In: Finance
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%?
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%