Each part (subquestion) is worth 4 points
Not all questions are of equal difficulty
The total time for the examination is 2 hours
Please allocate your time efficiently
1. The Russell Frank Company, a revered consulting firm, has made the following predictions for the expected returns and risks (standard deviations) of several mutual funds over the next year:
Fund Exp Return Std Dev A 5.0 0.0 B 7.0 10.0 C 10.0 20.0 D 8.5 15.0
1a. (4 points) Bob Jung has decided to invest all his money in one (and only one) of these funds. Without knowing anything about his preferences, can you advise him to choose a specific fund, or at least not to consider one or more funds? Use indifference curves to defend your answer.
There is no dominance in this set of funds, so any one could be optimal, depending on Bob's preferences. It is possible to draw a set of indifference curves that have the standard properties (upward-sloping with increasing slopes) that put any one of the four funds on the highest feasible curve.
1b. (4 points) Becky Morgan has decided to select one fund and put all the rest of her money in the bank, which pays 5% interest with no risk. She also can borrow money from her financial aid counselor at a rate of 5%. Can you recommend a fund (A,B, C or D) to her or at least rule out one or more funds? Why or why not?
If she is going to put any money at risk you can advise her to use fund C plus borrowing or lending. It has the highest excess return Sharpe Ratio, as shown below.
Fund Exp Return Std Dev SR A 5.0 0.0 -- B 7.0 10.0 0.20 C 10.0 20.0 0.25 D 8.5 15.0 0.23
This means that for any desired level of risk, a combination of C plus borrowing or lending will provide a higher expected return than any combination of B plus borrowing or lending or D plus borrowing or lending. Fund A (or equivalently, putting all the money in the bank) would be a possibility only if she were very risk averse, with an indifference curve at point A with a slope greater than 0.25.
1c. (4 points) Bob Fargo is considering investing in either (1) a portfolio of funds B and C or (2) putting all his money in fund D. He would like a risk equal to that of fund D. He believes that the returns on funds B and C are not perfectly positively correlated. Should he choose a portfolio of B and C or investment solely in fund D? If he should choose a portfolio of B and C, can you say whether the optimal combination will include more B or more C, or is it impossible to determine without knowing the precise value of the correlation coefficient?
Fund D lies on the line connecting funds B and C. As long as the latter are not perfectly positively correlated, the plot of combinations of risk and return that can be obtained with combinations of them will lie to the left of the line segment (except at the end points). Thus for a risk of 15, it is possible to obtain a higher expected return with a portfolio of B and C than with fund D.
Since the curve showing the risk and return of combinations of B and C will lie to the left of the line connecting their points, we know that a portfolio of B and C with the same expected return as D will lie to the left of point D (that is, have a lower standard deviation). Since D's expected return is half-way between that of B and C, this portfolio will have equal amounts of the two funds. To increase the standard deviation to equal that of D will require larger amounts of fund C. Thus we can say that the optimal portfolio will have more than half invested in in Fund C. To be more precise, of course, we would have to know the precise value of the correlation coefficient.
2. Merrill Chase, 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; cost today = $1
S: invest $1 in a stock today, cash it out at the end of year 1; cost today = $1
Bu: invest $1 in a one-year bond next year if the market goes up in the first year; cost today = $0
Su: invest $1 in a stock next year if the market goes up in the first year; cost today = $0
Bd: invest $1 in a one-year bond next year if the market goes down in the first year; cost today = $0
Sd: invest $1 in a stock next year of the market goes down in the first year; cost today = $0
The following matrix, M, shows their assumed binomial process (blanks are zero).
B S Bu Su Bd Sd u 1.04 1.25 -1 -1 d 1.04 0.95 -1 -1 uu 1.04 1.25 ud 1.04 0.95 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 in each year.
2a. (4 points) What is the one-year interest rate?
4%
2b. (4 points) What is the annual interest rate, compounded annually, on a two-year zero coupon bond?
4%
2c. (4 points) A two-year bond with 4% coupon is currently selling at par. The bond pays $4 at the end of year 1 and $104 at the end of year 2. It currently sells for $100. If the market is up at the end of year 1, what will be the price of this bond?
