Use either Excel or Word to answer these questions.
All calculated answers in Excel or Word should contain the formulas that are used to obtain them
1. Mark is 45 years old, earns $50,000 per year, and expects that his future earnings will keep pace with inflation, but will not exceed inflation. He has not yet saved anything towards his retirement. His company does not offer any pension plan. Mark pays Social Security taxes equal to 7.5% of his salary, and he assumes that when he retires at age 65, he will receive $12,000 per year in inflation-adjusted Social Security benefits for the rest of his life. His life expectancy is age 85.
Mark buys a book on retirement planning that recommends saving enough so that when private savings and Social Security are combined, he can replace 80% of his preretirement salary. Mark buys a financial calculator and goes through the following calculations:
First, he computes the amount he will need to receive in each year of retirement to replace 80% of his salary: .8 x $50,000 = $40,000.
Since he expects to receive $12,000 per year in Social Security benefits, he calculates that he will have to provide the other $28,000 per year from his own retirement fund. (Hint: Mark’s calculation so far is correct.)
Using the 8% interest rate on long-term default-free bonds, Mark computes the amount he will need to have at age 65 as $274,908 (the present value of $28,000 for 20 years at 8% per year). Then he computes the amount he will have to save in each of the next 20 years to reach that future accumulation as $6,007 (the annual payment that will produce a future value of $274,908 at an interest rate of 8% per year). Mark feels pretty confident that he can save 12% of his salary (i.e., $6,007/$50,000) in order to insure a comfortable retirement.
a. If the expected long-term real interest rate is 3% per year, what is the long-term expected rate of inflation?
b. Has Mark correctly taken account of inflation in his calculations? If not, how would you correct him? How much percent of salary does he need to save each year?
c. What is Mark’s permanent income? How much should Mark save in each of the next 20 years (until age 65) if he wants to maintain a constant level of consumption (that equals his permanent income) over the remaining 40 years of his life (from age 45 to age 85)? Ignore income taxes but not Social Security tax. (Note: Ignore the savings in part b, because those were used to replace 80% of preretirement salary, but not a constant level of consumption over his lifetime.)
2. Alexis Enterprises is evaluating alternative uses for a three-story manufacturing and warehousing building that it has purchased for $1,450,000. The company can continue to rent the building to the present occupants for $61,000 per year. The present occupants have indicated an interest in staying in the building for at least another 15 years. Alternatively, the company could modify the existing structure to use for its own manufacturing and warehousing needs. Alexis’s production engineer feels the building could be adapted to handle one of two new product lines. The cost and revenue data for the two product alternatives are as follows:
|Product A||Product B|
|Initial cash outlay for building modifications||$95,000||$125,000|
|Initial building outlay for equipment||$195,000||$230,000|
|Annual pretax revenues (generated for 15 years)||$180,000||$215,000|
|Annual pretax expenditures (generated for 15 years)||$70,000||$90,000|
The building will be used for only 15 years for either Product A or Product B. After 15 years the building will be too small for efficient production of either product line. At that time, Alexis plans to rent the building to firms similar to the current occupants. To rent the building again, Alexis will need to restore the building to its present layout. The estimated cash cost of restoring the building if Product A has been undertaken is $55,000. If Product B has been manufactured, the cash cost will be $80,000. These cash costs can be deducted for tax purposes in the year the expenditures occur.
Alexis will depreciate the original building shell (purchased for $1,450,000) over a 30-year life to zero, regardless of which alternative it chooses. The building modifications and equipment purchases for either product are estimated to have a 15-year life. They will be depreciated by the straight-line method. The firm’s tax rate is 34 percent, and its required rate of return on such investments is 12 percent.
For simplicity, assume all cash flows occur at the end of the year. The initial outlays for modifications and equipment will occur today (Year 0), and the restoration outlays will occur at the end of Year 15. Alexis has other profitable ongoing operations that are sufficient to cover any losses. Which use of the building would you recommend to management?
