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ECON6015 Finance
EXERCISE 1 - Notes on Answers
mpton
Note that these are outline answers only - full answers will include definitions and expla-
nations as appropriate.
1. VEVOX exercises
(a) True/False: You are told that U(W ) = W 2 is a utility function with dimin-
ishing marginal utility.
Outline answer: False. For a utility function to exhibit diminishing marginal
utility, the second derivative of the utility function must be negative.
(b) A person has an expected utility function of the form U(W ) =
√
W . He
initially has wealth of $ 4. He has a lottery ticket that will be worth $12 with
probability 0.5 and will be worth $ 0 with probability 0.5.
i. What is his expected utility? What is the value of certainty equivalent
wealth? What is the value of risk premium?
Outline answer: EU=0.5(U(12+4))+0.5U(4+0)=3; U(Wce)=3, Wce = 9
E(W)= 0.5*16+0.5*4 = 10 Risk premium = E(W)-Wce=1
2. Find the coefficients of relative risk aversion for the following utility functions where
W denotes the investor’s wealth and β > 0.
(a)
U(W ) =
1− e−βW
β
Outline answer:
U(W ) =
1− e−βW
β
U ′(W ) = e−βW
U ′′(W ) = −βe−βW
ρr = −WU
′′(W )
U ′(W )
= βW
(b)
U(W ) =
W 1−β − 1
β − 1
1
Outline answer:
U(W ) =
W 1−β − 1
β − 1
U ′(W ) = −W−β
U ′′(W ) = βW−β−1
ρr = −WU
′′(W )
U ′(W )
= β
3. Rank the following risky investment projects based on stochastic dominance.
Project A Project B Project C
Payoff Probability Probability Probability
5 0.2 0 0.1
10 0.4 0.1 0.4
20 0.4 0.5 0.2
30 0 0.4 0.2
40 0 0 0.1
Outline answer :
You should define all notation, FSD, SSD etc. and arrange your calculations of
FA(X),
∫
FA(X)dX etc. in a table.
From the table, note that FB(X) ≤ FA(X) and FC(X) ≤ FA(X) for all X ∈ [5, 40],
so that Project B and Project C each FSD Project A.
Comparing Projects B and C, FB(X) and FC(X) intersect, so neither B nor C FSD
the other. However, observe that
∫
FB(X) dX ≤
∫
FC(X) dX for all X ∈ [5, 40], so B
SSD C.
In conclusion, Project B SSD Project C, both of which FSD Project A.