DEPARTMENT OF FOIS
FACULTY OF BUSINESS
BROCK UNIVERSITY
SOLUTION MIDTERM EXAMINATION I
Course: MBAB5P42 – Spring 2005
Instructor: M. A. Ayadi
Question 1. (10 marks)
Briefly define each of the following terms:
1. Main properties of financial assets.
Financial assets are claims to the returns generated by real assets and have the following
properties: (i) Moneyness, (ii) Divisibility, (iii) Reversibility, (iv) Term of Maturity, (v) Liquidity,
(vi) Convertibility, and (vii) Tax Status: rates at which the income generated by ownership of the
asset is taxed.
2. Limited liability asset.
A limited liability asset is one whose price is never negative. If you hold a limited liability asset, the
worst thing that can happen is that the price goes to zero, in which case, the simple gross return is
zero and the simple net return is –1 = –100%
3. Buying on the margin.
If I want to buy a stock, but I do not have the funds, I need to buy on margin. When buying on
margin, the broker (who, in turn, is borrowing from a bank, or in the money market) lends me part
of the funds I need for the purchase, while the stocks purchased are the collateral to the loan. Still,
there is a minimum percentage of the margin purchase that I must finance with my own funds, this
is the initial margin requirement.
4. Auction markets.
Auction markets are centralized, and buyers and sellers interact closely with each other. The
presence of intermediaries is limited. Example of an auction market: the NYSE and major
Canadian markets. If the auction is continuous, trades may occur at any time. Technically, only
final traders are needed for such a market to operate, although it would not be very effective.
Intermediaries, market makers or specialists, smooth out the transitory variations in supply and
demand, standing ready to buy and sell securities at a price that is similar to the price of the last
previous trade. Whereas, in a call market (or batch) auction buyers and sellers are pooled together
at a given time. An aggregate price is determined and all trading takes place at that price. Trading
can only take place at these times. A call auction is used to open daily transactions on the NYSE
5. Stop-loss orders.
The broker is instructed to buy (sell) a stated number of shares immediately after the price has
reached higher (lower) stop price, Pstop. This can be viewed as a conditional market order
Stop-sell: sell immediately after the price has reached a lower stop price. Stop-buy: buy immediately
after the price has reached a higher stop price
Question 2. (20 marks)
As a CEO of a newly founded mutual fund, you have $100 in cash. Suppose that you can buy shares of
WWW Inc. for $50 each and shares of PPP Inc. for $25 each right now. You expect the price of WWW to
go up to $60 in one year, and you believe that PPP shares should go down to $15,
1. Calculate the net return on these two stocks. (6 marks)
R1,t +1 =
60 − 50
15 − 25
= 0.20 for WWW and R2,t +1 =
= −0.40 for PPP
50
25
2. What is the best return you can get by the end of the year if you beliefs are correct and you are not
allowed to short? (6 marks)
Using portfolios:
R1,t +1 = 0.20 for WWW and
R2,t +1 = −0.40
for PPP. To maximize the net return
R p ,t +1 = w1t R1,t +1 + w2t R2,t +1 = 0.2 w1t − 0.4 w2t you should choose w1t as large as possible and w2t
as small as possible. You are not allowed to short, so the weights of the portfolio w1t and w2t
cannot be negative. Therefore, the optimal weights are w1t = 1 and w2t = 0 , which means investing
all of your money in WWW ($100 for 2 shares WWW). The optimal
R p ,t +1 = 0.2 × 1 − 0.4 × 0 = 0.20 .
3. What is the return if you are able to short 4 shares of PPP? If you beliefs about PPP are incorrect, and
PPP shares appreciate to $40, what is the return on your portfolio after you shorted 4 PPP shares? (6
marks)
If you are allowed to short PPP, you should short as much as possible your net return. Shorting 4
shares of PPP at $25 gives you an extra $100 worth of cash, so you have $200 to invest in 4 shares of
WWW. The optimal portfolio weights are w1t = 2 and w2t = −1
The net return is R p ,t +1 = 2 × 0.2 + ( −1) × ( −0.4) = 80%
If the PPP shares turn out to be $40 instead of $15, then R2,t +1 =
40 − 25
= 0.60 and
25
R p ,t +1 = 2 × 0.2 + (−1) × (0.6) = −20% . If your beliefs run out to be incorrect, you face a loss.
