代写FN2190 Asset pricing and financial markets帮做Python语言程序

日期：2024-09-29 06:01

FN2190 Asset pricing and financial markets

Question 1

Assume that the CAPM holds. Consider a stock market that consists of only two risky securities, Stock 1 and Stock 2, with the following expected returns (µ) and standard deviations of returns (σ): µ1 = 0.25, σ1 = 0.20; µ2 = 0.15, σ2 = 0.10. Table 1 shows the expected returns and standard deviations for six assets, A to F (the entries for asset A are left intentionally blank). These assets are portfolios consisting of the risk-free asset, Stock 1, and Stock 2 in varying quantities (which may be positive, negative, or zero).

(a) Asset D has zero investment in the risk-free asset. Determine the portfolio weights of Stock 1 and Stock 2 in asset D.        (3 marks)

(b) What is the correlation between the returns of Stock 1 and Stock 2?            (3 marks)

(c) Asset A is the minimum variance portfolio formed from Stocks 1 and 2 only.

i. Show that the weight of Stock 2 in asset A is given by:

where σ1,2 denotes the covariance between Stocks 1 and 2.               (4 marks)

ii. Complete the entries for asset A in Table 1.                 (3 marks)

(d) One of the assets in Table 1 is the market portfolio. Identify the market portfolio, clearly explaining/justifying your choice.     (6 marks)

(e) Show how you can construct an efficient portfolio with the same expected return as Stock 1 but a lower volatility, illustrating your argument by sketching a graph. Assuming you intend to borrow/lend $10,000 risk-free to create this portfolio, how many dollars would need to be invested in each stock?               (8 marks)

(f) Determine the CAPM beta of the portfolio you constructed in (e). Given your answers to the rest of this question, if we observe a rational investor holding 100% of his wealth in Stock 1, what can we conclude about financial constraints faced by the investor?          (6 marks)

Question 2

(a) Consider a two-period consumption model consisting of now, t = 0, and next year, t = 1. There are two investors, Investor A who is patient and wants to wait and consume the maximum amount possible at t = 1, and Investor B who is impatient and wants to consume the maximum amount possible now. Both investors have an income of $200,000 today and no income at t = 1. Both investors have access to a real investment opportunity costing $200,000 now and returning a guaranteed $215,000 at t = 1. They also have access to risk-free borrowing and lending at an annual rate of 10%. Explain the investment decisions taken by each investor (both real and financial) and the resultant cash flows and consumption, including a brief discussion of the optimality of the net present value (NPV) rule.       (7 marks)

(b) You have just purchased a house in Sydney for $2 million, using your own savings to make a down payment equal to 20% of the house’s value, with the remainder financed via a 25-year mortgage. The mortgage has fixed monthly payments, with the first payment due in exactly one month. The stated annual interest rate on the loan is 6% with monthly compounding. How much of the loan principal will you repay in the first year of the loan, expressed as a percentage of your total annual mortgage payment. Will this percentage increase, decrease, or stay the same in subsequent years? Explain.            (6 marks)

(c) The term structure of interest rates is flat with all spot rates equal to 10%. Consider the following bonds:

• Bond A: a ten-year 10% coupon bond with a face value of $100.

• Bond B: a four-year zero coupon bond with a face value of $100.

• Bond C: a four-year 5% coupon bond with a face value of $100.

All coupons are paid annually. Using the bond duration concept, explain which bond will experience a smaller percentage price change if the term structure shifts upwards by 100 basis points (i.e. to 11%).            (5 marks)

(d) Comment on the validity of the following statement: ‘According to the CAPM, an in-the-money put option on a zero beta stock should have an expected return equal to the risk-free rate’.            (5 marks)

(e) Suppose that in any given year, there is a 50% chance that a mutual fund will outperform. the market simply by chance. If there are N mutual funds, what is the smallest number N such that the probability of observing at least one out of these N funds outperform. the market for 10 consecutive years by chance exceeds 99%? Noting that there are around 8,000 mutual funds in the US, comment on the implication(s) of your answer for empirical tests of efficient markets.           (5 marks)

