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 P it = E t [ m t+1 X it+1 ] where P it = price of asset i at time t E t = expectation conditional on investors’ time t information X it+1 = asset i’s payoff at t+1 m t+1 = stochastic discount factor

The stochastic discount factor m t+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: p it = E t [ m t+1 X it+1 ] = E t [m t+1 ]E t [X it+1 ] + cov t (m t+1,X it+1 ) Repeated substitution gives p it = E t [  m t,t+j X it+j ] (if no bubbles)

Stochastic discount factor & prices If a riskless asset exists which costs $1 at t and pays R f = 1+r f at t+1 1 = E t [ m t+1 R f ] or R f = 1/E t [m t+1 ] So our risk-adjusted discounting formula is p it = E t [X it+1 ]/R f + cov t (X it+1,m t+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 : p it = E t [ m t+1 X it+1 ] iff law holds Absence of arbitrage: there are no arbitrage opportunities iff  m > 0 : p it = E t [m t+1 X it+1 ]

Stochastic discount factors For stocks, X it+1 = p it+1 + d it+1 (price + dividend) For riskless asset if it exists X it+1 = 1 + r f = R f Since p t is in investors’ information set at time t, 1 = E t [ m t+1 ( X it+1 /p it ) ] = E t [m t+1 R it+1 ] This holds for conditional as well as for unconditional expectations

Stochastic discount factor & returns If a riskless asset exists 1 = E t [m t+1 R f ] or R f = 1/E t [m t+1 ] E t [R it+1 ] = ( 1 – cov t (m t+1,R it+1 )/E t [m t+1 ] E t [R it+1 ] – E t [R zt+1 ] = -cov t (m t+1,R it+1 )E t [R zt+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., m t+1 =  ∂U/∂C t+1 /E t ∂U/∂C t from first-order condition of investor’s utility maximization problem –E.g., m t+1 = a + bf t+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) = R f m(s)  (s) = R f c(s) = c(s)/E t (m) Then p t = E* t (x)/R f ( 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(c 0, c 11, c 12 ) = Σπ s (lnc 0 + lnc 1s ) –i.e. u(c 0, c 1s ) = lnc 0 + lnc 1s (additive) time separable utility function m= ∂u 1 /E∂u 0 =(c 0 /c 11, c 0 /c 12 )=(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[m t+1 (R it+1 – R f ) ] =0 So E[R it+1 -R f ] = -cov(m t+1,R it+1 )/E[m t+1 ] E[R it+1 -R f ] = -  (m t+1,R it+1 )  (m t+1 )  (R it+1 )/E[m t+1 ] Rewritten in terms of the Sharpe ratio E[R it+1 -R f ]/  (R it+1 ) = -  (m t+1,R it+1 )  (m t+1 )/E[m t+1 ]

Hansen-Jagannathan bound Since -1 ≤  ≤ 1, we get  (m t+1 )/E[m t+1 ] ≥ sup i | E[R it+1 -R f ]/  (R it+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 R f ) we calculate E(m)S*(R f ); 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 satisfy E( 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