Econ Homework
zon5008
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ECON 417: Economics of Uncertainty
Contents
I Expected Utility Theory 3
1 Lotteries 3
2 St. Petersburg’s Paradox 3
3 von Neumann and Morgenstern Axioms and Expected Utility Form 4
4 Risk Attitudes 5
5 Risk Premium and Certainty Equivalent 6
6 Measures of Risk Aversion 6
II Mean-Variance Optimization 8
7 One Riskfree Asset, One Risky Asset 8
8 Many Risky Assets 9
9 One Riskfree Asset, Many Risky Assets 9
10 Diversification 10
11 Capital Asset Pricing Model 10
III Insurance 11
12 Utility Maximization 11
12.1 Tangency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
12.2 Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
12.3 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1
13 State Preference Approach to Insurance 13
14 Overview of Insurance 15
15 Demand for Insurance 15
15.1 Mossin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
15.1.1 Actuarially Fair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
15.1.2 Not Actuarially Fair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
15.2 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
15.3 Coinsurance and Deductibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
16 Supply of Insurance 18
16.1 Risk pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
16.2 Risk spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
16.3 “Undersupply” of Full Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
17 Asymmetric Information 20
17.1 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
17.1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
17.1.2 Tangency Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
17.1.3 Two types of consumers, symmetric information . . . . . . . . . . . . . . . . 22
17.1.4 Two types of consumers, asymmetric information . . . . . . . . . . . . . . . 23
17.1.5 Pooling Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
17.1.6 Separating Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
17.2 Moral Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
IV After the Midterm 27
17.3 Insurance (cont.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
18 The Value of Information 27
19 Options 27
19.1 Financial Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
19.2 Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2
Part I
Expected Utility Theory
1 Lotteries
For decision making under uncertainty, we consider lotteries.
Lotteries are representations of risky alternatives.
Definition 1 (Simple Lottery). A simple lottery L is a list L = (p1, ..., pN ) with pi ≥ 0 for all N and∑N i=1 pi = 1, where pi is the probability of outcome i occurring. The outcomes that may result
are certain: x1, ..., xN .
Definition 2 (Compound Lottery). A compound lottery is a lottery in which the outcomes are simple lotteries.
We can find the probability of each outcome (terminal node) in a compound lottery by multi-
plying the probabilities of the branches leading to the outcome.
Definition 3 (Reduced Lottery). The overall probability measure R on X is a simple lottery that we call the reduced lottery of the compound lottery.
R = x1P (x1)+ ...+xN P (xN )
The theoretical analysis of expected utility rests on the consequentialist premise: We assume that for any risky alternative, only the reduced lottery over final outcomes is of rel-
evance to the decision maker.
2 St. Petersburg’s Paradox
Suppose someone offers to toss a fair coin repeatedly until it comes up heads, and
to pay you $1 if this happens on the first toss, $2 if it takes two tosses to land a head,
$4 if it takes three tosses, $8 if it takes four tosses, etc.
Question: How much is the lottery worth? How much are you willing to pay to play this lottery?
3
Expected value of the St. Petersburg problem:
E(X ) = ∞∑
i=1 pi xi
= (1 2
) ·1+ (1 2
)2 ·2+ (1 2
)3 ·4+ ... = ∞
Paradox: If I charged $1 million to play the game, I would surely have no takers, despite the fact
that $1 million is still considerably less than the expected value of the game.
Bernoulli:
• argued that individuals do not care directly about the dollar prizes of a game; rather they
care about the utility of prizes • considered u(x) = log x, which exhibits diminishing marginal utility
3 von Neumann and Morgenstern Axioms and Expected Utility
Form
von Neumann and Morgenstern describe necessary and sufficient conditions for the represen-
tation of a utility function.
In expected utility theory, preference relations, %, are characterized by 3 axioms:
1. Weak-Order requires that the preference relation be complete and transitive
• Completeness requires that all elements are comparable. For L1,L2 ∈L , the preference relation is complete if either L1 % L2 or L2 % L1
• Transitivity requires that choices be consistent. For L1,L2,L3 ∈L , the preference relation is transitive if L1 % L2 and L2 % L3 implies L1 % L3
2. Continuity means that small changes in probabilities do not change the nature of the ordering between two lotteries
If L1,L2 ∈L are such that L1 % L2, then for all L3 ∈L , there is an α such that 0 <α< 1 and L1 Â (1−α)L2 +αL3 and there is a β such that 0 <β< 1 and (1−β)L1 +βL3 Â L2
4
For example, if a “beautiful and uneventful trip by car” is preferred to “staying home,”
then a mixture of the outcome “beautiful and uneventful trip by car” with a sufficiently
small but positive probability of “death by car accident” is still preferred to “staying home.”
