Name: ..................................................
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Assignment 1, Financial Economics ECN 3350, Due: 11 October 2013
Short Essay Questions: Answer the Following Two Questions (in the
exam you will be given a choice TWO out of THREE)
Part A: Short essay questions (10 marks each)
A1 Define a random walk process. Explain the relationship between the efficient market
hypothesis and the claim that asset prices follow a random walk! Contrast random walk and
AR(1) process! Which are the implications of the latter? Which is the evidence that stock
prices follow either a random walk or an AR(1) process?
If an asset price X follows a random walk, the price at time t equals the price at time
Xt = Xt-1 + t where t is noise which is not forecastable. The efficient market
hypothesis says that all publicly available information is immediately reflected in the
asset price.1 So the best estimate of the future asset price is today’s asset price. Only
news which are not forecastable will change the asset price. 4 marks
If an asset price follows an (autoregressive) AR(1) process, the price at t equals Xt =
(1) a +  Xt-1 + t where a is a constant. With (0,1), the asset price is predicted to
return back to its “natural” value a whenever it deviates from it. In practice, that
means that assets can be overvalued or undervalued at some time. 5 marks
Empirics: We did not really cover that in class except that we showed that actual
asset prices could either be simulated as random walk or AR(1) processes with 
close to 1. In both cases one has to keep in mind that only after a number of
simulations one may end up with the right graph. Whilst a successful simulation
may be visually convincing, it is not a proper empirical test.
For those interested: The efficient market hypothesis and its implication – random
walk – has often been confirmed in empirical testing. Yet there are anomalies – we
came across patterns between related asset prices which Long Term Asset
Management successfully exploited to make money for its first few years. Also,
stories of asset managers which have successfully employed fundamental analysis
such as Warren Beatty refute the efficient market hypothesis: If EMH did hold
generally true, there could not be any undervalued assets to make money from. 1
mark
1
At some point we may distinguish between different formulations of the efficient market hypothesis, but in its
most prevalent form it claims that all information on historic prices and all publicly available information is fully
reflected in the market price.
A 2 What does the efficient market hypothesis state?
Define, using the discounted dividend model: The fundamental value of a common
stock! How do you interpret the discount rate? Demonstrate that if you expect that a
buyer in t=1 only wants to pay the fundamental value in that time, your own
willingness to pay for the stock has to be equal to the fundamental value in t=0. Why
must, more generally, in an efficient market, market price and fundamental value
coincide?
The efficient market hypothesis states that all relevant information is reflected in
asset prices, so one cannot make extra profits by looking at past price movements or
trying to assess the fundamental value because the market price already contains that
information. 2 marks (if you refer to the joint hypothesis, the better so)
An implication is that the stock price coincides with the fundamental value which, by
the discounted dividend model is

1
1
1
D1 
D

...

Dt where Dt is the expected dividend payment

2
2
t
1 k
(1  k )
t 1 (1  k )
in t and k is the return on comparable assets.
P0 
6 marks
Say the buyer in t=1 purchases after dividend in t=1 has been paid, so he pays the

1
1
1
D2 
D

...

Dt
fundamental value P1 

3
2
t 1
1 k
(1  k )
t  2 (1  k )
You would be willing to pay
P0 
1
1
D1 
P1
1 k
1 k
anticipating the selling price and the dividend payment. Inserting the expression for

1
1
1
D1 
D

...

