Consumption Risk In Modern Macro-Finance Models

Stocks returns are $8\%$ per year higher than bond returns on average. It’s hard to explain such a large equity premium using the standard consumption-based model because consumption growth isn’t risky enough. So, to fix this problem, modern macro-finance models introduce new state variables capturing other kinds of risk investors might care about, such as the surplus consumption ratio (habits; Campbell-Cochrane 1999) and news about long-run consumption growth (long-run risks; Bansal-Yaron 2004).

Because exposure to one of these new state variables typically explains most of the $8\%$ per year equity premium in modern macro-finance models, there’s a sense among researchers that it doesn’t matter whether investors are trying to insure themselves against shocks to consumption growth.

Not so!

This post shows why. The new state variables used in modern macro-finance models are not separate from consumption risk. They’re ways of amplifying the effects of consumption risk. Arguing that investors don’t care about consumption risk because exposure to the surplus consumption ratio explains most of the $8\%$ per year equity premium is like arguing that Hollywood doesn’t care about beauty because plastic surgery has a bigger effect on casting decisions than the face actors are born with. It’s nonsense.

Consumption CAPM

Investors in the consumption capital asset-pricing model (CCAPM; Lucas 1978) try to maximize their expected discounted utility by choosing how much to consume, $C_t$ , how much to invest in the stock market, $S_t$ , and how much to invest in riskless bonds, $B_t$ :

(1) $\begin{equation*} \begin{array}{rl} \text{maximize} & \Exp\left[ \, \sum_{t=0}^\infty \, \beta^t \cdot U_t \, \right] \\ \text{subject to} & \quad U_t = C_t^{1-\gamma} / \, (1-\gamma) \\ & W_{t+1} = (W_t - C_t) \cdot (1 + R_f) + S_t \cdot (R_{t+1} - R_f) \\ & \,\,\,\,\,W_t = C_t + B_t + S_t \end{array} \end{equation*}$

$\beta \in (0, \, 1)$ is investors’ subjective time preference, $U_t$ is their utility from consumption, $W_t$ represents their wealth, and $\gamma > 0$ is their coefficient of risk aversion. $(1+R_{t+1}) = (P_{t+1} + D_{t+1})/P_t$ is the gross return on the value-weighted stock market, and $(1+R_f)$ is the return on a riskless bond.

The CCAPM predicts that the expected excess return on stocks, $\Exp[R_{t+1}] - R_f$ , will be proportional to the covariance between consumption growth and stock returns:

(2) $\begin{equation*} \Exp[R_{t+1}] - R_f \approx \gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}] \end{equation*}$

The price of stocks is inversely related to the expected market return, $\Exp[1+R_{t+1}] = \Exp[P_{t+1} + D_{t+1}]/P_t$ . So, the CCAPM says that investors pay more for stocks when stock returns tend to offset negative consumption shocks, $\Cov[\Delta \log C_{t+1}, \, R_{t+1}] < 0$ . In other words, Equation (2) says CCAPM investors view the stock market as a way to insure future consumption shocks.

Equity Premium Puzzle

Unfortunately for the CCAPM, investors’ desire to hedge future consumption shocks can’t explain the entire $\Exp[R_{t+1}] - R_f \approx 8\%$ per year equity premium that we observe in the data on its own. To see why, consider rewriting the right-hand side of Equation (2) using the definition of a covariance:

(3) $\begin{equation*} \gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}] = \gamma \times \big( \, \rho \cdot \sigma_{\Delta \log C} \cdot \sigma_R \, \big) \end{equation*}$

The parameter $\rho = \Corr[\Delta \log C_{t+1}, \, R_{t+1}]$ is the correlation between consumption growth and stock returns, $\sigma_{\Delta \log C} = \Sd[\Delta \log C_{t+1}]$ is the volatility of consumption growth, and $\sigma_R = \Sd[R_{t+1}]$ is the volatility of stock returns. In the data, we observe values of roughly $\rho \approx 0.20$ , $\sigma_{\Delta \log C} \approx 1\%$ per year, and $\sigma_R \approx 16\%$ per year. Thus, investors would need a risk aversion of $\gamma = 250$ for the CCAPM to explain an $8\%$ equity premium.

A risk aversion of $\gamma = 250$ seems too high. But, if we assume a lower risk aversion of merely $\gamma = 10$ , the equity premium should only be $10 \times (0.20 \cdot 1\% \cdot 16\%) \approx 0.32\%$ per year according to the CCAPM. Given the power of compounding to increase investor wealth over long horizons, a difference of $8\% - 0.32\% = 7.68\%$ per year is a big deal. CCAPM investors with a $\gamma = 10$ should be putting much more money in stocks than they actually are, which would drive up stock prices and thereby lower the expected returns. So, if the basic logic of the CCAPM is correct, there must be something else about stock returns scaring investors away.