3. Independence (of irrelevant alternatives) means that if we mix each of the two lotteries with a third one, then the preference ordering of the two resulting mixtures is indepen-
dent of the particular third lottery used
For all lotteries L1,L2,L3 ∈L and all α such that 0 <α< 1, L1 Â L2 ⇐⇒ αL1 + (1−α)L3 %αL2 + (1−α)L3
The Allais Paradox is a violation of the Independence Axiom.
Theorem 1 (Expected Utility Theorem). If the decision maker’s preferences over lotteries satisfy the weak-order, continuity, and independence axioms, then her preferences are representable by
a utility function with the expected utility form.
EU (L) = N∑
i=1 pi u(xi ) = p1u(x1)+ ...+pN u(xN )
Criteria for Maximization
L1 % L2 ⇐⇒ EU (L1) ≥ EU (L2)
4 Risk Attitudes
Risk aversion captures the idea that individuals dislike risk and uncertainty.
Definition 4 (Fair bet). A fair bet is a random game with a specified set of prizes and associated probabilities that has an expected value of zero.
Definition 5 (Concavity). The function f (x) is concave if a straight line joining any two points on it lies entirely below the function itself. In other words, the function f (x) is concave if for
any x1 and x2, and λ : 0 ≤λ≤ 1,
f (λx1 + (1−λ)x2) ≥λ f (x1)+ (1−λ) f (x2)
If f (x) is a concave (twice differentiable) function, then f ′′(x) < 0.
A decision maker is risk averse if
5
1. at any level of wealth, he rejects every fair bet
2. he strictly prefers a certainty consequence to any risky prospect whose mathematical ex-
pectation of consequences equals that certainty
3. u(x) is concave. In other words, for every lottery with outcomes x1, ..., xN and probabili-
ties p1, ..., pN , respectively
u( N∑
i=1 pi xi ) ≥
N∑ i=1
pi u(xi )
Concavity of the utility function implies diminishing marginal utility.
A decision maker is risk loving if
1. at any level of wealth, he accepts every fair bet
2. he strictly prefers the lottery to its mathematical expectation
3. u(x) is convex. In other words, for every lottery with outcomes x1, ..., xN and probabilities
p1, ..., pN , respectively
u( N∑
i=1 pi xi ) ≤
N∑ i=1
pi u(xi )
A decision maker is risk neutral if
1. at any level of wealth, he is indifferent to every fair bet
2. he is indifferent between the lottery and its mathematical expectation
5 Risk Premium and Certainty Equivalent
Definition 6 (Certainty Equivalent). The amount of money, C E(L), when obtained for certain, provides the same expected utility as the lottery
Definition 7 (Risk Premium). The maximum amount, π, that an individual is willing to forego in order to receive the expected value of the lottery with certainty
The risk premium is the difference between the expected value of the lottery and the certainty
equivalent of the lottery.
π= E(L)−C E(L)
6 Measures of Risk Aversion
The Arrow-Pratt measures of risk aversion are quantitative measures of how averse to risk a
person is. It provides a way to measure the degree of concavity of the utility function (hence,
the strength or intensity of risk aversion).
6
Definition 8 (Absolute Risk Aversion). The absolute risk aversion measure A(x) for a utility function u(x) is
A(x) ≡−u ′′(x)
u(x)
Definition 9 (Relative Risk Aversion). The relative risk aversion measure R(x) for a utility func- tion u(x) is
R(x) ≡−x u ′′(x)
u(x) = x · A(x)
Definition 10 (DARA, CARA, IARA). The utility function u(·) has decreasing (constant, increas- ing) absolute risk aversion if A(x,u) is a decreasing (constant, increasing) function of x. This
depends on the sign of the derivative of A(x) with respect to x, i.e. d A(x)d x .
Empirical evidence supports DARA. The power utility function exhibits DARA.
Definition 11 (DRRA, CRRA, IRRA). The utility function u(·) has decreasing (constant, increas- ing) relative risk aversion if R(x,u) is a decreasing (constant, increasing) function of x. This
depends on the sign of the derivative of R(x) with respect to x, i.e. dR(x)d x .
7
Part II
Mean-Variance Optimization
V (µ,σ)
• µ is the expected return of the asset
• σ is the standard deviation of the asset. Risk is measured by the standard deviation.
Risk attitudes are determined by the partial derivatives with respect to risk
• δVδσ < 0 risk averse • δV
δσ > 0 risk loving
• δVδσ = 0 risk neutral Typically, financial economists think of investors as being risk averse, thus investors trade off
risk and return.
