with Answers (and additional comments)
(points shown in blue)
1. In the small country of petitpays there are only four stocks. The number of shares outstanding and last night's price per share of each are shown below:
Stock Shares Price Alpha Industries 41,250 40 Beta Telecom 30,000 30 Gamma Technology 25,000 12 Delta Toys 30,000 5
Stock |
Shares |
Price |
Value |
PctMkt |
PctLg |
PctSml |
Alpha Industries |
41,250 |
40 |
1,650,000 |
55% |
65% |
|
Beta Telecom |
30,000 |
30 |
900,000 |
30% |
35% |
|
Gamma Technology |
25,000 |
12 |
300,000 |
10% |
67% |
|
Delta Toys |
30,000 |
5 |
150,000 |
5% |
33% |
|
3,000,000 |
Despite the small number of stocks, there are 25 mutual funds in petitpays. You have been asked to evaluate six of them. For each one, the composition of last night's portfolio is shown using relative values, based on last night's stock prices.
Stock Brigantine Complet Fido Quasar Rabba Westwood Alpha Industries 64.7 55.0 75.0 0.0 57.0 0.0 Beta Telecom 35.3 30.0 25.0 0.0 29.0 0.0 Gamma Technology 0.0 10.0 0.0 50.0 8.0 66.7 Delta Toys 0.0 5.0 0.0 50.0 6.0 33.3 Total 100.0 100.0 100.0 100.0 100.0 100.0
1a. Which (if any) of the funds are:
[1] a. Active Large Cap Fido
[1] b. Active Small Cap Quasar
[1] c. Enhanced Index (high R-squared) Rabba
[1] d. Large Cap Index Brigantine
[1] e. Small Cap Index Westwood
[1] f. Market Index Complet
Assume that at the close of the market today the return on the entire market was 2.5%.
[4] 1b. The Wall Street Journal reports the performance of the mutual fund industry by averaging the total returns of all the funds, with each fund given the same weight. What was the resulting average return? (circle one)
More than 2.5%
2.5%
Less than 2.5%
You can't sayWhy?
Assuming that the 2.5% is the return on the average dollar invested in the market (or equivalently, the return on a market-capitalization weighted index of all stocks), this may be more, less or equal to a simple average of the returns on funds, each of which has a different market value. If funds were representative and their returns were weighted by fund size, the gross returns would equal the market-capitalization weighted average for the stocks in the market. However, even in this case, the net return would be less. And, since the reported fund average is not weighted by fund size, anything could happen.
2. The following table shows the styles of four mutual funds, expressed in terms of Cash, Bonds and Stocks.
Cash Bonds Stocks Aggressive Growth Fund 0.05 0.00 0.95 Balanced Safety Fund 0.10 0.45 0.45 Conservative Investors' Fund 0.20 0.70 0.10 Responsible Investors' Fund 0.20 0.40 0.40
2a.Doctor Watson has just invested 20% of her money in the Aggressive Growth Fund, 30% in the Balanced Safety Fund, 25% in the Conservative Investors' Fund, and 25% in the Responsible Investors' Fund. She has come to you for advice. In particular, she wants to know her current asset allocation. In order to avoid doing this in detail over and over in the future, you have decided to construct matrices, vectors and such so that you can write the task as a simple matrix expression. Show the contents of each matrix or vector that you would use and write an expression, such as a=b*c, that could be interpreted by a programming language such as MATLAB and that would provide the answer.
[2] Matrices and Vectors:
b = a vector with 1 row and funds columns {1*f}
b = [ 0.2000 0.3000 0.2500 0.2500 ]
c = a matrix with funds rows and assets columns {f*a}
c = [ 0.0500 0 0.9500
0.1000 0.4500 0.4500
0.2000 0.7000 0.1000
0.2000 0.4000 0.4000 ]
[1] Expression: a = b*c
[1] How big would your answer matrix or vector be? 1 row and 3 (assets) columns
[1] How would you interpret the contents of each cell?
The first entry shows the proportion of Watson's fund invested in cash, the second the proportion invested in bonds, and the third the proportion invested in stocks.
