Risk, Insurance and Information Lecture Notes for Financial Economics II
University of Adelaide
This version : February, 2022
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Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
1 Introduction 1
2 Expected Utility 3
2.1 RandomOutcomesandLotteries …………………. 3
2.1.1 TheExpectedUtilityHypothesis ……………… 6
2.1.2 The Theory of von Neumann and Morgenstern (Optional) . . . . 8
Preferences Over Non-Stochastic Alternatives and Utility Functions 8
Preferences Over Lotteries and Expected Utility Functions . . . . 10
2.2 Summary …………………………….. 12
3 Risk Aversion 13
3.1 AnIntroductiontoRandomVariables……………….. 13 3.1.1 DiscreteRandomVariables…………………. 13 3.1.2 ContinuousRandomVariables……………….. 16
3.2 RiskAversion…………………………… 19 3.2.1 ConcavityofUtilityFunctions……………….. 20
3.3 RiskAversion…………………………… 22 3.3.1 MeasuresofRisk-Aversion…………………. 23
Deriving the CARA and CRRA measures of risk (Optional Leisure Reading) …………………….. 24
3.4 Summary …………………………….. 30
4 Consumption Under Risk 31
4.1 ATwo-PeriodProblem………………………. 32 4.1.1 LinearUtility ……………………….. 35 4.1.2 ConcaveUtility ………………………. 37
4.2 ConsumptionandSavingsUnderUncertainty . . . . . . . . . . . . . . . 39
4.2.1 ATwo-PeriodProblemwithIncomeRisk. . . . . . . . . . . . . . 39 State-ContingentClaims………………….. 40 ClosingtheModel …………………….. 43 Non-Contingent Savings: Partial Insurance . . . . . . . . . . . . . 43
4.2.2 ThePrecautionarySavingsMotive …………….. 45 4.3 ChapterSummary ………………………… 48
5 Asset Pricing 50
5.1 ASimpleModelofExchange……………………. 50
5.1.1 A Two Individual Endowment Economy With Trade in a Spot Market 50
5.1.2 Problems …………………………. 53
5.1.3 A Two Individual Endowment Economy With Trade in a Market
forClaims…………………………. 53
5.2 EconomicExchangeinanEconomywithRisk . . . . . . . . . . . . . . . 58
5.3 TheConsumptionCAPM …………………….. 61
5.4 Summary …………………………….. 67
6 Adverse Selection in Markets 69
6.1 TheMarketsforLemons ……………………… 69
6.2 TheMarketforLemonsProblem …………………. 70
6.2.1 TheModel…………………………. 70 TheSeller’sProblem:……………………. 72 TheBuyer’sProblem: …………………… 75 ACompetitiveEquilibrium ………………… 76
6.2.2 FullInformationCase …………………… 77
6.3 Summarizing:MarketforLemons…………………. 78
6.4 Adverse Selection in Competitive Insurance Markets . . . . . . . . . . . . 79
6.4.1 Markets with Divisible Insurance Contracts . . . . . . . . . . . . 79 DistinctInsuranceMarkets…………………. 79
6.4.2 Single-Priced Insurance Markets and the Adverse Selection Problem 84
6.4.3 Competitive Markets in Insurance Contracts . . . . . . . . . . . . PoolingEquilibria……………………… SeparatingEquilibria…………………….
6.5 Summarizing ……………………………
7 Herd Behaviour
7.1 BinaryLearningModels ……………………… 7.1.1 BinaryLearningwithBinarySignals …………….
7.2 TheFullInformationModel …………………….
7.3 TheHerdBehaviourModel ……………………. 7.3.1 BeliefDynamics……………………….
7.4 Summary ……………………………..
8 Learning From Prices and Aggregates
8.1 TheSignal-ExtractionProblem ………………….. 8.2 A Simple Quadratic Model with Costly Information Acquisition . . . . . 8.3 Summarizing ……………………………
89 91 93 94
102 103 107
108 110 116
A A Short Note on Differentiation, Differentials and Maximization 117
A.1 ConcavityofaFunctionanditsDerivatives . . . . . . . . . . . . . . . . 117 A.2 SomeRulesofDifferentiation …………………… 120 A.3 MultivariateFunctionsandPartialDerivatives . . . . . . . . . . . . . . . 121 A.4 QuickReviewofTotalDifferentials ………………… 122 A.5 EqualityConstrainedOptimization ………………… 123 A.6 InequalityConstrainedOptimization ……………….. 126
Bibliography
Chapter 1 Introduction
The purpose of this course and these lecture notes is for the student to build an understanding of why and how risk averse individuals use financial instruments to hedge payoff risk, to provide a theory about how asset prices are determined in a competitive environment and how informational asymmetries can result in problems in economic markets.
