Asset Pricing - Consumption-Based Asset Pricing II

The Basic Consumption-Based Model's Drawbacks

• The basic consumption-based asset pricing model links consumption growth and risk premia directly.
• Its simplicity leads to discrepancies with stylized facts like the equity premium puzzle, risk-free rate puzzle, and volatility puzzle.
• The consumption-based approach isn't invalidated by these issues.
• Empirical problems might make the basic model's failure seem worse than it is.
• The tested model is a simplified version of the general consumption-based model.

Advanced Consumption-Based Asset Pricing

• Over the last two decades, alternative specifications of the general consumption-based asset pricing model have been developed.
• These specifications can be categorized as:
  • Alternative representation of preferences (not time-additive CRRA).
  • Alternative aggregate consumption dynamics (not lognormal consumption growth).
  • Market incompleteness (not first welfare theorem).

Utility with a Benchmark

• An alternative is allowing the utility at t to depend on some benchmark $X_t$: state-dependent utility function.
  • Example: $X<em>t$ as others’ consumption levels (keeping up with Joneses) leads to $\zeta</em>t = e^{-\delta t} \frac{u<em>c(c</em>t, X<em>t)}{u</em>c(c<em>0, X</em>0)}$.
• Campbell and Cochrane (1999) use $X_t$ as past habit (weighted average of the previous consumption rate).
  • $E \left[ \sum<em>{t=0}^T e^{-\delta t} u(c</em>t, X<em>t \equiv h</em>t) \right]$, where $u(c, h) = \frac{(c - h)^{1-\gamma}}{1 - \gamma}$ (\gamma > 0, $c \geq h$), and $h<em>t = h</em>0 e^{-\beta t} + \alpha \sum<em>{s=1}^{t-1} e^{-\beta(t-s)} c</em>s$.

Campbell & Cochrane (1999): Habit Formation

• The RRA is no more constant & state-dependent:
  • \gamma(ct, ht) = -\frac{c \cdot u{cc}(ct, ht)}{uc(ct, ht)} = \frac{\gamma}{1 - \frac{ht}{ct}} > \gamma.
  • Bad state: $c<em>t \approx h</em>t \Rightarrow \gamma(c<em>t, h</em>t)$ is high.
  • Good state: $c<em>t \gg h</em>t \Rightarrow \gamma(c<em>t, h</em>t)$ is low.
  • Under habit formation, there's a counter-cyclical relative risk aversion (↔ CRRA).
• How does it affect the risk-free rate & asset returns?
• What empirical observation does it explain?

Campbell & Cochrane (1999): Habit Formation (2)

• Ceteris paribus, in equilibrium, the agent will invest more in the risk-free asset than in the CRRA model:
  • To ensure that future consumption does not get close to the future habit level; It partially resolves the risk-free rate puzzle.
• In the basic consumption-based model, the risk premium is increasing in $\gamma$; We have similar relations in the habit formation model, but now with a state-dependent RRA.
  • Bad state: $c<em>t \approx h</em>t \Rightarrow \gamma(c<em>t, h</em>t)$ is high $\Rightarrow$ the risk premium is high.
  • Good state: $c<em>t \gg h</em>t \Rightarrow \gamma(c<em>t, h</em>t)$ is low $\Rightarrow$ the risk premium is low.
  • Explains the countercyclical risk premium puzzle (Harvey, 1989).

Two Types of Risk Attitudes

• Suppose that there is a fair coin with 1/2 chances of having either H or T
• When the coin toss gives you $i \in {H,T}$, your consumption level is $c<em>i$ with cH > c_L
• Which gamble do you prefer?
  1 Toss a coin only at $t = 0$ and determine your consumption levels for all time periods
  2 Determine your consumption level every time by tossing a coin in each period
• The time-additive expected utility’s weakness: It does not reflect her preferences for the timing of resolution of uncertainty.

Risk Aversion and EIS

• The time-additive expected utility’s weakness: It does not reflect preferences for the timing of resolution of uncertainty.
• RRA reflects aversion to consumption variation across states at a particular time (atemporal risk).
• Elasticity of intertemporal substitution (EIS) reflects attitude towards consumption variation over time.

Elasticity of Intertemporal Substitution (EIS)

• In equilibrium, it measures responsiveness of consumption growth to the risk-free rate:
  • $EIS \equiv \frac{d(c<em>{t+1}/c</em>t) / (c<em>{t+1}/c</em>t)}{d R<em>f^{t+1} / R</em>f^{t+1}} = \frac{d \ln(c<em>{t+1}/c</em>t)}{d \ln R_f^{t+1}}$.
• The problem of the CRRA time-additive expected utility:
  • $\text{EIS} = \frac{d \ln(c<em>{t+1}/c</em>t)}{d \ln R_f^{t+1}} = \frac{1}{\gamma}$. That is, EIS and RRA are intertwined in the basic model.

