ECMC49Y Midterm
Instructor: Travis NG
Date: Jun 21, 2006
Duration: From 9am to 10:55am
Total Marks: 100
[1] [25 marks] Decision-making under certainty
(a) [10 marks] (i) Graphically demonstrate the Fisher Separation Theorem. (ii) Briefly describe what it
means. (iii) And briefly explains its implication to a firm’s “project selection” decision.
(b) Suppose that Jean’s utility function for inter-temporal consumption is
U(C0,C1) = [√(C0)] + [√(C1)]/(1+ρ)
where C0 is his current period consumption, C1 is his future period consumption and ρ is his subject
rate of time preference. Let ρ be 7%.
If Jean is endowed with $500 this period and $500 in the next period. And suppose the risk-free
interest rate is 5%.
i.
[5 marks] Draw the capital market line that Jean is facing. States the precise X-Intercept and
Y-Intercept. What does the value of X-Intercept mean?
ii.
[5 marks] What is his optimal consumption path (i.e., the optimal level of current and future
consumption) if he can only allocate wealth through lending and borrowing?
iii.
[5 marks] Suppose now he discovers an investment opportunity. This is a risk-free project
that needs initial investment of $400 and it generates $425 in the next period. What is his
optimal consumption path now?
[2] [25 marks] Expected Utility Theory
(a) [5 marks] State the assumptions needed to obtain the indifference curves on the “expected returnstandard deviation of return” diagram we used in class.
(b) Suppose Louis’s utility function is U(W)=ln(W). His current wealth is $5,000. He is now given a
chance to buy a futures contract on Nickel that gives him 75% chance of winning $5,000, and 25%
chance of losing $4,000.
i.
[2 marks] Is Louis risk-averse? Why?
ii.
[5 marks] Evaluate the following statements, “As Louis gets richer, he becomes more and
more risk-averse.”
iii.
[3 marks] What is his expected utility if he does not hold the futures contract? What is his
expected utility if he is already holding the futures contract? (that means he has bought the
futures contract in the past to engage in the gamble)
iv.
[5 marks] If the futures contract is selling at $200, will Louis buy it? Why or why not?
(c) [5 marks] Use English but not Mathematics to explain what is risk-aversion.
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[3] [25 marks] Portfolio Theory
Let Rx and Ry be the returns from two risky assets x and y respectively. Assume each of the two
returns is normally distributed with expected returns and standard deviations as followings:
E(Rx) = 5% and E(Ry) = 12%, σ(Rx) = 20% and σ(Ry) = 30%.
Assume these 2 risky assets are the only investment vehicles available.
(a) [2 marks] If the two risky assets have returns that are perfectly positively correlated, plot the feasible
mean-variance investment opportunity set in an “expected return-standard deviation” space.
(b) [4 marks] If the two risky assets have returns that are perfectly negatively correlated, plot the feasible
mean-variance investment opportunity set on a NEW graph.
(c) [6 marks]Continuing from (b), what is the expected return of a completely hedged investment
portfolio?
(d) [8 marks] Assume now that the two risky assets have returns that have a correlation between -1 and 1.
(i) Suppose everyone can borrow and lend unlimited amount of money at risk-free rate of 3%, draw
the corresponding diagram in a NEW graph. (ii) Which risky portfolio from the minimum variance
opportunity set is now “mean-variance” efficient? (iii) Why other risky portfolios on the minimum
variance opportunity set are not efficient?
(e) [5 marks] Suppose the Two-Fund Separation Theorem holds, explain why everyone is made better
off with the introduction of the capital market.
[4] [25 marks] Capital asset pricing model
(a) [5 marks] List the four equilibrium conditions for CAPM. (Some explanations are needed for full
marks)
(b) [5 marks] The following table is the stock quotation for Toyota from Google.
If the annualized T-bill rate is 5%; the NYSE International 100 Index is expected to generate 15%
return in the coming year and its variance is 16%; and Toyota’s beta is given in the table as 0.7.