External Habit

Campbell-Cochrane (1999) argue that the something else is captured by a variable called the surplus consumption ratio. Their starts with Problem (1) and plugs in a modified utility function:

(4) $\begin{equation*} U_t = (C_t-X_t)^{1-\gamma} / \, (1-\gamma) \end{equation*}$

In this specification, investors care about their consumption in excess of the level of consumption they have become accustomed to, $X_t$ . This level corresponds to a weighted average of past consumption:

(5) $\begin{equation*} \log X_t = \lambda \cdot {\textstyle \sum_{\ell=0}^{\infty}} \, \phi^{\ell} \cdot \log C_{t-\ell} \qquad \qquad \lambda > 0, \, \phi \in (0, \, 1) \end{equation*}$

So, drops in consumption following prolonged periods of high consumption are extremely painful for investors. Conversely, an increase in consumption following a long hungry spell will be really enjoyable.

The surplus consumption ratio is $Z_t = (C_t - X_t) / C_t$ . The model says expected stock returns could be high either because stock returns covary with consumption growth or because they covary with growth in the surplus consumption ratio:

(6) $\begin{equation*} \Exp[R_{t+1}] - R_f \approx \gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}] + \gamma \times \Cov[\Delta \log Z_{t+1}, \, R_{t+1}] \end{equation*}$

Yes, the covariance between consumption growth and stock returns isn’t strong enough explain the $8\%$ equity premium on its own. But, stock market crashes tend to sucker punch investors, occurring just when investors’ consumption falls following a prolonged boom. $\Cov[\Delta \log Z_{t+1}, \, R_{t+1}] > 0$ explains nearly all of the $8\%$ per year equity premium according to this habit-formation model.

Long-Run Risk

Bansal-Yaron (2004) add a different state variable to the CCAPM. This model also starts with Problem (1) and plugs in a new utility function based on Epstein-Zin (1989) recursive preferences rather than habit formation:

(7) $\begin{equation*} U_t = \left\{ \, (1 - \beta) \cdot C_t^{1-\alpha} + \beta \cdot \big(\Exp_t[U_{t+1}^{1-\gamma}]^{\frac{1}{1-\gamma}}\big)^{1-\alpha} \, \right\}^{\frac{1}{1-\alpha}} \end{equation*}$

These preferences are recursive because they indicate that investors care not only about their consumption today, $(1 - \beta) \cdot C_t^{1-\alpha}$ , but also the present value of their expected future consumption, $\beta \cdot \big(\Exp_t[U_{t+1}^{1-\gamma}]^{\frac{1}{1-\gamma}}\big)^{1-\alpha}$ .

$\beta$ and $\gamma$ represent investors’ time preferences and risk aversion just like in the original power utility specification. The only new parameter is $\alpha > 1$ . The ratio $1/\alpha$ represents investors’ elasticity of intertemporal substitution (EIS). This parameter captures how much investors want to resolve future uncertainty about consumption, not because they want to do something with the information but because resolving uncertainty as soon as possible makes them happy. The guy on the subway platform who’s leaning dangerously out onto the tracks staring down the tunnel so that he can be the first to spot the next train is someone with a very high EIS. He wants to know as soon as possible if the next train is immanent, not because it will allow him to board sooner (everyone boards at the same time) but because knowing the train is about to arrive makes him happy. For a long-run risk model to work, we need $\gamma \cdot (1/\alpha) > 1$ .

Let $P_t$ denote the current price of an asset whose payout is aggregate consumption in the following period. The key new state variable in the long-run risk model is $\log Z_{t+1} = \log (P/C)_{t+1}$ . The model says that the equity premium will be determined as follow:

(8) $\begin{equation*} \Exp[R_{t+1}] - R_f \approx \gamma \times \Cov[ \, \Delta \log C_{t+1}, \, R_{t+1} \, ] + \mathrm{f}(\gamma, \, \alpha) \times \Cov[ \, \log Z_{t+1}, \, R_{t+1} \, ] \end{equation*}$

$\mathrm{f}(\gamma, \, \alpha) \leq 0$ is a function that stems from a Campbell-Shiller (1988) approximation of $Z_t$ . Thus, the long-run risk model says expected stock returns could be high either because future stock returns tend to covary with consumption growth or because these stock returns tend to covary with the future price-to-dividend ratio of the aggregate consumption claim. This price-to-dividend ratio will partly reflect current changes in consumption, $\Delta \log C_{t+1}$ . But, since consumption growth is persistent and investors have recursive preferences, it will also reflect future consumption shocks as well. In calibrations, most of the $8\%$ per year equity premium is explained by variation in $\log Z_{t+1}$ coming from consumption shocks far off in the future.

Source Of Confusion

What would it mean for investors not to care about consumption risk in one of these models? $\rho = \Corr[\Delta \log C_{t+1}, \, R_{t+1}]$ captures the stock market’s exposure to consumption risk. When $\rho \approx 1$ , stock market booms always coincide with increases in consumption. When $\rho = 0$ , knowing that the stock market is booming tells you nothing about whether aggregate consumption is increasing or decreasing. So, investors in a particular model would be indifferent to changes in consumption risk if

(9) $\begin{equation*} \partial_{\rho} (\Exp[R_{t+1}] - R_f) = 0 \end{equation*}$

In other words, they wouldn’t care about consumption risk if increasing the amount of consumption risk had no effect on their demand and thus no effect on equilibrium prices.

