Decisions Under Risk Flashcards
what is a simple lottery
L = [p1, . . . , pN ; x1, . . . , xN ]
. . . where xi ∈ X , pi ≥ 0 and ∑ pi = 1
What is the expected value of a lottery?
EV (L) =N∑i=1 pi xi
(EV = p1x1 +p2x2 + …)
What is expected utility of a lottery?
EU(L) = N∑i=1 pi u(xi )
What is the certainty equivalent?
[1; CE (L)] is as good as Ls
u(CE (L)) = N∑i=1 pi u(xi )
What is the risk premium?
RP(L) = EV (L) − CE (L)
What are the properties satisfied by an individual using EU?
An individual using EU satisfies the basic rationality properties of preferences: Completeness and Transivitity. Overall, EU is a utility function over lotteries
What are compound lotteries
C = [p1, . . . , pN ; L1, . . . , LN ]
. . . where Li are simple lotteries
What does the property of continuity mean?
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
What is expected Utility Theorem?
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)
When do we say that an individual is risk averse?
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
What is the Absolute Arrow-Pratt Measure for u?
-(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
What is the Relative Arrow-Pratt Coefficient?
the Absolute Arrow-Pratt Coefficient multiplied by wealth
What is First order Stochastic Dominance?
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
What is Second Order Stochastic Dominance?
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)
What is a mean-preserving spread?
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