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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

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lecture 5 arrow debreu economy
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

arrow debreu assets
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
  • Agents don’t know which state will occur (in a future time) but they know state price given their expectations.
  • 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.

L5: Arrow-Debreu Securities

complete market
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

deriving pure security prices
Deriving Pure Security Prices

Security state 1 state 2 security price

i $10 $20 pj=$8

j $30 $10 pk=$9

Compute the prices for two pure securities.

See page 80, CWS.

L3: Arrow-Debreu Securities

determinants of the price of a pure security
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

slide6

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 Ps, s= 0, 1. P0<1+r<P1
  • We can Replicate 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

ad example w two states
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 Πs in slide 6 is same as ps in slide 5; ps refers to probability of state s in EGS.

L5: Arrow-Debreu Securities

option and ad securities
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
  • 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 P1 – 1 has the same payoff as an AD security
    • Buying a call option with an exercise price equal to P1 – 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)

L5: Arrow-Debreu Securities

s of pure security vs security price p
Пs 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

risk neutral probability and p
Risk Neutral Probability and P

Solving exercise 5.1

L3: Arrow-Debreu Securities

optimal portfolio of ad assets
Optimal Portfolio of AD Assets
  • Go through the example in CWS (page 84-88)
  • Let csdenote the investment in AD asset with state s

L3: Arrow-Debreu Securities

when s stay constant across states
When πs stay constant across states
  • If πs (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: cs=w(1+r)

L3: Arrow-Debreu Securities

when s differs across states
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

graphic illustration of the s 2 case
Graphic Illustration of the s=2 Case

L3: Arrow-Debreu Securities

analogy to intertemporal consumption decisions
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

then with the capital market
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

examples exercise 5 4 a
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?

L5: Arrow-Debreu Securities

implications of arrow debreu securities
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

exercises
Exercises
  • EGS, 5.2
  • CWS, 4.4

L3: Arrow-Debreu Securities