Lecture 3: Arrow-Debreu Economy

1. Lecture 3: Arrow-Debreu Economy • The following topics are covered: • Arrow-Debreu securities • Optimal portfolios of Arrow Debreu securities • How Arrow-Debreu securities differ framework differs from the standard utility maximization? • Implications in asset pricing • Example L3: Arrow-Debreu Securities

2. Arrow-Debreu Assets • There are S possible states indexed by s=0, 1, …, S-1. • A pure security (or say an Arrow-Debreu asset) stays in each state, paying $1 if a given state occurs and nothing if any other state occurs at the end of the period • Let Пs denote the price of the Arrow-Debreu security associated with s, i.e., the price to be paid to obtain one monetary unit if state s occurs • State price Пs can be decomposed into the probability of the state, ps, and the price of an expected dollar contingent on state s occurring (or say the state price per unit of probability of associated contingent claims), πs. L3: Arrow-Debreu Securities

3. Complete Market • The market is considered to be complete when investors can structure any set of state-contingent claims by investing in the appropriate portfolio of Arrow-Debreu securities • In other words, (1) there are enough independent assets to “span” the entire set of all possible risk exposures; (2) The market will be complete if there are at least as many assets who vectors of state-contingent payoffs are linearly independent as there are number of states • Can we say the market is complete? • Applying this idea, we have the following example L3: Arrow-Debreu Securities

4. An Example of Two State • Two assets, (a) risk free bond: with r in both states; (b) risky asset: final value is Ps, s= 0, 1. initial price of the risky asset is 1 • P0<1+r<P1 • Replicating the Arrow-Debreu security associated with state s=1 by purchasing alpha units of the risky asset and by borrowing B at the risk-free rate, in such a way that L3: Arrow-Debreu Securities

5. Option and AD Securities • Assuming P0 < P1 – 1, then the AD asset associated with state s=1 is a call option with strike price P1 – 1 • In other words, buying a call option with an exercise equal to P1 – 1 has the same payoff as an AD security • Why? • Exercise 5.4 (b) L3: Arrow-Debreu Securities

6. Пs vs Security Price P • Pure assets can be replicated by market security • On the other hand, each market security may be considered as a specific set of payoff combination of AD assets. In other words, it represents a particular investment choice in AD assets. • A particular example is the risk-free asset • It has a payoff of 1 in each state of nature at the end of the period • We have the following: L3: Arrow-Debreu Securities

7. Risk Neutral Probability and P Solving exercise 5.1 L3: Arrow-Debreu Securities

8. Optimal Portfolio of AD Assets • Go through the example in CW • Let csdenote the investment in AD asset with state s L3: Arrow-Debreu Securities

9. When πs stay constant across states • If πsstays the same across all states, then it is optimal for the agent to purchase the same quantity of these claims. • It can be shown that • Thus the above condition is equivalent to the case that a risk is actuarially priced, we need to purchase full insurance on it • The asset is a risk free asset: cs=w(1+r) L3: Arrow-Debreu Securities

10. When πs differs across states • When πs differs across states, then • We have different level of consumption of each value of πs • Note cs is what we want to solve for • Figure 5.1 • Exercise 5.2 • Key: the consumption curve changes from 2 to 1 L3: Arrow-Debreu Securities

11. Graphic Illustration of the s=2 Case L3: Arrow-Debreu Securities

12. Analogy to Intertemporal Consumption Decisions • The idea here is consistent with the choice between consumption and investment discussed in CW Chapter 1. It incorporates: • Utility function • Indifference curve • Maximization under constraint – a decreasing return investment function only; i.e., consumption and investment without capital market • Consumption is about the choice between consuming now and the future • Investment is about choosing optimal investment return, which affects consumption pattern • Here investment and consumption decisions are not separatable L3: Arrow-Debreu Securities

13. Then with the Capital Market • Fisher separation theorem: Given perfect and complete capital markets, the production decision is governed solely by an objective market criterion (represented by maximizing attained wealth) without regard to individuals’ subjective preferences which enter into their consumption decisions • Choose optimal production first, • Choose optimal consumption pattern (C0, C1) based on each individual’s utility function (indifference curve) • Less risk averse individual will consume more today • Transaction costs break down the separation theorem L3: Arrow-Debreu Securities

14. Examples: Exercise 5.4(a) E5.4: Three assets: asset A (2, 5, 7); asset B (2, 4, 4); asset C (1, 0, 2) How to construct AD in state 1? How to construct a risk-free asset? L3: Arrow-Debreu Securities

15. Implications of Arrow Debreu Securities • A means to model uncertainty • About consumption in different states • The same idea can be applied to the consumption over time • Allocate wealth over time versus allocate wealth across states • Easy to achieve any payoff, such as option payoffs L3: Arrow-Debreu Securities

16. Exercises • EGS, 5.2 L3: Arrow-Debreu Securities