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 6%. The probability distribution of the risky funds is as follows:

	Expected Return			Standard Deviation
Stock fund (S)		21	%		28	%
Bond fund (B)		12			18

The correlation between the fund returns is 0.09.

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

Return on stocks ( E(s) ) = 21%

Standard devaition of stocks (SDs)= 28%

Return on bond ( E (b) ) = 12%

Standard deviation of bonds (SDb) = 18%

Risk free rate (rf) = 6%

Correlation (correl) = 0.09

Weight of investment in stock is given by :

ws = ( ( E(s) - rf) * SDb^2 - ( E(b) - rf) * SDs * SDb * Correl ) / ( E(s) - rf) × SDb^2 + (E(b) - rf) × SDs^2 - ( E(s) - rf + E(b) - rf) * SDs * SDb * correl

This is little confusing, so lets solve one by one

Numerator = ( ( E(s) - rf) * SDb^2 - ( E(b) - rf) * SDs * SDb * Correl )

Numerator = (0.21 - 0.06) × 0.18 ^2  - (0.12 - 0.06) * 0.28 * 0.18 * 0.09
= 0.46%

Denominator = ( E(s) - rf) × SDb^2 + (E(b) - rf) × SDs^2 - ( E(s) - rf + E(b) - rf) * SDs * SDb * correl

Denominator = (0.21 - 0.06) × 0.18 ^2 + (0.12 - 0.06) × 0.28^2 - (0.21 - 0.06 + 0.12 - 0.06) * 0.18 * 0.28 * 0.09

=0.86%

So

wS = 0.46 / 0.8611 = 0.5328

So

wB = 1 - 0.5328 = 0.4672

Now lets calculated mean and standard devaition of this optimal portfolio

Mean = Weight of S * Return of S + Weight of B * Return of B

Mean E(P) = (0.5328 × 0.21) + (0.4672 × 0.12) = 16.79%

Standard deviation = Square root of ( ( Weight of S * SD of S)^2 + ( Weight of B * SD of B)^2 + (2 * Weight of S* Weight of B * Correlation) )

Standard deviation = square root ( (0.5328 × 0.28^2) + (0.4672 × 0.18^2) + (2 × 0.5328 × 0.4672 × 0.09) )
= 17.72%

The reward-to-volatility ratio of the optimal CAL is:

E(P) - rf / standard deviation

= (16.79% - 6%) / 17.72%
= 0.61

Let me know if you have any doubts

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