We saw above that, if we assume a risk aversion coefficient of $\gamma = 10$ , then the first term in Equations (6) and (8) is very small. $\gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}] \approx 0.3\%$ . And, as a result, the effect of an increase in consumption risk on asset prices coming from this first term is quite small as well:

(10) $\begin{equation*} \partial_{\rho} (\gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}]) = \gamma \times \sigma_{\Delta \log C} \cdot \sigma_R \approx 0.016 \end{equation*}$

Judged only by the effect of this initial term, an increase in consumption risk from $\rho = 0.00$ to $\rho = 0.40$ would only increase the expected excess return on the stock market by $0.016 \times 0.40 = 0.64\%$ per year. We observe a correlation between stock returns and consumption growth of $\rho = 0.20$ in the data. So, these numbers imply that a $2\times$ swing around the mean $\rho$ would explain less than a tenth of the total $8\%$ per year equity premium puzzle if consumption risk only affected asset prices via the $\gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}]$ term.

However, consumption risk doesn’t only affect asset prices via the $\gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}]$ term in Equations (6) and (8). Therefore

( $!!!$ ) $\begin{equation*} \partial_{\rho} (\gamma \times \Cov[\Delta \log C_{t+1}, \, R_{t+1}]) \approx 0 \qquad \text{does \underline{\textbf{not}} imply} \qquad \partial_{\rho} (\Exp[R_{t+1}] - R_f) \approx 0 \end{equation*}$

Such a conclusion would only be valid if the new state variables introduced in Campbell-Cochrane (1999) and Bansal-Yaron (2004) happened to be unrelated to consumption growth. This is absolutely not the case! Changes in the surplus consumption ratio are highly correlated with consumption growth. And, the long-run risk model assumes that consumption growth is very persistent, so price changes due to anticipated consumption shocks in the far distant future will be highly correlated with consumption growth today too.

Plugging In Numbers

How much does the Campbell-Cochrane (1999) model suggest expected excess returns should increase in response to a move from $\rho = 0$ to $\rho = 0.40$ ? Campbell-Cochrane (1999) talk about habit formation as “amplification mechanism for consumption risks in marginal utility. (page 240)” Mathematically, this shows up as a scaling up of the risk-aversion coefficient from $\gamma$ to $\gamma / \Exp[Z_t]$ . The authors use $\phi = 0.87$ . With $\sigma_{\Delta \log C} = 1\%$ per year and $\gamma = 10$ , the average surplus consumption ratio is $\Exp[Z_t] = \sigma_{\Delta \log C} \cdot \sqrt{\frac{\gamma}{1 - \phi}} \approx 0.088$ . So, in the external habit model, the effect of consumption risk on asset prices will be:

(11) $\begin{equation*} \partial_{\rho} (\Exp[R_{t+1}] - R_f) = (\gamma / 0.088) \times \sigma_{\Delta \log C} \cdot \sigma_R \approx 0.18 \end{equation*}$

Because increasing the stock market’s correlation with consumption growth must also increase its correlation with growth in the surplus consumption ratio, a $\Delta \rho = 0.40$ increase in consumption risk will increase the annual expected excess return on the stock market by $0.18 \times 0.40 = 7.30\%$ in a habit model.

How much does the Bansal-Yaron (2004) model suggest expected excess returns should increase in response to a $\Delta \rho = 0.40$ increase in consumption risk? Cochrane (2017) describes how this model “ties its extra state variables… to observables by the assumption of a time-series process in which short-run consumption growth is correlated with… long-run news.” When $1/\alpha \approx 1$ , the function $\mathrm{f}(\gamma, \, \alpha) = 1 - \gamma$ and Equation (8) can be re-written as:

(12) $\begin{equation*} \Exp[R_{t+1}] - R_f \approx \gamma \times \Cov[ \, \Delta \log C_{t+1}, \, R_{t+1} \, ] + (1 - \gamma) \times \Cov[ \, \log (P/C)_{t+1}, \, R_{t+1} \, ] \end{equation*}$

Changing an asset’s correlation with consumption growth also changes its correlation with the future log price-to-consumption ratio. I estimate $\log (P/C)_{t+1} = 3.61 - 30.26 \cdot \Delta \log C_{t+1} + \varepsilon_{t+1}$ , which would imply that $\partial_{\rho} (\Exp[R_{t+1}] - R_f) = [\gamma - (1-\gamma) \cdot 30.26] \cdot \sigma_{\Delta \log C} \cdot \sigma_R \approx 0.45$ . Thus, a $\Delta \rho = 0.40$ increase in consumption risk will increase annual expected excess returns by $0.45 \times 0.40 \approx 18\%$ in the long-run risk model!