Question

Your portfolio has consisted of a single stock, IBM, which you inherited from your grandmother. You realize that such a portfolio is not diversified and you would like to start on the path to fiscal stability. You are going to purchase Proctor & Gamble stock as a step in the right direction toward diversification. IBM has an annualized standard deviation of 0.54 and P&G 0.28. The correlation coefficient between the two stocks is 0.65. IBM has a beta of 1.1 and P&G 0.6. You are also considering investing in a 10 year Treasury bond that has a yield to maturity of 3.15%. This Treasury bond has a semi-annual coupon rate of 2.85% and a face value of $10,000. The market risk premium is 5.2%.

Part a.
If you choose a portfolio that is 70% invested in IBM and 30% invested in P&G stock, what are the portfolio return and standard deviation?

Part b.
What price should you pay for the Treasury bond if you choose to invest in it?

Part c.
Suppose you create a portfolio of 50% IBM stock, 30% P&G and 20% Treasury bond. What are the beta and expected return of this portfolio?

Answer 1

Risk free rate = Yield to Maturity on Treasury bonds = 3.15%

So, as per CAPM

Rate of Return for IBM = Risk free rate + Beta of IBM * market risk premium

=3.15%+1.1*5.2%

=8.87%

Rate of Return for P&G = Risk free rate + Beta of P&G * market risk premium

=3.15%+0.6*5.2%

=6.27%

a) The return of a portfolio is the weighted return of the two stocks

So Return of this portfolio = 0.70 * 8.87% +0.30 *6.27% = 8.09%

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

So, standard deviation of portfolio =sqrt (0.7^2*0.54^2+0.3^2*0.28^2+2*0.7*0.3*0.54*0.28*0.65)

=sqrt(0.1912176)

=0.43728 =43.73%

b) Coupon amount = $10000 *2.85%/2 = $142.50

Semiannual YTM = 3.15%/2 = 0.01575

No of periods = 10*2 = 20

Bond price = 142.5/0.01575*(1-1/1.01575^20)+10000/1.01575^20 = $9744.36

Price of $9744.36 should be paid for the bond

c)

Beta of a riskfree security = 0

As the Beta and expected return of a portfolio is the weighted average Beta and weighted average return of individual components

Beta of portfolio = 0.5*1.1+0.3*0.6+0.2*0

=0.73

Beta of the Portfolio is 0.73

Expected Return = 0.5*8.87%+0.3*6.27%+0.2*3.15% = 6.946%