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ECON 4751: Final practice

July 28, 2022

Long Answer (25 points each)

Question 1. Par values of both the 1-Year zero and the 2-Year zero are $1,000. Suppose that the short rate today is 2 percent and the expected short rate next year is 7 percent.

• (3) Find the price of the 2-Year zero under the Expectations Hypothesis.

• (3) Find the price at which the 2-Year zero can be sold after holding it for one year under the Expec-tations Hypothesis

• (4)Suppose a short term-investor (one that is interested only in 1-Year investments) is willing to pay $800 for the 2-Year zero. Find the expected holding period return given this price that the investor is willing to pay.

• (4) Find the risk-premium implied by the expected holding period return

• (4)Find the yield to maturity of the 2-Year zero given the par value and the price that the investor demands.

• (4)Find the forward rate implied by this yield to maturity

• (3)Find the Liquidity Premium resulting from this forward rate

Question 2. This question asks you to apply your knowledge on The Markowitz Portfolio model.

Suppose we have an economy with three financial assets. There is a risk-free asset F that provides an annual return of rf = 0.02. There is a mutual fund D that specializes in long-term debt securities. It has an expected return E(rD) = 0.05 and a standard deviation σ(rD) = 0.1. There is also a mutual fund E that specializes in equity securities. It has an expected return E(rE) = 0.1 and a standard deviation σ(rE) = 0.2.

There is an investor with utility function given by

U(r) = E(r) − 2/1AV ar(r).

The investor has a risk aversion coefficient of 6. His goal is to maximize utility and he is able to invest his wealth across different assets. We will denote by y the fraction of total wealth that he will invest in risky assets D and E. Call the amount of wealth invested in risky assets "risky wealth". We will denote by wD and wE, where wD +wE = 1, the fraction of "risky wealth" that the investor will invest in assets D and E respectively.

• (1)Find the risk premium for each asset.

• (1)Draw and label each asset on a graph that has expected return on the vertical axis and standard deviation on the horizontal axis.

• (3)Describe in words what the set of all assets resulting from different choices of wD and wE will look like on the graph if ρ(rD, rE) = 1 and if ρ(rD, rE) = −1.

• (2)Now suppose the returns of the risky assets are uncorrelated. Sketch roughly on the graph the set of all risky assets resulting from different choices of wD and wE

• (3) Again, assuming that returns of risky assets are uncorrelated, write down the maximization problem that the investor will solve when choosing the optimal risky portfolio. Make sure you write the objective function in an expanded form. How will the investor be able to identify the optimal risky portfolio on the graph? Explain.

• (3) Suppose the optimal weights are w∗D = 0.65 and w∗E = 0.35. Find the risk and the expected return of the optimal risky portfolio

• (3) Write down the maximization problem that the investor is trying to solve when choosing the optimal complete portfolio. Make sure you write the objective function in an expanded form. How will the investor be able to identify the optimal complete portfolio on the graph? Explain.

• The investor will be able to identify the optimal complete portfolio on the graph by drawing indifference curves tangent to his CAL (the line defined by the risk-free asset and asset P).

• (3) Find the risk and the expected return of the optimal complete portfolio.

• (3) Suppose that the market portfolio is a portfolio with weights 0.5 and 0.5 on assets D and E. Find the β for assets D and E

• (3) Draw the security market line. Plot assets D and E on the graph if αD = 0.02 and αE = −0.02. Are stocks overpriced?

Question 3. This question asks you about options and option pricing models. Suppose there exists a stock A currently selling at price X.

Consider a portfolio that consists of: 1) a put option bought at strike price X − 10, 2) a put option written at strike price X − 5, 3) a call option written at strike price X + 5, and 4) a call option bought at strike price X + 10.

The premia for the above options are given by P1, P2, C1, and C2, respectively.

• (6)Draw the payoffs and profits for this portfolio

• (4) Write the piece-wise function that defines this portfolio’s profits

• (2)Why would an investor pursue this strategy?

• (1) Draw and fully label the stock price tree

• (1) Draw and fully label (as much as you can at this point) the call value tree

• (5) Using the binomial pricing model, how much should I pay today for a call option?

• (2) How much should an investor pay for a put option with the same exercise price?

• (4) You currently have a portfolio worth $100,000. Suppose an at-the-money put option has a delta of -0.4. Suppose value of the portfolio drops by 6%. What is the total loss of the portfolio if you were able to use the put? If the put didn’t exist, how could you replicate this outcome? Show your work.

True or False (5 points each). To receive full credit you must explain your answer, regardless of whether the statement is true or false.

Question 4. True or False: While it is possible to perfectly hedge away future risk using futures contracts, it isn’t possible to do so with options.

Question 5. True or False: The current price of Apple stock is S0. A futures contract on Apple stock with futures price F0 has a delivery date of 1 year. One year Treasury Bills are assumed to be risk-free and pay a (net) interest rate given by F0S0 − 1. Then Apple stock pays no dividends.

Question 6. True or False: It is possible to have an infinitely negative margin when buying on the margin regardless of going short or long in the security.

Question 7. True or False: Both a call option and a put option on Apple stock have the same expiration date and are at the money. The put option is cheaper than the call option.

Question 8. True or False: It is essential to put extra weight on high beta stocks in the optimal risky portfolio.