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 sure rate of 4.2%. The probability distributions of the risky funds are:

	Expected Return	Standard Deviation
Stock fund (S)	12%	33%
Bond fund (B)	5%	26%

The correlation between the fund returns is 0.0308.

What is the Sharpe ratio of the best feasible CAL?

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 = 12% or 0.12

E (rB) = Expected return on Bond B = 5% or 0.05

And r is the risk free (T-bills) rate of return = 4.2% or 0.042

σS is the standard deviation of stock S = 33% or 0.33

σB is the standard deviation of Bond B = 26% or 0.26

Cov (S,B) is the correlation coefficient between the returns of S and B =ρ*σS*σB, where ρ= 0.0308

Therefore Cov (S, B) = 0.0308 *0.33 *0.26 = 0.002643

Now putting all the values into formula, we get

WS = [(0.12 – 0.042) *0.26^2 - (0.05 – 0.042) * 0.002643] / [(0.12– 0.042) *0.26^2 + (0.05 – 0.042)*0.33^2] - [{(0.12-0.042) + (0.05-0.042)}* 0.002643]

= 0.005253/0.005916702 = 0.8878 or 88.78%

And WB = 1 – WS = 1 – 0.8878 = 0.1122 or 11.22%.

The proportion of the optimal risky portfolio that should be invested in bond B is approximately 11.22%.

The return and standard deviation of the optimal risky portfolio -

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

= 88.78 % * 12% + 11.22% * 5% = 11.22%

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

=√ [0.8878^2*0.33^2+0.1122^2*0.26^2 +2* 0.8878*0.1122*0.002643]

=0.2953 or 29.53%

The Sharpe ratio of the best feasible CAL

= (E (rp) –rf)/σp

= (11.22% -4.2%) /29.53%

=0.2377

The Sharpe ratio of the best feasible CAL is 0.2377

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