2b. Impressed with your skills, Dr. Watson recommends your services to her friend Mr. Holmes. Holmes has a modest amount of money saved and wishes to invest it. He has deduced that the appropriate asset allocation for him requires 10% in cash, 30% in bonds and 60% in stocks. Your task is to find a fund or combination of two or more of the funds that will accomplish his goal. You want to be able to do this for others in an efficient manner, so you have decided to construct matrices, vectors, and such so you can write the task as a simple matrix expression. Show the contents of each matrix and vector that you woul use:
[7] There are more funds than assets in this case. Thus it should be possible to obtain a desired asset allocation with only three of them (barring degeneracy). For example, Holmes could use the first three funds
Cash Bonds Stocks Aggressive Growth Fund 0.05 0.00 0.95 Balanced Safety Fund 0.10 0.45 0.45 Conservative Investors' Fund 0.20 0.70 0.10
We now have a square matrix that is {f*a}. Call if cc:
cc = [ 0.0500 0 0.9500
0.1000 0.4500 0.4500
0.2000 0.7000 0.1000 ]
We seek a vector of fund holdings (bb) that is {1*f} such that:
aa = bb*cc
where aa = the desired asset mix in a vector that is {1*f}:
aa = [ 0.1000 0.3000 0.6000 ]
[1] Write a MATLAB expression that would provide the answer: bb = aa*inv(cc)
[1] How big would the answer matrix or vector be? 1*3, showing the proportions in each of the three selected funds
[1] How would you interpret each cell?
The first cell would indicate the proportion to invest in fund 1, the second would indicate the proportion to invest in fund 2, and the last would indicate the proportion to invest in fund 3, where the fund numbers refer to the chosen funds.
[6] 2c. After you report your solution to Mr. Holmes he says "Aha -- you must have made an error, since Mr. Callan, with whom I consulted yesterday, recommended a different combination of funds and showed me that his combination gave the required asset allocation!". How do you deal with this criticism?
You point out to Holmes that he doesn't understand financial economics. With more funds than assets, there will typically be alternative ways to get the same asset allocation. For example, using the first three funds in the procedure above gives a result, stated in tems of the four funds of:
b1 = [ 0.4615 0.3077 0.2308 0 ]
while using funds 2 through 4 gives:
b2 = [ 0 1.0000 -0.5000 0.5000 ]
But both portfolios offer the same asset allocation (although the latter involves a short position in one fund, and hence may not be feasible)..
3. In a highly advanced society there are two securities. One is called rf (for riskfree), the other s (for stock). There are only two possible future states of the world next year (sunny and rainy). The proceeds obtained by selling $1 worth of each of the securities next year in each of the two states are as shown below:
State rf s sunny 1.04 1.29 rainy 1.04 0.89
You have asked a mathematican friend to invert this matrix. Uncertain about what you really want, she provides two answers:
M1 = [ 1.04 1.29
1.04 0.89 ]
inv(M1) = [ -2.1394 3.1010
2.5000 -2.5000 ]
M2 = [ 1.04 1.04
1.29 0.89 ]
inv(M2) = [ -2.1394 2.5000
3.1010 -2.5000 ]
You believe that it is as likely to be sunny as to be rainy.
3a. Given this information:
[2] What is the expected return on stocks (s)?
1+e = .5*1.29 + .5*.89 = 1.09
expected return = 9%
[3] What is the standard deviation of stocks (s)?
v = .5 * ((1.29 -1.09)^2) + .5 * ((0.89 - 1.09)^2) = .04
sd = sqrt(v) = 0.20
standard deviation of return = 20%
[2] What is the equity risk premium?
9% - 4% = 5%
[3] What is the Sharpe Ratio for equities?
SR = (9 -4) / 20 = 0.25
3b. You are running a financial intermediation firm in this country. An umbrella manufacturer asks you if you would issue a sunshine insurance policy which would pay $100 if the weather turns out to be sunny. The manufacturer is willing to pay $50, which seems to him to be a fair price. Would you write the policy? If so, how much profit would there be for you in the deal?
To answer this question (and the next one), we need to find the prices for the pure securities (one for each state). If we had a portfolio of securities x {securities*1}, the payoff pattern across states would be:
z = M1*x
(note that M1 is {states*securities}. Thus z will be {states*1}, as desired.
To find a portfolio x that will give a desired payoff pattern z, we can solve:
x = inv(M1)*z
In this case, we want z to equal [1 0]. This means that the first column of inv(M1) gives the composition of the portfolio that will provide $1 if the weather is sunny. Since each security costs $1, the total cost is simply the sum of the entries in the first column of inv(m1) times 100. Thus
Cost = .3606*100 = $ 36.06
[3] So you should definitely issue the policy.
[4] The profit will be $13.94.
[3] 3c. If a sunglass manufacturer came to you and asked for a rain policy on the same terms (i.e. for $50 now you would pay $100 if it rained) would you write the policy? If not, why not?
