Consider the following information:
Rate of Return if State Occurs
State of Probability of
Economy State of Economy Stock A Stock B Stock C
Boom .62 .10 .19 .37
Bust .38 .16 .08 − .04
a.
What is the expected return on an equally weighted portfolio of these three stocks?
b.
What is the variance of a portfolio invested 16 percent each in A and B and 68 percent in C?
2 Answers
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Return Variance
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a.
To find the expected return of the portfolio, we need to find the return of the portfolio in each state of the economy. This portfolio is a special case since all three assets have the same weight. To find the expected return in an equally weighted portfolio, we can sum the returns of each asset and divide by the number of assets, so the expected return of the portfolio in each state of the economy is:
Boom: RP = (.10 + .19 + .37) / 3 = .2200, or 22.00%
Bust: RP = (.16 + .08 −.04) / 3 = .0667, or 6.67%
To find the expected return of the portfolio, we multiply the return in each state of the economy by the probability of that state occurring, and then sum. Doing this, we find:
E(RP) = .62(.2200) + .38(.0667) = .1617, or 16.17%
b.
This portfolio does not have an equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get:
Boom: RP = .16(.10) +.16(.19) + .68(.37) =.2980, or 29.80%
Bust: RP = .16(.16) +.16(.08) + .68(−.04) = .0112, or 1.12%
And the expected return of the portfolio is:
E(RP) = .62(.2980) + .38(.0112) = .1890, or 18.90%
To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance of the portfolio is:
σp2 = .62(.2980 − .1890)2 + .38(.0112 − .1890)2 = .019379
