Question

Tom has $10,000. He can invest the money in (1) a corporate bond, (2) a stock, and (3) the risk-free T-bill. The table below provides these assets’ expected returns and standard deviations: Bond (D) Stock (E) T-Bill (F) Expected Return 5% 10% 2% Standard Deviation 10% 20% 0 The coefficient of correlation between the corporate bond and the stock (ρDE) is 30%. Tom has a risk aversion coefficient of A=5. To construct the optimal portfolio with two risky assets and one risk-free asset, Tom first constructs an optimal risky portfolio with the two risky assets. He then decides how to allocate his money between the optimal risky portfolio and the risk-free asset. (a) What are the weights of the bond and stock in the optimal portfolio of two risky assets? (20 points) (b) What are the expected return, standard deviation, and Sharpe ratio of the optimal risky portfolio in (a)? (15%) (c) Tom wants to include both risky assets and the risk-free asset in his portfolio. What are the optimal weights that the investor choose for the optimal risky asset portfolio from (a) and for the risk-free T-bill? (10 points) (d) How many dollars should the investor invest in the corporate bond, the stock, and the Tbill, respectively? (5 points) (e) What is the Sharpe ratio of the optimal portfolio in (c)? Is it higher or lower than the Sharpe ratio of the optimal risky portfolio in (a)? (5%) (f) Douglas has a risk aversion coefficient of A=3. Would Tom and Douglas choose the same or different optimal risky portfolio in (a)? Would the two investors choose the same optimal portfolio in (d)? (5%)

Answer 1

Answer.

Investment Option	Corporate Bond (D)	Stock (E)	T-Bill (F)
Expected Return (R)	5%	10%	2%
Standard Deviation (σ)	10%	20%	0

The coefficient of correlation between the corporate bond and the stock (ρDE) = 30%

From Formula

Percentage of investment in Corporate Bond (D)= (σE2 - ρDE* σD*σE )/ (σD2 +σE2 - 2* ρDE* σD*σE)

= (20^2-0.30*20*10)/(10^2+20^2-2*0.30*20*10)= (400-60)/(100+400-120)= 0.8947 i.e. 89.47%

Percentage of investment in Stock (E)= 1-0.8947= 0.1053 i.e..10.53%

Answer (a)Weight of Bond( WD) = 89.47% and Weight of Stock (WE )= 10.53%

Answer (b) Expected Return from portfolio= WD*RD+WE*RE= .8947*5+0.1053*10= 5.53%

Standard deviation of portfolio(σDE)=( (WD*σD)^2+(WE*σE)^2+2* ρDE* σD*σE*WD*WE )^1/2

  = ((0.8947*10)^2+(0.1053*20)^2+2*0.30*0.8947*0.1053*10*20)^1/2

=(89.04881+4.435236+11.30543)^1/2= 9.79%

Sharp Ratio= (RDE- Rf)/σDE= (5.53-2)/9.79= 0.3606 i.e..36.06%

Answer (c) Tom's Risk aversion coefficient= 5,

Optimal Investment in Risk free and Optimal Portfolio= WDE*σDE

5= WDE*9.79= WDE= 5/9.79= 0.5107

So, Investment in Risk Free T-Bill (WF)= 1-0.5107= 0.4893

Answer (d) Investment in Corporate Bond = $10000*0.5107*0.8947= $4,569.23

Investment in Stock (E)= $10000*0.5107*0.1053= $537.77

Investment in T-Bill (F)= $10000*0.4893=$4893

Answer 2(e).

Expected Return of portfolio in part c = 0.6329*5.53+0.3671*2= 4.23%

Standard Deviation of portfolio in part c= 5 (Refer note given below)

Sharp Ratio of Portfolio in part (c)= (RDEF-RF)/σDEF= (4.23-2)/5= 0.4468 i.e.. 44.68%

Answer 2(f).

In the case of portfolio of risky project both "Tom" and "Douglas" select same proportion of stocks if they want to be include both the stocks (For part (a)).

Now, if risk aversion coefficient of Douglas is less than the Tom then, Douglas would invest more in risk free T-Bill and Less in Risky Portfolio i.e.. (portfolio would be different in the case of part d)

Note:- It is assume that Tom 's expected risk is equal to risk aversion coefficient so S.D.. of portfolio in part c is equal to 5.