What is the expected share price of Toyota about 1 year from now (i.e., share price on Jun 16, 2007)?
(c) [5 marks] Evaluate the following statement. Explain whether it is true or false.
“Given Toyota’s beta is 0.7, it is considered lower risk than holding the market portfolio.”
(d) [5 marks] Write down the precise expression for the Capital Market Line and for the Security Market
Line using the data given in part (b).
(e) [5 marks] State the intuition of “higher risk-higher return” that CAPM provides. Use English but not
Mathematics to state your intuition.
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Cheat sheet for this Midterm:
[1]
Suppose your utility has the following structure
U(C0,C1) = U(C0) + U(C1)/(1+ ρ)
To derive the slope of the corresponding indifference curve, we take total derivative:
U'(C0)dC0 + [1/(1+ ρ)]U'(C1)]dC1 = 0
Rearranging, we get the slope as:
dC1/dC0 = -(1+ ρ) [U'(C0)/ U'(C1)] =MRS
[2]
Multi-period Present value:
PV = C0 + C1/(1+R) + C2/(1+R)2 + ….
Where Ct = cash flow at time t, can be positive or negative, and assuming a constant
discount rate R. If the project is risk-free, R is the risk-free interest rate.
Perpetuity (You don’t need this, but I write it down here so that you feel more comfortable):
PV = C/R
Annuity (You don’t need this, but I write it down here so that you feel more comfortable):
PV = C[1/R – 1/(R(1+R)t)]
, where t = the period the annuity ends
[3]
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.
[4]
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.
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).
[5]
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Suggested Solutions
Question 1:
C1
Individual 2
P*1
Individual 1
Y1
P*0
Y0
W*0
C0
[1] (a)
(i) The above graph demonstrates the Fisher Separation Theorem.
(ii)The pink point is the optimal production/investment point. If the two individuals are endowed with the
same initial wealth (as represented by the (Y0,Y1) point, are faced with the same interest rate (as represented
by the same capital market line) and are faced with the same investment opportunities (as represented by the
same production opportunity set), they will both choose the pink point as their optimal production point.
They, however, choose different consumption points along the capital market line so as to maximize their
utility.
(iii) Imagine both individuals are shareholders of one company. According to the Fisher Separation Theorem,
they will both agree on the same investment point of their company. The projects that they both agree the
company to take would be those that earn positive NPV. Any projects with negative NPV will not be
suggested by either of them to undertake.
(b)
(i) The graph is given below. The X-intercept represents the present value of Jean’s initial wealth.
(ii) There are two conditions for getting the optimal consumption path:
MRS = (1+R) and
Present value of initial wealth = present value of lifetime consumption.
MRS = MU(C0)/MU(C1) = [0.5/√(C0)]/{0.5/[(1+ρ)√(C1)]} = [(1+ρ)√(C1)]/[√(C0)]= [(1.07)√(C1)]/[√(C0)]
1+R = 1.05
Present value of initial wealth = 500 + 500/(1+R) = 500 + 500/(1.05) = 976.1905
Present value of lifetime consumption = C0 + C1/(1+R) = C0 + C1/(1.05)
Solutions are: C0 = $509.1988 and C1 = $490.3412.
(iii) First of all, we have to determine whether this project is worth undertaking. Using the NPV rule, we can
compute its NPV. NPV = -400 + 425/(1.05) = 4.7619 > 0. This project thus generates positive NPV and thus
is worth undertaking.
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We then think of what is the production point if Jean undertakes the project. His first period consumption
point without capital market becomes (500-400) = 100, and his second period consumption point without
capital market is (500+425) = 925.
Thus, we can compute the new present value of wealth after he has undertaken the project. It is given as:
100 + 925/(1+0.05) = 980.9524.
Apply the same algorithm as in part (b ii), and we will get the following solutions:
C0 = $511.6827 and C1 = $492.7332.