The risk-return tradeoff:
• A risk averse, mean-variance optimizing investor will only accept a riskier portfolio if the
expected return of that portfolio is appropriately higher
• A risk averse, mean-variance optimizing investor will only accept a portfolio that has a
lower expected return if the risk of that portfolio is appropriately lower
Consider the particular functional form for a mean-variance optimizer:
V (µ,σ) =µ− 1 2
A ·σ2
where µ is expected return, σ2 is the variance, and A is the coefficient of risk aversion (A > 0).
7 One Riskfree Asset, One Risky Asset
Assume that an investor must decide how to invest all of her wealth and has only two options:
a riskfree asset, R f and a risky asset. The expected return of the risky asset is E(Ri ) and its
variance is V ar (Ri ) = σ2i . To determine the optimal fraction of wealth an investor will allocate to a risky asset, k∗, consider the following maximization problem
max k
V (µp ,σp ) = µp − 1 2
A ·σ2p
= E(Rp )− 1 2
A ·V ar (Rp )
= R f +k(E(Ri )−R f )− 1
2 A ·k2V ar (Ri )
8
Solve the optimization problem. The first-order condition requires that the derivative, with
respect to k, is equal to zero. We find
k∗ = (E(Ri )−R f ) A ·V ar (Ri )
= S A ·σi
Definition 12 (Sharpe Ratio). The Sharpe Ratio, S of the risky asset is the expected excess re- turn of the risky asset per unit of its standard deviation. It is the reward-to-variability ratio of
investing in the risky asset.
S = E(Ri )−R f σi
Definition 13 (Capital Allocation Line). A graph of all possible expected returns and standard deviations of a portfolio formed by combining the risky asset with the riskfree asset.
8 Many Risky Assets
Consider a portfolio of two assets with weights k1 and k2, expected returns E(R1) and E(R2),
and return variances σ21 and σ 2 2.
The portfolio expected return is
E(Rp ) = k1E(R1)+k2E(R2)
The portfolio variance is
V ar (RP ) = k21σ21 +k22σ22 +2 ·k1 ·k2 ·Cov(R1,R2)
The graph is a hyperbola when volatility is plotted on the x-axis and expected returns are plotted
on the y-axis.
Definition 14 (Efficient Frontier). A graph of the feasible investments with the highest expected returns for all possible portfolio standard deviations. It is the top part of the graph above the
minimum variance portfolio.
9 One Riskfree Asset, Many Risky Assets
Definition 15 (Capital Market Line). The line from the riskfree investment through the efficient portfolio of risky assets when volatility is plotted on the x-axis and expected returns are plotted
on the y-axis.
9
Definition 16 (Tangency Portfolio). The portfolio of risky assets with the highest Sharpe Ratio. It is an efficient portfolio and it generates the steepest line combined with the riskfree asset.
Theorem 2 (Mutual Fund (Separation) Theorem). Investors with the same beliefs about expected returns, risks, and correlations all will invest in the portfolio or “fund” of risky assets that has the
highest Sharpe Ratio, but they will differ in their allocations between this fund and the riskfree
asset based on their risk tolerance.
10 Diversification
The risk of a stock includes idiosyncratic risk and market risk. Idiosyncratic risk is also known as firm-specific, unique, stand-alone, or diversifiable risk. Market risk is also known as system- atic or undiversifiable risk.
To limit your exposure to idiosyncratic risk, you can diversify your portfolio. This means choos-
ing stocks that are imperfectly correlated, i.e. ρ → −1, where ρ is the correlation coefficient. The benefit of diversification will increase the further away from ρ = 1. Definition 17 (Risk premium). It represents the additional return that investors expect to earn to compensate them for a security’s risk. It is the difference between the expected return of the
security minus the riskfree rate of return.
E(Ri )−R f
11 Capital Asset Pricing Model
Intuition for the Capital Asset Pricing Model (CAPM)
1. Because diversification does not reduce market risk, the risk premium of a security should
be determined by its market risk.
2. To measure market risk, we need a market portfolio. If all investors are mean-variance
optimizers, by the Mutual Fund Theorem, they should be holding the Tangent Portfolio.
Let the Market Portfolio be the Tangent Portfolio.
CAPM relates the security’s risk premium to the market risk premium.