The sum of the entries in the second column of inv(m1) indicates the cost of providing $1 if the wealther is rainy. Thus the cost of issue this insurance policy is:
Cost = .6010*100 = $ 60.10
So you should tell the manufacturer to go to another insurance company (preferably one that doesn't know matrix algebra).
4. You have spent the better part of the weekend making estimates of next year's returns for two mutual funds -- A and B. You believe that Fund A has an expected return of 8% and a standard deviation of return of 10%, while Fund B has an expected return of 10% and a standard deviation of return of 15%. You estimate that the correlation between the returns of the two funds is 0.70. The risk-free rate of interest is 5%. You have two clients: X and Y. Client X has a risk tolerance of 25, while client Y has a risk tolerance of 75.
[3] 4a. If you had to invest all of Client X's money in either Fund A or Fund B, which would you choose? _______
Utility = e - v/tUa = 8 - (10^2)/25 = 4 Ub =10 - (15^2)/25 = 1So -- choose Fund A
[3] 4b. If you had to invest all of Client Y's money in either Fund A or Fund B, which would you choose? _______
Utility = e - v/tUa = 8 - (10^2)/75 = 6.67 Ub =10 - (15^2)/75 = 7So -- choose Fund B
[5] 4c. Client X can borrow or lend at 5%. You need to help her choose between (1) a combination of Fund A plus borrowing or lending and (2) a combination of Fund B plus borrowing or lending. Which fund would you choose and how much would you advise her to borrow or lend (if any)?
Given the ability to borrow or lend, the goal is to choose the fund with the largest Sharpe ratio.
SRa = (8 - 5) / 10 = 0.30SRb = (10 - 5) /15 = 0.33Hence, choose fund B.
To find the optimium amount to invest in fund B, let x equal the proportion invested in B. Then:
ep = 5 + 5*xvp = (x^2)*(15^2) = 225*(x^2)U = e - v/t = 5 + 5x - (225*(x^2)) / t
To maximize utility, we set marginal utility (dU/dx) to zero:
5 - ( 2*x*225) /t = 0
So that:
x = (5*t) / 450
(a formula that we have derived in other contexts)
For client X, t = 25, so that:
x = 5*25/450 = .278
So client X should put 27.8% of her money in Fund B and the remaining 72.2% in the bank.
[2] 4d. Client Y can also borrow or lend at 5%. You need to help him choose between (1) a combination of Fund A plus borrowing or lending and (2) a combination of Fund B plus borrowing or lending. Which fund would you choose and how much would you advise him to borrow or lend (if any)?
Again, the better fund is Fund B, since it has the higher Sharpe ratio.
Again, the optimum amount to invest in fund B is equal to:
x = (5*t) / 450
Since Client Y has a risk tolerance of 75, the optimum investment in the fund is
x = (5*75)/450 = 0.8333
So he should put 83.33% of his money in Fund B and the remaining 16.67% in the bank.
[5] 4e. An investment management company has introduced a "fund of funds" that has 50% invested in Fund A and 50% in Fund B. This new fund (called, somewhat unimaginatively) Fund AB, has no additional expenses other than those associated with Funds A and B. Client X can borrow or lend at 5% and you now have to help her choose either (1) a combination of Fund A plus borrowing or lending or (2) a combination of Fund B plus borrowing or lending, or (3) a combination of Fund AB plus borrowing or lending. Which alternative would you choose?
Fund AB has an expected return of:
e = 0.5*8 + 0.5*10 = 9.0
a variance of::
v = ((0.5^2)*(10^2)) + (2*0.5*0.5*0.70*10*15) + ((.5^2)*(15^2)) = 133.75
and a standard deviation of:
s = sqrt(v) = 11.565
Its Sharpe ratio is thus:
SRab = (9 - 5) / 11.565 = 0.3459
Since this is greater than the Sharpe ratios of the other funds, the best alternative is (3) -- a combination of Fund AB plus borrowing or lending.
[2] 4f. Client Y can also borrow or lend at 5% and you also have to help him choose either (1) a combination of Fund A plus borrowing or lending or (2) a combination of Fund B plus borrowing or lending, or (3) a combination of Fund AB plus borrowing or lending. Which alternative would you choose?
The best alternative is still (3) -- Fund AB plus borrowing or lending, although the amount borrowed or lent will differ for the two clients.
