108156.pdf

54 Mathematics for Finance

scenarios are−6%, 4%, 30%, respectively, then the expected annual return is

−6%×

1

4 +4%×

1

2 + 30%×

1

4 =8%.

Exercise 3.9

With the probabilities of recession, stagnation and boom equal to 1/2,

1 /4, 1/4 and the predicted annual returns in the first two of these scenar-

ios at−5% and 6%, respectively, find the annual return in the remaining

scenario if the expected annual return is known to be 6%.

Exercise 3.10

Suppose that the stock prices in the following three scenarios are

Scenario S(0) S(1) S(2)

ω 1 100 110 120

ω 2 100 105 100

ω 3 100 90 100

with probabilities 1/4, 1/4, 1/2, respectively. Find the expected returns

E(K(1)),E(K(2)) andE(K(0,2)). Compare 1 +E(K(0,2)) with (1 +

E(K(1)))(1 +E(K(2))).

The last exercise shows that the relation established in Proposition 3.1 does
not extend to expected returns. For that we need an additional assumption.

Proposition 3.3

If the one-step returnsK(n+1),...,K(m) are independent, then

1+E(K(n, m)) = (1 +E(K(n+ 1)))(1 +E(K(n+ 2)))···(1 +E(K(m))).

Proof

This is an immediate consequence of Proposition 3.1 and the fact that the
expectation of a product of independent random variables is the product of
expectations. (Note that if theK(i) are independent, then so are the random
variables 1 +K(i)fori=n+1,...,m.)