3. Trident Inc.’s current business generates a constant stream of earnings per share of $5 currently, if no new investment is undertaken. Suppose the management will retain 40% of its earnings at Year 1, and invest the retained earnings in a project. For each dollar invested, the new investment will generate a return of 30% per year for only the next three years (Year 2, 3, and 4). Then the new project will end. The discount rate is 10%.
a. Suppose the company doesn’t take any new investment. What is the stock price?
b. What are the dividends at Year 1, 2, 3, 4, and 5? What is the dividend growth rate for Year 2? And for Year 3?
c. The company just announced to undertake the new project. Calculate the new stock price using the dividend discount model.
d. Calculate the new stock price using the NPVGO model.
e. Suppose you are the company’s CEO, who can change the scale of investment for the new project. How much earnings would you retain and invest at Year 1, if your goal is to maximize all shareholders’ value? Explain.
4. Vice Corp. issued 12-year coupon bonds 2 years ago at a coupon rate of 8%. The bond was issued at par and pays semiannual coupon payments. Hardy Corp. has 8% coupon bond outstanding, with semiannual coupon payments. The Hardy Corp. bonds currently have 3 years to mature. The interest rate has been unchanged, and both bonds have been priced at par value until just now, when the Fed announced to cut the annual interest rate by 1 percent.
(a) Walter bought one Vice Corp. bond at the time of issuing 2 years ago. He has already received four semiannual coupon payments. And he decides to sell the bond after the Fed’s announcement. Assume that bond prices change instantaneously to reflect the new interest rate. What is Walter’s annualized holding period return? Why is it higher or lower than 8%?
(b) What is the percentage change in the price of these bonds?
(c) Suppose instead the Fed announced to raise the interest rate by 1 percent. What is the percentage change in the price of these bonds? Illustrate your answers in (b) and (c) by graphing bond prices versus YTM. What does this problem tell you about the interest rate risk of longer-term bonds?
5. There are two states of nature (s1, s2) with equal probabilities. Suppose there is a representative agent who is endowed with 1 unit of consumption today, and (2, 1) tomorrow. The agent has quadratic utility
u(c) = – (c – 3)2 , c≤3
and β=1. In the asset market, there are three securities. Security 1 is a comprehensive stock index that is a claim to the entire endowment of the economy. By definition, the stock index is the market portfolio with payoffs (2,1). Security 2 has payoffs (1,0), security 3 has payoffs (0,1), and security 4 is a riskfree bond that pays (1, 1).
(a) What are the risk-neutral probabilities? Explain intuitively why the risk- neutral probability in one state is higher than that in the other state.
(b) What is the expected gross return (i.e. expected payoffs over price) of security 2? Is the expected risk premium (E[R2]-Rf ) of this security positive or negative? Please explain intuitively why this is the case.
(c) What is the expected gross return of the market portfolio? What is the expected risk premium of the market portfolio (aka. the market risk premium)?
(d) The Capital Asset Pricing Model holds when the representative agent has quadratic utility. Its formula is
E[Ri] – Rf = i (E[RM] – Rf)
where E(RM) is the expected gross return of market portfolio, which we calculate in part (b), and i is a measure of systematic risk for security i. The formula says that the expected risk premium of any security i is proportional to its measure of systematic risk i. Show that the beta of Security 2 is 4, and the beta of Security 3 is -2.
6. Suppose there are two states of nature in the future. In the asset market, there are the two contingent claims, one for each state. There’s a third security with payoffs x=(4,1). Suppose the prices of the contingent claims are both 1/2: p(c(s1))= p(c(s2))=1/2. The price of the third security is 3.
(a) What is the payoff matrix X of the asset market? Is the market complete or incomplete? Why? Is there any redundant security?
(b) What is the price vector p of the securities market?
(c) The law of one price says that the securities with the same payoffs must have the same price. Does the LOOP hold here? Why or why not?
(d) Find an arbitrage portfolio. Show that one can possibly receive some positive payoffs without any cost or risk using the arbitrage portfolio.