Dt . Note that your discount
P1 above gives P0 

2
2
t
1 k
(1  k )
t 1 (1  k )
rate is k because the next best alternative would be to invest in some other stock
which gives you a return of k. 2 marks
Part B: Problem Sets (10 marks each)
Solve the following two Problems (in the midterm you will be given a choice of
two out of three problems)
B1: A decision maker has the wealth scaling function U(w) = w, an initial wealth
w0=1,000$ and decides by expected utility.
a) Anita R. inherits a beauty shop which gets her a profit of 1,500$ with probability p=0.80
and nothing with probability (1-p)=0.2.
What is her expected utility? Which is the minimum amount of money at which she would
want to sell the business? Which is, therefore, her risk premium? Using a graph, relate risk
premium and certainty value!
The guideline answer here ignores the information about initial wealth of 1,000.
See the appendix on the last page for a variant with initial wealth!
EU  0.8 1,500  0.2 0  0.8  38.73  0  30.9839
Let P* be the minimum price at which she would be willing to sell. This price must
give her a utility to match her expected utility with the business, hence
U (P* )  P*  30.9839  P*  960
Her certainty value of the risky business is thus P*=960 which is less than its expected
monetary value EV=1,200. Her risk premium is the difference, i.e. 240.
6 marks
b) Investor Leonard O’Real considers buying Anita’s business. Leonard has a wealth
of 1,000,600$ and his wealth scaling function is U(w) = w. Would he be willing to
pay Anita’s minimum price? What if she asked a price of 1,000?
At a price of 960$, Leonard would want to buy if
1, 000, 600  0.2 1, 000, 600  960  0.8 1, 000, 600  960  1,500
1, 000, 600  0.2 999, 640  0.8 1, 001,140
1, 000, 600  0.2 999, 640  0.8 1, 001,140
1, 000.30  0.2  999.82  0.8 1, 000.57
1,000.30 < 1,000.42.
If he had to buy at a price of 1,000, his expected utility would be
0.2 1, 000, 600  1, 000  0.8 1, 000, 600  1, 000  1,500 = 1,000.40 > 1,000.30, so he would
still be willing to buy!
2 marks
c) Now suppose that Anita’s behavior satisfies the postulates of prospect theory, in particular
her psychological value (PsyVal) from suffering a loss L is PsyVal =– 2 L. If she is threatened
with a certain loss of 400: Would she rather go for a lottery where she loses 900 with a
probability of 4/9? Use a graph to explain!
The psychological value of losing 400 for certain is lower (i.e. – 40) than her
psychological value of losing 900 with probability 4/9 (i.e. – 26.67) even though the
expected loss of the lottery is also 400. 2 marks
B2: An investor puts $1,000 into a business project with the following returns:
t=1
200
t=2
300
t=3
400
t=4
500
His alternative is to put his money into a bank account which would pay him a return of 8%
per annum.
a) What is Net Present Value of the investment project? Should it be carried out?
Net present value is $127.43, so he should realize the investment project!
(sorry for the misprint in the text of the assignment. If you had used 1,000,000$, NPV
would indeed have been negative)
b) The investor would like to retire immediately after having made the investment which is
left in the hands of an associate. He plans to use all the returns of the investment project and
pay himself a constant annual grant over the next ten years. If he calculates the present value
of his income stream and the annual grant using an interest rate of 8%: What is the value of
his annual grant? What is the role of the interest rate in doing all these calculations?
Explain!
The present value of the stream of return payments is $1,127.43. He converts into an
annuity using the formula
(1  i)n  1
(1  i)n i

PMT
PMT

PV . Inserting PV = $1,127.43 and i = 0.08 gives
(1  i)n i
(1  i)n  1
PMT = $168.02.
PV 
The interest rate is equal to the interest rate at which he can reinvest the return
payments: The first return payment of 200 exceeds his annual living expenses of
$168.02, so the investor does not need to take out a loan to realize his consumption
plan.
c) Now assume that the investor faces a cash flow tax for his investment of 40%. In
particular, assume that he receives a tax credit on his initial expense of 40% which is directly
paid to him. His savings are taxed at a rate of 25%. What is the net present value of the
investment project now?
The net stream of payments is
t=0: - 600, t= 1: 120, t=2: 180, t=3: 240, t=4: 300.
At a net of tax interest rate of 6%, NPV = $112.54.
The present value of the return payments starting is $712.54 + $400 = $1,112.54. Don’t
forget the tax credit which must also count as a return payment on the investment of
$1,000!
So if he converts $1,112.54 into a yearly annuity, he gets $168.02
If he takes the 1,000$ which he has and puts it into his bank account, he can get a
yearly payment over 10 years of $135.87. The latter is smaller, as it must if the
investment increases his net wealth!
Appendix: This appendix presents the solution taking into account an initial
wealth for Anita of 1,000.
B1: A decision maker has the wealth scaling function U(w) = w, an initial wealth w0=1,000$
and decides by expected utility.
a) Anita R. inherits a beauty shop which gets her a profit of 1,500$ with probability p=0.80
and nothing with probability (1-p)=0.2.
What is her expected utility? Which is the minimum amount of money at which she would
want to sell the business? Which is, therefore, her risk premium? Using a graph, relate risk
premium and certainty value!
EU  0.8 2,500  0.2 1, 000  0.8  38.73  0.2  31.62  46.32
Let P* be the minimum price at which she would be willing to sell. This price must
give her a utility to match her expected utility with the business, hence
U (w 0  P* )  w 0 +P*  46.32  w0  P*  2,145.96
with w0=1,000, P*=1,145.96.
Her certainty value of the risky business is thus P*=960 which is less than its expected
monetary value EV=1,200. Her risk premium is the difference, i.e. 240.6 marks
b) Investor Leonard O’Real considers buying Anita’s business. Leonard has a wealth of
1,000,600$ and his wealth scaling function is U(w) = w. Would he be willing to pay Anita’s
minimum price? What if she asked a price of 1,000?
At a price of 1,145.96 $, Leonard would want to buy if
1, 000, 600  0.2 1, 000, 600  1,145.96  0.8 1, 000, 600  1,145.96  1,500
1, 000, 600  0.2  999.73  0.8 1, 000.48
1,000.30 < 1,000.32.
So he would just so be willing to buy. Same holds, of course, at a lower price of 1,000.
2 marks