712 CHAPTER 19. SETTING THE COST OF INTERNATIONAL CAPITAL
returns are related to risk in an efficient portfolio. This relationship is due to Sharpe
(1964), Lintner (1965), and Mossin (1965).
19.2.1 How Asset Returns Determine the Portfolio Return
The model is typically derived in terms of returns rather than prices: academics use
returns in empirical work, and practitioners want a formula for the expected return
to be used forNPVapplications. The key relation is that the realized return on the
portfolio (subscriptp) can always be written as (i) the risk-free return over that
period, plus (ii), for all risky assets in the portfolio, the weighted average of the
returns over and above the risk-free rate:
̃rp−r =∑N
j=1xj( ̃rj−r), (19.7)with a weightxjdefined as the initial amount invested in assetj, divided by total
initial investment. A return over the risk-free rate is called anexcess return, and its
expected value is called therisk premium.
Example 19.2
You have 1000 to invest. Below, we show for three risky assets (denoted as 1, 2,
3) an initial price, the number of shares you buy, your total initial investment per
asset, the asset weight, a possible time-1 price, the corresponding return, and the
weighted return. The risky assets take up 900 of the money, so the balance, 100, is
invested risk-free at, say, 5 percent. In the table we see the weights,^4 and how they
sum to unity. We next compute the value of the portfolio at time 1, and see that
it has gone up to 1105, implying a (net rate of) return of 0.105,i.e. 10.5 percent.
The excess return is 10. 5 −5 = 5.5%, and this is exactly what you get by summing
the value-weighted “excess” returns on the three risky assets.
time-0 data and decisions time-1 result (excess) rates of return
j Vj, 0 nj njVj, 0 xj Vj, 1 njVj, 1 rj rj−r xj( ̃rj−r)
risky: 1 100 4 400 0.40 120 480 0.20 0.15 0.060
2 50 4 200 0.20 70 280 0.40 0.35 0.070
3 25 12 300 0.30 20 240 -0.20 -0.25 -0.075
subtotal =900 =0.90 =0.055
risk-free 0 +100 +0.10 105 +0.05
total =1000 =1.00 =1105 rp=0.105(^4) Note that the weights we need in the formula are initial weights, determined by time-0 numbers,
meaning that they are not stochastic.