Utility-based asset pricing

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Utility-based asset pricing

• HKUST
• FINA790C Spring 2006

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Intertemporal utility theory and asset pricing

• mt,tj intertemporal marginal rate of
  substitution between consumption at time t and
  consumption at time tj
• Rit,tj gross j-period holding period rate of
  return for asset I starting at date t
• xt,j ln(mt,tj) lnR1t,tj lnRNt,tj
• Suppose xt,j -8lt t lt 8 is normally
  distributed, stationary stochastic process

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Intertemporal asset pricing

• If Etmt,tjRit,tj 1 then
• Take logarithms of both sides

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Time-separable power utility

• One way to specify form of mt,tj time-separable
  power utility for representative consumer
• Max ?? ?j C1-?tj/(1-?)
• Constant relative risk aversion with ? the
  coefficient of relative risk aversion
• Properties of power utility scale invariant
  allows for aggregation to get representative
  consumer with aggregate consumption

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Intertemporal MRS

• In this case the MRS is
• mt,tj ?j Ctj /Ct -?
• Substituting in above gives

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Time-series cross-section implications

• Time-series
• ln(Rft) ln( 1/Et(mt1) )
• -ln? - ½ ?2 vart(?ct1) ?Et?ct1
• One-period riskless rate is linear in expected
  consumption growth, with slope coefficient of
  relative risk aversion

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Time-series cross-section implications

• Cross-section of excess log returns
• Et lnRit1 lnRft -½ vart(lnRit1)
• ?covt( lnRit1, ?lnct1 )
• Logarithmic risk premia are linear in covariances
  with consumption growth, with slope coefficient
  of relative risk aversion

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How does the power utility specification do?

• Equity risk premium puzzle
• Riskfree rate puzzle

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Alternatives to power utility

• Utility is non-separable between consumption and
  leisure
• Utility is non-separable between current and
  future consumption
• Durable goods
• Habit persistence
• Non-expected utility