C1
500(1+5%) + 500
500
500
500 + 500/(1+5%)
C0
Question 2:
(a) The assumptions are:
(i) 5 axioms of decision making under uncertainty: comparability, transitivity, strong independence,
measurability and ranking.
(ii) Everyone is greedy
(iii) Everyone is risk-averse
(iv) Risky assets are jointly normally distributed.
(b)
(i) He is risk-averse because his utility function is concave as proven by his U’(W)>0 and U’’(W)<0.
(ii) This statement concerns whether his ARA and RRA increases as W increases. The ARA is:
ARA = - U’’(W)/ U’(W)=1/W. The RRA is: RRA=W(ARA) = 1.
Since ARA decreases as W increases, we say that Jean is actually less and less risk-averse in absolute term as
he gets richer. Since RRA is constant as W increases, we say that Jean is actually about as risk-averse in
relative term as he gets richer. Both statements contradict with the one in the question.
(iii)
Expected utility without the option = 100%U(5000) = ln(5000) = 8.517193.
Expected utility with the option = 75%U(5000+5000) + 25%U(5000-4000) = 8.634694.
(iv)
Expected utility if buy at $200 = 75%U(5000+5000-200) + 25%U(5000-4000-200) = 8.563756.
Since he is still better off in expected utility term when the futures contract is selling at $200, he will
buy it. This does not contradict to his risk-preference whatsoever.
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(c) This is what I get from Wikipedia:
“Risk aversion is a concept in economics, finance, and psychology explaining the behaviour of consumers
and investors under uncertainty. Risk aversion is the reluctance of a person to accept a bargain with an
uncertain payoff rather than another bargain with a more certain but possibly lower expected payoff.”
I am looking for some concepts similar to this one. We have more solid mathematical definitions of riskaversion, but the world is communicated with understandable English but not Mathematics. Your ability to
understand Economics depends crucially on how well you transform concepts learnt in Mathematics to
English that everyone can understand.
The simplest way is to say that an individual will avoid playing a fair gamble. Suppose someone offers to
play a coin-flipping game with you. If “head” turns up, you win $x, if “tail” turns up, you lose $x. You don’t
need to pay anything to play with him. If you are risk-averse, you will refuse to play with him even if the
expected payoff of the game is $0. If you are obligated to play the game but are given an option to pay some
money to avoid the game, you will be willing to pay a positive amount of money to avoid. The more riskaverse you are, the larger the amount of money you will be willing to pay to avoid.
Question 3:
(a)
The graph is the following:
E(Rp)
12%
Min-var. opp. set
5%
σp
20%
(b)
(c)
30%
The graph is as follows. Since the two assets are perfectly negatively correlated in terms of their
returns, you can carefully choose a weight across the two assets so that your overall portfolio is riskless. This is called fully-hedged. The precise hedging ratio is solved by the following formula from
cheatsheet:
Var(aũ+bẽ)=a2Var(ũ) + b2Var(ẽ) + 2abCov(ũ,ẽ)
Substituting correlation equals to -1 as given in the question, it gives you the formula for finding the
portfolio’s variance:
Var(Rp)=a2(20%)2 + (1-a)2(30%)2 - 2a(1-a)(20%)(30%)
Finding the fully-hedged ratio, a*, is simply to set Var(Rp)=0. And solving, you get a* = 0.6. Thus,
the expected return of the risk-free portfolio is:
E(Rp)=a*(5%) + (1-a*)(12%) = 7.8%
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E(Rp)
12%
7.8%
Min-var. opp. set
5%
σp
20%
30%
(d) (i) The following graph depicts the situation:
E(Rp)
12%
Capital Market Line
Point M
Min-var. opp. set
5%
3%
σp
20%
30%
(ii) The risky portfolio as represented by point M on the graph is now mean-variance efficient.
