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Stochastic discount factors

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)

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Stochastic discount factors

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  1. Stochastic discount factors HKUST FINA790C Spring 2006

  2. 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)

  3. 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

  4. 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[ S mt,t+j Xit+j ] (if no bubbles)

  5. 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)

  6. 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]

  7. 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

  8. 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

  9. 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 = b∂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

  10. 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

  11. 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

  12. 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) )

  13. 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

  14. 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

  15. 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]

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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’ ]

  21. 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)

  22. HJ bounds • These are the weakest bounds on the sdf (additional restrictions delivered by the specific theory generating m) • Tighter bound: require m>0

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