Presentation on theme: "L5: Arrow-Debreu Securities 1 Lecture 5: Arrow-Debreu Economy The following topics are covered: –Arrow-Debreu securities –Pricing Arrow-Debreu securities."— Presentation transcript:
L5: Arrow-Debreu Securities 1 Lecture 5: Arrow-Debreu Economy The following topics are covered: –Arrow-Debreu securities –Pricing 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
L5: 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 П s 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 Agents don’t know which state will occur (in a future time) but they know state price given their expectations. State price П s can be decomposed into the probability of the state, p s, 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), π s.
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 Example: for the case of 3 states of nature, is the market complete with 3 assets: (1,1,1), (1,0,0), and (0,1,1)? How about adding an additional asset (0,1,3)? Given a complete security market, we could theoretically reduce the uncertainty about our future wealth to zero – page 78 CWS.
L3: Arrow-Debreu Securities 4 Deriving Pure Security Prices Security state 1 state 2security price i $10 $20 p j =$8 j $30 $10 p k =$9 Compute the prices for two pure securities. See page 80, CWS.
L3: Arrow-Debreu Securities 5 Determinants of the Price of a Pure Security Time preferences for consumption and the productivity of capital (p82 CWS) Expectations as to the probability that a particular state will occur given homogeneous belief: Individuals’ attitudes towards risk (e.g., risk aversion), given the variability across states of aggregate end-of-period wealth –Pure security prices must be adjusted to make the state 1 security relatively expensive and the state 3 security relatively cheap (page 83). –Equation on page 84.
L3: Arrow-Debreu Securities 6 An Example of Two States Two assets, (a) risk free bond: with r in both states; (b) risky asset initial price of the risky asset is 1 final value is P s, s= 0, 1. P 0 <1+r
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"description": "P 0 <1+r
L5: Arrow-Debreu Securities 7 AD example w/ two states Which state would be more expensive? What decides the cost of the state contingent security? Try the sum of state prices 0 and 1. What do you get? Note Π s in slide 6 is same as p s in slide 5; p s refers to probability of state s in EGS.
L5: Arrow-Debreu Securities 8 Option and AD Securities –Assuming P 0 < P 1 – 1, then the AD asset associated with state s=1 is a call option with strike price P 1 – 1 –Premium of the call option is the price for pure security of state 1 –In other words, buying a call option with an exercise price equal to P 1 – 1 has the same payoff as an AD security Buying a call option with an exercise price equal to P 1 – 1 has the same payoff as the mutual fund portfolio (long alpha share of stocks and borrow money – you can view this portfolio as a synthetic call option) –Exercise 5.4 (b)
L3: Arrow-Debreu Securities 9 П s of Pure Security 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 (see slide 4). In other words, it represents a particular investment choice in AD assets. A particular example is the risk-free asset (this replicates the first point of slide 5) –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 10 Risk Neutral Probability and P Solving exercise 5.1
L3: Arrow-Debreu Securities 11 Optimal Portfolio of AD Assets Go through the example in CWS (page 84-88) Let c s denote the investment in AD asset with state s
L3: Arrow-Debreu Securities 12 When π s stay constant across states If π s (state price per unit of of probability) stays 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: c s =w(1+r)
L3: Arrow-Debreu Securities 13 When π s differs across states When π s differs across states, then We have different level of consumption of each value of π s Note c s 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 14 Graphic Illustration of the s=2 Case
L3: Arrow-Debreu Securities 15 Analogy to Intertemporal Consumption Decisions The idea here is consistent with the choice between consumption and investment discussed in CW Chapter 1. It incorporates: (1)Utility function (2)Indifference curve (3)Maximization under constraint – a decreasing return investment function only; i.e., consumption and investment without capital market (1)Consumption is about the choice between consuming now and the future (2)Investment is about choosing optimal investment return, which affects consumption pattern (3)Here investment and consumption decisions are not separatable
L3: Arrow-Debreu Securities 16 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 (C 0, C 1 ) based on each individual’s utility function (indifference curve) –Less risk averse individual will consume more today –Transaction costs break down the separation theorem
L5: Arrow-Debreu Securities 17 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 18 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