* * * very representative
* * fairly representative
* not very representative
Saturday June 6, 1998
1:00 - 5:00 P.M.
GSB 70 and GSB 71
Please pick up and leave your exam in GSB 70
Please write your answers on the examination paper. If you need more space, place additional material on the pages in the back, with a suitable reference in the appropriate space for the question.
Each separately identified question (e.g. 2a, 3b, ...) is worth 2 points. The total number of points is 138. The final score on the examination will be determined by dividing the sm of the points received by 1.38.
You may consult any desired written material during the examination but should not discuss the examination with anyone prior to the conclusion of the designated time.
Student Name ___________________________
(printed)
I acknowledge and accept the Honor Code
(signed) _________________________________
* * * 1. Bob and Myron have formed a new company called Medium-term Capital Management (MTCM). The firm will run 15 to 20 different hedge strategies and employ leverage. They claim that they will produce the same expected return and risk as a diversified portfolio of US stocks but that their returns will be uncorrelated with those of US stocks. You believe them.
Before MTCM entered the picture, you had decided to put $500,000 of your $1,000,000 in the bank and the rest in a portfolio of US stocks. You did so on the assumption that you could earn 5% from the bank and that the US stock market had an expected return of 11% and a standard deviation of 15%. You see no reason to change those estimates.
a. Of any money that you do not put in the bank, what proportion should be in MTCM (that is, MTCM as a percent of MTCM plus Stocks)?
b. If you put $500,000 in the bank, $250,000 in MTCM and $250,000 in Stocks what wille the expected return on your overall portfolio? What will be its risk?
c. If you wanted your portfolio to have the same risk as it did before MTCM came along, how would you allocate your assets? What would be the expected return on your portfolio?
d. Assume that in choosing your initial portfolio (pre-MTCM) you had maximized ep - vp/rt, where ep is the expected portfolio return, vp is the portfolio risk, and rt is your risk tolerance. If you wanted to do so again (with the same rt), how would you allocate your assets? What would be your expected return? What would be your risk?
e. Assume that the Capital Asset Pricing Model holds and that US Stocks are "the market". If Bob and Myron were willing to work for nothing, what would be your estimate of the expected return of MTCM? If the risk of their portfolio is the same as that of the US stock market, what would be your asset allocation?
f. Bob and Myron want to receive a fee based on their "value added". They propose that you pay them 20% of the difference between the MTCM return and that of a pre-specified benchmark if the difference is positive, with any shortfalls credited against future fees earned. What is the lowest-expected return asset or mix of assets that would you accept as a benchmark?
* 2a. Would you expect securities that provide a good hedge against inflation to have higher expected returns, lower expected returns, or the same expected returns as others that are similar in every other respect? Why?
b. Assume that you have been appointed to the board of Leland University, which spends an inordinate amount of its budget on faculty salaries. The staff has proposed investing a small percentage of the endowment in fully collateralized commodity futures. Would you expect the return on such an investment to be higher, lower, or equal to the return on Treasury bills? Would you endorse investment in such a "commodity play" Why or why not?
* 3. You have been analyzing two international mutual funds - the Romer International Financial Fund and Roberts Global Corporate Fund.. Both compare their returns with those of the Morgan Stanley Capital International Europe, Australia and Far East index (EAFE). Both have significantly underperformed the index over the last six years. On further examination you find that both were heavily concentrated in the stocks of Japanese corporations over this period. Digging back into earlier data you discover that Romer was heavily invested in Europe in the previous six years while Roberts was primarily in Japanese stocks at that time as well.
a. Which fund probably had the better performance in the prior period?
b. If you had to choose an index with which to compare Romer's future returns, would you choose EAFE, a Japanese index, or a European index?
c. If you had to choose an index with which to compare Roberts' future returns, would you choose EAFE, a Japanese index, or a European index?