5. The President of a large endowment fund has completed a detailed analysis of the fund's investments. The fund's board has let the President select any allocation among six major asset classes as long as no more than 50% of the fund is invested in any given asset class and there are no short positions in any of the asset classes. The President has engaged a consulting firm run by several recent graduates of a prestigious MBA program to recommend an asset allocation. The table below shows the recommended allocation, the expected returns used in the analysis, and the associated "marginal utilities" of the asset classes, given the risk tolerance of the trustees of the foundation.
Allocation (%) Expected Return Marginal Utility Cash 0 5 5.0 Government Bonds 0 6 5.3 Corporate Bonds 10 7 5.5 Stocks 20 11 5.5 Real Estate 20 9 5.5 Venture Capital 50 15 6.1
[3] 5a. Assuming that all the inputs used by the consulting firm were correct, have they recommended the best possible allocation among the asset classes? If not, what changes would you recommend?
The recommendation is optimal, since the marginal utilities of all the assets that are "in the solution" (between their bounds are the same (5.5), all those for assets that are at their lower bounds ("down") are less (5.0 and 5.3), and the marginal utility for the asset that is at its upper bound (Venture Capital) is more (6.1).
[3] 5b. A recent change in interest rates has raised the best estimate of expected return for government bonds from 6.0% to 6.1%. What, if any, change should be made in the foundation's asset allocation?
Note that the marginal utility of an asset is:
mu(i) = e(i) - 2*c(i,p)/t
Since only the asset's expected return has changed, the marginal utility will increase by the same amount as the increase in the expected return. This increases the Government Bond marginal utility from 5.3 to 5.4. Since this is still below the common marginal utility for the "in variables", the current asset mix is still optimal and no changes should be made.
[4] 5c. Government interest rates have fallen back to their original levels, but the President is convinced that the expected return on Real Estate is 9.5% rather than 9.0% and that Venture Capital has an expected return of 14% rather than 15%. She doesn't want to attract the attention of the board, but would like to make at least a small change in the allocation by lowering the amount invested in one asset and increasing the amount invested in another one. Would you recommend that she make such a change? If so, which one?
The situation is now:
Allocation (%) Expected Return Marginal Utility Cash 0 5 5.0 Government Bonds 0 6 5.3 Corporate Bonds 10 7 5.5 Stocks 20 11 5.5 Real Estate 20 9.5 6.0 Venture Capital 50 14 5.1
Real Estate has the highest marginal utility and is below its upper bound, so it is the most attractive candidate for an increase. Venture capital has a lower marginal utility than any other variable that can be decreased, so it is the most attractive candidate for a decrease. Hence the best single change would involve selling Venture Capital and purchasing Real Estate with the proceeds from the sale.
6. The table below provides spaces for you to write your estimates of the style of each of 10 mutual funds. In each style box write one number, constituting your best estimate of the percentage (blanks are treated as zero and the sum for each fund should equal 100). All estimates should be forward-looking for next year.
All asset classes are represented by index funds, with asset returns based on the net returns to investors in the index funds.
[2] for each fund
Some of the numbers below are guesses, others are relatively precise.
| Fund 1 | Fund 2 | Fund 3 | Fund 4 | Fund 5 | Fund 6 | Fund 7 | Fund 8 | Fund 9 | Fund 10 | |
| Cash | 5 | -25 | -20 | |||||||
| Intermediate Govt. Bonds | 10 | 16 | ||||||||
| Long-term Govt. Bonds | 60 | 10 | ||||||||
| Corporate Bonds | 50 | 80 | 40 | 6 | ||||||
| S&P500 Value | 47.5 | 17.5 | 62.5 | 40 | 30 | 17 | ||||
| S&P500 Growth | 47.5 | 100 | 17.5 | 62.5 | 40 | 50 | 30 | 17 | ||
| Wilshire 4500 | 100 | 15 | 20 | 14 | ||||||
| MSCI EAFE | 20 | 20 |
The descriptions of the funds follow:
