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 5.9%. The probability distributions of the risky funds are: Stock fund (S) Bond fund (B) Expected Return 20% 9% Standard Deviation 49% 43% The correlation between the fund returns is .0721. Suppose now that your portfolio must yield an expected return of 18% and be efficient, that is, on the best feasible CAL. a. What is the standard deviation of your portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Standard deviation b-1. What is the proportion invested in the T-bill fund? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Proportion invested in the T-bill fund

Answer 1

To find the fraction of wealth to invest in stock fund that will result in the risky portfolio with minimum variance
the following formula to determine the weight of stock fund in risky portfolio should be used
w(*d)= ((Stdev[R(e)])^2-Stdev[R(e)]*Stdev[R(d)]*Corr(Re,Rd))/((Stdev[R(e)])^2+(Stdev[R(d)])^2-Stdev[R(e)]*Stdev[R(d)]*Corr(Re,Rd))
							Where
						stock fund	E[R(d)]=	20.00%
bond fund	E[R(e)]=	9.00%
stock fund	Stdev[R(d)]=	49.00%
bond fund	Stdev[R(e)]=	43.00%
	Var[R(d)]=	0.24010
	Var[R(e)]=	0.18490
T bill	Rf=	5.90%
Correl	Corr(Re,Rd)=	0.0721
Covar	Cov(Re,Rd)=	0.0152
stock fund	Therefore W(*d)=	0.4301
bond fund	W(*e)=(1-W(*d))=	0.5699
	Expected return of risky portfolio=	13.73%
	Risky portfolio std dev (answer Risky portfolio std dev)=	33.45%
Where
Var = std dev^2
Covariance = Correlation* Std dev (r)*Std dev (d)
Expected return of the risky portfolio = E[R(d)]*W(*d)+E[R(e)]*W(*e)
Risky portfolio standard deviation =( w2A*σ2(RA)+w2B*σ2(RB)+2*(wA)*(wB)*Cor(RA,RB)*σ(RA)*σ(RB))^0.5
Desired return = tbill return*proportion invested in tbill+risky portfolio return *proportion invested in risky portfolio
= tbill return*proportion invested in tbill+risky portfolio return *(1-proportion invested in tbill)
0.18=0.059*Proportion invested in Tbill+0.1373*(1-Proportion invested in Tbill)
Proportion invested in Tbill (answer b-1) = (0.1373-0.18)/(0.1373-0.059)
=-0.5453 (-54.53%)
proportion invested in risky portfolio = 1-proportion invested in tbill
=1.5453 (154.53%)
Proportion invested in bond fund (answer proportion invested in bond fund) =proportion invested in risky portfolio *weight of bond fund
=0.8807 (88.07%)
Proportion invested in stock fund (answer b-2) =proportion invested in risky portfolio *weight of stock fund
=0.6646 (66.46%)
std dev of portfolio (answer a) = std of risky portfolio*proportion invested in risky portfolio
1.5453*0.3345=51.69%