6.

Expected Portfolio Return

(1) (2) (3) (4)

State of Probability of Portfolio Return if State Occurs Product

(2) — (3)

Economy State of

Economy

Bust .30 20% .06

Boom .70 26% .182

E(RP) = 24.2%

B-62 CORRADO AND JORDAN

7.

Calculating Portfolio Variance

(1) (2) (3) (4) (5)

State of Probability of Portfolio Squared Deviation from Product

(2) — (4)

Economy State of Returns if Expected Return

Economy State Occurs

Bust .30 .05 .053361 .016008

Boom .70 .38 .009801 .006861

σ2P = .022869

σP = 15.1225%

8. Based on market history, the average annual standard deviation of return for a single,

randomly chosen stock is about 50 percent. The average annual standard deviation for an

equally-weighted portfolio of many stocks is about 20 percent, or 60 percent less.

9. If the returns on two stocks are highly correlated, they have a strong tendency to move up

and down together. If they have no correlation, there is no particular connection between

the two. If they are negatively correlated, they tend to move in opposite directions.

10. An efficient portfolio is one that has the highest return for its level of risk.

11. Notice that we have historical information here, so we calculate the sample average and

sample standard deviation (using n - 1) just like we did in Chapter 1. Notice also that the

portfolio has less risk than either asset.

Annual Returns on Stocks A and B

Year Stock A Stock B Portfolio AB

1995 15% 55% 39%

1996 35 -40 -10

1997 -15 45 21

1998 20 0 8

1999 0 10 6

Avg returns 11% 14% 12.8%

Std deviations 19.17% 37.98% 18.32%

FUNDAMENTALS OF INVESTMENTS B-63

Intermediate Questions

12. Given the following information, calculate the expected return and standard deviation for

portfolio that has 40 percent invested in Stock A, 30 percent in Stock B, and the balance

in Stock C.

State of Probability Returns

Economy of State of Stock A Stock B Stock C Portfolio

Economy

Boom .40 15% 18% 20% 17.4%

Bust .60 5 0 -5 .5%

E(RP) = .4 — (.174) + .6 — (.005) = 7.26%

σ 2P = .4 — (.174 - .0726)2 + .6 — (.005 - .0726)2 = .006855; taking the square root, σP =

8.2793%.

E(RP) = .4 — (.30) + .6 — (.26) = 27.6%

13.

σ 2P = .42 — .652 + .62 — .452 + 2 — .4 — .6 — .65 — .45 — .3 = .18626; σP = 42.73%.

σ 2P = .42 — .652 + .62 — .452 + 2 — .4 — .6 — .65 — .45 — 1 = .2809; σP = 53%.

14.

σ 2P = .42 — .652 + .62 — .452 + 2 — .4 — .6 — .65 — .45 — 0 = .1405; σP = 37.48%.

σ 2P = .42 — .652 + .62 — .452 + 2 — .4 — .6 — .65 — .45 — (-1) = .0001; σP = 1%.

(.452 - .65 — .45 — .3)/(.452 + .652 - 2 — .65 — .45 — .3) = .255

15.

E(RP) = .255 — (.30) + .745 — (.26) = 27.02%

σ 2P = .2552 — .652 + .7452 — .452 + 2 — .255 — .745 — .65 — .45 — .3 = .1732

σP = 41.6%.

B-64 CORRADO AND JORDAN

16.

Risk and Return with Stocks and Bonds

Portfolio Weights Expected Standard

Stocks Bonds Return Deviation

1.00 0.00 14.00% 20.00%

0.80 0.20 12.20% 15.92%

0.60 0.40 10.40% 12.11%

0.40 0.60 8.60% 8.91%

0.20 0.80 6.80% 7.20%

0.00 1.00 5.00% 8.00%

17. True.

18. False.

Look at σp2 :

19.

σp2 = (xA — σA + xB — σB)2

= xA2 — σA2 + xB2 — σB2 + 2 — xA — xB — σA — σB — 1, which is precisely the

expression for the variance on a two-asset portfolio when the correlation is +1.

Look at σp2 :

20.

σp2 = (xA — σA - xB — σB)2

= xA2 — σA2 + xB2 — σB2 + 2 — xA — xB — σA — σB — (-1), which is precisely the

expression for the variance on a two-asset portfolio when the correlation is -1.

21. From the previous question, with a correlation of -1:

σp = xA — σA - xB — σB = x — σA - (1 - x) — σB

Set this to equal zero and solve for x to get:

0 = x — σA - (1 - x) — σB

x = σB/ (σA + σB)

This is the weight on the first asset.

22. If two assets have zero correlation and the same standard deviation, then evaluating the

general expression for the minimum variance portfolio shows that x = ½; in other words,

an equally-weighted portfolio is minimum variance.

FUNDAMENTALS OF INVESTMENTS B-65

Let ρ stand for the correlation, then:

23.

σp2 = xA2 — σA2 + xB2 — σB2 + 2 — xA — xB — σA — σB — ρ

= x2 — σA2 + (1 - x)2 — σB2 + 2 — x — (1 - x) — σA — σB — ρ

Take the derivative with respect to x and set equal to zero:

dσp2/dx = 2 — x — σA2 - 2 — (1 - x) — σB2 + 2 — σA — σB — ρ - 4 — x — σA — σB — ρ = 0

Solve for x to get the expression in the text.

B-66 CORRADO AND JORDAN