ECMC49S Final – Suggested Solution
(NOTE: TA’s marking scheme in general is consistent with this suggested set. However, he retains the
discretion of looking for more complete answers, or rewarding students fully with less complete answers, as
long as it is reasonable and consistent across ALL students. In addition, points are only given to answers
that are, at the very least, understandable. Further, there is no guarantee that the requirement for getting 1
point in any part of the exam has to be consistent with that of any other part. Consistency only applies to
the same part of the exam across ALL students, NOT across different parts of the exam for the same
student.)
Instructor: Travis NG
Date: Apr 24, 2007
Duration: From 7pm to 10pm
Location: MW170
Total Marks: 100
[1] [20 marks] Expected Utility Theory
(a) [5 marks] Other than the St. Petersburg’s paradox, please give another convincing real life example to
show that people are risk-averse AND that we differ in the degree of risk-aversion.
Any reasonable example will do. However, claiming things like buying insurance that is compulsory will not.
You simply cannot drive without getting insured. This is by law, not from your risk attitude.
(b) [5 marks] Your grandparents seem to have been running their business forever and they know how risky
it is to run a business. They love being praised as entrepreneurial and hate the term “risk-averse”. They
usually say,
“if you don’t take risk, you don’t live.”
You feel confused because in financial economics, most great minds have comfortably assumed everyone
is risk-averse. Suppose you buy this assumption, how do you justify the fact that they seem to love taking
risk, but that does not mean they AREN’T risk-averse?
Taking risk does not mean they are not risk-averse. This is the most important confusion people will have. As
long as the expected return is high enough, a risk-averse person WILL take some risk.
(c) [10 marks] Steve inherits the ownership of all wireless spectrum of a growing town in North Ontario,
which can potentially be used for wireless communication. Bell and Rogers both approach him and ask
him for a price of the spectrum. Steve’s utility function is U(W)=lnW, where W denotes his wealth. Steve
currently has wealth of $0.5 million. If he hires a team of executives to build his own wireless business,
he will be getting $2 million if everything goes smooth, or $1 Million if something goes wrong, and $0 if
something goes seriously wrong. The probability is 30%, 40% and 30% respectively for each scenario.
i.
(Technical problem) What is the minimum price p such that Steve would be better off accepting
than rejecting any price offer p > p?
ii.
(Economic problem) Do you expect him to get this minimum price p? Why or why not?
i. This is easy. Just calculate his expected utility as follows:
E[U(W)] = (30%)ln($2.5million) + (40%)ln($1.5 million) + 30%ln(0.5 million)
= 14.04464
U(CE) = 14.04463967
CE = $1,257,504.383
Therefore, to buy him out, the minimum amount p is $1,257,504 - $500,000 = $757,504.383.
ii. First of all, his utility function is a private information of him. It may not be precisely known by Bell or Rogers.
Therefore, they may not be able to price-discriminate him by charging such a precise price. Second, there is
competition among Bell and Rogers which increase Steve’s outside option of rejecting one’s offer. Therefore, it is
likely that Steve is getting way more than this minimum amount.
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[2] [30 marks] Capital Asset Pricing Model
(a) [6 marks] Two fund separation theorem states that everyone’s optimal portfolio is a combination of the
risk-free asset and a portfolio of risky assets. In fact, such a portfolio of risky assets is identical for
everyone. Explain why everyone’s portfolio of risky assets is identical.
In fact, we call the optimal portfolio of risky assets the market portfolio. Because it is the only efficient portfolio
on the minimum variance opportunity set that investors can potentially invest. Certainly, investors can combine
the risk-free asset with some other portfolio of risky assets, but such an investment portfolio is strictly dominated
by a corresponding combination of risk-free asset and the market portfolio, i.e., given a level of risk, any investor
can combine the risk-free asset with the market portfolio in such a way as to be able to generate higher expected
return. Therefore, everyone will be choosing the strictly dominating portfolio of risky assets, which is the market
portfolio.
(b) [6 marks] Wells Fargo, a bank in the States, is expected to have an annualized expected return of 12%.
Yahoo!, the dot.com giant, currently undergoing a huge corporate reorganization, is expected to generate
5% only. Comparing to Wells Fargo, Yahoo! is having a much more volatile return. How do you
reconcile the fact that Yahoo!, being riskier, yet is expected to generate lower return than Wells Fargo
does?
There is nothing wrong with a riskier stock having lower expected return since risk is composed of not only
systematic risk but also unsystematic risk. It may be that Wells Fargo is having a higher systematic risk than
Yahoo, and therefore higher expected return.
