Complete markets and Arrow-Debreu assets

State prices Equilibrium Risk neutral probabilities Optimal risk sharing Incomplete markets Default probabilities

Complete markets and Arrow-Debreu assets

Pierre [email protected]

July 6, 2011


States of the world

I A state of the world is defined as a given set of values for thestates variables, which describe the state of the economy,commodity prices, unemployment, natural disasters, etc.Denote by S the set of possible (future) states.

I In a perfect world, an individual could specify his consumptionin any possible state of the world. He would have astate-contingent consumption plan. Under which conditionscan he do this (at least in principle) with financial assets?

I Definition: Market are complete if any consumption plancan be spanned by an investment strategy.

I Markets are complete iff there exists S assets with linearlyindependent payoffs.


Arrow-Debreu assetsContingent claims

I An implication of a world of complete markets is that there isno need to consider existing financial assets. Consideringconsumption and states of the world suffices!

I An Arrow-Debreu asset (also known as a contingent claim)pays off one unit of the consumption good (the numeraire) ina given state of the world s, and in this state only.

I If markets are complete, we can suppose that investors cantrade contingent claims, because the existing assets span orsynthesize all contingent claims.

I This is not meant to be realistic. But if markets are complete,it is equivalent to work with Arrow Debreu assets or withexisting financial assets, and the former is easier! All theresults will be the same.

I Suppose that a riskfree asset does not exist. Are marketsincomplete?


State pricesI The price of the Arrow-Debreu asset which pays off 1 in state

s is called the state price, and is denoted by q(s).I Proposition: In complete markets without arbitrage

opportunities, the state price vector exists and it is unique.I Investors trade until MRS are equal across agents state by state

(if not, there are opportunities for mutually beneficial trades).I In incomplete markets, several different state price vectors can

be obtained in equilibrium (even in the absence of arbitrageopportunities).

I The investor portfolio is described by the number ofArrow-Debreu assets purchased, for any s ∈ S .

I We assume for simplicity that the investor does not have anyother contingent income, so that his consumption plan is fullydescribed by his portfolio.

I The number of state-s Arrow-Debreu assets purchased andconsumption in state s, which we denote by c(s), thereforecoincide.


Arrow-Debreu equilibriumComplete markets equilibrium

I Each agent i ∈ I receives a state-contingent endowmente i (1), e i (2), . . . , e i (S).

I Definition: An Arrow-Debreu equilibrium is a vector ofstate prices q(s)s={1,...,S} and consumption plans

{c(s)i}i={1,...,I},s={1,...,S} such thatI The budget constraint of each agent is satisfied (in

equilibrium):

c i ∈ B i (q(s)s={1,...,S}) ={

c ∈ RS+s.t.

∑s

q(s)c(s) =∑

s

q(s)e i (s)}

I Agents maximize their expected utility: U(c i ) ≥ U(c) for anyc ∈ B i (q(s)s={1,...,S}).

I All markets clear:∑

i ci (s) =

∑i e

i (s) for any s.


Asset pricing with contingent claims

I Consider any asset with next period payoff described by thevector x .

I This asset can be viewed as a package of contingent claims.

I Its price is equal to the sum of the prices of these contingentclaims:

p(x) =∑

s

q(s)x(s)


Risk-neutral probabilities

I Denote by π(s) the probability of the state of nature s, andby c(s) consumption in this state. Then

∑s π(s) = 1.

I Denote by q(s) is the price at t = 0 of one unit ofconsumption in state s. It is the state price.

I Define

q ≡∑

s

q(s) =1

1 + rf

I Define the risk-neutral probability as

π?(s) ≡ q(s)

q(1)


Asset pricing with risk neutral probabilities

I Consider any asset with next period payoff described by thevector x .

I Let’s price this asset using risk neutral probabilities instead ofstate prices:

p(x) =∑

s

q(s)x(s) =∑

s

π?(s)q x(s)

=∑

s

π?(s)x(s)

1 + rf=

E ?[x ]

1 + rf


The link between the SDF, risk-neutral probabilities, andstate prices

I The usual first-order conditions give the state prices:

q(s) = βπ(s)u′(c(s))

u′(c0)= π(s)m(s) (2)

I The relation between risk-neutral probabilities and the SDF m:

π?(s)

(1 + rf )= π(s)m(s) π?(s) =

m(s)

E [m(s)]π(s)

I Risk-neutral probabilities do not include time discounting,while the SDF does not include (physical) probabilities.

I The state price accounts for time discounting, physicalprobabilities, and risk aversion.

