Insurance + Risk Aversion

Question 1

What is a Risky Prospect?

Answer 1

Set of Payoffs (W1, … , Wn)

- Each Payoff occurs w/ Prob. (p1, … , pn, ∑pi =1)

Question 2

How can we Evaluate a Risky Prospect?

Answer 2

Calculating EU

EU = ∑ pi U(Wi)

Question 3

What does a Risk Neutral person’s Utility look like?

Answer 3

Straight 45º Line - EU(L) = U [E(W)]

Question 4

What does a Risk Averse person’s Utility look like?

Answer 4

Concave Utility Function

EU(L) < U [E(W)]

Question 5

What does a Risk Loving person’s Utility look like?

Answer 5

Convex Utility Function

EU(L) > U [E(W)]

Question 6

What is the Exp. Wealth?

Answer 6

E(W) = p(T - C) + (1 - p)T

= T - pC

Question 7

What is the Exp. Utility of Wealth?

Answer 7

EU(W) = p U(T - C) + (1 - p) U(T)

Question 8

What is the Utility of Exp. Wealth?

Answer 8

U [E(W)] = U(T - pC)

Question 9

What 2 things does an Insurance Contract specify?

Answer 9

Compensation - Y

Premium - X

Question 10

What is Compensation under Full Insurance and what level of Wealth does it provide?

Answer 10

Y = C

Certain Wealth : T - X

Question 11

When is Full Insurance acceptable to a buyer?

Answer 11

S: U(S) = p U(T - C) + (1 - p) U(T) => S = Value that gives EU
F.I Acceptable: U(T - X) ≥ U(S)

Question 12

When will a Risk Averse person NOT Buy Insurance?

Answer 12

If X > T - S

Question 13

What Premium will an Insurance Company accept?

Answer 13

X ≥ pY

Question 14

What is the Actuarially Fair Premium?

Answer 14

X = pY - due to Competition in Market

- Exp. Profits = 0

Question 15

What are Fair Odds?

Answer 15

X = pY

Question 16

What are Unfair Odds?

Answer 16

X > pY

Question 17

What are Favourable Odds?

Answer 17

X < pY

Question 18

What level of Wealth does Full Insurance provide?

Answer 18

Certain Wealth

Question 19

What is Wealth w/ No Theft + Insurance?

Answer 19

Wnt = T - X

Question 20

What is Wealth w/ Theft + Insurance?

Answer 20

Wt = (T - C) + (Y - X)

= (T - X) - (C - Y)

Question 21

When does Wnt = Wt?

Answer 21

When Y = C - Full Insurance

Question 22

When is E(W) the same, regardless of level of Insurance + Prove it?

Answer 22

Under Fair Odds
E(W) = p (T - C + Y - X) + (1 - p) (T - X)
 = T - pC + (pY - X)
Fair Odds - X = pY
=> E(W) = T - pC - NOT dependent on Y

Question 23

What is the Insurers’ Exp. Profit?

Answer 23

E(π) = p (X - Y) + (1 - p) X

Question 24

Find Insurers’ Zero Profit in terms of Wnt and Wt?

Answer 24

E(π) = p (X - Y) + (1 - p) X
- Wt = T - C + Y - X
- Wnt = T - X
=> E(π) = p (T - C - Wt) + (1 - p) (T - Wnt)

Question 25

How do you find the Insurers’ Zero Profit Line in State Space?

Answer 25

E(π) = p (T - C - Wt) + (1 - p) (T - Wnt)
Rearrange: Wt = [((T - pC) - E(π))/p] - [(1 - p)/p]Wnt
Take dWt/dWnt = - [(1 - p) / p] < 0
S.O.C: d2Wt / dWnt dp = 1 / p^2 > 0

Question 26

What happens to Zero Profit Line when p Increases?

Answer 26

Line becomes FLATTER

Question 27

How do we find the Insureds’ I.C?

Answer 27

EU = p U(Wt) + (1 - p) U(Wnt)
Total Diff.: dEU = [p U’(Wt)] dWt + [(1 - p) U’(Wnt)] dWnt
Rearrange: dWt / dWnt = - [(1 - p) / p] x [U’(Wnt) / U’(Wt)] < 0
S.O.C: d2Wt / dWnt dp = (1 / p^2) x [U’(Wnt) / U’(Wt)] > 0

Question 28

How does Increased p affect the I.C?

Answer 28

Higher p –> Higher Risk –> Flatter I.C

- Willing to give up Less in Theft State to have more in NT State than before

Question 29

What is the Insurance Outcome w/ Fair Odds?

Answer 29

I.C is Tangent to Zero Profit Line
- [(1 - p) / p] = - [(1 - p) / p] x [U’(Wnt) / U’(Wt)]
=> U’(Wnt) = U’(Wt)
=> Wnt = Wt - Full Insurance

Question 30

What is the Insurance Outcome w/ Unfair Odds?

Answer 30

Insurance sold using p’ > p - Exp. Profit Line becomes FLATTER
tangency now requires: - [(1 - p’) / p’] = - [(1 - p) / p] x [U’(Wnt) / U’(Wt)]
=> U’(Wnt) > U’(Wt) - Partial Insurance