Chapter 3 - Measures of investment risk Flashcards
Variance of return
(-inf) to (inf) ∫(mu - x)^2 * f(x)*dx
Why is the variance a good measure of risk?
-mathematically tractable
-mean-variance framework leads to elegant solutions for optimal portfolios (good approximation)
> > MVPT leads to optimum portfolios if investors can be assumed to have a quadratic utility functions or if returns can be assumed to be normally distributed (1st 2 moments)
Disadvantage of the variance
Most investors don’t dislike uncertainty of returns as such, but dislike the probability of low returns
Semi-variance of return
(-inf to mu) ∫(mu-x)^2 f(x)dx
Give an advantage & disadvantage of semi-variance
adv- quantifies downside risk
disadv- not easy to handle mathematically
-takes no account of variability above the mean
*if returns on assets are symmetrically distributed, semi-variance is proportional to variance
Shortfall probability
Probability of returns falling below a certain level
SFP= (-inf to L) ∫f(x)*dx
where L is the benchmark level
*benchmark level, L, is the return on a benchmark fund if this is more appropriate than an absolute level (relative to an index, median fund or some level of inflation)
Give an advantage & disadvantage of the SFP
adv- easy to understand & calculate
disav- gives no indication of the magnitude of any shortfall
Value at Risk (VaR)
likelihood of underperforming by providing a statistical measure of downside risk
VaR(X)= -t where P(X<t)= p
[represents the maximum potential loss on a portfolio over a given future time period with a given degree of confidence, where the latter is normally expressed as 1-p]
+VaR (-ve t) -> loss
-VaR(+ve t) ->profit
[monetary amount not %]
disadv- doesn’t quantify the size of the ‘tail’
Expected Shortfall (ESF)
ESF- expected shortfall below a certain level
ESF = E[max(L-X,0)] = (-inf to L) ∫(L-x)f(x)dx where L = benchmark
(if L is chosen to be a particular percentile point on the distribution, then the risk measure is known as the TailVaR)
Give 5 other similar measures of risk
1) expected tail loss
2) tail conditional expectation
3) conditional VaR
4) tail conditional VaR
5) worst conditional expectation
(all of the above measure risk of underperformance against some criteria)
For any given percentile, measures of expected shortfall will be higher than VaR measures, particularly if the distribution is fat-tailed.
Explain the relationship between risk measures & utility functions
Investors base decisions on available combos of risk & e(returns).
Given a particular utility function, the appropriate risk measure can be determined.
> > If E(RETURN) & VARIANCE are used as basis of decisions => QUADRATIC utility function
> > if E(RETURN) & SEMI-VARIANCE below the e(return) are used as a basis => QUADRATIC BELOW & LINEAR ABOVE e(return)
Explain how companies use insurance to manage risk
Individuals & corporations face risks resulting from unexpected events and therefore offer protection on certain events based on frequency and severity of such events.
POOLING OF RESOURCES can be used to reduce insurer’s risk (total cost increases but VaR doesn’t)
If insurer is risk averse, the insurance premium will need to include a margin to compensate the insurer for taking such risks.
Adverse / Anti-selection
fact that people who know they are particular bad risks are more inclined to take out insurance than those who know they are good risks.
(To reduce such problems, insurers try find out lots of info about potential policyholders. The PH can be put into small, reasonably homogenous pools & charged appropriate premiums)
Moral Hazard
policyholder may, because they have insurance, act in a way which makes the insured event more likely
(Moral hazard makes insurance more expensive)
