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 5.7%. The probability distributions of the risky funds are: Stock fund (S): Expected Return: 18% Standard Deviation: 47% Bond fund (B) Expected Return: 7% Standard Deviation: 41%

The correlation between the fund returns is 0.0317.

What is the Sharpe ratio of the best feasible CAL?

Answer 1

       Expected return       Standard deviation  
       Er          
Stock fund (s)        18 %   47   %
Bond fund(B)       7 %   41   %
T=bills rate (Rf) =       5.7   %      
Correlation between stock and bond fund = 0.0317  
                  
Covariance (CoV SB) = r * σS * σB                  
       0.0317*47*41 = 61.0859  

Weight of stock A as per Optimal Risky portfolio formula= ( ( Er S - Rf) * σB^2 - ( (Er B - Rf) * Cov SB )) / ((Er S - Rf)*σB^2 + ((Er B - Rf) * σS^2 )- ((Er S - Rf +ErB-Rf)* Cov SB ))

=(((18-5.7) * (41)^2 )- ((7-5.7) * 61.0859))/ (((18-5.7) * (41)^2)+ ( (7-5.7) * (47)^2)- ((18-5.7+7-5.7) * 61.0859))

= 20596.88833   /   22717.23176
              
So, weight of Stock fund =           90.67%  
weight of Bond fund =           9.33%  
              
Expected return = (weight of S * Expected return of S) + (Weight of B * Expected retun of B)              
= (90.67%*18%)+(9.33%*7%)          
= 16.9733   %      
              
expected retun of risky portolio is 16.9733   %
              
              
Standard deviation formula              
              
(σp) =   ( (wS * σS ) ^2 + (wB * σB ) ^2 + (2 * wB* wS*σB *σS* rSB) )^(1/2)          
= ((90.67%*47%)^2+(9.33%*41%)^2+(2*90.67%*9.33%*47%*41%*0.0317))^(1/2)          
= 42.9053   %      
              
Standard deviation of risky portfolio is            42.9053   %
              
Return to volatility ratio = (Expected return of portfolio - risk free rate of return) / Standard deviation              
= (16.9733%-5.7%) / 42.9053%          
= 0.2627482573          
              
So, return to volatility or sharpe ratio of best feasible cal is                0.2627