Question

Here are some historical data on the risk characteristics of Ford and Harley Davidson.

	Ford	Harley Davidson
β (beta)	1.62	0.80
Yearly standard deviation of return (%)	30.0	16.5

Assume the standard deviation of the return on the market was 17.0%.

1. a. The correlation coefficient of Ford’s return versus Harley Davidson is 0.30. What is the standard deviation of a portfolio invested half in each share?

   b. What is the standard deviation of a portfolio invested one-third in Ford, one-third in Harley Davidson, and one-third in risk-free Treasury bills?

   c. What is the standard deviation if the portfolio is split evenly between Ford and Harley Davidson and is financed at 50% margin, that is, the investor puts up only 50% of the total amount and borrows the balance from the broker?

   d-1. What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 1.62 like Ford?

   d-2. What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 0.80 like Harley Davidson? (For all requirements, use decimals, not percent, in your calculations. Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)

Here are some historical data on the risk characteristics of Ford and Harley Davidson. Ford Harley Davidson 0.80 16.5 ß (beta) Yearly standard deviation of return (%) 1.62 30.0 Assume the standard deviation of the return on the market was 17.0%. a. The correlation coefficient of Ford's return versus Harley Davidson is 0.30. What is the standard deviation of a portfolio invested half in each share? b. What is the standard deviation of a portfolio invested one-third in Ford, one-third in Harley Davidson, and one-third in risk- free Treasury bills? c. What is the standard deviation if the portfolio is split evenly between Ford and Harley Davidson and is financed at 50% margin, that is, the investor puts up only 50% of the total amount and borrows the balance from the broker? d-1. What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 1.62 like Ford? d-2. What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 0.80 like Harley Davidson? (For all requirements, use decimals, not percent, in your calculations. Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.)

Answer 1

Given,

sigma(F) = 30%

sigma(HD) = 16.5%

Beta(F) = 1.62

Beta(HD) = 0.80

sigma(market) = 17%

Corr(F,HD) = 0.30

a) We know, the sigma of the portfolio is given as:

Oportfolio = wigi + w202 + 2w1 W2P120102

Here w1 = 50%, w2 = 50%, rho = 0.30

So sigma(portfolio)2 = (0.5^2)(0.30^2) + (0.5^2)(0.165^2) + 2(0.5)(0.5)(0.30)(0.30)(0.165)

=> sigma(portfolio) = 19.16%

b) w1 = 1/3, w2 = 1/3

The treasury bill has no risk so it won't affect the calculations.

So, using the same formula as above:

sigma(portfolio) = 12.65%

c) The standard deviation of the portfolio does not change due to change in financing as long as the portfolio contains the same stock.

So, sigma(portfolio) = 19.16%

d) If the portfolio contains the stocks having the save beta, then its standard deviation will be equal to the standard deviation of that particular stock. So, here a portfolio containing 100 stocks of Ford, then the standard deviation will remain as the standard deviation of one Ford stock = 30%

e) If the portfolio contains the stocks having the save beta, then its standard deviation will be equal to the standard deviation of that particular stock. So, here a portfolio containing 100 stocks of HD, then the standard deviation will remain as the standard deviation of one HD stock = 16.5%