FM for Actuaries

Rates of Return 121
4.4 Portfolio Return
We now consider the return of a portfolio of assets. Suppose a portfolio consists of
N assets denoted by A 1 , ··· ,A N . Let the value of asset A j in the portfolio at time
0 be A 0j ,forj =1, ··· ,N. We allow A 0j to be negative for some j, so that asset
A j is sold short in the portfolio. 3 The portfolio value at time 0 is B 0 = ∑ N
j=1 A 0j.
Let the asset values at time 1 be A 1j , so that the portfolio value is B 1 = ∑ N
j=1 A 1j.
Denote R P as the return of the portfolio in the period from time 0 to time 1. Thus,
R P = B 1 − B 0
= B 1
− 1.
B 0 B 0
We define
w j = A 0j
,
B 0
which is the proportion of the value of asset A j in the initial portfolio, so that
N∑
w j =1,
j=1
and w j < 0 if asset j is short sold in the portfolio. We also denote
R j = A 1j − A 0j
A 0j
which is the rate of return of asset j. Thus,
= A 1j
A 0j
− 1,
which implies
1+R P = B 1
B 0
= 1 B 0 N ∑
=
=
R P =
N∑
j=1
j=1
A 1j
A 0j
B 0
× A 1j
A 0j
N∑
w j (1 + R j ),
j=1
N∑
w j R j , (4.13)
j=1
3 The mechanism of short selling will be discussed in the next section.