4. Comment on the advantages and disadvantages of shorting stock. (2 marks)
Shorting is the most natural way to express a bearish view on a stock. In practice, there are number
of difficulties associated with shorting. One is that the owner of the stock has the right to recall the
stock at any time, in which case the short-seller is forced to quickly find another lender to be able to
maintain his position. There are circumstances where this can become expensive or even impossible
because there are few willing lenders. In such a case, the investor may be forced out of a short
position. The margin requirements associated with short positions can also be cumbersome,
especially for small investors.
Question 3. (15 marks)
The endowment of Green University had the following returns over the last four years: -1%, 6%, 3%, 0%.
The endowment of Yellow University had the following returns: -3%, 8%, 2%, 1%.
1. Calculate the sample mean (arithmetic average) return for each endowment. (3 marks)
E(RG) = (-0.01+0.06+0.03+0)/4 = 0.02 = 2 %
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Solution Midterm 1
E(RY) = (-0.03+0.08+0.02+0.01)/4 = 0.02 = 2 %
2. Calculate the sample variance of return for each endowment. (3 marks)
Var(RG) = (1/4)*(-0.01-0.02)2 + (1/4)*(0.06-0.02)2 + (1/4)*(0.03-0.02)2 + (1/4)*(0-0.02)2 = 0.00075
Var(RY) = (1/4)*(-0.03-0.02)2 + (1/4)*(0.08-0.02)2 + (1/4)*(0.02-0.02)2 + (1/4)*(0.01-0.02)2 = 0.00155
3. Calculate the sample covariance and correlation between the Green and Yellow endowment returns. (6
marks)
Cov(RG,RY) = (1/4)*(-0.01-0.02)*(-0.03-0.02) + (1/4)*(0.06-0.02)*(0.08-0.02) + (1/4)*(0.030.02)*(0.02-0.02) + (1/4)*(0-0.02)*(0.01-0.02) = 0.001025
Corr(RG,RY) = Cov(RG,RY)/(σ(RG)* σ(RY)) = 0.951
4. Write a short paragraph comparing the two universities’ investment performance. Do their
endowments appear to be invested in similar ways? Can you say whether one university or the other has
better performance? (3 marks)
The correlation of the returns on the endowments of Green University and Yellow University is
very high. This suggests that they are invested in similar assets, perhaps in the same asset classes.
However, Yellow’s endowment seems to be invested in riskier assets than Green’s endowment.
Since Yellow doesn’t seem to earn a higher return than Green and Green seems to be exposed to
less risk it seems that Green’s investment strategy is preferable to Yellow’s.
Question 4. (20 marks)
1. You are able to invest in many stocks, each of which has a standard deviation of return of 40% and a
correlation of 0.25 with each other stock.
A. As the number of stocks increases, what happens to the variance of an equal-weighted portfolio of
stocks? In the limit where the number of stocks is infinite, what is the variance of the equal-weighted
portfolio? (5 marks)
We know that the variance of a portfolio with N equal-weighted stocks is
1 2
1
σ + (1 − ) ρσ 2
N
N
As long as the standard deviation of return and the correlation between stocks remain constant, the
variance of the equal-weighted portfolio decreases as the number of stock (N) increases. In the
limiting case, the variance of the portfolio is given by ρσ 2 .
Using the numbers given in the example, the limiting variance equals to 0.25 × 0.40 × 0.40 = 0.04
B. Suppose you require that the variance of your portfolio is no more than 0.01 greater than the variance
you calculated in (A). What is the smallest number of stocks you have to hold in your portfolio to achieve
this? (5 marks)
Using the information in part a) we want know that: 0.05 ≥
1 2
1
σ + (1 − ) ρσ 2
N
N
Substituting the numbers from the example, we have:
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Solution Midterm 1
0.05 ≥
1
1
0.16 + (1 − )0.04
N
N
Manipulating the equation so that N is isolated yields N ≥ 12
Question 5. (20 marks)
Consider the four stocks in the following table. P1 represents closing price on day 1, and Q1 represents the
total shares outstanding at the market close of day 1.