1. An S&P500 stock index fund that maintains a cash margin of approximately 5%.
2. A U.S. equity fund that specializes in medium and small stocks
3. A fund that buys large growth stocks
4. A convertible bond fund
5. A fund that buys junk bonds
6. A fund that buys large stocks on margin, with $125 invested for every $100 of capital.
7. A fund that specializes in stocks of large global corporations based in the United States.
8. A balanced fund that favors high-grade bonds and large stocks of admired companies
9. A fund that holds very long-term government bonds and stocks with betas (relative to the S&P500) greater than 1.
10. A fund that invests 80% of its money in market proportions of all U.S. securities and 20% in non-U.S. equities.
with Answers (and additional comments)
(points shown in blue)
1. In the small country of petitpays there are only four stocks. The number of shares outstanding and last night's price per share of each are shown below:
Stock Shares Price Alpha Industries 41,250 40 Beta Telecom 30,000 30 Gamma Technology 25,000 12 Delta Toys 30,000 5
Stock |
Shares |
Price |
Value |
PctMkt |
PctLg |
PctSml |
Alpha Industries |
41,250 |
40 |
1,650,000 |
55% |
65% |
|
Beta Telecom |
30,000 |
30 |
900,000 |
30% |
35% |
|
Gamma Technology |
25,000 |
12 |
300,000 |
10% |
67% |
|
Delta Toys |
30,000 |
5 |
150,000 |
5% |
33% |
|
3,000,000 |
Despite the small number of stocks, there are 25 mutual funds in petitpays. You have been asked to evaluate six of them. For each one, the composition of last night's portfolio is shown using relative values, based on last night's stock prices.
Stock Brigantine Complet Fido Quasar Rabba Westwood Alpha Industries 64.7 55.0 75.0 0.0 57.0 0.0 Beta Telecom 35.3 30.0 25.0 0.0 29.0 0.0 Gamma Technology 0.0 10.0 0.0 50.0 8.0 66.7 Delta Toys 0.0 5.0 0.0 50.0 6.0 33.3 Total 100.0 100.0 100.0 100.0 100.0 100.0
1a. Which (if any) of the funds are:
[1] a. Active Large Cap Fido
[1] b. Active Small Cap Quasar
[1] c. Enhanced Index (high R-squared) Rabba
[1] d. Large Cap Index Brigantine
[1] e. Small Cap Index Westwood
[1] f. Market Index Complet
Assume that at the close of the market today the return on the entire market was 2.5%.
[4] 1b. The Wall Street Journal reports the performance of the mutual fund industry by averaging the total returns of all the funds, with each fund given the same weight. What was the resulting average return? (circle one)
More than 2.5%
2.5%
Less than 2.5%
You can't sayWhy?
Assuming that the 2.5% is the return on the average dollar invested in the market (or equivalently, the return on a market-capitalization weighted index of all stocks), this may be more, less or equal to a simple average of the returns on funds, each of which has a different market value. If funds were representative and their returns were weighted by fund size, the gross returns would equal the market-capitalization weighted average for the stocks in the market. However, even in this case, the net return would be less. And, since the reported fund average is not weighted by fund size, anything could happen.
2. The following table shows the styles of four mutual funds, expressed in terms of Cash, Bonds and Stocks.
Cash Bonds Stocks Aggressive Growth Fund 0.05 0.00 0.95 Balanced Safety Fund 0.10 0.45 0.45 Conservative Investors' Fund 0.20 0.70 0.10 Responsible Investors' Fund 0.20 0.40 0.40
2a.Doctor Watson has just invested 20% of her money in the Aggressive Growth Fund, 30% in the Balanced Safety Fund, 25% in the Conservative Investors' Fund, and 25% in the Responsible Investors' Fund. She has come to you for advice. In particular, she wants to know her current asset allocation. In order to avoid doing this in detail over and over in the future, you have decided to construct matrices, vectors and such so that you can write the task as a simple matrix expression. Show the contents of each matrix or vector that you would use and write an expression, such as a=b*c, that could be interpreted by a programming language such as MATLAB and that would provide the answer.
[2] Matrices and Vectors:
b = a vector with 1 row and funds columns {1*f}
b = [ 0.2000 0.3000 0.2500 0.2500 ]
c = a matrix with funds rows and assets columns {f*a}
c = [ 0.0500 0 0.9500
0.1000 0.4500 0.4500
0.2000 0.7000 0.1000
0.2000 0.4000 0.4000 ]
[1] Expression: a = b*c
[1] How big would your answer matrix or vector be? 1 row and 3 (assets) columns
[1] How would you interpret the contents of each cell?
The first entry shows the proportion of Watson's fund invested in cash, the second the proportion invested in bonds, and the third the proportion invested in stocks.
