代写Econ 136 Final Spring 2024帮做R程序

Econ 136 Final

Spring 2024

1. True or false. (25 points, 5 points each)

Are the following statements true or false? Explain your answer in no more than two sentences. You will be graded on your explanation.

(i)  Historically, low dividend-price ratios of the S&P 500 have predicted high  (positive) subsequent price growth and essentially no subsequent change in dividends.

(ii)  The  following fact violates the semi-strong form of the efficient markets hypothesis: prices of companies tend to increase a few days before public announcement of good news.

(iii) When the risk-free rate increases, the optimal portfolio share of risky assets for a mean- variance investor declines.

(iv) In a CAPM equilibrium, since investors are compensated for holding risk, two securities with the same standard deviation should have the same expected return.

(v)  Consider a European call and a European put for the same non-dividend-paying stock, same expiration T, and same strike X, where X = F and F is the forward price of the underlying for delivery date T. Under no arbitrage, the two options have the same price today.

2. Welfare efects of risk. (20 points, 5 points each)

Consider an economy where CAPM holds, the risk-free rate is Rf   = 2%, and the return of the market portfolio has expectation E[Rm] = 10% and standard deviation 40%.

(a) Draw the capital allocation line.  What is the optimal portfolio of a mean-variance investor with risk aversion A = 2?  What is the mean and standard deviation of this portfolio? Show this portfolio in the figure. What is the value of this investor’s mean- variance utility function if he invests in this optimal portfolio (i.e., what is E[Rp] - (A/2)Var(Rp))?

(b) What is the optimal portfolio of an investor with risk aversion A = 4? What is its mean and standard deviation? Show this portfolio in the figure (from part (a)) as well. What is the value of this investor’s mean-variance utility function when investing optimally?

(c) Now suppose that due to a reduction in uncertainty, the standard deviation of the mar- ket return falls to 20%, while other parameters are unchanged.  Draw the new capital allocation line (in a new figure).  What are the new optimal portfolios of the two in- vestors? What are these portfolios’ means and standard deviations? Show them in the figure. Which investor’s portfolio share of risky assets changes by more after the change in the standard deviation of the market return? Why?

(d) What are the values of the two investors’ utility functions now, given their new optimal investments? Which investor’s utility increases by more after the change in the standard deviation of the market return? Why?

Comment on this statement:  “Reductions in risk are most beneficial to conservative investors who are highly sensitive to fluctuations in their wealth.”

3. Capital budgeting. (30 points, 5 points each)

Consider an economy where the risk-free rate is Rf  = 4%, the expected return on the market portfolio is E[Rm] = 12%, and the standard deviation of the return on the market portfolio is 20%.   The covariance between the return on ABC stock and the return on the market portfolio is 0.06.  All of this data refers to annual returns.  Suppose that ABC stock pays a dividend of $10 per share next year, and dividends are expected to grow at a rate of 2% per year.

Recall that under the Gordon Growth Model (GGM), we can write the price of a stock at any time t as

Pt = R−g/Dt+1

where R is the discount rate and g is the growth rate of dividends.

(a)  ABC’s manager argues that according to the Gordon Growth Model (GGM) his shares should sell for a price of $10/(.04 - .02) = $500.  Explain why this valuation is inappro- priate.

(b)  Assuming that CAPM holds, compute ABC’s beta with respect to the market portfolio, and the expected rate of return of ABC stock.

(c)  Given your answer to (b), what price does the GGM imply for a share of ABC stock?

(d) It turns out that the market price of ABC is $50, which is diferent from what you computed in (c) [if not, you made a mistake!]. However, you realize that ABC has only narrowly avoided bankruptcy last year, and has a very high book-to-market ratio.  Is the fact that ABC has a lower price than what’s predicted by part (c) consistent with what you know about the expected return of stocks with high book-to-market ratios? Why?

(e)  Now you want to apply a more sophisticated asset pricing model than CAPM to price ABC. Let HML = RH -RL denote the excess return of value stocks over growth stocks, and SMB = RS  - RB  the excess return of small stocks over big stocks.  Suppose that ABC has a beta of 1.5 with respect to HML, and a beta of zero with respect to SMB. The beta of ABC with respect to the market portfolio is still what you computed in part (b). If the expected excess return of value stocks over growth stocks is E[HML] = 4%, what should be the expected return of ABC according to the Fama-French model? Is it higher or lower than the expected return you computed in (b)? Why?

(f)  Using the expected return you computed in part (e), what should the price of a share of ABC stock be according to the GGM? Does your answer justify the market price of $50?

4. Options. (25 points, 5 points each)

Consider an economy in three periods, t = 0, t = 1 and t = 2.  Suppose that a stock index behaves as follows: the initial index value at time t = 0 is 100, and each period the index either rises by 15 or falls by 5 with equal probability (so for example at t = 2, the highest possible index value is 100+15+15=130). The index does not pay dividends during these two periods. The riskfree rate of return each period is Rf  = 0%.

Now consider a European call option on the index, with expiration date t = 2, and strike price X = 100.

(a) Draw the event tree for this economy.  For each node in period t = 1 and t = 2, write St, the current price of the index.  For each node at t = 2, write the payof of the call option.

(b) Consider the node where the stock price has gone up to 115 in period 1.  Construct a portfolio at this node that replicates the payof of the option in both possible states at t = 2.  Specifically, assume that at this node, you purchase x  shares of the index and y shares of the riskfree asset. Solve for x and y from the assumption that this is a replicating portfolio. What is the price of this portfolio at t = 1 (in the event when the stock price is S1  = 115)? What is the price of the option, C1?

(c) Following a similar procedure as in (b), now solve for the price of the option at t = 1 in the event when the stock price is S1  = 95.

(d) Now go back to period t = 0. To compute C0, construct a portfolio of the index and the riskfree asset that pays C1  in period t = 1 (that is, the number you obtained in (b) if the price goes up in period 1, and the number you obtained in (c) otherwise). What is the price of this portfolio? What is the price of the call option?

(e) Suppose that a European put option on the index with expiration t = 2 and strike price X = 100 is traded at a price of P0  = 5.425.  Is there an arbitrage opportunity in this economy?  If yes, construct a portfolio of the put, the call, the index and the riskfree asset to exploit it. If not, why?

5. Spot-forward parity. (15 points, 5 points each) Use the following notation:

Symbol	Description
T t S ST K f F r	time when forward contract matures (years) current time (years), where t ∈ [0, T] price of stock underlying forward contract at time t price of stock underlying forward contract at time T delivery price of the forward contract value of a long forward contract at time t forward price at time t risk-free rate of interest per annum (cts comp)

(a)  For a non-dividend paying stock, construct two portfolios at time t, that have the same payof at time T, and use the law of one price to establish the following relationship:

f = S — K · e-r(T-t)

(b)  Noting that when a forward contract is initiated, the forward price equals the delivery price, which is chosen so that the value of the contract is zero, show that:

F = S · er(T-t)

(c) At maturity, a forward contract pays of ST — K.  By risk-neutral valuation we can write f = e-r(T-t)ERN[ST — K]

where ERN[·] indicates we are taking the expectation using the risk-neutral probabilities.

Recall that in a“risk-neutral world”(i.e., when the objective probabilities and the risk- neutral probabilities coincide) the expected return of all risky assets is r. Use this fact to carefully derive the relationship from part (a).