These notes lean on mathematics to ensure that the stories being told are internally consistent. Towards this end, students are encouraged to spend significant time work- ing through the mathematics in each chapter if only to gain an appreciation of how economists use mathematics as a language to tell their stories just as a fiction writer uses English, French, Chinese, Japanese, etc. The main mathematical tool that is used in these notes is constrained optimization. For those who require an introduction, please consult Appendix A. There the most basic explanation is provided for the use of the Method of Lagrange to solve equality constrained optimization problems and the Method of Kuhn-Tucker (which is only used once in these lecture notes for something that ends up being shown diagramatically) for inequality constrained optimization problems. Two ex- cellent mathematics references for economics students are Chiang and Wainwright (2005) and Simon and Blume (1994). Both should be accessible for the undergraduate student with Simon and Blume’s textbook being more comprehensive.
It is important to acknowledge that there are many competing views regarding how individuals behave in making choices, how individuals interact in markets to determine the prices and allocations that are observed, how information is processed by individuals, etc. As such, it is not the objective of these lecture notes to push one view rather than others. Instead, the material here can be viewed as a launching point for the interested student to explore other theories and world views.
Finally, these notes are meant to complement lecture material and vice versa – neither is a perfect substitute for the other. In order to use these lecture notes as they are intended, the advice to students is as follows. First, read the assigned sections once quickly to identify the main punchline. Second, with pencil and paper in hand, work through the mathematics of the sections ensuring that you are able to move from one
line of math to the next. Finally, without the aid of the lecture notes, rewrite the punchline of the sections and try to explain exactly how that result is obtained given the assumptions of the models being used. This last step means that you understand exactly how the model works to deliver its economic story. You are best working through the readings on your own and then consulting with a group of study buddies to see what you might have missed. It is likely that you will learn more working with peers than without so make an effort to find at least one other person with whom to study and learn.
Have fun reading and learn something!
Chapter 2 Expected Utility
Much of the financial economics is related to mitigating risk faced by individual decision makers. The purpose of this chapter is to provide a simple framework that will enable the study of individual decision making in the face of risky outcomes. To begin, we will describe the expected utility hypothesis which is a standard representation of capturing the manner in which individuals value different bundles of risky outcomes. Once we are able to describe the manner in which an individual decision maker assigns value to different bundles of risky outcomes (which can be referred to as lotteries), we will then be able to formalize what we mean when referring to an individual as “risk-averse” or “risk-loving”. This will be sufficient for us to move forward to studying the economic topics that are the focus of this course.
2.1 Random Outcomes and Lotteries
While not the only representation of describing how individuals value risky alterna- tives, the expected utility framework has become the standard modelling framework to which alternative modelling frameworks can be compared. Thus, even if you end up being skeptical of the results that obtain using the expected utility framework, when faced with other frameworks for decribing valuation of uncertain alternatives, you will have a solid understanding of why such alternative approaches might generate different outcomes.
In order to describe how individuals will value lotteries over potential outcomes, let’s first remind ourselves about how we apply probabilities to calculate expected values of a random variable. Suppose we have a set of events or, alternatively, a set of states- of-nature Ω = {ω1,ω2,…,ωN}. The set ω consists of N events that can be chosen by nature.1 Each event is denoted by ωi where variation in the subscript i captures the idea that there can be different outcomes across these distinct events or states-of-nature.
Let π denote a probability function and use the notation π(si) to denote the proba- bility that the event si is chosen by nature for any i ∈ {1,…,N}.2. When the notation is
1The term states-of-nature is often replaced with the term “states-of-the-world”.
2The character ∈ means “in the set” so that “for any i ∈ {1,…,N}” translates to “for any value i
obvious, it might be the case that we replace π(si) with πi. As π represents a probability function over a finite set of possible outcomes, it must be the case that if we sum across all the possible outcomes, i ∈ {1,…,N}, Ni=1 πi = π1 + π2 + π3 + … + πN = 1, that is, summing across the probabilities of all possible outcomes must equal one.