2b. Impressed with your skills, Dr. Watson recommends your services to her friend Mr. Holmes. Holmes has a modest amount of money saved and wishes to invest it. He has deduced that the appropriate asset allocation for him requires 10% in cash, 30% in bonds and 60% in stocks. Your task is to find a fund or combination of two or more of the funds that will accomplish his goal. You want to be able to do this for others in an efficient manner, so you have decided to construct matrices, vectors, and such so you can write the task as a simple matrix expression. Show the contents of each matrix and vector that you woul use:
[7] There are more funds than assets in this case. Thus it should be possible to obtain a desired asset allocation with only three of them (barring degeneracy). For example, Holmes could use the first three funds
Cash Bonds Stocks Aggressive Growth Fund 0.05 0.00 0.95 Balanced Safety Fund 0.10 0.45 0.45 Conservative Investors' Fund 0.20 0.70 0.10
We now have a square matrix that is {f*a}. Call if cc:
cc = [ 0.0500 0 0.9500
0.1000 0.4500 0.4500
0.2000 0.7000 0.1000 ]
We seek a vector of fund holdings (bb) that is {1*f} such that:
aa = bb*cc
where aa = the desired asset mix in a vector that is {1*f}:
aa = [ 0.1000 0.3000 0.6000 ]
[1] Write a MATLAB expression that would provide the answer: bb = aa*inv(cc)
[1] How big would the answer matrix or vector be? 1*3, showing the proportions in each of the three selected funds
[1] How would you interpret each cell?
The first cell would indicate the proportion to invest in fund 1, the second would indicate the proportion to invest in fund 2, and the last would indicate the proportion to invest in fund 3, where the fund numbers refer to the chosen funds.
[6] 2c. After you report your solution to Mr. Holmes he says "Aha -- you must have made an error, since Mr. Callan, with whom I consulted yesterday, recommended a different combination of funds and showed me that his combination gave the required asset allocation!". How do you deal with this criticism?
You point out to Holmes that he doesn't understand financial economics. With more funds than assets, there will typically be alternative ways to get the same asset allocation. For example, using the first three funds in the procedure above gives a result, stated in tems of the four funds of:
b1 = [ 0.4615 0.3077 0.2308 0 ]
while using funds 2 through 4 gives:
b2 = [ 0 1.0000 -0.5000 0.5000 ]
But both portfolios offer the same asset allocation (although the latter involves a short position in one fund, and hence may not be feasible)..
3. In a highly advanced society there are two securities. One is called rf (for riskfree), the other s (for stock). There are only two possible future states of the world next year (sunny and rainy). The proceeds obtained by selling $1 worth of each of the securities next year in each of the two states are as shown below:
State rf s sunny 1.04 1.29 rainy 1.04 0.89
You have asked a mathematican friend to invert this matrix. Uncertain about what you really want, she provides two answers:
M1 = [ 1.04 1.29
1.04 0.89 ]
inv(M1) = [ -2.1394 3.1010
2.5000 -2.5000 ]
M2 = [ 1.04 1.04
1.29 0.89 ]
inv(M2) = [ -2.1394 2.5000
3.1010 -2.5000 ]
You believe that it is as likely to be sunny as to be rainy.
3a. Given this information:
[2] What is the expected return on stocks (s)?
1+e = .5*1.29 + .5*.89 = 1.09
expected return = 9%
[3] What is the standard deviation of stocks (s)?
v = .5 * ((1.29 -1.09)^2) + .5 * ((0.89 - 1.09)^2) = .04
sd = sqrt(v) = 0.20
standard deviation of return = 20%
[2] What is the equity risk premium?
9% - 4% = 5%
[3] What is the Sharpe Ratio for equities?
SR = (9 -4) / 20 = 0.25
3b. You are running a financial intermediation firm in this country. An umbrella manufacturer asks you if you would issue a sunshine insurance policy which would pay $100 if the weather turns out to be sunny. The manufacturer is willing to pay $50, which seems to him to be a fair price. Would you write the policy? If so, how much profit would there be for you in the deal?
To answer this question (and the next one), we need to find the prices for the pure securities (one for each state). If we had a portfolio of securities x {securities*1}, the payoff pattern across states would be:
z = M1*x
(note that M1 is {states*securities}. Thus z will be {states*1}, as desired.
To find a portfolio x that will give a desired payoff pattern z, we can solve:
x = inv(M1)*z
In this case, we want z to equal [1 0]. This means that the first column of inv(M1) gives the composition of the portfolio that will provide $1 if the weather is sunny. Since each security costs $1, the total cost is simply the sum of the entries in the first column of inv(m1) times 100. Thus
Cost = .3606*100 = $ 36.06
[3] So you should definitely issue the policy.
[4] The profit will be $13.94.
[3] 3c. If a sunglass manufacturer came to you and asked for a rain policy on the same terms (i.e. for $50 now you would pay $100 if it rained) would you write the policy? If not, why not?
