Question

Risk averse investors require a positive risk premium to compen- sate for risks they take while risk neutral investors donít. Consider a risky portfolio. The end-of-year cash áow derived from the portfolio will be either $42,000 with probability 0.6 or $140,000 with probability 0.4. Your friend is risk neutral. She is willing to pay $80,000 for this risky portfolio today.

1. (a) What is the risk free rate implied? Show your work.

2. (b) You are risk averse, you require a risk premium of 2.5% to invest in this portfolio. How much are you willing to pay for this portfolio? Show your work.

3. (c) Another friend is more risk averse than you. If she were to consider the same portfolio, is she willing to pay more or less than you? Explain.

Answer 1

a). Let the risk-free rate be rf. Then,

PV of risky portfolio = sum of PV of cash flows at the end of the year

80,000 = 0.6*42,000/(1+rf) + 0.4*140,000/(1+rf)

(1+rf) = (0.6*42,000+0.4*140,000)/80,000

1+rf = 1.015

rf = 0.015 or 1.50%

b). Your required return (r) = rf + risk premium = 1.50%+2.5% = 4.0%

PV of risky portfolio = [(0.6*42,000) + (0.4*140,000)]/(1+4%)

= 78,076.92 (You would be willing to pay this amount for the risky portfolio)

c). If your friend is more risk averse than you then she will be wiling to pay less than what you are paying for the risky portfolio as her risk premium will be greater than yours.