* * 4. On Jan 1, 2000 the exchange rate of U.S. dollars for Japanese Yen was Y100/$1. At the time the one-year interest rates were 6% in the U.S. and 2% in Japan. The level of the Nikkei 225 was 20,000 at the time. Due to continuing austerity, Japanese firms had announced that they would pay no dividends until the year 2001.
a. On Jan 1, what was the yen/dollar forward rate for delivery on Dec. 31, 2000?
b. On Jan. 1, 2000 what was the futures price in yen for a one-year contract for a unit of the Nikkei 225 index (that is, a price agreed upon today to be paid on Jan. 1, 20001 for delivery of a unit of the index)?
c. On Jan. 1, 2001, the Nikkei 225 stood at 25,000. What was the return (in yen) for a Japanese investor who purchased stocks in the index on Jan. 1, 2000?
d. On Jan. 1, 2001 the exchange rate was Y90/$1. What was the profit or loss (in dollars) for a U.S. investor who converted dollars to yen, purchased stocks in the N225 index, then sold the stocks at the end of the year and converted the proceeds to dollars?
e. What was the profit or loss (in dollars) for a U.S. investor who put $200 in a U.S. bank and took a position in the Nikkei 225 futures contract in Japan, using the U.S. bank account as collateral?
f. What was the magnitude (in dollars) of the difference between the profit for the investor in (d) and that of the investor in (e)? To what would you attribute this difference?
* * * 5. The managers in charge of the endowments of West Coast University (WCU) and East Coast University (ECU) have both recently completed asset allocation studies. They each estimate that stocks have an expected return of 10% and a standard deviation of 17% while bonds have an expected return of 7% and a standard deviation of 10%. However, they differ in their assessment of the correlation between bond and stock returns. WCU believes it will equal 0.70, while ECU believes that it will equal 0.30.
a. If both WCU and ECU plan to have 60% in stocks and 40% in bonds, will their projected expected returns be the same or differ, and if the latter, which will project the greater expected return?
b. If both WCU and ECU plan to have 60% in stocks and 40% in bonds, will their projected risks be the same or differ, and if the latter, which will project the greater expected return?
c. If both WCU and ECU wish to have an overall risk of 14%, will their projected expected returns be the same or differ, and if the latter, which will project the greater expected return?
* * * 6. Tacoma Partners, a well-known financial planning firm, has recently concluded an asset allocation study for a high net worth individual. Based on their analyses, the optimal mix of assets would be 60% in the Vanguard Total Stock Market Fund and 40% in the Vanguard Total Bond Market Fund, if only index funds were utilized. However, the planner who has conducted the analysis has recommended that the client consider investing at least some money in one or both of two actively managed funds -- Convertible Opportunities Fund (COF) and Utilities R'Us (URU). Based on style analyses and projections of future performance, the planner has made the following preductions for the two funds:
| Exposure to Bonds | Exposure to Stocks | Expected Selection Return (%/yr) | Std. Dev. of Selection Return (%/yr) | |
| COF | .60 | .40 | 0.50 | 5.00 |
| URU | .30 | .70 | 1.20 | 10.00 |
The client's risk tolerance is 25.
a. What is the Selection Sharpe ratio of COF?
b. What is the Selection Sharpe ratio of URU?
c. If positions of equal dollar value are to be taken in both COF and URU, which position would have the greater selection risk?
d. If optimal positions are to be taken in both COF and URU, which position should be greater in dollar terms?
e. If the client has $100,000 to invest, how much should be invested in each of the four funds?
* * * 7. Darrell Finch, a well-known investment bank with a high-quality financial engineering department, has recently built a model of the bond and stock market for use in pricing and hedging derivative securities. For two-year securities they have designated six possible contingent payment states
u: pay $1 at the end of year 1 if the market goes up in the first year
d: pay $1 at the end of year 1 if the market goes down in the first year
uu: pay $1 at the end of year 2 if the market goes up in the first year and again in the second year
ud: pay $1 at the end of year 2 if the market goes up in the first year and down in the second year
du: pay $1 at the end of year 2 if the market goes down in the first year and up in the second year
dd: pay $1 at the end of year 2 if the market goes down in the first year and again in the second year
They have also identified six different investment decision variables:
B: invest $1 in a one-year bond today, cash it out at the end of year 1
S: invest $1 in a stock today, cash it out at the end of year 2
Bu: invest $1 in a one-year bond next year if the market goes up in the first year
Su: invest $1 in a stock next year if the market goes up in the first year
Bd: invest $1 in a one-year bond next year if the market goes down in the first year
Sd: invest $1 in a stock next year of the market goes down in the first year
The following matrix shows their assumed binomial process.