The sum of the entries in the second column of inv(m1) indicates the cost of providing $1 if the wealther is rainy. Thus the cost of issue this insurance policy is:
Cost = .6010*100 = $ 60.10
So you should tell the manufacturer to go to another insurance company (preferably one that doesn't know matrix algebra).
4. You have spent the better part of the weekend making estimates of next year's returns for two mutual funds -- A and B. You believe that Fund A has an expected return of 8% and a standard deviation of return of 10%, while Fund B has an expected return of 10% and a standard deviation of return of 15%. You estimate that the correlation between the returns of the two funds is 0.70. The risk-free rate of interest is 5%. You have two clients: X and Y. Client X has a risk tolerance of 25, while client Y has a risk tolerance of 75.
[3] 4a. If you had to invest all of Client X's money in either Fund A or Fund B, which would you choose? _______
Utility = e - v/tUa = 8 - (10^2)/25 = 4 Ub =10 - (15^2)/25 = 1So -- choose Fund A
[3] 4b. If you had to invest all of Client Y's money in either Fund A or Fund B, which would you choose? _______
Utility = e - v/tUa = 8 - (10^2)/75 = 6.67 Ub =10 - (15^2)/75 = 7So -- choose Fund B
[5] 4c. Client X can borrow or lend at 5%. You need to help her choose between (1) a combination of Fund A plus borrowing or lending and (2) a combination of Fund B plus borrowing or lending. Which fund would you choose and how much would you advise her to borrow or lend (if any)?
Given the ability to borrow or lend, the goal is to choose the fund with the largest Sharpe ratio.
SRa = (8 - 5) / 10 = 0.30SRb = (10 - 5) /15 = 0.33Hence, choose fund B.
To find the optimium amount to invest in fund B, let x equal the proportion invested in B. Then:
ep = 5 + 5*xvp = (x^2)*(15^2) = 225*(x^2)U = e - v/t = 5 + 5x - (225*(x^2)) / t
To maximize utility, we set marginal utility (dU/dx) to zero:
5 - ( 2*x*225) /t = 0
So that:
x = (5*t) / 450
(a formula that we have derived in other contexts)
For client X, t = 25, so that:
x = 5*25/450 = .278
So client X should put 27.8% of her money in Fund B and the remaining 72.2% in the bank.
[2] 4d. Client Y can also borrow or lend at 5%. You need to help him choose between (1) a combination of Fund A plus borrowing or lending and (2) a combination of Fund B plus borrowing or lending. Which fund would you choose and how much would you advise him to borrow or lend (if any)?
Again, the better fund is Fund B, since it has the higher Sharpe ratio.
Again, the optimum amount to invest in fund B is equal to:
x = (5*t) / 450
Since Client Y has a risk tolerance of 75, the optimum investment in the fund is
x = (5*75)/450 = 0.8333
So he should put 83.33% of his money in Fund B and the remaining 16.67% in the bank.
[5] 4e. An investment management company has introduced a "fund of funds" that has 50% invested in Fund A and 50% in Fund B. This new fund (called, somewhat unimaginatively) Fund AB, has no additional expenses other than those associated with Funds A and B. Client X can borrow or lend at 5% and you now have to help her choose either (1) a combination of Fund A plus borrowing or lending or (2) a combination of Fund B plus borrowing or lending, or (3) a combination of Fund AB plus borrowing or lending. Which alternative would you choose?
Fund AB has an expected return of:
e = 0.5*8 + 0.5*10 = 9.0
a variance of::
v = ((0.5^2)*(10^2)) + (2*0.5*0.5*0.70*10*15) + ((.5^2)*(15^2)) = 133.75
and a standard deviation of:
s = sqrt(v) = 11.565
Its Sharpe ratio is thus:
SRab = (9 - 5) / 11.565 = 0.3459
Since this is greater than the Sharpe ratios of the other funds, the best alternative is (3) -- a combination of Fund AB plus borrowing or lending.
[2] 4f. Client Y can also borrow or lend at 5% and you also have to help him choose either (1) a combination of Fund A plus borrowing or lending or (2) a combination of Fund B plus borrowing or lending, or (3) a combination of Fund AB plus borrowing or lending. Which alternative would you choose?
The best alternative is still (3) -- Fund AB plus borrowing or lending, although the amount borrowed or lent will differ for the two clients.