$100
2d. (4 points) Is Merrill Chase assuming that stock returns are identically and independently distributed over time? Please explain.
yes
2e. (4 points) Ken Gerard is being interviewed for a position in the financial engineering department at Merrill Chase. He has been asked to evaluate a proposal for a security that would be issued by Merrill that would promise the holder $1000 in one year if and only if the market is up. His task is to consider the appropriate price to charge for such a security and the strategy required to completely hedge the firm's obligation. To begin, he has computed the inverses of two matrices, as follows:
m1 = [ 1.04 1.25
1.04 0.95 ]
inv(m1) = [ -3.0449 4.0064
3.3333 -3.3333 ]m2 = [ 1.04 1.04
1.25 0.95 ]
inv(m2) = [ -3.0449 3.3333
4.0064 -3.3333 ]
Ken is a bit confused -- he doesn't know if he needs the inverse of m1, the inverse of m2, or neither. In any event, he must answer the question soon or run the risk of being rejected for the job. How should he proceed? What are the answers to the questions? How can he be sure? In your answer, show the dimensions of each matrix you use stated in terms of content (for example, {assets*states}).
Matrix m1 is {states*assets}. If the desired portfolio, n, is {assets*1}, then the cash flows will be:
c = m1*n
where c is {states*1}. Here we know c = [1000 0]' and need to find n and its cost. To do this we multiply each side by the inverse of m1, giving:
inv(m1)*c = n
In this case: the required strategy is to invest $3,333.33 in a stock and short $3,044.90 of bonds. The net cost is thus 3,333.33 - 3,044.90, or $288.50. The latter could have been obtained more elegantly by using a vector of prices for the assets, pa = [1 1] and computing the cost as:
cost = pa*n
2f. (4 points) Having successfully answered the prior questions in his interview, Ken is posed another set of questions. He is to determine the Arrow-Debreu state prices for (1) $1 if the market is up and (2) $1 if the market is down. Then he is to show that these are consistent with the one-year discount factor. Worse yet, he has only a few minutes to provide the answers. What are the two prices? Show that they are consistent with the discount factor.
We already know the state price for the up-state. It is $0.2885. Similar calculations would give the second (most easily obtained by summing the two entries in the second column of inv(m1). It is $0.6731. The two state prices sum to $0.9615. The discount factor is 1/1.04, which is also $.9615. Q.E.D.
2g. (4 points) The mutual fund department of Merrill Chase would like to issue a certificate with "upside potential and downside protection". In particular, this certificate would promise the holder $1,000 two years hence if the market is up two years in a row (upside potential) and $800 otherwise (downside protection). What would be the cost of hedging such a claim? To help you make any needed calculations, we provide below the inverses of M and M'.
inv(M) = [ -3.0449 4.0064 -0.8783 -2.0494 1.1557 2.6966
3.3333 -3.3333 0.9615 2.2436 -0.9615 -2.2436
0 0 -3.0449 4.0064 0 0
0 0 3.3333 -3.3333 0 0
0 0 0 0 -3.0449 4.0064
0 0 0 0 3.3333 -3.3333 ]
inv(M') = [ -3.0449 3.3333 0 0 0 0
4.0064 -3.3333 0 0 0 0
-0.8783 0.9615 -3.0449 3.3333 0 0
-2.0494 2.2436 4.0064 -3.3333 0.0000 0
1.1557 -0.9615 0 0 -3.0449 3.3333
2.6966 -2.2436 0 0 4.0064 -3.3333 ]
The easiest way to deal with this problem is to find the state prices. Since only the first two columns (B and S) cost money today, the price for each state will simply equal the sum of the first two elements in the inverse of M. This gives
ps = [ 0.2885 0.6731 0.0832 0.1942 0.1942 0.4530 ]
The desired payments are c = [ 0 0 1000 800 800 800 ]'
Multiplying these two vectors gives the cost:
cost = 756.29
2h. (4 points) The head of the mutual fund department has found through market research and focus groups that the proposed certificate (in 2g) could easily be sold for $800 each. She asks you to determine whether or not to issue such a certificate and, if so, to provide instructions for the trading department (1) today, (2) next year if the market is up, and (3) next year if the market is down. You plan to have an assistant do all the needed calculations. He is proficient in linear algebra, MATLAB and Excel but knows nothing else. Provide the instructions he needs to find the answers in a form that he can understand and the information he needs to prepare the instructions for the trading department.
We have the formulas already. Strategy n will provide cash flows c if:
c = M*n
To find the strategy we solve for n in
n = inv(M)*c
The first two entries of n indicate the dollars to be invested (shorted, if negative) in bonds and stocks today. The next two indicate the dollars to be invested (shorted, if negative) in bonds and stocks next year if the market is up. The final two indicate the dollars to be invested (shorted, if negative) in bonds and stocks next year if the market is down.
2i. (4 points) Assume that Merrill is correct in its assessment that the odds are 50/50 that the market will go up or down in each year.
What is the expected growth of $1 invested in the new
certificate over the two years (that is, the expected ending value per dollar invested)?
