformance of the benchmark.

Question 14

Tail value at risk (TailVaR or TVaR)

Answer 14

If the value of L chosen for the expected shortfall is a particular percentile point on the distribution, then the risk measure is known as the TailVaR.
However, TailVaR can also be expressed as the expected shortfall conditional on there being a shortfall.

Question 15

5 other similar measures of risk

Answer 15

1. Expected tail loss
2. Tail conditional expectation
3. Conditional VaR
4. Tail conditional VaR
5. Worst conditional expectation
   They all measure the risk of under-performance against some set criteria

Question 16

The relationship between risk measures and utility functions in general

Answer 16

An investor using a particular risk measure will base his decisions on a consideration of the available combinations of risk and expected return. Given a knowledge of how this trade-off is made it is possible, in principle, to construct the investor’s underlying utility function.
Conversely, given a particular utility function, the appropriate risk measure can be determined.

Question 17

To which measure of investment risk does a quadratic utility function correspond?

Answer 17

If an investor has a quadratic utility function, the function to be maximised in applying the expected utility theorem will involve a linear combination of the first two moments of the distribution of return. Thus, variance of return is an appropriate measure of risk in this case.

If expected return and variance are used as the basis of investment decisions, it can be shown that this is equivalent to a quadratic utility function.

Question 18

To what form of utility function does semi-variance correspond?

Answer 18

If expected return and semi-variance below the expected return are used as the basis of investment decisions, it can be shown that this is equivalent to a utility function that is quadratic below the expected return and linear above.

Question 19

To what form of utility function does the use of a shortfall probability correspond?

Answer 19

Use of a shortfall risk measure corresponds to a utility function that has a discontinuity at the minimum required return.