[PPT] - Lecture 2: Stochastic Discount Factor Simon Gilchrist Boston PowerPoint Presentation

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Lecture 2: Stochastic Discount Factor

Simon Gilchrist Boston Univerity and NBER

EC 745

Fall, 2013

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Stochastic Discount Factor (SDF)

A stochastic discount factor is a stochastic process {Mt,t+s} such that for any security with payoff xt+1 at time t + 1 the price

f that security at time t is

Pt = EtMt,t+1xt+1 Equivalently EtMt,t+1Rt+1 for Rt+1 = xt+1 Pt Ex: representative agent Mt+1 = β u′ (Ct+1) u′ (Ct)

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Issues to think about:

Heterogeneous agenst with complete vs incomplete markets Hansen-Jagannathan volatility bounds (JPE, 1991). SDF must have (roughly) permanent innovations (Alvarez and Jermann, Econometrica 2004)

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Basic properties of SDF:

Risk free rate: Rf

t+1 = EtMt+1

CAPM – SDF is linear function of market return: log Mt+1 = a − bRm

t+1

CCAPM – SDF is a function of consumption growth: log Mt+1 = log β − γ∆ log Ct+1 Compensation for risk: EtRe

t+1 − Rf t+1 = σt (Mt+1)

EtMt+1 corrt

Mt+1, Re

t+1

σt
Re

t+1

where σt(Mt+1)

EtMt+1 may be interpreted as the market price of risk.

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Constructing SDF under complete markets:

One period world, s = 1 : S possible states with probability πs. Let qs be the state-contingent price – price of an Arrow-Debreu security that payoffs one unit in state s. Then the price of a security with payoffs {ds}S

s=1 is

p ({d}) =

S

s=1

qsds =

S

s=1

πs qs πs ds then we have defined the SDF as ms = qs πs so that p ({d}) =

S

s=1

πsmsds = Emsds

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Incomplete markets: an example.

One period economy with N agents There is uncertainty about the state s Trade occurs before uncertainty is resolved. Securities: L securities with prices ql and payoffs dls in state s.

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Agents problem:

Max max U (ci1, ..ciS) subject to cis = yis +

L

i=1

qlθil ≤ wi where yis is the endowment in state s, θil is number of shares in security l purchased by agent i and wi is initial wealth of agent i. FOC:

S

s=1

∂ui ∂cs dls = qlλi where λi is agent i’s Lagrange multiplier on the wealth constraint.

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Equilibrium:

A price vector q = {ql} and shares {θil}i,l such that each agent maximizes utility and markets clear:

L

i=1

θil = 1, l = 1..L SDF: Consider any agent with λi > 0 then the Euler equation is ql =

S

s=1

misdls where mis =

∂ui ∂cs

λi We can use the marginal rate of substitution of any consumer who is unconstrained as an SDF. Agents may not be unconstrained in each state but as long as there are no arbitrage

pportunities there exists an SDF.

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Incomplete markets: no arbitrage pricing.

The system of securities characterized by {dls} and {ql} is arbitrage free if there is no vector of portfolio choices {θl} such that both

L

l=1

qlθl ≤ 0 and for all s = 1..S

L

l=1

dlsθl ≥ > 0 for some state s. In word: there does not exist a portfolio with a zero (or negative) price that gives positive payoffs in all states of the world.

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Implications

If prices {ql} result from a competitive equilibrium then there will be no arbitrage – otherwise demand an infinite amount of the security. More generally we can use no arbitrage theory to place restrictions on prices. Example if asset x pays more dividends than asset y in all states then the price of asset x must be (weakly) greater than the price of asset y. Assets that are redundant, i.e. can be replicated or “spanned” by

ther assets can be priced even if markets are incomplete.

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Example

Example: S=6,l=3 D =   1 2 1 0 0 0 1 3 1 0 0 0 0 1 0 0 0 0   Since the third asset is the first minus the second it can be priced. Black and Scholes – under NAO, options can be priced as linear combinations of equity and debt payouts.

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Some general results on SDF:

The law of one price holds iff there is (at least) one stochastic discount factor.

LOOP implies prices are linear functions of payoffs: P (x + y) = P (x) + P (y) . This implies that for a security that payoffs x P(x) =

wsxs

Define ms = ws πs then P (x) =

s

πsmsxs so that ms is an SDF. Converse is obvious.

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Some general results on SDF:

There are no arbitrage opportunities iff there exists (at least) one strictly positive SDF.

NAO: if x ≥ 0, p(x) ≥ 0 and if xs > 0 for some s then p (x) > 0, use separating-hyperplane argument to argue that SDF exists.

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Some general results on SDF:

The stochastic discount factor is unique iff markets are complete.

Suppose markets are complete then there is an asset that pays off

ne unit in state s where the price of that asset ps satisfies:

ps = msπs This is true for any SDF: ps = m∗

sπs

therefore m∗

s = ms

Suppose the SDF ms is unique, assume that the market is

incomplete. Then there is a state j which is non-tradeable. Define

m∗

s

= ms if s = j = ms + 1 if s = j The SDF m∗

s will price all tradeable securities hence ms is not

unique.

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Hansen-Jagannathan Bounds

Consider any risky return R and risk-free return Rf then E (mR) = 1 E (m) Rf = 1 E (mR) = E (m) Rf This implies: cov (m, R) = E (mR) − E (m) E (R) = E (m) Rf − E (m) E (R)

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Hansen-Jagannathan Bounds

Now use bound (−1 ≤ corr(x, y) ≤ 1): −σ (x) σ (y) ≤ cov (x, y) ≤ σ (x) σ (y) to obtain −σ (m) σ (R) ≤ E (m) Rf − E (m) E (R) ≤ σ (m) σ (R) Reverse inequality to obtain: − σ (m) E (m) ≤ E (R) − Rf σ (R) ≤ σ (m) E (m)

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Comments:

E(R)−Rf σ(R)

is the Sharpe ratio and σ(m)

E(m) is the market price of risk.

The Sharpe ratio for the market is 0.06/0.17=35%. This applies to any asset – Sharpe ratios can easily be on the

rder of 100% so the price of risk must be very high.

To obtain tight bounds, consider a portfolio with very high sharp ratio. Under complete markets, the price of risk is the slope of the mean-variance frontier – this defines parabola in E(m), σ(m) space.

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Price of risk under CRRA

For CRRA and random-walk consumption: ∆ ln ct = µ + εt where εt ∼ N(0, σ2

ε)

Results from log-normal: E(Ct+1 Ct = exp

µ + 1

2σ2

ε

var(Ct+1

Ct ) =

eσ2

ε − 1

e2µ+σ2

ε

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Price of risk under CRRA

These results imply: E (m) = β exp

γ
−µ + γ σ2

ε

2

and

σ (m) E (m) =

exp
σ2