Question

In: Finance

Q2) A pension fund manager is considering three mutual funds. The first is a stock fund,...

Q2) 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 3%. The probability distribution of the funds is as follows: Show all formulas and provide step by step calculations. Do NOT use Excel formulas!

Expected Return

Standard Deviation

Stock Fund

20%

40%

Bond Fund

10%

15%

Risk-free

3%

Correlation

20%

  1. Find the investment proportions in the minimum variance portfolio (MVP) of the two risk asset (5 points)
  2. Find the expected return and standard deviation for the minimum variance portfolio (5 points)
  3. What portion of your wealth should go to S and B respectively to achieve the tangent portfolio (i.e., the portfolio with the highest Sharpe ratio) (5 points)

Solutions

Expert Solution

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)]= 10.00%
stock fund Stdev[R(d)]= 40.00%
bond fund Stdev[R(e)]= 15.00%
Var[R(d)]= 0.16000
Var[R(e)]= 0.02250
T bill Rf= 3.00%
Correl Corr(Re,Rd)= 0.2
Covar Cov(Re,Rd)= 0.0120
stock fund Therefore W(*d)= 0.0662
bond fund W(*e)=(1-W(*d))= 0.9338
Expected return of risky portfolio= 10.66%
Risky portfolio std dev (answer Risky portfolio std dev)= 14.77%
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
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
w(*d)= ((E[Rd]-Rf)*Var(Re)-(E[Re]-Rf)*Cov(Re,Rd))/((E[Rd]-Rf)*Var(Re)+(E[Re]-Rf)*Var(Rd)-(E[Rd]+E[Re]-2*Rf)*Cov(Re,Rd)
Where
stock fund E[R(d)]= 20.00%
bond fund E[R(e)]= 10.00%
stock fund Stdev[R(d)]= 40.00%
bond fund Stdev[R(e)]= 15.00%
Var[R(d)]= 0.16000
Var[R(e)]= 0.02250
T bill Rf= 3.00%
Correl Corr(Re,Rd)= 0.2
Covar Cov(Re,Rd)= 0.0120
stock fund Therefore W(*d)= 0.2458
bond fund W(*e)=(1-W(*d))= 0.7542

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