You are considering investing $1,000 in a complete portfolio. The complete portf

Question

You are considering investing $1,000 in a complete portfolio. The complete portfolio is composed of Treasury bills that pay 5% and a risky portfolio, P, constructed with two risky securities, X and Y. The optimal weights of X and Y in P are 60% and 40%, respectively. X has an expected rate of return of 14%, and Y has an expected rate of return of 10%. The risky portfolio, P, has a standard deviation of 0.7. Assuming you decide to hold a complete portfolio that has an expected return of 8%:

a. What is the expected return of your risky portfolio?

b. What is the weight you invested in Treasury bills? What is the weight you invested in risky portfolio? (hint: rc = wp*rp + wf*rf; and wf = 1-wp)

c. What is the variance of the complete portfolio?

d. What is the price of risk of the complete portfolio?

e. What is the sharp ratio of the CAL?

** You must show calculation steps not using excel*

Explanation / Answer

a. Expected return of risky portfolio

Expected Return = R1P1 + R2P2

R = Expected rate of return

P = Probability

= 0.14*0.60 + 0.10*0.40

= 0.084+0.04

= 12.4%

b. You have an expected return of 8%.

Expected return = WP * RP + WF * RF

WP = Weight of portfolio

RP = Return on portfolio

WF = Weight of T-bills

RF = Return on T-bills

8% = WP * 0.124 + WF * 0.05

Now WF = 1-WP

0.08 = WP*0.124 + (1-WP)*0.05

= 0.08 = 0.124WP + 0.05 - 0.05WP

= 0.08 - 0.05 = 0.074WP

= 0.03 = 0.074WP

WP = 0.4054

WF = 1-0.4054

= 0.5946

So, weight of t-bills = 59.46% (approx.)

Weight of risky portfolio = 40.54% (approx.)

c. Variance of complete porifolio

SD of Risky portfolio = 0.7

Variance = 0.7^2 = 0.49

SD of treasury bills = 0 (Becasue t-bills are risk free)

Variance = Variace of P * WP + Variance of F * WF

= 0.49 * 0.4054 + 0 * 0.5946

= 0.1986 + 0

= 19.86%

d. What is the price of risk of the complete portfolio?

Under root of 0.1986

= 0.4456

e. Sharpe ratio

Sharpe ratio = (Mean portfolio return - Risk free rate)/Standard deviation of portfolio

= (0.08 - 0.05)/Under root of 0.1986

= 0.03/0.4456

=6.73%

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