Now suppose that we define an act, a(·), as a mapping from the possible states- of-nature and its associated probability distribution, (Ω, π), into a set of possible conse- quences C, a : (Ω, π) → C. For most of our applications, the set of possible consequences, C, will be the set of real numbers so we can use the notation, C = R when C = (−∞,∞) or C = R+ when the set of consequences is given by the set of non-negative real numbers.
By defining a state-space (the set of possible states-of-nature), an associated proba- bility function (a mapping from states-of-nature to numbers on the interval [0,1] such that Ni=1 πi = 1) and an “act” that maps states-of-nature into consequences define in the set of feasible consequences C, we have the foundations upon which to construct lotteries.
Definition 1 A lottery is an act that maps random realizations of states-of-nature into a set of consequences.
Example: As an example, consider a lottery that results from flipping a coin that has the image of a head on one side of the coin and a tail on the other side. In this example, the set of states-of-nature is given by Ω = {H, T } where “H” denotes the outcome that nature reveals that the coinflip results in a “head” and “T” represents the outcome that the coinflip results in a “tail” being revealed. The associated probabilities for outcomes under this random coinflip are πH = 1/2 and πL = 1/2.3 Define the set of consequences as the set of non-negative real numbers, C = [0,∞), so that the set of consequences includes the value zero (squared bracket means includes the value while curved bracket means not including the value).
An particular example of a lottery is then given by the act, $100 ifω=H
a(ω) = $0 if ω = T.
an alternative lottery can be constructed by assigning different monetary outcomes to
Head and Tail:
$10 ifω=H a ̃(ω)= $5 ifω=T.
Task: Construct a monetary lottery that is defined on a state-space consisting of three possible outcomes. In other words, construct a lottery that has three possible monetary
can take in the set {1,…,N}.
3Alternatively we can write Ω = {ω1,ω2} and assign values ω1 = H and ω2 = T. Then we would
have written the associated probabiities of this coinflip draw as π1 = 1/2 and π2 = 1/2. As long as you are clear with your use of notation, either way is fine.
outcomes which depends on the realization of a random variable that can only take on three possible values.
Now we can build upon our definition of a lottery to model the way in which an individual assigns value to any particular lottery. Suppose that the individual derives utility (happiness or felicity) from a consequence as given by a utility function u(c) where c comes from the set of consequences C, written mathematically as c ∈ C. Then the we can model the value that an individual assigns to a specific lottery, a, as the expected utility that the individual would obtain from such a lottery. Consider a state-space that has N possible states-of-nature, Ω = {ω1, ω2, ω3, …, ωN }. Associated with this is a set of probabilities detailing the likelihood of that each particular state-of-nature will occur, π = {π(ω1), π(ω2), π(ω3), …, π(ωN )}. Then define the “expected utility” that the individual assigns to a particular lottery, before knowing the outcome that will arise, to be
V (a) = π(ω1)u(c(ω1)) + π(ω2)u(c(ω2)) + π(ω3)u(c(ω3)) + … + π(ωN )u(c(ωN )) N
= π(ωi)u(c(ωi)). (2.1.1) i=1
The sum on the righthand side of equation (2.1.1) is comprised of N individual terms, π(ωi)u(c(ωi), i ∈ {1,…,N}. How is each of these terms interpreted? Well, c(ωi) is the consequence enjoyed by the individual when ωi is drawn by Nature from the set of possible states-of-nature Ω. The happiness that the individual enjoys from consequence c(ωi) is u(c(ωi)). As we are assigning ex ante value to the lottery, we know that the individual obtains the happiness u(c(ωi)) with probability π(ωi). So the probability weighted utility (or return) from state-of-nature ωi is π(ωi)u(c(ωi)). Summing probability weighted utility terms across all potential states-of-nature gives use the expected utility from the lottery – it is the expected payoff that the individual obtains from facing the lottery prior to having Nature revealing the outcome of the lottery.
It turns out that it is possible to find utiity functions that, when paired with a set of consequences, can result in ill-defined expected utilities from lotteries. As an example, consider a utility function u(c) = c. This says that if the individual receives c units of consumption goods then the utility is simply c units of happiness. Note that in this example the utility function converts measured units of consumption into units of happiness. Now consider the following game. A coin is flipped repeatedly. Each time it is flipped, the individual checks to see if a head is revealed. Once a head is revealed, the coin is no longer flipped – the game is complete. If it takes n flips for the coin to reveal a head for the first time, the individual receives 2n units of the consumption good.