(c) [6 marks] What is the benefit of diversification? How are you going to fully diversify in the real world if
it is almost impossible to buy many different stocks with a limited amount of money due to all sorts of
fixed transaction costs and the fixed lot size of stock purchases.
The benefit of diversification is that you can diversify away un-systematic risk, and therefore, lowering risk
without sacrificing expected return. To do it, you spread your portfolio over many risky assets to avoid excessive
exposure to any one source of risk. In practice, one does not usually diversify efficiently if he/she acts alone.
Therefore, mutual funds emerge as instruments to pool resources of many people to allow a bigger pool and more
efficiently diversification to many risky assets.
(d) [6 marks] Why is CAPM so important if it merely shows risk is directly proportional to expected return?
It is not only showing risk is directly proportional to expected return, it shows what kind of risk is, what kind of
risk is not. It is the quantifying of risk of a risky asset to an index called beta and the simplicity of the model that
are fascinating. The insight it offers makes it widely used by people in a variety of areas including corporate
finance and asset pricing of both marketable and non-marketable risky assets. That, however, does not mean the
world stops at where CAPM predicts. It acts as a fundamental pillar for subsequent research to build on.
(e) [6 marks] Suppose the market is expected to pay out 10% with a standard deviation of 8%. T-bill annual
rate is 3.5%. If Tim has $200,000 to invest and he borrows an additional $150,000 to invest, what is his
portfolio’s expected return and standard deviation? If part of his portfolio consists of a mutual fund that is
expected to pay 10%, exactly as the market is expected to pay, what would be the standard deviation of
the mutual fund’s return?
Rm=10%, Rf=3.5%, σm = 8%
Tim invests -150,000/200,000 or -3/4 into risk-free asset, implying that he borrows. He then invests 7/4 into the
market. His portfolio’s expected return is (7/4)(10%) + (-3/4)(3.5%) = 14.875%.
His portfolio should be well-diversified and is on the capital market line. We therefore just have to plug in to the
CML equation to find the standard deviation. (Alternatively, use the variance equation will do too.)
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E[Rp]= 3.5% + (10%-3.5%)/(8%-0%)σp,
E[Rp]= 3.5% + 6.5/8σp,
14.875% = 3.5% + 6.5/8σp,
=> σp = 14%.
But suppose we know that there is a mutual fund involved in her portfolio. That means the market portfolio itself
has a portion of holding of that mutual fund. We know the portfolio is well-diversified, and thus individual assets
may have their risks hedged by other assets inside the market portfolio. We actually have no information at all to
tell how risky is the return of that any mutual fund. Therefore, the answer is, “we do not know.”
[3] [20 marks] Options
(a) [5 marks] Explain how you will use “open interest” to infer any information about option trading?
CBOE definition of Open Interest: The number of outstanding option contracts in the exchange market or in a
particular class or series.
It gives you an idea of how thick the market for trading that particular option. The thickness of the market has
great importance in the real world because if there is relative small open interest, it may be hard to close-out a
position. Therefore, it gives you an idea of liquidity.
Comparing daily trading volume with the open interest also gives you an idea about the trading activity of that
particular option. For more info, please go to:
http://www.investopedia.com/articles/optioninvestor/04/060904.asp
(b) [5 marks] What is the effect of the volatility of the underlying asset on the price of a put option? Your
mark will be based mainly on the explanation of your answer.
The payoff of a put option is zero if the stock price is higher than the exercise price, and positive if the stock price
is lower than the exercise price. A cent decrease in stock price below exercise price leads to a cent increase in the
payoff of put. Therefore, extremely high stock price would not worsen the payoff, while extremely low stock
price would increase payoff. Therefore, with larger volatility of the underlying asset, the price of put option
increases because it enhances its payoff.
An easier and more intuitive answer is the following: options give flexibility and therefore with more
uncertainty/volatility, the value of flexibility increases. Therefore the put option should be valued more. However,
if a student answers this, I can hardly differentiate whether or not he/she really understand the mechanism. I will
give maximum 3.5 out of 5 for the intuitive answer.
(c) [5 marks] Assume that the stock price of Advanced Micro Devices (NYSE: AMD) half a year from today
is either $17 with probability α and $12 with probability (1-α), where 0<α<1. Today, AMD stock trades
for $14.59. The annual risk-free interest rate is 3 percent. If there are trades for its European put with
strike price of $15 and it expires exactly half a year from today, how much are you willing to pay for one
such European put option? How would its price change if α increases (i.e., it is more likely that half a year
from now, AMD share is sold at $17.)
This is a standard binomial option pricing question. First, we want to find the ratio m*.
If AMD sells at $17, then the call option is worth $2. If AMD sells for $12, it is worth nothing. Therefore
(m*)($17) - $2 = (m*)($12),
=> m* = 2/5.