E(Ri )−R f =β · (E(Rm)−R f )
and β, which measures the sensitivity of the security’s return to the return of the overall market
is
β= Cov(Ri ,Rm) V ar (Rm)
10
Part III
Insurance
12 Utility Maximization
There are three methods you can use to solve the utility maximization problem:
max x1,x2
u(x1, x2) subject to their budget constraint: I = p1x1 +p2x2
12.1 Tangency Condition
The slope of the indifference curve and the slope of the budget line should be equal at the point
of tangency. It is the point at which the consumer maximizes his or her utility, given his or her
budget constraint.
slope of indifference curve = slope of budget line MRSx1x2 =
p1 p2
MU1 MU2
= MRSx1x2 = p1 p2
Example 1. Suppose we had the following utility function
max x1,x2
u(x1, x2) = log x1 + log x2 subject to their budget constraint: I = p1x1 +p2x2
slope of indifference curve = slope of budget line 1
x1 1
x2
= p1 p2
⇒ x2 = p1 p2
x1
Plug into the budget constraint
I = p1x1 +p2x2 I = p1x1 +p2 p1
p2 x1
I = 2p1x1 ⇒ x∗1 =
I
2p1 and x∗2 =
I
2p2
11
12.2 Substitution Method
Example 2. Consider the following constrained problem with two variables
max x1,x2
log x1 + log x2 s.t
p1x1 +p2x2 = I
The idea of the substitution method is to use the constraints to get rid of some variables. In the
example above we can use the constraint to obtain that x2 = I−p1x1p2 , and after we plug this into the objective function we get
ũ(x1) = log x1 + log I −p1x1 p2
This becomes an unconstrained maximization problem for a function of one variable x1. Using
the chain rule we obtain the following first order condition (FOC)
0 = ũ′(x1) = 1 x1
+ p2 I −p1x1
(−p1 p2
) = 1
x1 − p1
I −p1x1 which yields I −p1x1 = p1x1, the solution of this equation is x∗1 = I2p1 . By the chain rule and the power rule we have
ũ′′(c1) =− 1 x21
− (−1)(−p1) p1 (I −p1x1)2
=− 1 x21
− p 2 1
(I −p1x1)2
Clearly ũ′′(c1) < 0 for any x1 and so the sufficient condition for a local maximum is satisfied. Finally, using the constraint p1x∗1 +p2x∗2 = I we get x∗2 = I2p2 , so the solution of the problem is the consumption bundle (x∗1 , x
∗ 2 ) = ( I2p1 ,
I 2p2
).
12.3 Lagrange Multipliers
Theorem 3. Let f and g be two real-valued continuously differentiable functions of two vari- ables. Suppose that (x∗1 , x
∗ 2 ) is a solution to the following maximization problem
max x1,x2
f (x1, x2)
subject to
g (x1, x2) = 0
and that (x∗1 , x ∗ 2 ) is not a critical point of g . Then there exists a real numberλ
∗ called the lagrange multiplier, such that (x∗1 , x
∗ 2 ,λ
∗) is a critical point of the following function, called a Lagrangian
L (x1, x2,λ) = f (x1, x2)+λg (x1, x2)
12
i.e. all three partial derivatives of L are zero
∂L
∂x1 (x∗1 , x
∗ 2 ,λ
∗) = 0 ∂L
∂x2 (x∗1 , x
∗ 2 ,λ
∗) = 0 ∂L
∂λ (x∗1 , x
∗ 2 ,λ
∗) = 0
Example 3. Let’s apply the Lagrange Theorem to the consumer’s problem from previous sec- tion.
max x1,x2
log x1 + log x2 s.t. p1x1 +p2x2 = I
The objective function is f (x1, x2) = log x1+log x2, the constraint function is g (x1, x2) = I−p1x1− p2x2, and the Lagrangian function is
L (x1, x2,λ) = log x1 + log x2 +λ ( I −p1x1 −p2x2
) From the Lagrange Theorem, the First Order Necessary Condition is that all partial derivatives
of the Lagrangian are zero, i.e.
∂L
∂x1 (x∗1 , x
∗ 2 ,λ
∗) = 0 ⇒ 1 x∗1
−λ∗p1 = 0 ∂L
∂x2 (x∗1 , x
∗ 2 ,λ
∗) = 0 ⇒ 1 x∗2
−λ∗p2 = 0 ∂L
∂λ (x∗1 , x
∗ 2 ,λ
∗) = 0 ⇒ I −p1x∗1 −p2x∗2 = 0
Note that last equation simply says that the constraint in the maximization problem has to hold.