B S Bu Su Bd Sd u 1.05 1.20 -1 -1 d 1.05 1.00 -1 -1 uu 1.06 1.19 ud 1.06 1.01 du 1.04 1.25 dd 1.04 .95
They believe that in each year the odds are 50/50 that the market will go up or down. By investing $100 in a two-year zero coupon bond, an investor can be guaranteed to receive $109.72 at the end of two years.
a. What is the return in year 1 on a 2-year zero coupon bond if the market goes up?
b. What is the return in year 1 on a 2-year zero coupon bond if the market goes down?
c. Is the correlation of year 1 returns for stocks and 2-year zero coupon bonds positive, negative, or zero?
d. Do stock market expected returns (1) follow a random walk, (2) exhibit mean reversion or (3) exhibit the converse of mean reversion? Does the risk premium depend on investor's wealth? If so, in what way?
e. Is the risk of the stock market (1) greater, (2) lower, or (3) the same after an down market as it is after an up market?
Honeydew Capital Management Company has proposed a strategy that begins with $50 invested in stock and $50 invested in a one-year bond. If the market goes up, the strategy calls for rebalancing to a combination with 30% invested in stock and 70% invested in a one-year bond. If the market goes down, the strategy calls for a combination with 70% in stock and 30% in a one-year bond.
Allstate Consultants has proposed a strategy that also begins with $50 invested in stock and $50 invested in a one-year bond. If the market goes up, the strategy calls for rebalancing to a combination with 70% invested in stock and 30% invested in a one-year bond. If the market goes down, the strategy calls for a combination with 30% in stock and 70% in a one-year bond.
Tacoma consultants has recommended a strategy that involves putting $50 in stock and $50 in a one-year bond, then reinvesting any money received from either the bond or the stock in the same security (that is bond money in another one-year bond and any stock dividends reinvested in the stock).
f. If the market goes up both years, which advice (Honeydew, Allstate or Tacoma) would have been best? Which would have been worst?
g. If the market goes down both years, which advice (Honeydew, Allstate or Tacoma) would have been best? Which would have been worst?
h. Could all investors follow Honeydew's advice? Allstate's? Tacoma's?
i. If someone wanted to get $10,000 two years hence if the market goes up two years in a row but $5,000 otherwise, could Darrell Finch issue a security with such payments? If so, how would they price and hedge it? Show the formulas that they could use to answer these questions, with the matrix shown above represented as M and any other matrices or vectors shown explicitly in your answer.
* * 8. NT Fund Management has just introduced a market neutral fund. Management claims that it will carefully match its long and short positions so that there will be no net market, sector or factor exposure. You believe them and estimate that they will beat an appropriate benchmark by an expected 200 basis points(2%) per year with a standard deviation of 500 basis points (5%).
a. What is the appropriate benchmark for this fund?
b. What is the fund's annual excess return Sharpe ratio?
c. What is the fund's Selection Sharpe ratio?
Before the introduction of the NT market neutral fund you had chosen to put all your money in an S&P500 Index fund. You estimate that this strategy has an expected annual return 600 basis points (6%) greater than that of Treasury bills with a standard deviation of 1500 basis points (15%).
d. What is your risk tolerance?
NT also has introduced a fund that combines a swap and investment in the NT market neutral fund. For every $100 received, this new fund (alpha alpha, or AA) invests $100 in the market neutral fund, then takes a position with a notional value of $100 in a swap in which the fund promises to pay the return on Treasury bills to the counterparty, in return for which the counterparty promises to pay the return on the S&P500 to the fund.
e. What is the standard deviation of the AA fund's return?
f. What is the expected excess return on the AA fund?
g. If you could (1) invest all your money in the SP500 Index fund or (2) invest all your money in the AA fund, which would you choose?
h. If you could borrow money to invest in the AA fund,or put money in Treasury bills and invest in the AA fund, how would you invest your money?
