FN2190 Asset Pricing and Financial Markets 2020
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FN2190 Asset Pricing and Financial Markets
2020
Question 1
You observe that the current three-year discount factor for default-risk free cash flows is 0.68. Remember, the t-year discount factor is the present value of $1 paid at time t, i.e. = (1 + )− , where is the t-year spot interest rate (annual compounding). Assume all bonds have a face value of $100 and that all securities are default-risk free. All cash flows occur at the end of the year to which they relate.
a) What is the price of a zero-coupon bond maturing in exactly 3 years? (5 marks)
b) Your friend makes the following observation about the above bond: “Since there is no risk of default and there are no coupons to re-invest, buying the 3-year zero coupon bond today is a risk-free investment; that is, you are guaranteed to earn an annual return of 13.72% (i.e. 3-year spot rate)”. Explain why your friend is not entirely correct and how you would modify the statement to make it correct.
c) In addition to the bond in (a), you observe the following: a 2-year coupon bond paying 10% annual coupons with a market price of $97, and two annuities that are trading at the same market price as each other. The first annuity matures in 3 years and pays annual cash flows of $20, while the second annuity pays annual cash flows of $28 and matures in 2 years. Using this information:
i. Complete the term structure of interest rates, i.e. determine the one- and two-year discount factors, d1 and d2 , respectively. (6 marks)
ii. Determine the price of the annuities. (2 marks)
d) Assuming annual compounding, determine the implied one-period forward rates 2 (i.e. between year 1 and 2) and 3 (i.e. between year 2 and 3) in this economy. What inference can you make about the market’s estimate of the one-year spot interest rate at = 1 if the liquidity preference theory is correct? (5 marks)
e) Suppose you decide to purchase a 1-year zero-coupon bond today and also contract today to re-invest the proceeds from the bond for the following two years at 16.5% per year. Show that this arrangement presents an arbitrage opportunity. Demonstrate how you would take advantage of this opportunity. (6 marks)
f) Consider discount factors such that d1 < d2 < d3 . Explain why it would be odd to observe such a situation in a competitive market. (4 marks)
Question 2
Assume the CAPM holds. Consider three feasible portfolios of stocks X, Y and Z with the following return characteristics:
X
Y
Z
7.5%
5%
10%
5%
10%
15%
a) Explain why beta is the appropriate measure of risk in this world. (5 marks)
b) Portfolio Y is known to be uncorrelated with the market. Explain why this property implies that the risk-free rate in the economy is 5%. (5 marks)
c) It is known that one of the portfolios X, Y, Z lies on the efficient frontier (which includes the risk-free asset). Which portfolio is efficient? Explain/justify your answer. (5 marks)
d) An investment manager approaches you and offers you an investment product with a claimed expected return of 12% and standard deviation of 20%. Should you accept this investment? Why/why not? If not, show how the manager can optimally create a portfolio with an identical return volatility to his proposed portfolio but with a superior expected return. Illustrate your answer graphically, making sure to label all relevant elements of your picture. (6 marks)
e) Consider an investor who invests $50,000 in a portfolio consisting of X and Z. $10,000 of that investment was funded with risk-free borrowing. The expected return of the investor’s portfolio is 9.375%.
i. Calculate the dollar amounts invested in each of X and Z. (4 marks)
ii. If the correlation between X and Z is 2/3, what is the standard deviation of the investor’s portfolio? (2 marks)
f) Show that any portfolio on the Capital Market Line (CML) with a positive weight in the market portfolio is perfectly correlated with the market portfolio. Interpret this result. (6 marks)
Question 3
Assume all options are European, and that the underlying asset is a non-dividend paying stock, unless otherwise specified.
(a) A ‘protective put’ strategy involves buying both a put option on a stock and the underlying stock itself.
i. Draw a payoff diagram (not a profit-and-loss diagram) for a protective put strategy. Make sure to label all relevant parts of the diagram. Why do you think this strategy has its name? (5 marks)
ii. Using put-call parity, explain why the payoff of a protective put resembles the shape of the payoff of a call option. (5 marks)
(b) You observe two call options, A and B, with the same exercise price, written on the same underlying asset. Option A matures in one year, while B matures
(c) The value of a European put option must satisfy the following restriction:
0 ≥ − − 0
where 0 is the current put price, 0 is the current price of the underlying stock, is the exercise price, > 0 is the annualised continuously compounded risk-free rate, and is the time till expiration. Prove by contradiction that the above arbitrage restriction must hold, i.e. show that if
(d) It is also known that the value of a European put cannot be greater than the present value of its exercise price, i.e. 0 ≤ − . This restriction, along with the one in (c), suggests that the price of a European put can fall below its exercise value prior to maturity. When is this situation likely to arise? Give an intuitive explanation as to why its value is below its exercise value in such circumstances. (4 marks)
(e) Stock K currently sells for $120. After one year, its price will either increase by 10% or fall by 10%. The annual risk-free interest rate is 5%.
i. Calculate the current value of an at-the-money European call option on stock K maturing in one year. (5 marks)
ii. Now assume that the volatility of Stock K increases so that if the stock price increases, it will still increase by 10% but if it falls, it will fall by
more than 10%. Everything else including the expiration date, current stock price, exercise price, and interest rate stays the same. Show
mathematically that the current value of the call option is higher than
Question 4
(a) The table below lists some partial information about a firm. Assume that the number of shares outstanding stays constant forever.
|
Year |
||||
|
0 |
1 |
2 |
3 |
4 |
Book equity per share |
100 |
|
132.25 |
|
|
Earnings per share (EPS) |
|
|
|
|
21.325 |
Return on equity (ROE) |
|
0.20 |
0.20 |
0.15 |
0.15 |
Payout ratio |
|
0.25 |
0.25 |
0.50 |
0.50 |
Dividends per share |
|
5 |
|
|
|
Dividend growth rate |
|
- |
|
|
0.075 |
i. Fill in the missing values in the table from years 1 to 4. (6 marks)
ii. Assume that after year 4, the company maintains its ROE and payout ratio at year 4 levels. The cost of capital is 12.5%. What is the fair price of the company’s stock today (i.e. at t = 0)? (3 marks)
iii. Suppose the company announces today that it expects any new investments made in or subsequent to year 4 to only earn the cost of capital. The values you calculated in part (i) will be unaffected by this announcement. By how much will the share price change today after the announcement? (4 marks)
(b) Consider the following statement: “For markets to be informationally efficient, all investors must be rational”. Is this statement true? Why/why not? (you will
(c) Companies sometimes try to match the duration of their assets and liabilities. Explain how you think this approach may be useful in protecting net worth from interest rate risk (net worth = A – L, the difference between the market value of assets and liabilities). When is this approach likely to prove less effective? Why? (5 marks)
(d) Assume that the CAPM holds. Consider two firms X and Y, and the risk-free asset, with the following return characteristics:
|
|
( , ) |
( ) |
X |
225 |
200 |
7.5 |
Y |
1600 |
600 |
|
rf |
0 |
|
5 |
What is the expected return of stock Y? (5 marks)
(e) Exactly one year ago, you entered into a forward agreement to purchase one unit of a commodity for $F in exactly T years from now. The current price of the commodity in the spot market is $S. The risk-free continuously compounded interest rate in the market is currently r per year. There are no
convenience yields or storage costs associated with holding the commodity. Using a replicating portfolio approach, show that the current value of your forward agreement, f, is:
= − −
(6 marks)
2022-05-12