Weil (1989): Epstein-Zin Recursive Utility

• In this model, $f(c, q)$ has a CES functional form:
  • $f(c, q) = \left[ a c^{\frac{\psi-1}{\psi}} + b q^{\frac{\psi-1}{\psi}} \right]^{\frac{\psi}{\psi-1}}$,
  • $q<em>t (U</em>{t+1})^{1-\gamma} = E<em>t \left[ U</em>{t+1}^{1-\gamma} \right]$,
  • $\Rightarrow U<em>t = \left[ a c</em>t^{\frac{\psi-1}{\psi}} + b \left( E<em>t \left[ U</em>{t+1}^{1-\gamma} \right] \right)^{\frac{1}{1-\gamma} \cdot \frac{\psi-1}{\psi}} \right]^{\frac{\psi}{\psi-1}}$.
• The RRA ($\gamma$) and the EIS ($\psi$) are distinguished in this model.
  • \gamma \psi > 1: individual prefers early resolution of uncertainty.
  • \gamma \psi < 1: individual prefers late resolution of uncertainty.
• How can it explain the previous empirical puzzles?

Weil (1989): Epstein-Zin Recursive Utility (2)

• In the basic model, the risk premium increases in the RRA $\gamma$:
  • $r<em>f^t = \delta + \gamma \left( \mu</em>c - \frac{\sigma<em>c^2}{2} \right) - \frac{\gamma^2 \sigma</em>c^2}{2}$.
• To explain high risk premia, we needed a high level of $\gamma$ (the equity premium puzzle).
• Given a high level of $\gamma$, we have a too high risk-free rate (the risk-free rate puzzle).
• With the EZ preferences:
  • $r<em>f = \delta + \frac{\mu</em>c}{\psi} - \theta \frac{\sigma<em>c^2}{2 \psi^2} - (1 - \theta) \frac{\sigma</em>w^2}{2}$.
• In equilibrium, the EIS $\psi$ determines how $\mu<em>c$ affects $r</em>f$.
• In this case, we can separate the two puzzles from each other and explain the latter with a high $\psi$.

Weil (1989): Epstein-Zin Recursive Utility (3)

• Intuitive explanation:
  1. When high consumption growth is expected, an investor would want to borrow more to increase consumption today.
  2. To clear the market, the risk-free rate should increase.
  3. If an investor has a high $\psi$, their exchange rate between current and future consumption is highly sensitive to a change in the risk-free rate.
  4. Thus, when $\psi$ is high, a small change in the risk-free rate is sufficient to clear the market.

Bansal & Yaron (2004): Long-Run Risk

• Previous researchers had applied EZ preferences, but only in i.i.d. environments.
• EZ preferences + a long-run risk component in aggregate consumption.
• In this setup, the volatility ($\sigma<em>t$) in the consumption growth ($g</em>t$) is a time-variant & random stochastic process.
  • That is, $\sigma_t$ reflects the time-variant economic uncertainty.
• Thus, a small change in the long-run consumption growth component can lead to large changes in asset prices.
• In equilibrium, an increase in economic uncertainty leads to relatively large drops in asset prices and risk premia.

Bansal & Yaron (2004): Long-Run Risk (2)

• Example: \gamma \psi > 1 \Rightarrow the representative agent prefers the early resolution of uncertainty.
  • When the growth rate uncertainty ($\sigma_t$) is high, it implies that the risk is resolved slowly.
  • Hence, an agent with \gamma \psi > 1 requires a high return premium for bearing such risk: That is, the equity premium puzzle is (partially) explained when the EIS and the uncertainty are high.

Market Incompleteness

• If the markets are incomplete, the traditional representative-agent approach may break down.
• Remarks:
  1. Optimal MRS of each individual = SPD
  2. A weighted average of SPD is also a SPD
  3. Market Completeness $\Rightarrow$ Efficient risk sharing
• Hence, given an economy with homogeneous agents with CRRA $\gamma$ and $\delta$, the agents’ equally-weighted SPD is $\zeta<em>t = e^{-\delta t} \cdot \frac{1}{L} \sum</em>{l=1}^L \left( \frac{c<em>{l,t}}{c</em>{l,0}} \right)^{-\gamma}$.

Market Incompleteness Matters (2)

• If the market is complete, consumption growth $c<em>{l,t}/c</em>{l,0}$ would be the same across individuals:
  • $\Rightarrow \zeta<em>t = e^{-\delta t} \cdot \frac{1}{L} \left[ \sum</em>{l=1}^L c<em>{l,t} / \sum</em>{l=1}^L c_{l,0} \right]^{-\gamma}$, which implies that a representative agent also has CRRA utility with $\gamma$ and $\delta$.
• However, this is inconsistent with data except for unreasonably high $\gamma$: That is, financial markets may be incomplete and do not allow individuals to align their MRS.