Question

As a portfolio manager, you are considering three mutual funds: a stock fund, a bond fund, and a money market bond that yields a sure rate of        2.75      %. Below is the information concerning the two risky funds: (Total 50 points)

	Expected return	Standard deviation
Stock fund	0.28	0.43
Bond fund	0.06	0.11

The correlation between fund returns is    0.08    .

1. Find the weights of the minimum-variance portfolio and calculate its expected return and its standard deviation. Show all work. (You can use Solver to verify your solution, but you need to present formulas with plugged-in numbers to receive full credit.) (12 points)

ω_mv(stocks)=

ω_mv(bonds)=

E_mv =

σ_mv =

2. Calculate the weights of the optimal risky portfolio and compute its expected return and standard deviation. You can use Solver to compute the optimal weights. To receive full credit, show the formulas with plugged-in numbers only for the expected return and standard deviation. (12 points)

ω_op(stocks)=

ω_op(bonds)=

E_op =

σ_op =

3. Using the obtained valued from previous two steps, show on the graph in the space provided: (a) efficient frontier, (b) optimal risky portfolio, (c) risk-free asset, (d) capital allocation line, (e) minimum variance portfolio. Don’t worry about scaling properly. Make sure to label the axes, as well as corresponding values for expected returns and standard deviations. (7 points)

4. What is the equation of the CAL? What is the Sharpe ratio of the CAL? (3 points)

5. Suppose your portfolio must reach a target of 10% and remain efficient, that is on the best feasible CAL.

1. What is the standard deviation of this portfolio? (3 points)

2. What is the proportion invested in the money market fund and each of the two risky funds (stock fund and bond fund separately)? (6 points)

6. If you were to use only the two risky funds and still require 10% return,
   1. Calculate the investment proportions of this portfolio. (3 points)

2. Calculate the portfolio standard deviation. Compare the obtained standard deviation to that of the optimal portfolio in step 5a. Explain why they are the same or different. (4 points)

Answer 1

Required E(Rc)	10%
T Bill rate	2.75%
Correlation ρ	0.08
	Expected Return (R )	Standard Deviation (σ)
Stock Fund (S)	28%	43%
Bond Fund (B)	6%	11%

Minimum Variance Portfolio
Wmin (S) =	σ2B - Cov (B,S)
	σ2S + σ2B - 2Cov (B,S)
Cov (B,S) =	ρ *σS *σB =0.08 * 0.43*0.11
Cov (B,S) =	0.003784

Wmin (S) = (0.11*0.11-0.003784) / (0.43*0.43+0.11*0.11-2*0.003784) =	0.0439
Wmin (B) = 1- Wmin (S) = 1-0.0439	0.9561

E(Rmin)	Wsmin*Rs+WBmin*Rb = 0.0439*0.28 + 0.9561*0.06 =	6.9658%
σmin	√(W2sminσ2S + W2Bminσ2B + 2WsminWBminCov (B,S)) =sqrt(0.04392*0.432+0.95612*0.112+2*0.0439*0.9561*0.003784	= 10.8328%

Proportion of Stocks in optimal risky portfolio
Ws=	(Rs-rf)*σ2B - (RB-rf)Cov (B,S)	= (0.28-0.0275)*0.112 - (0.06-0.0275)*0.003784)
	(RB-rf)*σ2S + (RS-rf)σ2B -(RS-rf+RB-rf) (Cov (B,S)	(0.06-0.0275)*0.432 + (0.28-0.0275)*0.112 - (0.28-0.00275-0.6-0.00275)*0.003784
Ws=	36.7173550%
Wb= (1-36.717%)	63.2826450%
Portfolio Return E (R )=	Ws*Rs+WB*Rb =0.36717*0.28+0.63282*0.06 =	14.0778%
Portfolio Std Dev σp	√(W2sσ2S + W2Bσ2B + 2WsWBCov (B,S)) = sqrt(0.36722	*0.282 + 0.632832 *0.0.62 +2*0.36717*0.63283*0.003784)=17.7572%

Sharpe Ratio (Risk to Reward)	E(Rp)-rf
	σp
SR= (0.140778-00275) / 0.17752 =	0.6379
If portfolio is required to yield return of given percentage, then CAL is used to find Std dev
σc is given by, for E(Rc) =	10%
To find Proportion of investment in T bills
Let y be proportion invested in T bills
E(Rc) = (1-y) rf+y E(Rp) =y* (E(Rp)-rf)+rf
y = (0.1-0.0275) / (0.140778-0.0275) =	0.6400
1-y =1-0.64 =	0.3600
Proportion of stocks in complete portfolio
y*WS = 0.64*0.36717 =	0.2350
Proportion of Bonds in complete portfolio
y*WB = 0.64*0.63282 =	0.4050

Proportion in market fund = 1-y = 0.36 = 36%