Transcript Asset_Pricing_Theori..

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[ S 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 = 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
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 (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 riskneutral 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 HansenJagannathan 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 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