Department of Economics
University of Oslo
14 February 2001
Optional term paper, SØK460 (ECON460) Finance Theory, spring 2001
(“Semesteroppgave.”)
Your answers should be handed in (room 1240) before 15:00
hrs. on 6 March. I shall give my suggested answers in a lecture on
Thursday, 5 April at 12:15—14:00 in Auditorium 1 (ES)
and you will get your answers back at that time with some comments. You may write in
English or Norwegian.
The purpose of this is to give the students an opportunity to write an exam-like paper,
and receive written comments to it, as a preparation for the exam. The evaluation of the
paper does not affect your course grade in any way, but the paper is a strongly recommended
exercise.
Please write legibly on one side of the paper, and leave a margin for notes. Please
restrict yourself to less than 24 hand written pages, or less than 15 typed pages.
The questions are previously given as exam questions, but make up a little (about 35
percent) more than a six-hour exam.
Question 1 was given at a time when electronic calculators were allowed at the exam.
They are not allowed any more, but I have included this question anyhow, and you may
find a calculator or computer helpful. Since then the use of numbers in exam problems has
been somewhat restricted, and we try to find numbers which are easy to work with. That
is, hopefully, reflected in Question 2.
It is better to hand in incomplete answers than nothing. It is a good idea to work on
your own as much as possible. It is also a good idea to try to solve the other exercise
problems available on the course web site. Good luck!
Question 1
Consider an economy in which the Capital Asset Pricing Model holds. The economy has
only two types of shares with risky future values. Share i has a rate of return of R̃i , for
i = 1, 2. In addition the agents may save or borrow at a risk free interest rate Rf . It is not
necessary to derive the model results.
(a) Find the market portfolio’s expected rate of return, and the variance of this rate of
return, when the following information is given:
E(R̃1 ) = 0.1, E(R̃2 ) = 0.2, var(R̃1 ) = 0.062 = 0.0036,
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var(R̃2 ) = 0.182 = 0.0324, cov(R̃1 , R̃2 ) = 0.002,
the total market value of all shares of type 1 is equal to the total market value of all
shares of type 2.
(b) Show that the given information is sufficient to determine the magnitude of Rf , and
try to find this magnitude.
(c) Find the minimum-variance portfolio within the set of portfolios composed from the
two shares. Determine the composition of this portfolio, its expected rate of return,
and the variance of this rate of return.
Question 2
(a) Consider a risk averse person wishing to invest today to have money available for
consumption one period from now. There are only two investment opportunities, X
and Y . Both have normally distributed rates of return, R̃X and R̃Y , with expected
values E(R̃X ) = 0.18 and E(R̃Y ) = 0.15, and variances var(R̃X ) = 0.42 = 0.16 and
var(R̃Y ) = 0.32 = 0.09, and covariance cov(R̃X , R̃Y ) = 0.11. From this information,
can you narrow down what kind of investment the person will make?
(b) Consider a risk averse person wishing to invest today to have money available for
consumption one period from now. There are only two investment opportunities, V
and W . Both have normally distributed rates of return, R̃V and R̃W , with expected
values E(R̃V ) = 0.08 and E(R̃W ) = 0.11, and variances var(R̃V ) = 0.42 = 0.16 and
var(R̃W ) = 0.32 = 0.09, and covariance cov(R̃V , R̃W ) = 0.11. From this information
(and what you found under part (a)), can you narrow down what kind of investment
the person will make?
Question 3
As a point of departure consider the formula
P0 =
E(P̃e ) − λ cov(P̃e , R̃m )
.
1 + Rf
(a) Explain what the symbols in the formula mean.
(b) Show how the formula can be derived from the standard version of the Capital Asset
Pricing Model (CAPM). You are not asked to derive the CAPM.
(c) Give a verbal interpretation of what the formula expresses. You may well discuss
why one or more variables, which an uninformed person could have expected to see,
does not appear in the formula.
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Consider a corporation (joint-stock, limited company) which shall invest in a production
process which only needs input in the form of investment now, and which produces only
one period from now. If the invested money amount is I, then the produced quantity will
be f(I) with certainty. Here f is an increasing, concave function. The produced quantity
can be sold at a price per unit, P̃ , which is normally distributed. You can disregard the
possibility that this price becomes negative, since we assume that the probability for this
is very small.
(d) Show how the formula above can be used to find the value today of the production
which will take place one period from now.
(e) Find the first order condition for a maximum net value of the corporation, i.e., the
value you found under (d) minus the invested amount. How will the uncertainty
affect the amount to be invested?
(f) Do the same analysis as under (d) and (e), but this time under the assumption that
the corporation is subject to a cash flow tax at a rate t. The corporation will be
refunded a fraction t of the invested amount in the same period as the investment.
It must pay the same fraction of the production value as a tax in the next period.
How will the tax affect the amount to be invested? Interpret the result.
(g) Do the same analysis as under (f), but this time under the assumption that the
corporation instead is subject to a profits tax. Nothing is refunded in the period of
investment. But the invested amount may be deducted in the production value in
the next period, and a tax at a rate t is paid out of the difference. If the difference is
negative, the state will refund a fraction t. How will the tax affect the amount to be
invested? Interpret the result. How could one have adjusted the deduction for the
invested amount in order to obtain the same effect on the investment of the profits
tax as of the cash flow tax?
Diderik Lund
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