The above is a system of 3 equations and 3 unknowns (x∗1 , x ∗ 2 ,λ
∗) and is quite easy to solve. We get:
x∗1 = I
2p1 x∗2 =
I
2p2 λ∗ = 2
I
13 State Preference Approach to Insurance
Goal: To show that when faced with fair markets in contingent claims on wealth, a risk averse person will choose to ensure that he has the same level of wealth regardless of which state oc-
curs.
Categorize all of the possible things that might happen into a fixed number of states. We say that contingent commodities are goods delivered only if a particular state of the world occurs.
13
Consider the following expected utility model of two contingent goods: Wg is wealth in good
times and Wb is wealth in bad times.
max EU (Wg ,Wb) = pu(Wb)+ (1−p)u(Wg )
Initial wealth is W . Assume that this person can purchase a dollar of wealth in good times for
qg and a dollar of wealth in bad times for qb . 1 The price ratio
qg qb
shows how this person can
trade dollars of wealth in good times for dollars in bad times.
W̃ = qbWb +qg Wg
We say that prices are actuarially fair if the price ratio reflects the odds ratio:
qg qb
= 1−p p
Example 4. Consider the following expected utility maximization problem:
max EU (Wg ,Wb) = p log(Wb)+ (1−p) log(Wg ) subject to W̃ = qbWb +qg Wg
We can use the tangency condition to solve.
slope of indifference curve = slope of budget line 1−p Wg p
Wb
= qg qb
1−p p
Wb Wg
= qg qb
Use the condition that insurance is actuarially fair to simplify, and we get:
Wg =Wb
The individual is willing to pay an indemnity or cover for reduced wealth in the “good state" so
that he can have the same level of wealth in the event of a loss.
Wg = Wb W −qC = W −L−qC +C
where L is the loss, C is the cover, qC is the premium expressed as the the product of the cover
and a premium rate.
1I use q for prices because I don’t want to confuse it with p for probability.
14
14 Overview of Insurance
Insurance occurs when one party agrees to pay an indemnity (a promise to pay for the cost of
possible damage, loss, or injury) to another party in case of the occurrence of a pre-specified
random event generating a loss for the initial risk-bearer.
Definition 18 (Risk transfer). Insurance is the most common form of risk transfer. The shifting of risk is of considerable importance for the functioning of our modern economies.
• Insurance is a particular example of a type of risk-transfer strategy known as hedging. Hedging strategies typically involve entering into contracts whose payoffs are negatively
related to one’s overall wealth or to one component of that wealth. Thus, for example, if
wealth falls, the value of the contract rises, partially offsetting the loss in wealth.
The basic characteristics of all insurance contracts are:
• specified loss events
• losses, L
• cover (indemnity), C
• premium, Q. Q = qC is a common, but not universal, way of expressing the insurance premium.
15 Demand for Insurance
Question: How much insurance would a risk-averse person buy? What is the demand for cover?
Answer: max
C EU = pu(W −L−qC +C )+ (1−p)u(W −qC )
The first order condition is:
dEU
dC = pu′(W −L−qC +C )(1−q)− (1−p)u′(W −qC )q = 0
u′(W −L−qC +C ) u′(W −qC ) =
(1−p)q p(1−q) > 1
15
15.1 Mossin’s Theorem
15.1.1 Actuarially Fair
We say that insurance is actuarially fair if the expected payout of the insurance company just equals the cost of the insurance.
expected payout is probability of loss times the cover = expected cost is the insurance premium pC = qC
p = q
We might expect a competitive insurance market to deliver actuarially fair insurance. In this
case, the first order condition simplifies to:
u′(W −L−qC +C ) = u′(W −qC )
The consumer should fully insure and set the cover equal to the loss C∗ = L (full cover). Mossin’s Theorem states that a risk averse individual offered insurance at a fair premium will always choose full cover.
q = p ⇐⇒ u′(W −qC∗) = u′(W −L−qC∗+C∗) ⇐⇒ C∗ = L
15.1.2 Not Actuarially Fair
Question: What happens if the price of insurance is above the actuarially fair price, i.e. q > p?
u′(W −L−qC +C ) u′(W −qC ) =
(1−p)q p(1−q) > 1
Mossin’s Theorem With a positive loading, the buyer chooses partial cover;
q > p ⇐⇒ u′(W −qC∗) < u′(W −L−qC∗+C∗) ⇐⇒ C∗ < L
With a negative loading the buyer chooses more than full cover;
q < p ⇐⇒ u′(W −qC∗) > u′(W −L−qC∗+C∗) ⇐⇒ C∗ > L
where the last two results follow from the fact that u′(·) is decreasing in wealth, i.e., from risk aversion.
