Insurance + Risk Aversion Flashcards

1
Q

What is a Risky Prospect?

A

Set of Payoffs (W1, … , Wn)

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

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2
Q

How can we Evaluate a Risky Prospect?

A

Calculating EU

EU = ∑ pi U(Wi)

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3
Q

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

A

Straight 45º Line - EU(L) = U [E(W)]

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4
Q

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

A

Concave Utility Function

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

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5
Q

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

A

Convex Utility Function

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

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6
Q

What is the Exp. Wealth?

A

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

= T - pC

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7
Q

What is the Exp. Utility of Wealth?

A

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

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8
Q

What is the Utility of Exp. Wealth?

A

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

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9
Q

What 2 things does an Insurance Contract specify?

A

Compensation - Y

Premium - X

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10
Q

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

A

Y = C

Certain Wealth : T - X

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11
Q

When is Full Insurance acceptable to a buyer?

A

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

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12
Q

When will a Risk Averse person NOT Buy Insurance?

A

If X > T - S

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13
Q

What Premium will an Insurance Company accept?

A

X ≥ pY

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14
Q

What is the Actuarially Fair Premium?

A

X = pY - due to Competition in Market

- Exp. Profits = 0

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15
Q

What are Fair Odds?

A

X = pY

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16
Q

What are Unfair Odds?

A

X > pY

17
Q

What are Favourable Odds?

A

X < pY

18
Q

What level of Wealth does Full Insurance provide?

A

Certain Wealth

19
Q

What is Wealth w/ No Theft + Insurance?

A

Wnt = T - X

20
Q

What is Wealth w/ Theft + Insurance?

A

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

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

21
Q

When does Wnt = Wt?

A

When Y = C - Full Insurance

22
Q

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

A
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
23
Q

What is the Insurers’ Exp. Profit?

A

E(π) = p (X - Y) + (1 - p) X

24
Q

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

A

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)

25
Q

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

A

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

26
Q

What happens to Zero Profit Line when p Increases?

A

Line becomes FLATTER

27
Q

How do we find the Insureds’ I.C?

A

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

28
Q

How does Increased p affect the I.C?

A

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

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

29
Q

What is the Insurance Outcome w/ Fair Odds?

A

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

30
Q

What is the Insurance Outcome w/ Unfair Odds?

A

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