Notice that the probability that a head occurs on the first toss is π1 = 21 . For the first
head to be revealed on the second toss, it requires the first toss to reveal a tail and the
second toss to reveal a head. The probability of such an outcome is π2 = 12 because 2
there is a probability 12 that a tail is revealed on the first toss and a probability 12 that a head is revealed on the second toss. Thus the probability of a tail on the first toss and a head on the second toss is 12 × 12. Continuing along this line of calculation, the
probability that the first head occurs on the nth coin flip is 1n−1 (the probability of 2
n − 1 consecutive flips revealing tails) times the probability that the nth flip reveals head – which is 12.
Prior to flipping the coin for the first time, we want to know how this individual would value such a lottery. We know that if the coin is flipped n times and no head has been revealed then the utility up to the nth flip is zero because the game continues and no consumption goods have been paid up to and including the nth flip. Ergo, in order to construct the expected utility of this lottery, we need only multiply the payoffs if the successful flips occur on the nth trial by the probability of the first head occuring on trial n and then sum across all the possible values of n. To do this calculation, first imagine that there is a maximum number of flips of N. If no head has been revealed by the Nth coin flip then no payment is made to the individual – the individual is a loser. Now consider pushing N larger and larger towards its upper bound of infinity. Doing so we have
πnu(cn) n=1
= 2 2+ 2 2 2 + 2
1n−1 1 n
+ 2 2 2 +…
1 11 2 12 1 3 13 1 4
2 22 = 2+ 2
23 24 + 2 + 2
= 1+1+1+1+…
One can see that in this problem that the expected utility is unbounded so confronted with such an offer, a rational individual should be willing to pay a large sum of wealth to obtain the ability to play this game. In what has become known as the St. Petersburg paradox, it is argued that no sane individual would pay an enormous sum of money to play such a game.
A way to sidestep this problem is to not that the infinite expected value of this sum arises because the reward of winning in each stage grows at a rate that is equal to or larger than the rate at which the probability of obtaining the reward at each successive state of nature decreases. If the payoff in any state-of-nature is bounded from above, then the expected utility will be finite.
Now we are able to state the framework for decision making under risk that we will use for the bulk of this course.
2.1.1 The Expected Utility Hypothesis
Consider the problem of an individual who is faced with a set of lotteries A = {a1, a2, …, al}. Each of these lotteries is a mapping from a set of states-of-nature and its associated probability function, into a set of consequences. The realized consequences
2 2 + 2 2n
+…+ 2 +…
ω Lottery A Lottery B ωL 0 20
Table 2.1: Consumption Payoffs Under Different Lotteries
are valued by a payoff or utility function. The individual’s problem is to select the most valuable lottery. Recall that an act assigns a consequence to each possible state-of-nature c(ωi) = a(ωi) and each state-of-nature is assigned a probability of occuring π(ωi). The rational individual understands the probability associated with each possible state-of- nature and also understands, for each available lottery, which consequence is assigned to each possible state-of-nature by the act.
The expected utility hypothesis states that the rational behaviour of an individual who is confronted with such a problem is to choose the lottery that provides the maximum expected utility from all the available lotteries,
max πiu(ci)
subject to the consequences ci = a(ωi) depending on the act that maps states-of-nature into consequences. This can be written as
max π(ωi)u(a(ωi)).
Example: Consider an individual who values consumption of a good in the amount c by a utility function cα. Let α = 21. Our individual is faced with the choice amongst two lotteries. There are three possible states-of-nature, Ω = {ωL, ωM , ωH }. Nature draws ωL and ωH with equal probabilites of πL = πH = 14 and draws ωM with probability 12 .
There are two lotteries from which the individual is charged with selecting one of the options. Table 2.1.1 displays the consequences attached to each state-of-nature by Lotteries A and B.
In order to determine which lottery the individual optimally selects, we must calculate the expected utility from each lottery and then compare the expected utilities across all available lotteries to find out which yields the highest expected utility. We use E[·] to denote the mathematical expectation of the arguments inside the square brackets and ci,j to denote consumptio
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