* * 9. The Rhode Island Public Employees' Retirement System (RIPERS) has recently concluded an asset allocation study. After running an optimizer, using the estimated risk tolerance of the citizens of the state (50), they concluded that the best allocation among five asset classes was:
Cash 5% US Govt. Bonds 35% US Large Cap Stocks 40% US Small Cap Stocks 15% Non-US Stocks 5%
At the hearings held in the state assembly, the leader of the majority party argued that this was done without taking liabilities into account. The staff agreed that this was the case and promised to do the study again with liabilities taken into account.
Fortunately, RIPERS benefits had recently been frozen, with all benefits fixed at their current levels; henceforth all state contributions would be placed in a brand-new defined contribution plan. Since the old benefits were not indexed to inflation, this meant that all liabilities were fixed in dollars, with the payments subject only to mortality risks.
Staff has estimated that the current value of assets is $1 billion and that the present value of liabilities, when discounted using the present values of corresponding zero-coupon government bonds, is $0.8 billion.
Using Monte Carlo analyses, the staff has estimated that a 1% change in government interest rates would change the present value of the liabilities by 12%.
a. What is the duration of the liabilities?
b. The duration of the US Government Bond asset class is estimated to be 10. If the fund were invested totally in this asset class, would the fund's surplus be likely to increase, decrease, or stay the same if interest rates were to fall?
Using Monte Carlo analysis, the staff has estimated that the present value of the liabilities is expected to increase 8% in the coming year (before any benefit payments are made after the year-end). However, the standard deviation associated with this expectation is 15%. Estimates of the correlations between each of the asset classes and the percentage changes in the liabilities, along with the risk of each asset class is shown below.
Correlation with Liability Standard Deviation Cash 0.00 0 US Govt. Bonds 1.00 18 US Large Cap Stocks 0.40 15 US Small Cap Stocks 0.35 20 Non-US Stocks 0.10 22
c. Assume that the goal is to maximize the utility of the risk and return of the fund's surplus, expressed as a percentage of the fund's assets (for example, the current value is 20), with a risk tolerance of 50. Is the mix that was optimal when only assets were considered optimal in this context? Why or why not?
d. If only a relatively small change could be made in the fund's asset allocation, with allocation to one asset increased and the allocation to another decreased, which asset would you choose for an increase and which would you choose for a decrease?
For the remaining questions, assume that the fund's risk tolerance had been 100 and the initial asset allocation had been optimal from an asset-only perspective for that risk tolerance.
e. Is the current asset allocation optimal, if the goal is to maximize the risk and return of the change in the fund's surplus, expressed as a percentage of the beginning assets, with the same risk tolerance (100)?
f. If only a relatively small change could be made in the fund's asset allocation, with allocation to one asset increased and the allocation to another decreased, which asset would you choose for an increase and which would you choose for a decrease?
g. Would the legislature be justified in being as concerned about the staff's decision to ignore liabilities if the risk tolerance had been 100 than if it had been 50? Why or why not?
* 10. Ocean Avenue Consultants has been hired by both the Delaware Public Employees' Retirement System (DELPERS) and the William David Company (WD) to advise on asset allocation. Ocean has decided to use the same estimates of asset risk, returns and correlations for each of its clients. Moreover, in each engagement, attention will be limited to four asset classes: Treasury bills, U.S. Treasury Inflation-protected bonds, U.S. Treasury bonds, and U.S. stocks. Both the WD pension plan and the DELPERS plan are defined benefit plans. DELPERS has a relatively old and experienced workforce, while WD has a young workforce. Both plans base pension payments on average salary during the last five years of service, and total years of service. The DELPERS plan includes indexing for inflation after retirement, while the WD plan does not. The actuaries at WD have used an actuarial rate of 4% for the valuation of the firm's liabilities, while those at DELPERS have used a rate of 8 1/4%. The yields of recently-issued Treasury bonds are 6% for regular long-term bonds and 3 1/2% for inflation-protected bonds. For each plan, the actuaries have estimated future cash payments for benefits on the assumption that everyone quits at once and that there is no inflation in the future. For each plan the reported value of the liabilities to pay benefits is calculated by discounting these future cash payments using the plan's actuarial rate.
a. How would you value the liabilities of the WD plan? What discount rate would you use? Why?
b.How would you value the liabilities of the DELPERS plan? What discount rate would you use? Why?
