In: Finance
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 5.5%. The probability distributions of
the risky funds are:
Expected Return | Standard Deviation | |||
Stock fund (S) | 15 | % | 32 | % |
Bond fund (B) | 9 | % | 23 | % |
The correlation between the fund returns is 0.15.
Suppose now that your portfolio must yield an expected return of
12% 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.)
b-1. What is the proportion invested in the T-bill
fund? (Do not round intermediate calculations. Round your
answer to 2 decimal places.)
b-2. What is the proportion invested in each of
the two risky funds? (Do not round intermediate
calculations. Round your answers to 2 decimal
places.)
NUMBERS CANNOT BE ROUNDED UNTIL FINAL ANSWER OR ELSE IT WILL BE INCORRECT.
To find the fraction of wealth to invest in Stock fund that will result in the risky portfolio with maximum Sharpe ratio the following formula to determine the weight of Stock fund in risky portfolio should be used |
Where | |||||
Stock fund | E[R(d)]= | 15.00% | |||
Bond fund | E[R(e)]= | 9.00% | |||
Stock fund | Stdev[R(d)]= | 32.00% | |||
Bond fund | Stdev[R(e)]= | 23.00% | |||
Var[R(d)]= | 0.10240 | ||||
Var[R(e)]= | 0.05290 | ||||
T bill | Rf= | 5.50% | |||
Correl | Corr(Re,Rd)= | 0.15 | |||
Covar | Cov(Re,Rd)= | 0.0110 | |||
Stock fund | Therefore W(*d)= | 0.6466 | |||
Bond fund | W(*e)=(1-W(*d))= | 0.3534 | |||
Expected return of risky portfolio= | 12.88% | ||||
Risky portfolio std dev= | 23.34% |
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) | |||||
0.12=0.055*Proportion invested in Tbill+0.1288*(1-Proportion invested in Tbill) | |||||
Proportion invested in Tbill (answer b-1) = (0.1288-0.12)/(0.1288-0.055) | |||||
=0.1192 = 11.92% | |||||
proportion invested in risky portfolio = 1-proportion invested in tbill | |||||
=0.8808 | |||||
Proportion invested in Bond fund (answer b-2) =proportion invested in risky portfolio *weight of Bond fund | |||||
=0.3113 = 31.13% | |||||
Proportion invested in Stock fund (answer b-2) =proportion invested in risky portfolio *weight of Stock fund | |||||
=0.5695 = 56.95% | |||||
std dev of portfolio (answer a) = std of risky portfolio*proportion invested in risky portfolio | |||||
0.8808*0.233382=20.56% |