I q(s)π(s) , which is equal to the SDF (see equation (2)), is the priceof one unit of consumption in state s per unit of probability.It is also called a state price density, or a price kernel.


Risk-neutral probabilities and the state price density

I Rewrite the FOC in (2) as

u′(c(s))

u′(c0)=

1

β

q(s)

π(s)(3)

I Consumption in state s depends only on the state pricedensity in that state, q(s)

π(s) .

I Under risk neutrality, the LHS is a constant. The RHS istherefore also a constant ( q(s)

π(s) = cste): it follows from (1)

that the risk neutral probability π?(s) is equal to the physicalprobability π(s).

I If also follows from (2) that m is a constant (non-stochastic)under risk neutrality: the variation of m across states givesthe value of risk.



I Using (1), rewrite (3) as

π?(s) = β(1 + rf )u′(c(s))

u′(c0)π(s) (4)

π?(s) = ku′(c(s))π(s)∑

s

π?(s) = k∑

s

u′(c(s))π(s) = 1

I Under risk aversion, the RHS of (4) is decreasing inconsumption: given π(s), π?(s) is decreasing in c(s).

I If c(s) > c(τ), then π?(s)π(s) < π?(τ)

π(τ) .

I Since the ratio π?

π is decreasing in c , and since π and π? areprobabilities, we know that any state s with sufficiently highconsumption will be characterized by π?(s) < π(s)(underweight). Conversely, any state s with sufficiently lowconsumption will be characterized by π?(s) > π(s)(overweight).



I Since marginal utility is decreasing under risk aversion,consumption is decreasing in the state price density, which isintuitive.

I In the aggregate, the sum of individual consumptions must beequal to the total resources available in the economy in anygiven state (aggregate wealth).

I What does it suggest on the relation between the state pricedensity and aggregate wealth?


Consumption and state pricesOptimal risk sharing

I Normalize the utility function by setting u′(c0) = 1 fornotational convenience, differentiate both sides of (3) w.r.t.the state price density:

dc(s)

d q(s)π(s)

u′′(c(s)) =1

β(5)

I Combine with (3) to get

dc(s)

d q(s)π(s)

= −(A(c(s))

q(s)

π(s)

)−1< 0 (6)

I The LHS is the sensitivity of consumption to the state pricedensity, which is negative. This means that the demand for agiven Arrow-Debreu security (or equivalently for consumptionin a given state s) is decreasing in the state price density.


Consumption and state pricesOptimal risk sharing

I The higher the coefficient of absolute risk aversion A, the lessconsumption is sensitive to the state price density.

I An infinitely risk averse agent has a constant c , with the samelevel of consumption in any state of the world, regardless ofthe state prices. Smooth consumption at all costs!

I Tradeoff between smooth consumption and expectedconsumption. State prices are such that economic agentsshare risk optimally.

I dc(s)

d q(s)π(s)

is increasing in the state price if the agent is prudent. In

this case, the optimal consumption plan c(

q(s)π(s)

)is decreasing

and convex.


The mutuality principleOnly aggregate risk matters

I Since all agents face the same prices, equation (3) impliesthat, for any two investors i and j with the same beliefs:

βu′(c i (s))

u′(c i0)

= βu′(c j(s))

u′(c j0)

for any s (7)

I MRS between current and any future consumption areequalized across investors in equilibrium. NB: investors canhave different preferences, different levels of wealth, etc.!

I Investors are insured against idiosyncratic shocks inequilibrium (pooling of risks). Only aggregate/systematicshocks have an effect on individual consumption.

I Systematic risk should NOT be pooled! Ex: AIG.

I What happens if one investor is risk neutral?


The risk sharing ruleHow risk is allocated in an equilibrium with complete markets

I We already know that the marginal rate of substitutionbetween consumption in a given state at t = 1 andconsumption at t = 0 is the same for different agents (here iand j)):

βu′(c i

t=1(s = 1))

u′(c it=0)

= βu′(c j

t=1(s = 1))

u′(c jt=0)

βu′(c i

t=1(s = 2))

u′(c it=0)

= βu′(c j

t=1(s = 2))

u′(c jt=0)

I Therefore

u′(c it=1(s = 2))

u′(c it=1(s = 1))

=u′(c j

t=1(s = 2))

u′(c jt=1(s = 1))

I The MRS between consumption in any two future states (heres = 1 and s = 2) will also be the same for different agents.


The representative agent approachHow to derive the SDF

I Like Arrow-Debreu assets, the representative agent is a usefulabstraction.

I Like no one believes in father Christmas, no one believes thatthe representative agent actually exists.

