Solution to Problem 6: Dell vs. Microsoft
Problem 6: Dell vs. Microsoft
Historical data on the key risk characteristics of Dell and Microsoft stocks are shown in the following
table.
Stock
Beta
Standard Deviation
Dell
2.21
62.7%
Microsoft
1.81
50.7%
Correlation
coefficient
0.66
Assume the standard deviation of the return on the market portfolio to be 15%.
a) What is the standard deviation of a portfolio invested half in Dell and half in Microsoft?
P2 = x1212 + x2222 + 2x1x21212
Thus:
P2 = (0.52)(0.6272)+(0.52)(0.5072)+2(0.5)(0.5)(0.66)(0.627)(0.507)=0.26745
P = 0.517 = 51.7%
b) What is the standard deviation of a portfolio invested one-third in Dell, one-third in Microsoft
and one-third in Treasury Bills?
One of these securities, Treasury bills, has zero risk and, hence, zero standard deviation.
Thus:
P2 = (1/3)2(0.6272)+(1/3)2(0.5072)+2(1/3)(1/3)(0.66)(0.627)(0.507)
P2 = 0.11887
P = 0.345 = 34.5%
Another, completely equivalent, way to think of this portfolio is that it is comprised of one-third
Treasury Bills and two-thirds a portfolio which is half Dell and half Microsoft. Because the risk of
T-bills is zero, the portfolio standard deviation is two-thirds of the standard deviation computed in
part a) above, or
P = (2/3)(0.517) = 0.345 = 34.5%
c) What is the standard deviation if the portfolio is split evenly between Dell and Microsoft and is
financed with 50% margin?
With 50% margin, the investor invests twice as much money in the portfolio as he had to begin with.
Thus, the risk is twice that found in part a) when the investor is investing only his own money, hence :
P = 2  51.7% = 103.4%
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d) What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of
2.21 like Dell? How about 100 stocks like Microsoft?
With 100 stocks, the portfolio is well diversified (see Problem 5, Problem Set II), and hence the
portfolio standard deviation depends almost entirely on the average covariance of the securities in the
portfolio (measured by beta) and on the standard deviation of the market portfolio. Thus, for a portfolio
made up of 100 stocks, each with beta = 2.21, the portfolio standard deviation is approximately: (2.21 
15%) = 33.15%. For stocks like Microsoft, it is: 1.81  15% = 27.15%.
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