Stochastic discount factors

HKUST

FINA790C Spring 2006

Objectives of asset pricing theories

• Explain differences in returns across different assets at point in time (cross-sectional explanation)

• Explain differences in an asset’s return over time (time-series)

• In either case we can provide explanations based on absolute pricing (prices are related to fundamentals, economy-wide variables) OR relative pricing (prices are related to benchmark price)

Most general asset pricing theory

All the models we will talk about can be written as

Pit = Et[ mt+1 Xit+1] where Pit = price of asset i at time t

Et = expectation conditional on investors’ time t information

Xit+1 = asset i’s payoff at t+1

mt+1 = stochastic discount factor

The stochastic discount factor

• mt+1 (stochastic discount factor; pricing kernel) is the same across all assets at time t+1

• It values future payoffs by “discounting” them back to the present, with adjustment for risk:

pit = Et[ mt+1Xit+1 ]

= Et[mt+1]Et[Xit+1] + covt(mt+1,Xit+1)

• Repeated substitution gives

pit = Et[ mt,t+j Xit+j ] (if no bubbles)

Stochastic discount factor & prices

• If a riskless asset exists which costs $1 at t and pays Rf = 1+rf at t+1

1 = Et[ mt+1Rf ] or Rf = 1/Et[mt+1]

• So our risk-adjusted discounting formula is

pit = Et[Xit+1]/Rf + covt(Xit+1,mt+1)

What can we say about sdf?

• Law of One Price: if two assets have same payoffs in all states of nature then they must have the same price

m : pit = Et[ mt+1 Xit+1 ] iff law holds

• Absence of arbitrage: there are no arbitrage opportunities iff m > 0 : pit = Et[mt+1Xit+1]

Stochastic discount factors

• For stocks, Xit+1 = pit+1 + dit+1 (price + dividend)

• For riskless asset if it exists Xit+1 = 1 + rf = Rf

• Since pt is in investors’ information set at time t,

1 = Et[ mt+1( Xit+1/pit ) ] = Et[mt+1Rit+1]

• This holds for conditional as well as for unconditional expectations

Stochastic discount factor & returns

• If a riskless asset exists 1 = Et[mt+1Rf] or

Rf = 1/Et[mt+1]

• Et[Rit+1] = ( 1 – covt(mt+1,Rit+1 )/Et[mt+1]

Et[Rit+1] – Et[Rzt+1] = -covt(mt+1,Rit+1)Et[Rzt+1]asset’s expected excess return is higher the lower its covariance with m

Paths to take from here

• (1) We can build a specific model for m and see what it says about prices/returns– E.g., mt+1 = ∂U/∂Ct+1/Et∂U/∂Ct from first-order

condition of investor’s utility maximization problem– E.g., mt+1 = a + bft+1 linear factor model

• (2) We can view m as a random variable and see what we can say about it generally– Does there always exist a sdf?– What market structures support such a sdf?

• It is easier to narrow down what m is like, compared to narrowing down what all assets’ payoffs are like

Thinking about the stochastic discount factor

• Suppose there are S states of nature• Investors can trade contingent claims that pay

$1 in state s and today costs c(s)• Suppose market is complete – any contingent

claim can be traded• Bottom line: if a complete set of contingent

claims exists, then a discount factor exists and it is equal to the contingent claim prices divided by state probabilities

Thinking about the stochastic discount factor

• Let x(s) denote Payoff p(x) =Σ c(s)x(s)⇒

• p(x) = (s) { c(s)/(s) } x(s) , where(s) is probability of state s

• Let m(s) = { c(s)/(s) }

• Then p = Σ (s)m(s)x(s) = E m(s)x(s)So in a complete market the stochastic discount factor m exists with p = E mx

Thinking about the stochastic discount factor

• The stochastic discount factor is the state price c(s) scaled by the probability of the state, therefore a “state price density”

• Define *(s) = Rfm(s)(s) = Rfc(s) = c(s)/Et(m)

Then pt = E*t(x)/Rf ( pricing using risk-neutral probabilities *(s) )

A simple example

• S=2, π(1)= ½

• 3 securities with x1= (1,0), x2=(0,1), x3= (1,1)

• Let m=(½,1)

• Therefore, p1=¼, p2= 1/2 , p3= ¾

• R1= (4,0), R2=(0,2), R3=(4/3,4/3)

• E[R1]=2, E[R2]=1, E[R3]=4/3

Simple example (contd.)

• Where did m come from?• “representative agent” economy with

–endowment: 1 in date 0, (2,1) in date 1–utility EU(c0, c11, c12) = Σπs(lnc0+ lnc1s)–i.e. u(c0, c1s) = lnc0+ lnc1s (additive) time separable utility function

• m= ∂u1/E∂u0=(c0/c11, c0/c12)=(1/2, 1/1)• m=(½,1) since endowment=consumption• Low consumption states are “high m” states

What can we say about m?

• The unconditional representation for returns in excess of the riskfree rate is

E[mt+1(Rit+1 – Rf) ] =0

• So E[Rit+1-Rf] = -cov(mt+1,Rit+1)/E[mt+1]

E[Rit+1-Rf] = -(mt+1,Rit+1)(mt+1)(Rit+1)/E[mt+1]

• Rewritten in terms of the Sharpe ratio

E[Rit+1-Rf]/(Rit+1) = -(mt+1,Rit+1)(mt+1)/E[mt+1]

Hansen-Jagannathan bound

• Since -1 ≤ ≤ 1, we get

(mt+1)/E[mt+1] ≥ supi | E[Rit+1-Rf]/(Rit+1) |

• This is known as the Hansen-Jagannathan Bound: The ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe Ratio attained by any asset

Computing HJ bounds

• For specified E(m) (and implied Rf) we calculate E(m)S*(Rf); trace out the feasible region for the stochastic discount factor (above the minimum standard deviation bound)

• The bound is tighter when S*(Rf) is high for different E(m): i.e. portfolios that have similar but different E(R) can be justified by very volatile m

Computing HJ bounds

• We don’t observe m directly so we have to infer its behavior from what we do observe (i.e.returns)

• Consider the regression of m onto vector of returns R on assets observed by the econometrician

m = a + R’b + e where a is constant term, b is a vector of slope coefficients and e is the regression error

b = { cov(R,R) }-1 cov(R,m)

a = E(m) – E(R)’b

Computing HJ bounds

• Without data on m we can’t directly estimate these. But we do have some theoretical restrictions on m: 1 = E(mR) or cov(R,m) = 1 – E(m)E(R)

• Substitute back:

b = { cov(R,R) }-1[ 1 – E(m)E(R) ]• Since var(m) = var(R’b) + var(e)

(m) ≥ (R’b) = {(1-E(m)E(R))’cov(R,R)-1(1-E(m)E(R))}½

Using HJ bounds

• We can use the bound to check whether the sdf implied by a given model is legitimate

• A candidate m† = a + R’b must satisfyE( a + R’b ) = E(m†)E ( (a+R’b)R ) = 1

Let X = [ 1 R’ ], ’ = ( a b’ ), y’ = ( E(m†) 1’ )E{ X’ X - y } = 0

• Premultiply both sides by ’E[ (a+R’b)2 ] =[ E(m†) 1’ ]

Using HJ bounds

• The composite set of moment restrictions is E{ X’ X - y } = 0

E{ y’ - m†2 } ≤ 0

See, e.g. Burnside (RFS 1994), Cecchetti, Lam & Mark (JF 1994), Hansen, Heaton & Luttmer (RFS, 1995)

HJ bounds

• These are the weakest bounds on the sdf (additional restrictions delivered by the specific theory generating m)

• Tighter bound: require m>0