16
15.2 Comparative Statics
From the first order condition, we can in principle solve for the optimal cover as a function of
the exogenous variables of the problem: wealth, the premium rate, the amount of loss, and the
loss probability. The buyer’s demand for cover function can be expressed as:
C∗ =C (L, p,W,Q)
Question: How does the demand for cover change as wealth, the premium rate (price of insur- ance), the amount of loss, and the loss probability change?
• Amount of loss, L, i.e. δC ∗
δL . A ceteris paribus increase in L increases the demand for insur-
ance.
• The loss probability, p, i.e. δC ∗
δp . An increase in the risk of loss increases the demand for
cover.
• Wealth, W , i.e. δC ∗
δW
Proposition 1. If p = q , the agent will insure fully at C∗ = L for all wealth levels. If p < q , the agent’s insurance coverage as a function of wealth, C∗(W ) will decrease (increase) with wealth if the agent has decreasing (increasing) absolute risk aversion.
• Premium rate, Q, i.e. δC ∗
δQ . The total effect on insurance demand depends on the relative
magnitudes of the income and substitution effect.
15.3 Coinsurance and Deductibles
Proposition 2. Under a reasonable set of conditions, the optimal insurance contract always takes the form of a straight deductible.
Under proportional coinsurance we have cover
C =αL, α ∈ [0,1]
with α= 0 implying no insurance and α= 1 implying full cover. Under a deductible we have
C = {
0 for L ≤ D L−D for L > D
where D denotes the deductible and D = 0 implies full cover. Given the premium amount Q and wealth W in the absence of loss, the buyer’s state-contingent
wealth in the case of proportional coinsurance is
Wα =W −L−Q +C =W − (1−α)L−Q
17
and in the case of a deductible is
WD =W −L−Q +C =W −L−Q +max(0,L−D)
For losses above the deductible, her wealth becomes certain, and equal to
ŴD =W −L−Q + (L−D) =W −Q −D
A straight deductible insurance policy efficiently concentrates the effort of indemnification on
only the largest losses.
16 Supply of Insurance
16.1 Risk pooling
When an insurer enters into insurance contracts with a number of individuals, or a group of
individuals agrees mutually to provide insurance to each other, the probability distribution of
the aggregate losses they may suffer differs from the loss distribution facing any one individual.
Assume
• There are i = 1,2, ..., N individuals with identically and independently distributed risks • C̃i is the loss claim for each individual (the cover paid by the insurance company in the
event of a loss)
• µ is the expected claims cost (across the population) andσ2 is the variance of the expected
claims costs
• Each C̃i has the same probability distribution with mean µ and variance σ2
Let C̄N = 1N ∑N
i=1 Ci be the sample mean or average loss per contract (to the insurance com- pany).2
Proposition 3 (By the Law of Large Numbers).
lim N→∞
Pr[|C̄N −µ| < ²] = 1
In words, as N becomes increasingly large, the average loss per contract will be arbitrarily close
to the value µ with probability approaching 1.
2A sample is a (randomly) generated subset of the population under study. The parameters of the population in-
clude its mean, µ, variance,σ2, and its standard deviation,σ. The statistics of the sample include the sample mean
(or average), X̄ = 1N ∑N
i=1 Xi , the (unbiased) sample variance is s 2 = 1N−1
∑N i=1(Xi − X̄ )2, and the sample standard
error is the sample standard deviation divided by the square root of the sample size, i.e. sp N
.
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Stated differently,
for a sufficiently large number of insurance contracts, it is virtually certain that the loss per
contract is just about equal to the mean of the loss claims distribution.
Furthermore, the variance of the realized loss per contract about the mean of loss claims goes
to zero as N becomes increasingly large.
Var(C̄N ) = Var( 1 N
N∑ i=1
Ci ) = 1 N 2
·Nσ2 = σ 2
N
16.2 Risk spreading
When risks are not covered by insurance companies, the government can intercede by transfer-
ring money among parties. The government can spread the risk to increase social welfare.
As a risk is spread over an increasing number of individuals, the total cost of the risk tends to
zero and the price individuals are willing to pay for the risky prospect tends to the expected
value of the project.
Theorem 4 (Arrow-Lind). Under certain assumptions, the social cost of risk moves to zero as the population tends to infinity, so that projects can be evaluated on the basis of expected net
benefit alone. A necessary condition for the results is that the covariance between the individual’s
wealth from the insurance business and his marginal utility of wealth, if he does not share in this
business, must be zero.
The Arrow-Lind Theorem provides a basis for the assumption that the insurer is risk neutral.