I Under complete markets, we only need consider one agentwith the same preferences as the other agents1 who consumesaggregate consumption (which is also aggregate production).

I Thus, we only need to know individual preferences in the faceof risk and macroeconomic fundamentals (GDP) to derive themarginal rate of substitution of the representative agent forany two states.

I But knowing the MRS between consumption in each futurestate of the world at t = 1 and consumption at t = 0 gives usthe Stochastic Discount Factor!

1I’m simplifying here. Individuals do not need to have the same preferencesfor the representative agent approach to be valid.


TakeawayI With complete markets, idiosyncratic risk does not matter, in

the sense that individuals pool idiosyncratic risk in equilibrium,so that their consumptions are independent of their respectiveexposures to idiosyncratic risks. (cf. equation (7)).

I Only aggregate risk, i.e., fluctuations in aggregate endowmentor aggregate consumption, matter.

I The optimal allocation of this aggregate risk depends on therisk aversion of the different agents: less risk averse agentsbear a larger part of the risk (cf. equation (6)).

I Agents face a tradeoff between maximizing their expectedconsumption, or smoothing their consumption across states ofnature. The more risk averse they are, the more they leantoward the latter.

I In equilibrium, all aggregate risk must be borne! State pricesadjust so that, in any state, the sum of individualconsumptions (aggregate consumption) is equal to the sum ofindividual endowments (aggregate endowment).


Incomplete markets

I What if markets are incomplete? Two consequences:I Economic agents cannot conduct all mutually beneficial trades.I MRS are therefore not always equal across agents, which

indicates that risk sharing (the allocation of risks, i.e.variations in consumption across states of the world) issuboptimal.

I We lose the tractability and simplicity of the completemarkets framework:

I We cannot use Arrow-Debreu assets.I We cannot use the representative agent approach.

I We need to solve each investor’s problem individually, withthe existing financial assets.

I The resulting equilibrium is not Pareto efficient, but it isconstrained-Pareto efficient relative to the existing financialassets.


Incomplete marketsI Opportunities for efficiency improvements. Financial

innovation: create financial assets to complete markets.

“Risk sharing has been used primarily for certain narrow kinds of insurable

risks, such as stock market crashes or hurricanes, or for managing the

risks of conventional investments, such as diversifying investment

portfolios or hedging commodity risks. (. . . ) Finance has substantially

neglected the protection of our ordinary riches, our careers, our homes”

Robert Shiller, The New Financial OrderI In other words, financial assets have mainly been devised to

trade systematic risks, but it is still hard to trade (and pool)idiosyncratic risks.

I Efficient risk sharing already exists (almost) for someidiosyncratic risks: fire insurance, car insurance, medicalinsurance, even weather insurance!

I Some risks have yet to be efficiently shared with the creationof appropriate financial instruments. For example: real estatevaluation in a neighborhood, earnings prospects as a violinist,etc.


Application: fixed income pricing

I Consider a zero coupon bond with a repayment of 100 in tperiods in the absence of default. Default occurs withprobability π, and triggers a repayment equal to100× recovery ratio at maturity. The final payment is calledthe nominal value or face value of the bond.

I The bond price is equivalently given by:

P0 =100

(1 + rf + spread)t(8)

P0 =(1− π)100 + π(100× recovery ratio)

(1 + rf + risk premium)t(9)

P0 =(1− π?)100 + π?(100× recovery ratio)

(1 + rf )t(10)


Application: fixed income pricingCredit spreads and economic conditions (Source: Mishkin, JEP 2011)


Application: fixed income pricingCommon misconceptions

I NB: rf + spread ≡ YTM, yield-to-maturity. The YTM isusually derived from the observation of the bond price.

I The spread is not the same as the risk premium! Even in aworld where all investors are risk neutral or the default of thebond is not correlated with the SDF, the spread will bepositive if default is possible and entails a loss.

I A common mistake is to derive the physical probability π ofdefault from equation (10), while this equation gives the riskneutral probability π? of default! To the extent that defaulttends to occur in “bad” states of the world (with lowaggregate wealth), we expect that π? > π.

I This mistake, which does not take into account risk aversion,leads to the overestimation of the physical proba of default.

I It was very common in 2008, with people claiming that theprobabilities of default implied by observed bond prices wereimplausibly high.


I Acknowledgements: Some sources for this series of slidesinclude:

I The slides of Martin Boyer, for the same course at HECMontreal.

I Asset Pricing, by John H. Cochrane.I Finance and the Economics of Uncertainty, by Gabrielle

Demange and Guy Laroque.I The Economics of Risk and Time, by Christian Gollier.

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Complete markets and Arrow-Debreu assets