The correlation coefficient of Dell’s return versus McDonald’s was 0.34. What wa

Question

   

    

  

The correlation coefficient of Dell’s return versus McDonald’s was 0.34. What was the standard deviation of a portfolio invested half in Dell and half in McDonald’s? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

   

  

What was the standard deviation of a portfolio invested one-third in Dell, one-third in McDonald’s, and one-third in risk-free Treasury bills? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

  

  

What was the standard deviation if the portfolio is split evenly between Dell and McDonald’s 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? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

  

   

What was the approximate standard deviation of a portfolio composed of 100 stocks with betas of 1.28 like Dell? How about 100 stocks like McDonald’s? (Hint: Part (d) should not require anything but the simplest arithmetic to answer.) (Round your answers to 2 decimal places.)

  

Assume that Dell and McDonald’s had the following risk characteristics:

Explanation / Answer

Answer (a)

Standard Deviation = 19.65%

Answer (b)

Standard Deviation = 13.10%

Answer (c)

Standard Deviation = 19.65%

Answer (d)

Standard Deviation of

Portfolio of 100 stocks like Dell = 9.08%

Portfolio of 100 stocks like McDonalds = 5.95%

Standard deviation of Dell = 28.70%

Standard Deviation of McDonalds = 18.80%

Correlation coefficient = 0.34

Weight of Dell in portfolio =weight of McDonalds in Portfolio = 0.50

Covariance = correlation coefficient * standard deviation of Dell * standard deviation of McDonalds

                      = 0.34 * 28.70 * 18.80

                      = 183.4504

Variance of Portfolio = w1^2 * SD Dell^2 + w2^2 * SD McDonald^2 + 2*w1*w2*Covariance

= 0.5^2* 28.70^2 + 0.5^2*18.8^2 + 2*0.50*0.50*183.4504

= 0.25 * 823.69 + 0.25 * 353.44 + 91.7252

= 205.9225 + 88.36 + 91.7252

= 386.0077

Standard Deviation of Portfolio = Square Root (386.0077) = 19.6471 or 19.65% (rounded off)

  

Portfolio invested 1/3 in Dell, 1/3 in McDonalds and 1/3 in risk-free rate

Standard Deviation of risk-free asset = 0

Correlation coeffient of risk-free asset with that of stocks would be 0

Therefore the standard deviation of portfolio of 1/3 in Dell, 1/3 in McDonald and 1/3 in risk-free asset would be

Variance of Portfolio = (1/3)^2 * 28.70^2 + (1/3)^2 * 18.80^2 + 2*(1/3)*(1/3)*183.4504

Variance = 0.1111* 823.69 + 0.1111 * 353.44 + 40.766756

Variance = 91.511959 + 39.267184 + 40.766756

                  = 171.545899

Standard Deviation = Square Root (171.545899) = 13.09755 or 13.10% (rounded off)

By taking a margin finance and investing in stocks, the returns will be affected by the proportion of margin finance. However, as the amount is invested in the same stocks, then there would be no difference in standard deviation of the resultant portfolio. Hence the standard deviation would be 19.65%

If the portfolio consists of 100 stocks like Dell with Beta of 1.28, the correlation coefficient 1. Hence the standard deviation would be

Variance = (1/100)^2 * (28.70)^2 +-----+(1.100)^2*28.70^2

                = 0.01 * 823.69 +….+0.01*823.69 (100 times)

The term of 2*w1*w2*…w100*cov   will become infinitely small and hence can be ignored.

                  = 82.369

Standard Deviation = 9.07573 or 9.08%

Similarly for 100 stocks like McDonald

Variance = 0.10^2 * 18.80^2 +…+0.10^2*18.80^2 (100 terms)

                 = 0.001 *353.44 +---+0.001*353.44 (100 terms)

                = 0.35344 * 100

                = 35.344

Standard Deviation = 5.9450 or 5.95%

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