What would the ending value be if $1 were invested in a two-year
bond?
What would it be if $1 were invested in the stock market for two
periods?
In which ending states would $1 invested in the certificate outperform
$1 invested in the bond? The stock?
What are the risk premiums or discounts associated with each of the
three strategies?
Does the certificate offer a risk premium? If so, why? If
not, why not?
The certificate costs $756.29. It is equally likely to return $1,000, $800, $800 or $800. Thus the expected ending value is $850. This gives an expected value per dollar invested of 850/756.29, or 1.1239, for an expected two-year return of 12.39%.
For the bond $1 grows to 1.04^2, or 1.0816 for an expected two-year return of 8.16%.
An investment of $1 in stocks is equally likely to have an ending value of 1.25^2, 1.25*.95, 1.25*.95, or .95*.95. This gives an expected value of $1.21 for an expected two-year return of 21%.
Ending values per dollar invested in stocks,
bonds and the certificate are:
State Bond Stock Certificate uu 1.0816 1.5625 1.3222 ud 1.0816 1.1875 1.0578 du 1.0816 1.1875 1.0578 dd 1.0816 0.9025 1.0578
So the certificate outperforms the bond in
state uu and underperforms otherwise. It outperforms stocks in dd and underperforms
otherwise.
Since the bonds are riskless, they have no risk premium. The certificate has a risk
premium of 12.39-8.16, or 4.23% (per two years). Stocks have a risk premium of
21.0-8.16, or 12.84% per two years.
The certificate offers a risk premium because its highest payoff is in the state with the highest expected return (uu). Since all four states are equally likely, comparison of the state prices reveals this directly (they are 0.0832 0.1942 0.1942 and 0.4530). Thus the certificate does (relatively) badly in bad (not great) times. Presumably the reason for the high expected return in state uu is that goods are plentiful in that state with low marginal utilities for representative investors.
3. Assets R'Us (ARU) is a firm specializing in financial planning for individuals. After considerable research the firm has predicted the following for the returns in US dollars over the next year for four major asset classes:
Asset Expected Return Standard Deviation corr with
Cashcorr with
USBondscorr with
USStockscorr with
NonUSStocksCash 4.5 0.0 0 0 0 0 USBonds 5.5 5.0 0 1 0.4 0.2 USStocks 10.5 15.0 0 0.4 1 0.5 NonUSStocks 10.0 20.0 0 0.2 0.5 1
3a. (4 points) What are the covariances of the assets with one another? Please include a table showing your results.
Asset Cash USBonds USStocks NonUSStocks Cash 0 0 0 0 USBonds 0 25 30 20 USStocks 0 30 225 150 NonUSStocks 0 20 150 400
3b. (4 points) ARU has three clients, each of which has a numbered account. To make matters simple, each client is referenced by his or her risk tolerance. Thus the first client (client 25) has a risk tolerance of 25, the second (client 50) has a risk tolerance of 50, and the third (client 75) has a risk tolerance of 75. No client is allowed to take a short position in any asset.
At present, each client has the same portfolio. The proportions invested in the three assets in this portfolio are:
Asset Cash Cash 0.10 USBonds 0.30 USStocks 0.50 NonUSStocks 0.10
What is the expected return of this portfolio? What is its variance of return?
e = 8.35%
v = 87.70
3c. (4 points) What is the utility of the portfolio for client 25? for client 50? for client 75? If these differ, why do they differ? If not, why not?
u(25) = 4.842
u(50) = 6.596
u(75) = 7.1807
The higher risk tolerance investors do not subtract as large a risk penalty (v/rt) from expected return. Thus their certainty equivalent (or risk-adjusted return, or utility) is higher for the same portfolio.
3d. (4 points) Is this portfolio optimal for client 25? for client 50? for client 75?
To answer this question we need to compute marginal utilities. The marginal utility of an asset is its expected return minus its marginal risk divided by risk tolerance. The marginal risk of an asset is two times its weighted average of its covariances with the assets, using current portfolio weights. The results of the computations are shown below.
Asset Expected Return Marginal Risk MU(25) MU(50) MU(75) Cash 4.50 0 4.50 4.50 4.50 USBonds 5.50 49 3.54 4.52 4.8467 USStocks 10.50 273 -0.42 5.04 6.8600 NonUSStocks 10.00 242 0.32 5.16 6.7733
The portfolio is not optimal for any client since in each case there is at least one pair of "in" assets (between their upper and lower bounds) with different marginal utilities.