5. The President of a large endowment fund has completed a detailed analysis of the fund's investments. The fund's board has let the President select any allocation among six major asset classes as long as no more than 50% of the fund is invested in any given asset class and there are no short positions in any of the asset classes. The President has engaged a consulting firm run by several recent graduates of a prestigious MBA program to recommend an asset allocation. The table below shows the recommended allocation, the expected returns used in the analysis, and the associated "marginal utilities" of the asset classes, given the risk tolerance of the trustees of the foundation.
Allocation (%) Expected Return Marginal Utility Cash 0 5 5.0 Government Bonds 0 6 5.3 Corporate Bonds 10 7 5.5 Stocks 20 11 5.5 Real Estate 20 9 5.5 Venture Capital 50 15 6.1
[3] 5a. Assuming that all the inputs used by the consulting firm were correct, have they recommended the best possible allocation among the asset classes? If not, what changes would you recommend?
The recommendation is optimal, since the marginal utilities of all the assets that are "in the solution" (between their bounds are the same (5.5), all those for assets that are at their lower bounds ("down") are less (5.0 and 5.3), and the marginal utility for the asset that is at its upper bound (Venture Capital) is more (6.1).
[3] 5b. A recent change in interest rates has raised the best estimate of expected return for government bonds from 6.0% to 6.1%. What, if any, change should be made in the foundation's asset allocation?
Note that the marginal utility of an asset is:
mu(i) = e(i) - 2*c(i,p)/t
Since only the asset's expected return has changed, the marginal utility will increase by the same amount as the increase in the expected return. This increases the Government Bond marginal utility from 5.3 to 5.4. Since this is still below the common marginal utility for the "in variables", the current asset mix is still optimal and no changes should be made.
[4] 5c. Government interest rates have fallen back to their original levels, but the President is convinced that the expected return on Real Estate is 9.5% rather than 9.0% and that Venture Capital has an expected return of 14% rather than 15%. She doesn't want to attract the attention of the board, but would like to make at least a small change in the allocation by lowering the amount invested in one asset and increasing the amount invested in another one. Would you recommend that she make such a change? If so, which one?
The situation is now:
Allocation (%) Expected Return Marginal Utility Cash 0 5 5.0 Government Bonds 0 6 5.3 Corporate Bonds 10 7 5.5 Stocks 20 11 5.5 Real Estate 20 9.5 6.0 Venture Capital 50 14 5.1
Real Estate has the highest marginal utility and is below its upper bound, so it is the most attractive candidate for an increase. Venture capital has a lower marginal utility than any other variable that can be decreased, so it is the most attractive candidate for a decrease. Hence the best single change would involve selling Venture Capital and purchasing Real Estate with the proceeds from the sale.
6. The table below provides spaces for you to write your estimates of the style of each of 10 mutual funds. In each style box write one number, constituting your best estimate of the percentage (blanks are treated as zero and the sum for each fund should equal 100). All estimates should be forward-looking for next year.
All asset classes are represented by index funds, with asset returns based on the net returns to investors in the index funds.
[2] for each fund
Some of the numbers below are guesses, others are relatively precise.
| Fund 1 | Fund 2 | Fund 3 | Fund 4 | Fund 5 | Fund 6 | Fund 7 | Fund 8 | Fund 9 | Fund 10 | |
| Cash | 5 | -25 | -20 | |||||||
| Intermediate Govt. Bonds | 10 | 16 | ||||||||
| Long-term Govt. Bonds | 60 | 10 | ||||||||
| Corporate Bonds | 50 | 80 | 40 | 6 | ||||||
| S&P500 Value | 47.5 | 17.5 | 62.5 | 40 | 30 | 17 | ||||
| S&P500 Growth | 47.5 | 100 | 17.5 | 62.5 | 40 | 50 | 30 | 17 | ||
| Wilshire 4500 | 100 | 15 | 20 | 14 | ||||||
| MSCI EAFE | 20 | 20 |
The descriptions of the funds follow:
1. An S&P500 stock index fund that maintains a cash margin of approximately 5%.
2. A U.S. equity fund that specializes in medium and small stocks
3. A fund that buys large growth stocks
4. A convertible bond fund
5. A fund that buys junk bonds
6. A fund that buys large stocks on margin, with $125 invested for every $100 of capital.
7. A fund that specializes in stocks of large global corporations based in the United States.
8. A balanced fund that favors high-grade bonds and large stocks of admired companies
9. A fund that holds very long-term government bonds and stocks with betas (relative to the S&P500) greater than 1.
10. A fund that invests 80% of its money in market proportions of all U.S. securities and 20% in non-U.S. equities.