16.3 “Undersupply” of Full Insurance
1. Transactions (or insurance) costs include: drawing up and selling new insurance con-
tracts, administering the stock of existing contracts, processing claims, estimating loss
probabilities, calculating premiums, and administering the overall business. The Raviv
model shows how the existence of deductibles and coinsurance in the (equilibrium) in-
surance contract is related to the nature of insurance costs.
2. Nondiversifiable risks cannot be insured.
3. Adverse selection: individuals know their risk better than the insurance company
4. Moral hazard: individuals can take certain actions to reduce the probability of loss
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17 Asymmetric Information
Markets may not be fully efficient when one side has information that the other side does not
(asymmetric information). Carefully designed contracts may reduce such problems by provid-
ing incentives to reveal one’s information and take appropriate actions.
Principal-Agent Model There are two economic agents in this model: the informed party and the uninformed party.
One party will propose a “take it or leave it” (TIOLI) contract and therefore request a “yes or no”
answer; the other party is not free to propose another contract. The principal is the one who
proposes the contract and the agent is the party who just has to accept or reject the contract.
Hidden Type The uninformed party is imperfectly informed of the characteristics of the informed party; the
uninformed party moves first. The agent has private information about the state of the world
before signing the contract with the principal. The agent’s private information is called his type.
For historical reasons stemming from its application in the insurance context, the hidden-type
model is also called an adverse selection model.
Hidden Action The uninformed party moves first and is imperfectly informed of the actions of the informed
party. The agent’s actions taken during the term of the contract affect the principal, but the
principal does not observe these actions directly. The principal may observe outcomes that
are correlated with the agent’s actions but not the actions themselves. For historical reasons
stemming from the insurance context, the hidden-action model is called a moral hazard model.
17.1 Adverse Selection
Adverse selection is defined as the situation where the insured has better information about
her risk type than the insurer. We then say that the individual risk is her private information.
We will consider the Rothschild and Stiglitz (1976) model of adverse selection in competitive
insurance markets.
17.1.1 Basic model
Basic Model
• The individual is risk averse
• Individual is endowed with wealth, W
• In the event of a loss, the individual will have W −L • p is the probability of the loss
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• He can insure himself by paying a premium Q = qC in return for a cover C , if a loss occurs • The pair (Q,C ) completely describes the insurance contract
• Insurance contracts are exclusive: each individual can take on only a single insurance
contract
Demand for Insurance
EU = pu(Wb)+ (1−p)u(Wg ) where u(x) is the utility of money income; u(x) is an increasing concave function. An individual
chooses the insurance contract that maximizes his expected utility.
Supply of Insurance
• Companies are risk neutral and are concerned only about expected profits:
π=Q −pC
• A perfectly competitive market ⇒ zero economic profits • Zero administrative costs
• Free entry
• Each firm can offer only one contract
Equilibrium in a competitive insurance market is a set of contracts such that when individuals choose contracts to maximize expected utility
1. No contract in the equilibrium set makes negative expected profits
2. No contract outside the equilibrium set that, if offered, will make a nonnegative profit
Every firm makes zero profits and no firm (existing or new) can make positive profits by offering
a new contract.
17.1.2 Tangency Condition
Budget Line Final wealth in the two states of the world are
W̃ = {
Wg =W −qC in “good” state Wb =W −L+ (1−q)C in “bad” state
To find the budget line, multiply Wg by (1−q) and Wb by (q). Then add the two equations.
(1−q)Wg +qWb = W −qC −qW +q2C +qW −qL+qC −q2C = (1−q)W +q(W −L) = W −qL
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Solve for Wb and you’ll get
Wb = W −qL
q − 1−q
q Wg
This is a straight line passing through the point (W,WL), i.e. the no insurance point, and having
a negative slope equal to 1−qq in absolute value.
Marginal Rate of Substitution (MRS) Recall from microeconomics that the marginal rate of substitution is the slope of the indiffer-
ence curve, i.e. MRS = − x1x2 . It describes how much x2 a person is willing to give up in order to get more x1 and remain indifferent between the two consumption bundles. For example, if
MRS = 5 then the consumer is willing to give up 5 units of x2 to get one unit of x1. The MRS is also equal to the ratio of the marginal utilities.
From the expected utility maximization function, we find that
MRS = MUW g MUW b
= (1−p) p
u′(Wg ) u′(Wb)
17.1.3 Two types of consumers, symmetric information
Suppose that the market consists of two kinds of customers:
• low risk individuals with loss probability, pL • high risk individuals with loss probability, pH • Note that 1 > pH > pL > 0
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The MRS for each type is
MRSL = (1−pL) pL
u′(Wg ) u′(Wb)
andMRSH = (1−pH ) pH
u′(Wg ) u′(Wb)
The slope of the indifference curve of low risks is steeper than that of high risks.