We therefore know that if we long 2/5 share of AMD and short 1 call, the portfolio is risk-free. Half a year from
now, the payoff of such a risk-free portfolio is (2/5)($12) = $4.80
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Therefore, the price of a European call (C) on AMD with strike price of $15 is
(2/5)($14.59) – C = $4.80/[(1 + 3%)^(1/2)],
=> C = 1.105419.
By put-call parity, the price of a European put (P) on AMD with strike price of $15 is
($14.59) + P – C = $15/[(1 + 3%)^(1/2)],
=> P = $1.296359.
The final answer is $1.296359.
The option value stays constant when α changes.
(d) [5 marks] 3M share price is currently $77.58. One put option contract on 3M with strike price of $85 and
maturity date of Jul 2007 is currently selling at $730 at CBOE. What should you do? Explain in details.
There is an arbitrage opportunity and therefore you should exploit it. One transaction is to buy the put option
contract at $730 and also buy 100 shares of 3M at $77.58 for a total of $8,488. Immediately exercise the put
option and force someone to buy from you at $85 for 100 shares. The total proceeds would be $8,500. The gain
from this one transaction is $8,500 - $7,758 - $730 = $12. If the answer stops here, the student would not get full
marks. Because there is nothing to restrict one just to make one such transaction. In fact, anyone who spotted this
opportunity should gather as much money as possible and do this transaction many times. If anyone recognizes
this in their answer, give them full marks.
However, in reality, of course, it is not possible to do as many transaction as you can. Because you will bid up the
price. And the arbitrage opportunity would be destroyed by your own hand.
[4] [20 marks] Efficient Market Hypothesis
(a) [5 marks] Given that a lot of Latin American countries have gone through tough times in the past century
and are mostly in the stage of developing their financial markets, do you think that Latin American stock
markets are not efficient, while US stock market is efficient? Why or why not?
It is a wrong question to ask. It is the degree of efficiency that matters. Latin America, given it is still in the
developmental stage, is considered less efficient. It may reflect newly available information slower because of its
financial microstructure, or information infrastructure, or trading rules and regulations, or simply because market
participants may incur more relatively more transaction costs to execute. But it is unlikely that stock prices do not
reflect information in these countries.
On the other hand, US stock market is considered more developed. Yet, it is not as efficient in some area as in
others. At the very least, the fact that there are rules to restrict insider trading implies that it is not a strong-form
efficient market. Again, efficient or not is a matter of degree. It is tempting to say that developing financial market
is not efficient at all. But you will be surprised this is not a right way to describe these markets.
(b) [5 marks] It is empirically shown that on average, factoring in all costs, investing in index fund allows
you to get approximately the same level of net risk-adjusted return as compared to investing in funds that
actively trade. Suppose everyone takes it as a fact and start to invest in index funds only, how is it going
to affect market efficiency?
Mutual funds that actively trade, as well as investors that actively trade, make sure that the market is efficient. It is
because they exist that forces the market to be efficient. If they do not exist or people just buy index funds,
whenever there is new piece of news arrive, no one will be acting to adjust the price to the new information. The
market will not be efficient again.
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(c) [5 marks] Carefully explain why portfolio management is necessary if you believe that the market is
efficient and that the price you pay for a risky asset is always fair. One of your explanations should be
beyond the theoretic arguments that we have given in class.
This is a question that deserves repetition. I hope they can answer it well. First of all, from a static point of view,
portfolio management is necessary as you have your own risk-preference. Carefully selecting risky assets to
formulate your portfolio is necessary in order to fit your risk-preference and thereby maximizing your expected
utility. Also, it is hard to guarantee well diversification if we just randomly pick risky assets.
From a dynamic perspective, your situation changes over time. Your risk-attitude may change, your income or
profession, etc. Therefore, your preference (or risk-preference) also change over time. And the kind of investment
portfolio at any given moment may not be identical. Therefore, from time to time, you may want to re-adjust your
portfolio.
In reality, there is tax concern. And if you are within one profession, you may want to avoid heavily investing in
that industry. Or you may be completely financially-illiterate because you are just someone who plays piano.
However, that does not mean you don’t have a preference. Therefore, you should get financial consulting for
portfolio management.
(d) [5 marks] Do you think that stock price movements usually have momentum?
This question asks for the depth of student’s answer. As such, there is no model answer. Some of my thoughts are
as follows:
As shown in class, a random number generator can easily create patterns, i.e., a period of ups and a period of a
series of stock prices. This is called runs or momentum. If we believe stock price movements usually have
momentum, we may have overlooked the fact that pattern is so easily generated by pure randomness. We also
should not believe that the market is weak-form efficient because if there is momentum, it is enough to use past
price information to generate consistently abnormally high return. It is hardly the case in reality.
