Question

You have $100,000 invested in a complete portfolio that consists of a portfolio of risky assets (P) and T-Bills. The information below refers to these assets.

E(r_p)=10%

σ_p =20%

T-Bill rate=3%

Proportion of T-Bill in the complete portfolio: 30%

Proportion of risky portfolio P in the complete portfolio: 70%

Composition of P:

Stock X 30%

Stock Y 25%

Stock Z 45%

Total     100%

1. What is the expected return on your complete portfolio?
2. What is the standard deviation of your complete portfolio?
3. What are the dollar amounts of Stocks X, Y, and Z, respectively, in your complete portfolio?
4. If your degree of risk aversion is A=2, is your complete portfolio optimal? (Assume P is the optimal risky portfolio.)
   

Answer 1

(i) Expected Return on complete portfolio:

= Wp(Rp) + Wt(Rt)

= (0.7*10%) + (0.3*3%)

= 7.9%

(ii) Standard deviation of complete portfolio:

Since the T-bills are essentially risk-free, they have no standard deviation.

Thus, the complete portfolio standard deviation =

= Wp* (Rp)

= 0.7*(20%)

= 14%

(iii)

The given portfolio allocation can be visualised as below:

Complete Portfolio $100,000 Portfolio P $70,000 T-bills $30,000 Stock X (30% of Portfolio P) Stock Z (45% of Portfolio P) Stock Y (25% of Portfolio P)

Value of Portfolio X = 30%* 70,000 = $21,000

Value of Portfolio Y = 25%* 70,000 = $17,500

Value of Portfolio Z = 45%* 70,000 = $31,500

(iv) Optimal Portfolio:

The optimal allocation formula is given as:

EG)- 2

Thus,

Risk portfolio allocation = (0.10-0.03)/(2*0.20^2)

Risk portfolio allocation = 0.875

Thus, portfolio allocation to the risk portfolio should be 87.5% and to the risk-free portfolio should be 12.5% (100%-87.5%)