Portfolios are defined as follows (illustration will follow):

i where wi is the known investment weights in security i and ri is the security return on security i. Unlike the above, simpler definitions, portfolios are the weighted sum of multiple random variables.

Portfolio Expectations

Although the weights are fixed and known constants, they cannot be pulled out of the summation, because they are indexed by i (each could be different from the others).

Var I X Wi ■ ri I = X ] X Wi ■ wj ■ Cov(n,^)]

Here is an illustration. A coin toss outcome is a random variable, f, and it will return either $2 (head) or $4 (tail). You have to pay $2 to receive this outcome. This looks like a great bet: The mean rate of return on each coin toss, E(rT) is 50%, the variance on each coin toss is

Var (rx) = 1/2 ■ (0% - 50%)2 + 1/2 ■ (100% - 50%)2 = 0.50 (B.26)

Therefore, the standard deviation of each coin toss is $0.707.

Now, bet on two independent such coin toss outcomes. You have $10 invested on the first bet and $20 on the second bet. In other words, your overall actual and unknown rates of return are

(The second equation is in random variable terms.) Now, your investment portfolio consists of the following investments

We can now use the formulas to compute your expected rate of return (E (f)) and risk (Sdv(r)). To compute your expected rate of return, use

E(f) = X wi ■ E(ri) = w1 ■ E(r1) + w2 ■ E(r2)

722 ■ Chapter B MORE RESOURCES

To compute your variance, use

= w1 ■ w1 ■ Cov(f1,f1) + w1 ■ w2 ■ Cov(r1,r2) +w2 ■ w1 ■ Cov(r2,r1) + w2 ■ w2 ■ Cov(r2,r2) = w1 ■ Cov(r1,r1) + 2 ■ w1 ■ w2 ■ Cov(r1,r2)

+w2 ■ Cov(r2,r2) = w1 ■ Var(r1) + 2 ■ w1 ■ w2 ■ Cov(r1,r2)

+w2 ■ Var(r2) = (1/3)2 ■ Var(r1) + 2 ■ w1 ■ w2 ■ 0 + (2/3)2 ■ Var(r2) = (1/9) ■Var (r1) + (4/9) ■Var (r2)

The standard deviation is therefore V0.278 = 52.7%. This is lower than the 70.7% that a single coin toss would provide you with.

Solve Now!

QB.10 Repeat the example, but assume that you invest $15 into each coin toss, rather than $10 and $20 respectively. Would you expect the risk to be higher or lower? (Hint: What happens if you choose a portfolio that invests more and more into just one of the two bets.)

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