A deeper answer can extend to behavioral finance. There is some psychology of investors that is enough to
generate certain pattern of stock price movements. I am not familiar with this literature but it is definitely an area
that a lot of people have devoted into. Any reasonable argument from this discipline should be rewarded.
However, to merely claim that people are sometimes irrational and therefore generates momentum is not an
acceptable answer. This is because it is so easy to blame what we have observed on irrationality and giving such a
hypothesis is like cheating - it can never be proven right or wrong.
[5] [10 marks] Returns Measurement
(a) [5 marks] Your mother has just won the lottery prize of $100,000. Instead of spending it on a nice
vacation, she wants to invest the money into some mutual funds. What sort of information you will look
into to help her choose which fund(s) to invest?
The information you want includes past performance (i.e., past returns) of these mutual funds and each of their
risk levels. Then you compute their time-weighted average returns as estimates for their expected future returns. u
also need to know your own degree or risk-aversion. It is hard, but remember if you talk to a financial analyst,
he/she will probably ask you a number of questions to assess your risk tolerance. Something very similar to that
would do.
Finally, you need a benchmark to measure risk-adjusted performances. It is usually S&P 500. But if those mutual
funds contain concentrated content in a handful of industries or even oversea content, then there may be some
other proxies other than S&P 500 that is more appropriate. Both Treynor and Alpha will work, but not Sharpe's.
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(b) [5 marks] Peter, a friend of yours, is a member of a private investors association, a non-profit association
for investors to socialize and to exchange financial market opinions. The chairman drafted the following
description to introduce him before he gave a speech on an event.
“Peter has been a dedicated investor for years. Last 5 years, he has consistently outperformed the market
by having an average return of 20% per year on his portfolio. What a great showcase of his skills and
wisdom!”
Although Peter is your friend, you find it very disturbing about the chairman’s introduction. What is
wrong with his words?
The main point is only risk-adjusted measures give us a sense of skills. 5 years of return is a piece of evidence for
skills. This is true if we know about the risk-level. It is not even risk-adjusted, and what is the basis for comparing
his skills? Is the average return geometric or arithmetic? Is he taking very risky position?
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Some Mathematical formulas for your reference:
[1]
Arrow-Pratt Risk Premium:
π(W,z) = (1/2)σ2z(-U”(W)/U’(W)),
where z is the random payoff from gamble.
Degree of Absolute risk aversion:
ARA = (-U”(W)/U’(W))
Degree of Relative risk aversion:
RRA = W(-U”(W)/U’(W))=W(ARA)
Markowitz’s Risk Premium = Expected wealth given the gamble – Certainty Equivalent.
[2]
Asset j’s return in State s:
Rjs = (Ws – W0) / W0
Expected return on asset j:
E(Rj) = ∑sProb(State=s)Rjs
Asset j’s variance:
σ2j = ∑s Prob(State=s)[Rjs- E(Rj)]2
Asset j’s standard deviation:
σj = √σ2j
Covariance of asset i’s return & j’s return:
Cov(Ri, Rj) = E[(Ris- E(Ri)) (Rjs- E(Rj))]= ∑s Prob(State=s)[Ris- E(Ri)] [Rjs- E(Rj)]
Correlation of asset i’s return & j’s return:
ρij = Cov(Ri, Rj) / (σiσj),
where -1 ≤ ρij ≤ 1
Properties concerning mean and variance:
E(ũ+a) = a + E(ũ)
E(aũ) = aE(ũ)
Var(ũ+a) = Var(ũ)
Var(aũ) = a2Var(ũ)
Variance of 2 random variables (this is the equation we used in Portfolio theory too.):
Var(aũ+bẽ)=a2Var(ũ) + b2Var(ẽ) + 2abCov(ũ,ẽ),
where a & b are constants and ũ and ẽ are random variables.
[3]
CAPM Equation (Which I know you remember, but just in case you feel like to make sure)
E(Ri) = Rf + [E(RM)- Rf]βi
, where βi is Cov(Ri,RM)/Var(RM).
[4]
Put-Call Parity
P-C+S0=X/[(1+Rf) T]
[5]
Black-Scholes Equation
C = S0N(d1) – X•e-RfT•N(d2),
where d1 = [ln(S0/X) + (Rf + σ2/2)T] / σ√T and d2 = d1 - σ√T
[6]
Jensen’s measure:
αp = E(Rp) – {Rf + E(RM) – Rf]βp}
Treynor’s measure:
Tp = [E(Rp)– Rf]/βp
Sharpe’s measure:
Sp = [E(Rp)– Rf]/σ(Rp)
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