(iii) All other risky portfolios on the minimum variance opportunity set are not efficient because by
combining the risk-free asset with the efficient portfolio at point M, I can always create a portfolio of the
same risk-level and higher expected return than any of these risky portfolios. Therefore they are not efficient
anymore.
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(e) If Two Fund Separation Theorem holds, every individual investor will have in their optimal investment
portfolio two portfolios only, the risk-free asset and the M portfolio. This is given in the following diagram:
E(Rp)
CML
B
Q
M
A
rf
σp
Everyone is made better off because with the capital market line, everyone is able to get to a higher
indifference curve that gives him higher expected utility. There are two reasons for such welfare
enhancement: first, the capital market line gives everyone an objective guide to choose the only efficient
risky portfolio to be included in their investment, i.e., the M portfolio. Second, it gives everyone the ability to
lend and borrow to fine-tune the risk level of their own investment portfolio along the capital market line so
as to suit each individual’s specific risk-preference. This is analogous to the Fisher Separation Theorem in
the certainty case.
Question 4:
(a) Consult the class notes set 05.
(b) Risk-free rate is 5%, Expected market return is 15%, standard deviation of market is √16%=40%, beta is
0.7. Thus, employing the CAPM equation:
E(Ri) = Rf + [E(RM)-Rf] x βi
E(Ri) = 5% + (15% - 5%)0.7 = 12%
Expected share price of Toyota: $100.67 x (1 + 12%) = $112.7504.
(c) This is not a valid statement because we cannot compare apple to orange. A market portfolio consists of
all risky assets, while Toyota’s stock is just one risky asset. Toyota’s systematic risk is lower than the market
portfolio. But that does not mean Toyota’s total risk, which consists of both the systematic risk component
and the unsystematic risk component, is lower than that of the market portfolio. It is very likely that holding
Toyota alone is riskier than holding a market portfolio because of the substantial idiosyncratic shocks
specific to Toyota such as oil price fluctuations, competitions within the auto industry, suppliers’ conflicts,
etc. Those shocks affect the market portfolio as well, but on average, there will be other risky assets within
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the market portfolio that in fact take advantages of those shocks. In the end, those shocks will affect the
market portfolio less than Toyota.
In any case, it is unreasonable to assume the unsystematic risk of Toyota is zero. Therefore, they are not
comparable in broad sense, and it is not likely to be less risky for Toyota in terms of the intuitions we get
from CAPM.
(d) Equations for CML and SML are respectively
CML:
E(Rp) = Rf + [(E(RM)- Rf)/σM]σ(Rp)
SML:
E(Ri) = Rf + [E(RM)-Rf] x βi
Substitute these into the CML or SML equation to get:
CML:
E(Rp) = 5% + [(15%-5%)/40%]σ(Rp)
SML:
E(Ri) = 5% + [15%-5%] x βi
(e) CAPM tells us that if we bear higher risk, the market will compensate us with higher expected return.
However, not all risk will be rewarded for bearing. Only those that are correlated with the whole market,
namely, the systematic risk, will be rewarded. Beta captures the systematic risk component of every risky
asset. Unsystematic risk, as defined as idiosyncratic risk specific to a risky asset, is not rewarded by the
market for bearing. This is because there is nothing that restricts an individual to diversify by NOT investing
in one or a few risky assets, but a portfolio of many risky assets. Doing so actually diversifies away
unsystematic risk without cost. Since there is no free lunch in this world, if you can decrease unsystematic
risk costlessly, you will not be rewarded for bearing them.
Why would high beta corresponds to high expected return? Since holding investment portfolio that has high
beta implies such portfolio is likely co-vary with the market. If the market does badly, it is likely that such
portfolio is also doing badly. Everything else equal, individual investors need money the most when the
market is bad. Since holding such portfolio is not likely to give you the money when you need them the most,
there should be some compensation before it is rational for an individual to hold such portfolio. The
compensation comes with higher expected return.
The interview article of William Sharpe we posted in the website actually provides a lot more intuitions.
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