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A.3. UTILITY FUNCTIONS AND CHANGES IN RISK
A.3
Utility Functions and Changes in Risk
1. How might you create a realistic utility function using episodes of ‘Deal or No
Deal’ ? What are the main problems?
Try to limit your answer to no more than half a side of A4. Please start a new page
for question 2 (this is to allow question 1 to be marked separately from the other
questions).
2. Consider an individual with utility function u(w) = w1/2 . Her initial wealth is 10
and she faces the lottery x̃ :
6, 21 ; +6, 12 .
(a) Compute the exact values of this individual’s certainty equivalent and risk
premium for lottery x̃.
(b) Obtain the Arrow-Pratt approximation of the risk premium.
(c) Show that with this utility function absolute risk aversion is decreasing while
the relative risk aversion is constant in (initial) wealth.
(d) If the utility becomes v(w) = w1/4 again answer part (a). Are you surprised
by the changes in certainty equivalent and risk premium? Relate this change
to the notion of ‘more risk averse’.
(e) If the risk becomes ỹ :
3, 12 ; +3, 12 , compute the new (approximate) risk
premium. Why is the approximated risk premium four times smaller than the
risk premium for x̃.
3. If it held for all risks, the Arrow-Pratt approximation for a small zero mean risk
ỹ = kx̃, ⇡(w0 , u, ỹ) ' 0.5Eỹ 2 A(w0 ), would make the EUT model a particular case
of the mean-variance (MV) model. The theory of finance usually limits the analysis
to the mean and variance by restricting the set of utility functions and distributions that make the Arrow-Pratt approximation exact. Show that the Arrow-Pratt
approximation is exact when the utility function is CARA and ỹ is normally distributed.
4. Suppose that a decision maker has a utility function that satisfies constant relative
risk aversion. Show that the decision maker’s preference over certain wealth of w
and lottery w(1 + x̃) does not depend on w 0, where discrete random variable x̃
takes values {xi , xi > 1}i=1,...,n with probabilities {pi }i=1,...,n .
5. Consider the utility function
u(z) =
where 0 < a < 1.
(a) Show that u is concave.
(
z
z0 + a(z
if z  z0
.
z0 ) if z > z0
APPENDIX A. ASSIGNMENTS SB4B · HT 2017
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(b) Show that u exhibits first-order risk aversion at z = z0 , namely that the risk
premium ⇡(z0 , u, kx̃) tends to zero when k tends to zero as k, rather than as
k2.
(c) Show that a reduction in the constant a increases the degree of risk aversion.
(d) Does it satisfy DARA?
6. Consider the following two random variables: x̃ has a (continuous) uniform density on the interval [ 1, +1], while ỹ is a discrete random variable defined by
( 1, 12 ; +1, 12 ).
(a) Draw the cumulative distributions of x̃ and ỹ.
(b) By applying the Rothschild-Stiglitz ‘integral conditions’ or otherwise determine which random variable is riskier.
(c) Find the distributions of the ‘white noise’ that must be added to the less risky
lottery to obtain the riskier one.
7. A corporation must decide between two mutually exclusive projects. Both projects
require an initial outlay of £100 million, and they generate cash flows that are independent of the growth of the economy. Project A has an equal probability of four
gross payo↵s: £80 million; £100 million; £120 million; or £140 million. Project
B has a 50:50 chance of paying either £90 million or £130 million. Assuming that
shareholders are all risk averse, show that they unanimously prefer Project B to
Project A.
8. Consider two lotteries La and Lb . The outcomes of lottery La are uniformly distributed on the unit interval. The probability density function of Lb is given by
2
g(x) = c x 12 for x 2 [0, 1], and g(x) = 0 otherwise.
(a) Show that lottery Lb is a mean-preserving spread of lottery La .
(b) Does one of the lotteries La and Lb first-order stochastically dominate the
other?
9. An investor has wealth to invest in a set of n independent and identically distributed
lotteries, P
x̃1 , . . . , x̃n . Let ↵i 2 [0, 1] denote the share of wealth invested in lottery
i, where ni=1 ↵i = 1. Show that the distribution of final wealth generated by the
perfect diversification strategy ↵ = ( n1 , . . . , n1 ) second order stochastically dominates
the distribution of final wealth
by any feasible strategy.
Pn generated
1
[Hint: Consider adding i=1 (↵i n )x̃i to each potential outcome of the perfect
diversification strategy.]
10. Consider an economy where each agent faces a statistically independent risk of losing 100 with probability p. A pool of N agents decide to create a mutual agreement
where the aggregate loss in the pool will be equally split among its members.
(a) Describe the possible losses and associated probabilities for each member of
the pool when N = 2 and N = 3.
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A.3. UTILITY FUNCTIONS AND CHANGES IN RISK
(b) Show that an increase in the pool size from 2 to 3 would be preferred by all
members as long as they are risk averse
(Hint: Denoting x̃ as the loss borne by a member of a pool with N = 2 and ỹ
as the loss borne by a member of a pool with N = 3, show that x̃ ⇠ ỹ + "˜ for
some white noise "˜ where E[˜
"|ỹ] = 0.)
11. An agent with current wealth X has the opportunity to bet any amount on the
occurrence of an event that she believes will occur with probability p 2 (0, 1). If
she wagers w, she will receive (the gross amount) 2w if the event occurs and 0 if
it does not. Her utility function is given by u(x) = e rx with r > 0. How much
should she wager?