96 Mathematics for Finance
Exercise 5.4
Compute the value V(1) of a portfolio worth initially V(0) = 100
dollars that consists of two securities with weights w 1 = 25% and
w 2 = 75%, given that the security prices areS 1 (0) = 45 andS 2 (0) = 33
dollars initially, changing toS 1 (1) = 48 andS 2 (1) = 32 dollars.We can see in Example 5.4 and Exercise 5.4 thatV(1)/V(0) depends on
the prices of securities only through the ratiosS 1 (1)/S 1 (0) = 1 +K 1 and
S 2 (1)/S 2 (0) = 1 +K 2. This indicates that the return on the portfolio should
depend only on the weightsw 1 ,w 2 and the returnsK 1 ,K 2 on each of the two
securities.
Proposition 5.1
The returnKVon a portfolio consisting of two securities is the weighted average
KV=w 1 K 1 +w 2 K 2 , (5.2)wherew 1 andw 2 are the weights andK 1 andK 2 the returns on the two
components.
Proof
Suppose that the portfolio consists ofx 1 shares of security 1 andx 2 shares of
security 2. Then the initial and final values of the portfolio are
V(0) =x 1 S 1 (0) +x 2 S 2 (0),
V(1) =x 1 S 1 (0)(1 +K 1 )+x 2 S 2 (0)(1 +K 2 )
=V(0) (w 1 (1 +K 1 )+w 2 (1 +K 2 )).As a result, the return on the portfolio is
KV=V(1)−V(0)
V(0)=w 1 K 1 +w 2 K 2.Exercise 5.5
Find the return on a portfolio consisting of two kinds of stock with
weightsw 1 = 30% andw 2 = 70% if the returns on the components are