Asset prices in the representativeagent economy with background risk

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Asset prices in the representative-agent economy
with background risk

• Andrei Semenov (York University)

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Introduction

• The standard consumption CAPM
• Problems
• a) The equity premium puzzle
• b) The risk-free rate puzzle

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• Generalizations
• a) Preference modifications
• b) State-dependent parameters
• c) Psychological models of preferences
• d) Incomplete consumption insurance
• Brav et al. (2002), Balduzzi and Yao (2007), and
  Kocherlakota and Pistaferri (2009)

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Outline

• Consumption-Based Model with Background Risk
• The stochastic discount factor (SDF)
• Risk vulnerability and the asset pricing puzzles
• Risk aversion and the EIS under background risk
• Empirical Investigation
• The data
• The estimation procedure (conditional HJ
  volatility bounds)
• Estimation results
• Concluding Remarks

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1. A Consumption-based asset pricing model with
background risk

• 1.1 The Stochastic Discount Factor (SDF)
• The representative agent faces (Franke et al.
  (1998) and Poon and Stapleton (2005))
• The financial risk
• An independent, non-hedgeable, adverse
  background risk (the losses from domestic
  political turmoil, the inflation risk, an
  uncertain income tax rate, the risk from natural
  disasters, etc.)

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• In the presence of background risk, the
  representative agent maximizes
• is the representative-agents hedgeable
  consumption in period . The
  non-hedgeable consumption is
  independent of both optimal consumption and the
  risky payoff and has a non-positive expected
  value.

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• One of the first-order conditions
• or
• This is the consumption CAPM with background
  risk. The SDF
• In the absence of background risk,

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• 1.2 The precautionary premium
• Following Kimball (1990)
• where is an
  equivalent precautionary premium.
• Assume , then

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• We can write the SDF as
• Assume that the utility function is CRRA
• The precautionary premium for the agent with
  CRRA utility is hence

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• This implies that, with CRRA utility, for any t
• Where is the normalized variance of
  
• We need for marginal
  utility to be well-defined.
• The SDF is then

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• 1.3 Risk vulnerability and the asset pricing
  puzzles
• As introduced by Kihlstrom et al. (1981) and
  Nachman (1982), define the following indirect
  utility function
• Gollier (2001) argues that, in the case of the
  background risk with a non-positive mean
  preferences exhibit risk vulnerability if and
  only if the indirect utility function is more
  concave than the original utility function, i.e.,

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• As shown by Gollier (2001), this inequality holds
  if at least one of the following two conditions
  is satisfied
• (i) ARA is decreasing and convex and
• (ii) both ARA and AP are positive and decreasing
  in wealth (standard risk aversion (Kimball
  (1993)).

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• Risk vulnerability and the equity premium puzzle
• The consumption CAPM with background risk in
  terms of
• Assume
• Then
• If , then is less concave
  than utility
• and hence .

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• Risk vulnerability and the risk-free rate puzzle
• In the presence of background risk,
• Since and , then
• Because the agent is risk averse (i.e., utility
  is concave), from this it immediately follows
  that in each state the ratio of marginal
  utilities at t1 and t under background risk
  exceeds that in the no background risk case. As
  this is true in each state, this is true in
  expectation as well.

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• 1.4 Risk aversion and the EIS under background
  risk
• Suppose that at time t the agent faces the
  background risk and a lottery with an uncertain
  payoff .
• For any distribution functions and
• Since and are independent, then

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• A Taylor series expansion of
  around
• and hence
• It then follows that
• implying

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• The RRA coefficient of the agent with utility
  is then
• Since is independent of ,
• and hence

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• Denote as the risk premium we would observe
  if the utility curvature parameter were .
• The proportion of the risk premium due to the
  background risk in the total risk premium the
  representative agent is ready to pay to avoid the
  financial risk is then

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• Kimball (1992) defines the temperance premium
  by the following condition
• By analogy with the risk premium,
• The conditions and
  (the necessary conditions for risk
  vulnerability) imply that .

