92 Mathematics for Finance
out to be a convenient measure of risk.
Exercise 5.1
Compute the risk Var(K 1 ), Var(K 2 )andVar(K 3 ) in each of the following
three investment projects, where the returnsK 1 ,K 2 andK 3 depend on
the market scenario:
Scenario Probability ReturnK 1 ReturnK 2 ReturnK 3
ω 1 0. 25 12% 11% 2%
ω 2 0. 75 12% 13% 22%
Which of these is the most risky and the least risky project?Exercise 5.2
Consider two scenarios,ω 1 with probability^14 andω 2 with probability^34.
Suppose that the return on a certain security isK 1 (ω 1 )=−2% in the
first scenario andK 1 (ω 2 ) = 8% in the second scenario. If the return on
another security isK 2 (ω 1 )=−4% in the first scenario, find the return
K 2 (ω 2 ) in the other scenario such that the two securities have the same
risk.In some circumstances thestandard deviationσK=√
Var(K) of the return
is a more convenient measure of risk. If a quantity is measured in certain units,
then the standard deviation will be expressed in the same units, so it can be
related directly to the original quantity, in contrast to variance, which will be
expressed in squared units.
Example 5.1
Let the return on an investment beK=3%or−1%, both with probability
0 .5. Then the risk is
Var(K)=0.0004 or σK=0. 02 ,depending on whether we choose the variance or standard deviation. Now sup-
pose that the return on another investment is double that on the first invest-
ment, being equal to 2K=6%or−2%, also with probability 0.5each.Then
the risk of the second investment will be
Var(2K)=0.0016 or σ 2 K=0. 04.The risk as measured by the variance is quadrupled, while the standard devi-
ation is simply doubled.