c. Is the reported funded status (assets - liabilities) of the WD plan likely to be higher, lower, or the same as would be calculated by a financial economist? Why?
d. Is the reported funded status of the DELPERS plan likely to be higher, lower, or the same as would be calculated by a financial economist? Why?
e. If both DAPERS and WD wanted conservative (low asset/liability risk) asset mixes, would your recommendations for their optimal asset mixes be likely to be very different, different, or only slightly different? Why?
f. If both DAPERS and WD wanted aggressive (high asset/liability risk) asset mixes, would your recommendations for their optimal asset mixes be likely to be very different, different, or only slightly different? Why?
* * * 11. The small country of Nissan has zero inflation and zero interest rates. For this reason it has been studied extensively by financial analysts, who treasure the simplicity of calculations concerning financial products in the country. Another appealing feature of Nissan's economy is the fact that stocks can either go up by 20% or down by 10% each year, and that the odds are 50/50 that each year will be an "up-year" or a "down year". The currency of Nissan is the Nissan Dollar ($).
Mr.Buffet has $100 to invest. He would like to end up with twice as much money at the end of the year if the market is up as he will have if the market is down.
Mr. Kaufman also has $100 to invest but he would like to end up with twice as much money at the end of the year if the market is down as he will have if it is up.
Both Buffet and Kaufman have ample credit so that they can borrow all they want (at zero interest) if they wish to purchase stocks on margin. They may also take any short positions they desire and use any proceeds for other investments.
a. How much will Buffet have if the market is up? If it is down?
b. How much will Kaufman have if the market is up? If it is down?
c. How many dollars will Buffet invest in stocks? In Bonds? What orders will he give to his broker?
d. How many dollars will Kaufman invest in stocks? In Bonds? What orders will he give to his broker?
e. What is Buffet's expected return? What is the standard deviation of his return? What is the beta of his portfolio relative to that of the stock market?
f. What is Kaufman's expected return? What is the standard deviation of his return? What is the beta of his portfolio relative to that of the stock market?
g. Are the characteristics of the two portfolios consistent with the Capital Asset Pricing Model? Why or why not?
h. Is Buffet's portfolio efficient in the manner described by Markowitz? Is Kaufman's? Why or why not?
* * 12. Connie, Holden and Trent live in a country where by law the stock market can either go up by 25% or fall by 5% each year. Each of them has $100, with $60 invested in stocks and $40 in bonds. Bond return a guaranteed 5% per year. They all agree that the odds are 50/50 that the stock market will go up in the coming year. Each of them plans to retire in two years on whatever is left from their current investment portfolio. Connie has decided that if the market goes up in the coming year she will sell $10 of the stocks in her portfolio and put the proceeds into bonds; on the other hand, if the market goes down in the coming year she plans to sell $10 of the bonds in her portfolio and put the proceeds into stocks,.Holden has decided that whatever happens in the coming year, he will leave his portfolio alone. Trent has decided that if the market goes up he will sell $10 of bonds and use the proceeds to buiy stocks; on the other hand, if the market goes down he will sell $10 of stocks and use the proceeds to buy bonds.
a. Which, if any, of these three investors is a contrarian? A Trend-follower? A buy-and-hold investor?
b. Two years from now, what could happen? How rich would each investor be in each of the associated states of the world?
c. If the stock market follows a random walk, what will be each investor's expected retirement wealth?
d.Charlie, a well-known portfolio manager, has appeared as a guest lecturer in an investment class as a local university. He predicts that if the market goes up in the coming year there will be only a 40% chance that it will go up in the following year, but that if the market goes down in the coming year there will be a 60% chance that it will go up in the following year. If he is right, what will be each investor's expected retirement wealth?
e. Tom, who has appeared in the same class, feels that while Charlie may be right in the long term, he is wrong in the shorter term. Instead, Tom predicts that if the market goes up in the coming year there will be a 60% chance that it will go up in the following year, but that if the market goes down in the coming year there will be only a 40% chance that it will go up in the following year. If he is right, what will be each investor's expected retirement wealth?
f. Is there any possibility that Charlie could be right and that all three investors could agree with him and still plan to act in the manner described? If not, why not? If so, why?