3e. (4 points) If you were allowed to change the holdings in only two asset classes for client 25, and the amount of change was required to be very small, which two assets would you change, and in what manner, or would you recommend no changes?
For this client, sell US Stocks (with the lowest marginal utility) and buy cash (with the highest marginal utility).
3f. (4 points) If you were allowed to change the holdings in only two asset classes for client 50, and the amount of change was required to be very small, which two assets would you change, and in what manner, or would you recommend no changes?
For this client, sell cash (with the lowest marginal utility) and buy nonUS stocks (with the highest marginal utility).
3g. (4 points) If you were allowed to change the holdings in only two asset classes for client 75, and the amount of change was required to be very small, which two assets would you change, and in what manner, or would you recommend no changes?
For this client, sell cash (with the lowest marginal utility) and buy US stocks (with the highest marginal utility).
3h. (4 points) If all three clients were in the office when you made your recommendations how would you explain to them any differences in your advice? For each give a rationale (in english, using no technical terms) for your conclusions.
Client 25 has low tolerance for risk. You are recommending a more conservative portfolio.
Client 50 has medium tolerance for risk. The current portfolio is too conservative. Foreign stocks are especially attractive because they provide better diversification than adding to the already substantial holdings in US stocks. This more than offsets their somewhat lower expected returns in this case.
Client 75 has a high tolerance for risk. The current portfolio is far too conservative. While foreign stocks provide better diversification than US stocks, risk reduction is not important enough for this client to offset the higher expected returns for US stocks.
3i. (4 points) One of your colleagues has suggested the following portfolio for client 75. Compare its utility for this client with that of the current portfolio. Is this better? Could there be a better one? Please provide computations to defend your answers.
Asset Cash Cash 0.00 USBonds 0.00 USStocks 0.8269 NonUSStocks 0.1731
The marginal utilities are as follows:
Asset Expected Return Marginal Risk MU(75) Cash 4.50 0 4.5000 USBonds 5.50 56.538 4.7462 USStocks 10.50 424.035 4.8462 NonUSStocks 10.00 386.55 4.8460
It is definitely better. The characteristics are:
CHARACTERISTICS:
Current Proposed
ExpRet 8.350 10.413
StdDev 9.365 14.449
Utility 7.181 7.630
Is there a better one? Possibly, but it is not likely to be much better. The two down variables (cash and bonds) have marginal utilities below those of the two in variables (US stocks and nonUS stocks). The two in variables have very similar marginal utilities, so the likely improvement is very small.
4. A small investment consulting firm in the country of Transylvania is convinced that there are two key common factors affecting stock returns. One is associated with a stock's dividend yield, the other with its historic earnings growth rate. To this end, each of the 100 stocks in the Transylvanian market has been analyzed, and assigned two numbers. The first, y(i) is the relative yield of the stock. This is an integer number that ranges from a value of 100 (for the stock with the highest yield) to 1 (for the stock with the lowest yield). The stock with the second-highest yield has a y(i) of 99, and so on. The second number g(i) indicates the stocks's relative growth rate. Here, too the numbers are integers from 100 (for the stock with the highest historic growth rate) to 1 for the stock with the lowest historic growth rate.
The consulting firm has also classified each stock based on its economic activity. Every stock has been assigned to one (and only one) economic sector, either basic industries (B), consumer goods (C), finance (F), or technology (T).
The firm has hired you to implement this view in a factor model of security returns.
4a. (4 points) Write the equation that will characterize your model of the return-generating process. Please define each term in sufficient detail to avoid any confusion.
Six factors are needed. The first two can reflect the yield factor and growth factor, with the last four reflecting the industry factors. Let im1(i) equal 1 if security i is a member of industry 1 and zero otherwise, im2(i) equal 1 if security i is a member of indstry 2 and zero otherwise, etc. we have:
r(i) = y(i)*fy + g(i)*fg + im1(i)*if1 + im2(i)*if2 + im3(i)*if3 + im4(i)*if4 + e(i)
Here the factors are the yield factor (fy) the growth factor (fg), and industry factors 1 (if1) through 4 (if4) Alternatively, an intercept could be include (for example, a(i)) with the understanding that the expected value of e(i) is zero.
4b. (4 points) Assume that the yield factor was positive last month. Does it mean that every stock with a yield greater than the median yield outperformed every stock with a yield below the median yield? Suppose that there are two stocks in the same industry with the same historic earnings growth. Does this mean that the one with the higher yield outperformed the one with the lower yield? What, if anything, , can you say about the relative performance of high yield and low yield stocks, based on the fact that the yield factor was positive?
