Decisions Under Risk

Question 1

what is a simple lottery

Answer 1

L = [p1, . . . , pN ; x1, . . . , xN ]
. . . where xi ∈ X , pi ≥ 0 and ∑ pi = 1

Question 2

What is the expected value of a lottery?

Answer 2

EV (L) =N∑i=1 pi xi

(EV = p1x1 +p2x2 + …)

Question 3

What is expected utility of a lottery?

Answer 3

EU(L) = N∑i=1 pi u(xi )

Question 4

What is the certainty equivalent?

Answer 4

[1; CE (L)] is as good as Ls

u(CE (L)) = N∑i=1 pi u(xi )

Question 5

What is the risk premium?

Answer 5

RP(L) = EV (L) − CE (L)

Question 6

What are the properties satisfied by an individual using EU?

Answer 6

An individual using EU satisfies the basic rationality properties of preferences: Completeness and Transivitity. Overall, EU is a utility function over lotteries

Question 7

What are compound lotteries

Answer 7

C = [p1, . . . , pN ; L1, . . . , LN ]
. . . where Li are simple lotteries

Question 8

What does the property of continuity mean?

Answer 8

Given xH > xM > xL, there exists p∗ ∈ (0, 1) such that:
[1; xM ] ∼ [p∗, 1 − p∗; xH , xL]

EU also satisfies this property because EU is linear in p. The EU of the variable lottery is linear in p and must cross somewhere the value of xM

Question 9

What is expected Utility Theorem?

Answer 9

The following statements are equivalent:
1 The preferences >= are rational and satisfy reduction, continuity and independence
2 The preferences >= correspond to some expected utility EU(L) (that uses some function u)

Question 10

When do we say that an individual is risk averse?

Answer 10

We say that an individual is risk averse if:
- For every lottery, EU(L) ≤ u(EV (L)), or equivalently,
- For every lottery, CE (L) ≤ EV (L), or equivalently,
- For every lottery, RP(L) ≥ 0, or equivalently,
- u is concave

Question 11

What is the Absolute Arrow-Pratt Measure for u?

Answer 11

-(u’‘(x))/(u’(x))

with this mesure u and v+a+bu have the same absolute Arrow-Pratt Measure

The greater the Arrow-Pratt measure is, the more risk averse the individual is

Question 12

What is the Relative Arrow-Pratt Coefficient?

Answer 12

the Absolute Arrow-Pratt Coefficient multiplied by wealth

Question 13

What is First order Stochastic Dominance?

Answer 13

We say that lottery L1 first order stochastically dominates lottery L2 if, for every y :
FL1 (y ) ≤ FL2 (y )

(where FL is the CDF of lottery L)

If L1 FOSDs L2, every expected utility maximizer prefers L1 to L2

Question 14

What is Second Order Stochastic Dominance?

Answer 14

Given two lotteries L1 and L2 with the same expected value, we say that L1 second order stochastically dominates L2 if there exists a monetary value y ∗ such that
FL1 (y ) ≤ FL2 (y ) whenever y ≤ y ∗, and
FL1 (y ) ≥ FL2 (y ) whenever y ≥ y ∗

If L1 SOSDs L2, every risk-averse EU maximizer prefers L1 to L2

(FL is CDF of lottery L)
(safer lottery crosses once from below)

Question 15

What is a mean-preserving spread?

Answer 15

We have a compound lottery obtained by:
1 Playing first the simple lottery
2 Adding some mean-zero noise to the prize obtained
In the first example, flip a coin when 20 was obtained, adding or substracting 10
In the second example, flip a coin no matter what, adding or substracting 10
In such cases, we say that L1 is a mean-preserving spread of L2

If L1 is a mean-preserving spread of L2 and u is concave, the EU-individual prefers L2 to L1

if B is a mean-preserving spread of A, B spreads out one or more portions of A’s PDF or PMF whilst leasing the mean the same

Question 16

What is a risk pooling contract/contract with mutual risk?

Answer 16

• In the event that both of them, or none of them, suffer the loss, no transaction takes place.
• In the event that one, and only one of them, suffers the loss, the rich individual will compensate the poor individual by transferring half of the loss, y/2
• With probability p^2 both suffer the loss and each individual ends up with x − y
• With probability (1 − p)^2, no individual suffers the loss and each individual ends up with x
• With probability 2p(1 − p), one individual suffers the loss and each individual ends up with (x − y)/2

Question 17

Simple model of insurance

Answer 17

• Initial wealth: W
• This wealth includes a good subject to a risk of being lost
• Value of the good: H
• Probability of loss: p
• An insurance company offers insurance
• The consumer can select any amount Q to be insured (typically Q ≤ H). If the loss of the good happens, the company will compensate the consumer with Q
• The consumer must pay π units for each pound insured, for a total of πQ. This must be paid no matter whether the loss takes place or not

Question 18

What is the lottery generated by the choice of how much to insure?

Answer 18

LQ = [p, 1 − p; W − H + (1 − π)Q, W − πQ]

Question 19

What is the actuarially case?

Answer 19

When π = p. This is
called actuarially fair, as the insurance company has πQ − pQ = 0 expected profits. This may happen if the insurance company becomes large, manages a large number of (mutualised, independent) risks, and behaves in a (close to) risk-neutral way

In this case, the individual fully insures. This can be seen as an application of the SOSD principle. With an actually fair insurance, all lotteries have the same expected value and we have shown that are ordered by SOSD. Full insurance SOSDs all the other lotteries

Question 20

What happens in risk sharing?

Answer 20

If the investment is shared among n investors, or equivalently, a fraction 1/n is bought, the lottery is
Ln = [p1, . . . , pN ; x + z1/n , . . . , pN ; x + zN/n ]

EV (Ln) = x + EV (part project) = x + 1/n EV (project) ≥ x

Every potential payoff is now closer to the original wealth x, as x + zi/n is closer to x than x + zi is