Stochastic discount factors

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Stochastic discount factors

• HKUST
• FINA790C Spring 2006

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Objectives of asset pricing theories

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

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Most general asset pricing theory

• All the models we will talk about can be written
  as
• Pit Et mt1 Xit1
• where Pit price of asset i at time t
• Et expectation conditional on
  investors time t information
• Xit1 asset is payoff at t1
• mt1 stochastic discount factor

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The stochastic discount factor

• mt1 (stochastic discount factor pricing kernel)
  is the same across all assets at time t1
• It values future payoffs by discounting them
  back to the present, with adjustment for risk
• pit Et mt1Xit1
• Etmt1EtXit1 covt(mt1,Xit1)
• Repeated substitution gives
• pit Et S mt,tj Xitj (if no bubbles)

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Stochastic discount factor prices

• If a riskless asset exists which costs 1 at t
  and pays Rf 1rf at t1
• 1 Et mt1Rf or Rf 1/Etmt1
• So our risk-adjusted discounting formula is
• pit EtXit1/Rf covt(Xit1,mt1)

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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 mt1 Xit1 iff law holds
• Absence of arbitrage there are no arbitrage
  opportunities iff ? m gt 0 pit Etmt1Xit1

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Stochastic discount factors

• For stocks, Xit1 pit1 dit1 (price
  dividend)
• For riskless asset if it exists Xit1 1 rf
  Rf
• Since pt is in investors information set at time
  t,
• 1 Et mt1( Xit1/pit ) Etmt1Rit1
• This holds for conditional as well as for
  unconditional expectations

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Stochastic discount factor returns

• If a riskless asset exists 1 Etmt1Rf or
• Rf 1/Etmt1
• EtRit1 ( 1 covt(mt1,Rit1 )/Etmt1
• EtRit1 EtRzt1 -covt(mt1,Rit1)EtRzt1
  
• assets expected excess return is higher the
  lower its covariance with m

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Paths to take from here

• (1) We can build a specific model for m and see
  what it says about prices/returns
• E.g., mt1 b ?U/?Ct1/Et?U/?Ct from first-order
  condition of investors utility maximization
  problem
• E.g., mt1 a bft1 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

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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

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Thinking about the stochastic discount factor

• Let x(s) denote Payoff ? p(x) S 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 S ?(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

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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 Et(x)/Rf ( pricing using
  risk-neutral probabilities ?(s) )

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A simple example

• S2, p(1) ½
• 3 securities with x1 (1,0), x2(0,1), x3 (1,1)
• Let m(½,1)
• Therefore, p1¼, p2 1/2 , p3 ¾
• R1 (4,0), R2(0,2), R3(4/3,4/3)
• ER12, ER21, ER34/3

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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) Sps(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 endowmentconsumption
• Low consumption states are high m states

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What can we say about m?

• The unconditional representation for returns in
  excess of the riskfree rate is
• Emt1(Rit1 Rf) 0
• So ERit1-Rf -cov(mt1,Rit1)/Emt1
• ERit1-Rf -?(mt1,Rit1)?(mt1)?(Rit1)/Emt
  1
• Rewritten in terms of the Sharpe ratio
• ERit1-Rf/?(Rit1) -?(mt1,Rit1)?(mt1)/Emt
  1

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Hansen-Jagannathan bound

• Since -1 ? 1, we get
• ?(mt1)/Emt1 supi ERit1-Rf/?(Rit1)
  
• 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

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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

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Computing HJ bounds

• We dont 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 Rb 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

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Computing HJ bounds

• Without data on m we cant 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(Rb) var(e)
• ?(m) ?(Rb) (1-E(m)E(R))cov(R,R)-1(1-E(
  m)E(R))½

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Using HJ bounds

• We can use the bound to check whether the sdf
  implied by a given model is legitimate
• A candidate m a Rb must satisfy
• E( a Rb ) E(m)
• E ( (aRb)R ) 1
• Let X 1 R , ?? ( a b ), y (
  E(m) 1 )
• E X X ? - y 0
• Premultiply both sides by ?
• E (aRb)2 E(m) 1 ?

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Using HJ bounds

• The composite set of moment restrictions is E
  X X ? - y 0
• E y? - m2 0
• See, e.g. Burnside (RFS 1994), Cecchetti, Lam
  Mark (JF 1994), Hansen, Heaton Luttmer (RFS,
  1995)

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HJ bounds

• These are the weakest bounds on the sdf
  (additional restrictions delivered by the
  specific theory generating m)
• Tighter bound require mgt0