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• We have
• With power utility
• and therefore

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• The EIS in the model with an independent
  non-hedgeable background risk
• Since the representative agent with utility
  facing background risk has the same optimal
  consumption plan as the representative agent with
  utility in the no background risk
  environment, we can suppose that
• where and
• The model with background risk enables us to
  disentangle the coefficient of pure RRA and
  the EIS in the expected utility framework.

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2. Empirical investigation

• 2.1 The data
• The consumption data
• Quarterly consumption data (consumption of
  nondurables and services (NDS)) from the CEX (the
  US Bureau of Labor Statistics) from 1980Q1 to
  2003Q4.
• We drop households
• a) that do not report or report a zero value of
  consumption of food, consumption of nondurables
  and services, or total consumption,

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• b) nonurban households, households residing in
  student housing, households with incomplete
  income responses, households that do not have a
  fifth interview, and households whose head is
  under 19 or over 75 years of age.
• We consider four sets of households based on the
  reported amount of asset holdings at the
  beginning of a 12-month recall period in constant
  2005 dollars
• a) all households,
• b) households with total asset holdings gt 0,
• c) households with total asset holdings
  1000,
• d) households with total asset holdings 5000.

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• The returns data
• a) The nominal quarterly value-weighted market
  capitalization-based decile index returns
  (capital gain plus all dividends) on all stocks
  listed on the NYSE, AMEX, and Nasdaq are from the
  CRSP.
• b) The nominal quarterly value-weighted returns
  on the five and ten NYSE, AMEX, and Nasdaq
  industry portfolios are from Kenneth R. French's
  web page.
• c) The nominal quarterly risk-free rate is the
  3-month US Treasury Bill secondary market rate
  from the Federal Reserve Bank of St. Louis.
• d) The real quarterly returns are calculated as
  the quarterly nominal returns divided by the
  3-month inflation rate based on the deflator
  defined for NDS.

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• 2.2 The estimation procedure
• We use HJ (1991) volatility bounds to assess the
  empirical performance of three SDFs
• 1. Standard SDF
• 2. The SDF in Brav et al. (2002)
• where is the household i's consumption
  growth rate and

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• 3. The SDF in the consumption CAPM with
  background risk
• When estimating and
  for all t

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• For each of the above SDFs, we test the
  conditional Euler equations for the excess
  returns on risky assets
• and the risk-free rate
• Denote the error terms in the Euler equations as
• Thus, at the true parameter vector,
• where is a variable in the agent's time t
  information set.

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• A lower volatility bound for admissible SDFs
  , which have unconditional mean m and
  satisfy
• where is the unconditional variance-covariance
  matrix of .
• We look for the values of the SDF parameters, at
  which a considered SDF satisfies the volatility
  bound, i.e.,
• where

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• Denote as the utility curvature parameter
  in .
• The optimal value of the estimate of the
  curvature parameter is
• Denote as the instrument , for which
  
• Since
• or equivalently
• we estimate the subjective time discount factor
  as

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• In the model with background risk, the effective
  RRA coefficient differs from the utility
  curvature parameter (the coefficient of pure
  RRA) and is
• We follow Kimball (1993) and Franke et al. (1998)
  and assume that the background risk is
  actuarially neutral, i.e.,
• Under this assumption, the estimate of the
  effective RRA is

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Concluding remarks

• Empirical evidence that, in contrast to the
  previously proposed incomplete consumption
  insurance models, the asset-pricing model with
  the SDF calculated as the discounted ratio of
  expectations of marginal utilities over the
  non-hedgeable consumption states at two
  consecutive dates jointly explains the
  cross-section of risky asset excess returns and
  the risk-free rate with economically plausible
  values of the pure RRA coefficient and the
  subjective time discount factor.
• The results are robust across different sets of
  stock-returns and threshold values in the
  definition of asset holders.

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• Since the important components of the pricing
  kernel are the first two unconditional moments of
  the distribution of the non-hedgeable
  consumption, this supports the hypothesis that
  the independent non-hedgeable adverse background
  risk can account for the market premium and the
  return on the risk-free asset.