Question

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 7%. The probability distribution of the risky funds is as follows: Expected Return Standard Deviation Stock fund (S) 18 % 35 % Bond fund (B) 15 20 The correlation between the fund returns is 0.12. What is the Sharpe ratio of the best feasible CAL? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.)

Answer 1

To calculate the Sharpe ratio of the best feasible CAL, first we have to calculate the expected return and standard deviation of the optimal risky portfolio.

The formula for proportion of the optimal risky portfolio invested in the stock fund is as follows-

WS = [ (E (rS) – rf) *σB^2   - (E (rB) – rf) * Cov (S,B)] /[ (E (rS) σB^2 – rf) *σB^2   + (E (rB) – rf) * σS^2)]- [(E (rS) – rf) + (E (rB) – rf)] Cov (S,B)]

And WB = 1 – WS

Where WS = weight of stock S in portfolio

WB = weight of Bond fund B in portfolio

E (rS) = Expected return on stock S = 18% or 0.18

E (rB) = Expected return on Bond B = 15% or 0.15

And r is the risk free (T-bills) rate of return = 7% or 0.07

σS is the standard deviation of stock S = 35% or 0.35

σB is the standard deviation of Bond B = 20% or 0.20

Cov (S,B) is the correlation coefficient between the returns of S and B =ρ*σS*σB, where ρ= 0.12

Therefore Cov (S, B) = 0.12 *0.35 *0.20 = 0.0084

Now putting all the values into formula, we get

WS = [(0.18 – 0.07) *0.20^2 - (0.15 – 0.07) * 0.0084] / [(0.18– 0.07) *0.20^2 + (0.15 – 0.07)*0.35^2] - [(0.18-0.07) + (0.15-0.07)* 0.0084]

= 0.003728/0.012604 = 0.2958 or 29.58%

And WB = 1 – WS = 1 – 0.2958 = 0.7042 or 70.42%.

The proportion of the optimal risky portfolio that should be invested in bond B is approximately 70.42%.

The return and standard deviation of the optimal risky portfolio -

Expected return on the portfolio E (rp) = WS * E (rS) + WB * E (rB)

= 29.58 % * 18% + 70.42% * 15% = 15.89%

Standard deviation of portfolio σp = √ [(WS^2 * σS^2 + WB^2 * σFB^2 +2*WS*WB*Cov(S, B)]

=√ [0.2958^2*0.35^2+0.7042^2*0.20^2 +2* 0.2958*0.7042*0.0086]

=0.1845 or 18.45%

The Sharpe ratio of the best feasible CAL

= (E (rp) –rf)/σp

= (15.89% -7%) /18.45%

=0.4818

The Sharpe ratio of the best feasible CAL is 0.4818

(There could be minor difference between the answer due to rounding off the intermediate calculations)