Question

In: Finance

The risk-free rate is 1% while the expected return and standard deviation of the market portfolio...

The risk-free rate is 1% while the expected return and standard deviation of the market portfolio (S&P

500) are 9% and 19%, respectively.

(a) What is the standard deviation of a combination of risk-free security and S&P 500 that has an

expected return of 12%? What is its probability of loss? Assume that the S&P 500 returns have

a normal probability distribution.

(b) The optimal allocation to S&P 500 for an investor is 60%. What will be the optimal allocation to

S&P 500 for this investor if the standard deviation of S&P 500 returns were to increase to 25%?

Solutions

Expert Solution

a) What is the standard deviation of a combination of risk-free security and S&P 500 that has an expected return of 12%? What is its probability of loss? Assume that the S&P 500 returns have a normal probability distribution.

If w is the proportion invested in risk free security then,

Expected return, Rp = w x 1% + (1 - w) x 9% = 12%

Hence, w = (12% - 9%) /(1% - 9%) = -0.375 = - 37.50%

Standard deviation, σp = (1 - w) x σrisky = (1 + 37.50%) x 19% = 26.125%

Hence, portfolio return is normally distributed with mean of 12% and standard deviation of 26.125%

z = (X - mean) / std deviation = (0 - 12%) / 26.125% = -0.459330144

Probability of loss = Probability of X < 0 = Probability of Z < -0.459330144 (= -0.46) = 0.3228 = 32.28%

Part (b)

Allocation to S&P = 60%

Hence, w = allocation to risk free asset = 1 - 60% = 40%

Expected rerun = 40% x 1% + 60% x 9% = 5.8%

Stand dev = 60% x 19% = 11.40%

If the standard deviation of S&P 500 returns were to increase to 25%, the allocation has to be changed to 11.40% / 25% = 45.60%


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