Theory of Finance (BUSS 6201)

Theory of Finance (BUSS 6201)
Assignment #1 – Due October 10, 2023

1. A consumer has a utility function for a two-period horizon is

u(c1, c2) = ln c1 + 1/2 ln c2,

with an endowment of $2,000 for the current period and $3,000 for the next period. The interest rate is 8%.

(a) Draw the consumer’s indifference curve that passing through the point of his endowment. What is the utility of the consumer without borrowing or lending?

(b) Draw the market opportunity line. Is the consumption plan in Part (a) optimal?

(c) Find the optimal consumption plan for the consumer.

(d) Should the consumer borrow or lend to boost his utility? If yes, how much does the consumer borrow or lend?

2. The return/risk profiles of assets A and B are given below:
State
 1
  2
3
4
5
Probability
 0.10
 0.25
  0.35
 0.20
 0.10
Asset A
 -0.01
  0.05
 0.10
 0.20
 0.30
Asset B
 0.40
  0.30
  0.15
  -0.02
  -0.05
The risk-free rate is 3%.

(a) Is there a stochastic dominance between A and B?

(b) Under mean-variance criterion, which asset is preferred if an investor has a risk aversion index ρ = 5? What if ρ = 7?

(c) Given A and B are the only risky securities, estimate the market portfolio by maximizing the Sharpe ratio of portfolios of the two risky assets. Calculate the beta of A and B. Are assets A and B mean-variance efficient?

(d) If your target return is 12%, how much will you invest in the two risky assets and the risk free asset?

3. Bob and Ben are the only consumers/investors, and they prefer a smooth consumption over time with the natural logarithmic utility function. Each of them has $10 now. However, their future endowments depend on the outcome of the economic state. If the economic state is “good”, Bob will have $20 and Ben will have noth ing. If the economic state is “bad”, Bob will have nothing and Ben will have $30. Both “good” and “bad” states will occur with equal probabilities, and the two pure securities (digital options) are traded on the market.

(a) Formulate the utility maximization models for both Bob and Ben. Note, a smooth consumption preference means equal weighting on the time now and future consumption.

(b) What are the equilibrium prices of the two pure securities?

(c) What positions in the two pure securities should Bob take to finance his optimal consumption plan?

(d) If a zero-coupon bond is selling at $98 per $100 face value, is there an arbitrage opportunity?

4. Suppose there are two market states. You are given the following information:
Security
 State one
  State two
 Security price
A
  $10
  $4
 $7.78
B
  $12
  $5
 $9.40

(a) Is the market complete? Give a brief explanation for your answer.

(b) What are the prices of the pure securities?

(c) What is the price of a third security C which pays $6 in state one and $10 in state two?

(d) What should the risk-free rate be? If the risk-free is 4%, will there be an arbitrage opportunity? If yes, how would you construct an arbitrage portfolio using the risk-free asset, A, and B?

(e) What are the risk neutral probabilities? Explain how you would price the se curity in (c) with risk neutral probabilities.

5. Consider a two–factor model. Two well-diversified portfolios are given as

R1 = 0.17−0.8227F1+2.1024F2+ε1 and R2 = 0.13+3.3364F1−2.9707F2+ε2.

The risk-free rate is 2%.

(a) Calculate the hedging portfolio weights for the two factors and their risk premiums.

(b) What is the expected return on a security with β1 = 0.5 and β2 = 0.8?

(c) If the changes in the two risk factors are 0.02 and -0.01 in the two risk factors, how much is expected to change on the return of the security in (b)?

2023-10-13