(f) Consider a forward contract to deliver, in two years’ time, a two-year coupon bond with a face value of $100 and a coupon rate of 10%. The forward price F will be paid in year 2 while the bond payments occur in years 3 and 4. The current term structure of interest rates is given by: r1 = 6%, r2 = 5%, r3 = 4.5%, and r4 = 4%. Using a replicating portfolio approach, find the no-arbitrage forward price, F.        (5 marks)

Question 3

Consider the following information in Table 2, on three default-risk free bonds with annual coupon payments and face value of $1,000.

(a) Determine the prices of bonds A, B and C.                   (3 marks)

(b) Determine the current term structure of spot interest rates and briefly comment on the shape of the term structure.          (5 marks)

(c) Demonstrate how you can use bonds A, B and C to replicate a 3-year zero coupon bond with a face value of $1,000.     (6 marks)

(d) If the 3-year zero coupon bond in (c) has a market price of $780, show how you can earn an arbitrage profit. Make sure to clearly detail the arbitrage strategy.     (6 marks)

(e) Assume you can buy and sell zero coupon bonds (face value $1,000) of any maturity and that all bonds are trading at their no-arbitrage price implied by the term structure calculated in (b). An investor expects to receive a cash inflow of $5 million in one year’s time, which she then plans to lend out immediately. The term of the loan will be two years, with fixed coupon interest paid to the investor at the end of each year, along with the repayment of the entire principal amount at the maturity of the loan. Explain how the investor could arrange this loan today and ‘lock in’ the interest rate on the loan, stating exactly how many units of the required bonds need to be long/short and the resultant cash flows. What should the interest rate on this loan be?         (8 marks)

(f) Instead of paying fixed coupons, suppose Bond C pays a floating rate of coupon interest, whereby its coupon rate varies positively with the general level of interest rates. All else being the same, would Bond C’s modified duration be higher, lower, or the same as its fixed-interest counterpart? Explain your answer (you do not need to do any calculations).      (5 marks)

Question 4

(a) Consider the following two strategies:

• Strategy 1: short one call option with strike price X + 2a, short one call option with strike price X − 2a, and long two call options with strike price X.

• Strategy 2: long two put options with strike price X, short one put option with strike price X − 2a, and short one put option with strike price X + 2a.

Both strategies are based on the same underlying non-dividend paying stock and all options are European and have the same maturity date, T . Both X and a are positive constants, X − 2a > 0, and the stock price today is equal to X.

i. Which strategy has the higher cost to implement today? You must use a payoff table at maturity to comprehensively justify your answer, stating any assumptions.            (9 marks)

ii. Why would a trader use these strategies?           (2 marks)

(b) Comment on the validity of the following statement: ‘The risk-neutral pricing approach makes no assumptions about the nature of investors’ risk preferences’.           (3 marks)

(c) Consider a two-period binomial model (t = 0, 1, 2), with a risky non-dividend paying stock, BHP, which is currently trading for $4. In each period, the stock can go up by 20% or down by 5%. The risk-free interest rate for the second period (i.e. between t = 1 and t = 2) is 2%. An at-the-money European put option on BHP with maturity date t = 2 is currently trading at $0.20715.

i. What is the implied risk-free interest rate for the first period in this economy?             (6 marks)

ii. Using the replicating portfolio method, determine the value today of a European call option on BHP stock with maturity date t = 2 and exercise price of $3.85.            (6 marks)

iii. Consider a new European option in the market which is path-dependent, with underlying stock BHP. The new option has the following payoff function at the maturity date t = 2:

max(Savg − X, 0)

where X = $3.85 is the exercise price of the option and Savg is BHP’s average stock price during the life of the option, i.e. the arithmetic average of the stock price BHP realises on the path from t = 0 to t = 2. Explain intuitively whether you expect this new option to have a value today that is higher, equal, or lower than the option in part (ii)? What is the value today of this option?         (7 marks)