Chapter 3 - Measures of Investment Risk

Question 1

What do we use as measure of investment risk?

Answer 1

Variance

Question 2

What is the formula for the variance of return for a continuous distribution?

Answer 2

Variance =
int[-inf, inf] (mu-x)^2 f(x) dx

mu - mean return
f(x) - pdf of the return
Returns here means the proportion increase in the market value

Question 3

Define Skewness.

Answer 3

Skewness measure how symettrical the distibution aroun the mean
e.g. the normal distrubution has a skewness of 0 as its perfectly symetric about the mean.

Question 4

Define Kurtosis.

Answer 4

Kurtosis measure how fat the tails of a distribution are. i.e. how likely extreme values are likely to occur.

Question 5

What is the downside Semi-variance (aka semi-variance) of return?

Answer 5

The downside semi-variance measures the risk associated with negative returns below the mean. The formula is:

int[mu,-inf] (mu-x)^2*f(x) dx

Question 6

What is the main argument against the use of the variance as a measure of risk?

Answer 6

Most investors do not dislike uncertainty of returns as such; rather they dislike the possibility of low returns.

Question 7

What is the downside semi-variance of a normal distribution with mean 10 and variance 10?

Answer 7

Any normal distribution is symmetrical and therefore the downside variance is half of the variance.

i.e. downside semi-variance = 5

Question 8

Define the shortfall probability measure.

Answer 8

The shortfall probability measures the probability of returns falling below a certain benchmark (say, L). The formula is:

int[L, inf] f(x) dx

Question 9

What are the advantages/disadvantages of the shortfall probability?

Answer 9

Advantages:
Easy to understand and calculate
Disadvantages:
doesn’t give you any indications of the magnitude of shortfall

Question 10

Define Value at Risk (VaR) for a continuous distribution.

Answer 10

The VaR calculates the maximum potential loss om a portfolio over a given future time period with a given degree of confidence.

VaR(X)=-t where P(X<t) = p

Note: the value a risk is a ‘loss amount’.

If VaR(X) >0 (i.e -t) then this indicates a loss
If VaR(X) <0 (i.e +t) then this indicates a profit

Question 11

Define Value at Risk (VaR) for a discrete distribution.

Answer 11

The VaR calculates the maximum potential loss om a portfolio over a given future time period with a given degree of confidence.

VaR(X)=-t where t=max{x:P(X<x)<=p}

Question 12

Define Tail Value a Risk (TailVaR)

Answer 12

Expected shortfall/Probability of Shortfall

Question 13

Define Expected Shortfall.

Answer 13

Expected Shortfall = E[max(L-X,0)] - int[L,-inf (L-X)f(x) dx

Question 14

If an investor has a quadratic utility function, what does this means in respect of their attitude towards risk?

Answer 14

If a investor has a quadratic utility function then their attitude towards risk and return can be expressed purely in terms of the mean and variance of investment opportunities.

Question 15

Define adverse selection (aka anti-selection/self-selection).

Answer 15

Adverse selection describes the fact that people who know that they are particularly bad risks are more inclined to take out insurance than those who know that they are good risks.

Question 16

Define Moral Hazard.

Answer 16

Moral hazard describes the fact that a policyholder may, because they have insurance, act in a way which makes the insured event more likely.

Question 17

What impact does Moral Hazard have on insurance premiums?

Answer 17

Moral hazard makes insurance more expensive. It may even push the price of insurance above the maximum premium that a person is prepared to pay.

Question 18

What do insurance companies look at to decide which events they should cover protection for?

Answer 18

Freq & severity (ACPC)

Question 19

Discuss the usefulness of the downside semi-variance.

Answer 19

Only is concerned about the downside risk which is what investors are more concerned about that upside risk.
Not as mathematically tractable as variance
If a distribution is symmetric then the semi-var is proportional to var
Completely ignores anything above mean which is consistent with an investor being risk-neutral above the means which is unlikely to be the case in practice.
It does not give any indication how extreme the shortfalls are