In the first-best, symmetric information case, the insurance company can observe the individ-
ual’s risk type and offer a different policy to each. In the competitive market, each type can get
a separate contract with an actuarially fair premium and chooses full coverage.
17.1.4 Two types of consumers, asymmetric information
Question: What happens when the individual has private (not observable or verifiable) infor- mation about his type?
Intuition: If the same full insurance contracts for each group were offered, but types are not observable, then all individuals would choose the low type insurance contract. This could lead
to negative profits for the firm. Why? Insurers break even serving only the low-risk types, so
adding individuals with a higher probability of loss would push the company below the break-
even point. Therefore, we cannot offer full insurance to both types.
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There are two types of equilibria to consider: pooling and separating.
Definition 19 (Pooling equilibrium). Pooling equilibrium in a competitive screening model is an equilibrium where each type of agent chooses the same contract.
Definition 20 (Separating equilibrium). A separating equilibrium is a competitive screening model is where different types purchase different contracts.
17.1.5 Pooling Equilibrium
Proposition 4. There cannot be a pooling equilibrium.
Intuition: The pooling equilibrium cannot be a final equilibrium because there exist insurance policies that are unattractive to high-risk types, attractive to low-risk types, and profitable to in-
surers. These policies will involve “cream-skimming” behavior: the policies will attract low-risk
types away from the pooling contract. The insurers that continue to offer the pooling contract
are left with individuals whose risk is so high that insurers cannot expect to earn any profit by
serving them.
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17.1.6 Separating Equilibrium
Proposition 5. The separating equilibrium will involve actuarially fair full insurance for the high risk types and low risk individuals will be offered the best possible partial insurance con-
tract at a fair price, conditional on that contract being unattractive to high-risk individuals.
Definition 21 (Incentive compatibility constraints). The incentive compatibility (IC) constraints state that each consumer prefers the contract that was designed for him.
Intuition: We need to consider incentive compatibility constraints. There is no reason to distort the choice of insurance for the high-risk types, because low risk individuals do not have any
incentive to “pretend” to be high risk. But we need to make sure the high risk types don’t pretend
to be low risk types. The incentive compatibility constraint for the high type requires that the
contract designed for the low risk type be below or on the indifference curve of the high risk
type that goes through the full insurance contract.
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Existence of a separating equilibrium: “An equilibrium will not exist if the costs to the low-risk individual of pooling are low (because
there are relatively few of the high-risk individuals who have to be subsidized, or because the
subsidy per individual is low, i.e. when the probabilities of the two groups are not too different),
or if their costs of separating are high” (Rothschild and Stiglitz, 1976).
17.2 Moral Hazard
In the moral hazard model of insurance, the probability of the loss state may depend on the
behavior of the insured individual. This creates an incentive problem that leads to less than full
insurance, so that the insured retains some incentive to behave differently.
Suppose
• a risk-averse individual faces the possibility of incurring a loss, L, that will reduce his
wealth, W
• the probability of loss is p and is a decreasing convex function of effort, e (or level of care)
• exerting effort is costly, i.e. c(e) in an increasing function in effort; let c(e) = e (The insur- ance company cannot monitor the individual’s level of effort, e).
• u(x) is the individual’s utility given wealth
The individual’s expected utility as a function of the effort or level of care chosen is
EU = p(e)u(Wb)+ (1−p(e))u(Wg ) = p(e)u(W −e −Q −L+C )+ (1−p(e))u(W −e −Q)
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The expected profit of the (risk-neutral) insurance company is
π=Q −p(e)C
An actuarially fair insurance contract would set a premium equal to the expected coverage, i.e.
Q = p(e)C . The timing is as follows:
• The principal offers an insurance contract (Q,C )
• The individual decides to accept or reject the contract
• The individual chooses an effort level, e
Definition 22 (Participation Constraint). The participation, or individually rational (IR), con- straint guarantees that the consumer will accept the contract. The individual must be at least
as well off as he would be if he accepted the next best alternative. (No insurance may be the
next best alternative).
In our setting, the optimal contract must
• satisfy the zero-profit constraint (the IR constraint for the firm)
• satisfy the IR or participation constraint for the individual
• ensure that the effort level in the contract is credible in the sense that it will be chosen by
the agent under the incentives provided by the rest of the contract.
Part IV
After the Midterm
17.3 Insurance (cont.)
18 The Value of Information
19 Options
19.1 Financial Options
19.2 Real Options
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