The information that the yield factor was positive only indicates that on average, other things equal, high yield stocks outperformed low yield stocks. In fact there were undoubtedly many cases in which a high yield stock underperformed a low yield stock. This could be due to different exposures to other factors. However, even in the case in which two stocks had the same exposures to other factors (as in the second question asked), the residual returns could cause the high yield stock to underperform the low yield stock.
4c. (4 points) You have been presented with data for each of the 100 stocks for each of the preceding 60 months. For each month you have the following information for each stock:
y(i): the stock's yield ranking at the beginning of the month
g(i): the stock's historic growth ranking at the beginning of the month
sector(i): the stock's economic sector at the beginning of the month
What would you do to determine the realized values of the
relevant factors in a specific month?
What would you do to determine the historic covariance matrix for the factors and the
historic mean returns for the factors?
What would you do to determine the historic residual risks for each of the
securities?
Please answer in sufficient detail so that a statistician could perform the calculations
needed to provide the quantitive values in question.
What statistical measures might you use to see whether or not your factor model makes
sense?
For each month, you would perform a cross-section regression in which each stock was an observation, the stock returns (r(i)) are the dependent variable, and the six characteristics (y(i), g(i), im1(i), .. im4(i)) are the independent variables. This would give the realizations of the factors for the month and the residual return for each of the securities for that month.
Having done this for each month, you would compute the covariance matrix for the 60 months of factor values, the means of the 60 factor values, and the variances of the 60 residual returns for each of the stocks.
You could check the t-statistics for the cross-sectional regressions. For each factor the t-statistic should have an absolute value of, say, 2.0 in a significant number (more than 5%) of the months. You could also check the R^2 values for the overall regression in each month. These should be large enough to indicate significant explanatory power. Finally, you could check the correlations of the residal returns. These should average close to zero and there should be relatively few values that are large in absolute value.
5. Another consulting firm has decided that it is silly to worry about factors within the overall stock market since it is interested only in equity mutual funds. It has analyzed the behavior of three such funds by relating the excess return (return over the treasury bill rate) on each fund to the excess return on the overall stock market. They have termed the resultant sensitivity "stock exposure" -- thus a fund with a stock exposure of 0.80 would have an excess return 80% as great as that of the stock market on average. In addition to estimates of the fund's stock exposures, an estimate has been made for the added risk and return for each fund, over and above that attributable to its exposure to the stock market. The consulting firm assumes that each fund's added (residual) return is uncorrelated with the stock market and with the added (residual) return of each of the other funds. The results of this analysis are as follows:
Fund Stock exposure Added Expected Return (% per year) Added Risk (% standard deviation per year) A 0.80 1.0 5.0 B 1.00 2.0 10.0 C 1.20 1.0 5.0
Finally, the firm has made the following estimates for the stock market:
Expected Excess Return (%/yr) Std Dev of Return (%/yr) 5.0 15.0
The rate of interest on treasury bills is 4.0%.
5a. (4 points) What is the (total) expected return of fund A?
Expected return equals:
Treasury bill return:
4.0
Stock exposure * Stock expected excess : 0.80*5.0
= 4.0
+ Added expected return:
= 1.0
----------------
9.0 %
5b. (4 points) What is the variance of fund C?
The variance due to the stock market factors is:
(1.2^2)*225 = 324
To this we add the variance of the residual return to get total variance:
324 + 25 = 349
5c. (4 points) What is the covariance of the returns on funds A and B?
Here we multiply the exposure of A by the factor covariance matrix (which has only one element) and the exposure of B:
0.80*225*1.20 = 216
There is no residual term, since the residuals are assumed to be uncorrelated.
5d. (4 points) Assume that you had to choose between advising a friend to (1) put all of her money in fund B and (2) put half her money in fund A and half in fund C. Which would you advise her to take, or might your answer depend on her preferences?
The two alternatives have the same exposures to the stock market (1.00 and 1.00) and so are equivalent in terms of market (factor)-related risks and returns. B offers a residual expected return of 2.0% while AC offers only 1.0%. On the other hand, B has a residual standard deviation of 10.0 while AC has a residual standard deviation of 5.0/sqrt(2), since its variance will equal half that of each of the two components (given that they have equal risk, are uncorrelated with one another, and the portfolio has equal amounts invested in each). This implies that AC has half the expected added value of B but less than half the added risk. In terms of the Sharpe Ratio of the residual, AC is better than B. However, this is only relevant if she can separate out the factor-related components and lever or unlever the residual to a desired level of residual risk. As the question is stated, this does not appear to be possible, so the choice ultimately depends on her preferences -- a choice between a higher expected return and risk (B) and lower expected return and risk (AC).