Initially, the price-weighted index only includes stocks A, B, and C. But after the market closes on the
first day (at t = 1), stock C is removed from the index and stock D is added to the index. None of the stock
pays a dividend.
Stock
A
B
C
D
P0
92
23.25
145
90
Q0
137
58
102
70
P1
87
24.25
153
85
Q1
137
58
102
70
P2
89
23.25
54
65
Q2
137
58
306
70
1. Calculate the rate of return on a price-weighted index of three stocks (A, B, and C) for day 1. (7 marks)
RA = (87-92)/92 = -5.43%
RB = (24.25-23.25)/23.25 = 4.30%
RC = (153-145)/145 = 5.52%
wA = 92/(92+23.25+145) = 35.35%
wB = 23.25/(92+23.25+145) = 8.93%
wC = 145/(92+23.25+145) = 55.72%
It follows that the return on a price-weighted index of A, B, and C is 1.537%
2. Calculate the rate of return on the price-weighted index of the three stocks (now A, B, and D) for day 2.
(4 marks)
RA = (89-87)/87 = 2.30%
RB = (23.25-24.25)/24.25 = -4.12%
RD = (65-85)/85 = -23.53%
wA = 87/(87+24.25+85) = 44.33%
wB = 24.25/(87+24.25+85) = 12.36%
wD = 85/(87+24.25+85) = 43.31%
It follows that the return on a price-weighted index of A, B, and D is –9.682%
3. Calculate the rate of return on the value-weighted index for all four stocks for the two-day period that
includes day 1 and day 2. (4 marks)
RA = (89–92)/92 = -3.26%
MCA = 92 x 137 = 12604
RB = (23.25–23.25)/23.25 = 0%
MCB = 23.25 x 58 = 1348.5
RC = (54(3)–145)/145 = 11.72%
MCC = 145 x 102 = 14790
RD= (65 – 90) / 90 = -27.78%
MCD = 90 x 70 = 6300
Total MC = 12604 + 1348.5 + 14790 + 6300 = 35042.5
wA = 12604 / 35042.5 = 0.3597, wB = 0.0385, wC = 0.4221, wD =0.1798
So rvw = 0.3597(-3.26%) + 0.0385(0%) + 0.4221(11.72%) + 0.1798(-27.78%) = -0.0122
4. If the initial value of the equally-weighted index of stocks A, B and C was 1718 at the beginning of day
1 (at t = 0), what is the value of the equally-weighted index at the end of day 1. (at t = 1)? (4 marks)
RA = (87-92)/92 = -5.43%
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Solution Midterm 1
RB = (24.25–23.25)/23.25 = 4.3%
RC = (153-145)/145 = 5.52%
REW = (-5.43% + 4.3% + 5.52%)/3 = 1.46%
Ending = Beginning(1+ REW)
Ending = 1718(1+0.0146) = 1743
Question 6. (15 marks)
You have the following information on four different securities:
Investment
1
2
3
4
Expected Return
0.12
0.15
0.21
0.24
Std. Deviation
0.30
0.30
0.16
0.21
The utility function is given by: U = E (r ) − 0.005σ 2 (r )
1. Calculate the certainty equivalent for the four investments. (5 marks)
The certainty equivalent is given by the utility of the risky asset.
Investment
1
2
3
4
Certainty Equivalent
0.1196
0.1496
0.2099
0.2398
2. Which investment (only one position) would you select? Explain. (5 marks)
The fourth investment would be selected since it offers the highest utility.
3. Redo (1) and (2) if you’re a risk neutral investor. (5 marks)
In this case, the certainty equivalent is given by the expected return and the fourth investment
would be chosen since it has the highest expected return.
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Solution Midterm 1