Model Implications and Results

1.4.1 Equilibrium Discount Factors

The stochastic discount factor in Equation (1.5) is highly nonlinear becauseZa,t+1 will be

a recursive function of the model primitives, including the state variablest. As a result, we

do not have closed-form expressions for innovations in the SDF. However, our parsimonious setup allows us to solve the model exactly without invoking a log-linear approximation and simulation, which means we can look at the state contingent SDF being generated directly from the model. Step by step details of our numerical solution method are outlined in Appendix A.4. Let M denote the equilibrium stochastic discount factor generated by the model where each row and column corresponds to one of the model’s five states at time t

and t+ 1 respectively. Under our baseline calibration, we obtain:

Mt,t+1=                s1 s2 s3 s4 s5 s1 0.83 1.38 2.63 0.52 0.54 s2 0.68 1.13 2.15 0.42 0.44 s3 0.03 . 1.53 . 0.01 s4 1.46 . . 0.92 0.96 s5 2.08 . . . 0.73                . (1.9)

Recall the five states of the model:

{s1, s2, s3, s4, s5}={(gh, γ0, ψ0),(g`, γ0, ψ0),(g`, γelev, ψ0),(g`, γ0, ψdepr),(g`, γelev, ψdepr)}.

In consumption-based asset pricing models, the equilibrium discount factor has a direct relationship with investor marginal utility as discount factors for state contingent claims in a complete market will reflect the relative marginal utility the investor faces in each possible state of the world. More precisely, investors will pay a premium for state contingent assets

that insure them against bad states of the world that pay them when marginal utility is high. Likewise, investors will demand a discount for state contingent assets that pay them in good states of the world when marginal utility is low. Looking at Matrix (1.9), in times of regular preferences, and in recessions with only elevated risk aversion (state transition paths betweens1,s2ands3), discount factors behave as expected under the standard model.

Investors are willing to pay a premium (1.38, 1.13, 2.63, 2.15, 1.53) and demand a discount (0.83, 0.68, 0.03) for a claim in an expansion when marginal utility is low with the premiums and discounts being higher under elevated risk aversion.

However, in recessions with a depressed EIS parameter, the SDF seems to price assets counter to the standard model with discounts demanded to hold contingent claims paying off in states s4 and s5 when consumption growth is low. Although this result may seem

counterintuitive, it turns out that it is not because investors in the model consider states

s4 and s5 to be states oflow marginal utility from the perspective of time t, even though

consumption growth is low. To see why, we will compare states s2 = (g`, γ0, ψ0) ands4 =

(g`, γ0, ψdepr) that differ only in thats4is a state of depressed EIS. Recall that the smoothing

risk relates to uncertainty about what the investor’s optimal lifetime consumption profile should look like when making consumption and savings decisions at time t; whereas, the consumption risk channel affects the investor’s ability to smooth consumption conditional on having a particular preference for consumption smoothing. When consumption growth is more persistent, this increases the investor’s ability to plan for future periods relative to, say iid consumption. All else equal, an investor with a stronger preference for smoothing consumption should prefer these states of the world. Note that the transition matrix (1.2)

generated by the model calibration is given by Πt,t+1=                s1 s2 s3 s4 s5 s1 0.735 0.220 0.012 0.032 0.002 s2 0.265 0.611 0.032 0.087 0.005 s3 0.265 0 0.643 0 0.092 s4 0.265 0 0 0.698 0.037 s5 0.265 0 0 0 0.735                . (1.10)

From time tperspective, s2 and s4 have a probability of 73.5% of remaining in a low con-

sumption growth state, implying identical consumption persistence if the economy stays in these relative states. However, in spite of the fact that these states are identical aside from ψt and have identical consumption persistence over low consumption growth states,

the model implies from Matrix (1.9) thats2 is a state of high marginal utility whiles4 is a

state of low marginal utility. The only way this can happen is if investors are better off in utility terms when persistence in consumption growth is high when ψt =ψdepr relative to

states whereψt=ψ0. If this is the case, then we would expect discount factors for s2 and s4 to diverge as consumption persistence increases. We can test this directly by solving the

model for a grid of ρC and reporting the discount factors for s2 and s4 if the economy is

in states2 at timet(the second row of Matrix (1.9)). These results are shown in Figure 1

and we see that indeed, the discount factors diverge as consumption persistence increases. The intuition of this result is that if consumption is persistent and investors have a stronger preference for consumption smoothing, they will be better able to plan for future periods in a way that will give them higher overall utility relative to states where they have a weak preference for smoothing.

Given that statess4 = (g`, γ0, ψdepr) ands5 = (g`, γelev, ψdepr) are low marginal utility states

than whens1 = (gh, γ0, ψ0) for low marginal utility states since consumption growth is high

in this state. The only way for this to be true is if marginal utility is more sensitive to the smoothing risk than consumption risk. Or equivalently, the proportion of equilibrium asset prices explained by the smoothing risk investors face is larger than that of consumption risk. To determine if this is true, we solve for the equilibrium discount factors for s4 and s5 if the economy is in state s1 at time t (the first row of Matrix (1.9)) for values of con-

sumption growth volatility, the EIS offset parameter kψ and the risk aversion scale factor kγ holding the other parameters fixed under the baseline calibration in each case. Varying

the size of the parameterskψ andkγ will lead to larger fluctuations in risk aversion and EIS

in the model. We also normalize all discount factors by the discount factor in s1 for ease

of comparison. The results of this exercise are shown in Figure 2. It is clear from the fig- ure, comparing across panels, that discount factors, hence marginal utility and asset prices, are much more sensitive to fluctuations in the EIS parameter than shocks to consumption growth or fluctuations in risk aversion. This latter point is consistent with our previous discussion that risk aversion will just scale up the effect of EIS fluctuations in states where both shift to irregular levels. This can be seen in Panels (c) as risk aversion magnitudes increase, for a fixed value of the EIS offset parameter, discount factors are shifted up and the line fors5 lies strictly above the line for s4.

Given the discussion so far, it is clear where the smoothing risk enters discount factors. The EIS measures how averse an investor is at time t to consumption variations across future

time periods along a deterministic path of consumption growth. In our model, there is some

uncertainty about the “right amount” of consumption variation given that an investor’s preference for smoothing might increase with some small probability das their EIS falls in some future period. As argued above, from the perspective of timet, states where this event is realized are not “bad” states per-se because this increased desire to smooth consumption is complemented by persistence in consumption growth. However, once the investor is in this state of an increased desire to smooth consumption, uncertainty about their ability to

plan for the future will be seen as a risk. This is why investors are willing to pay a premium for the s1 state contingent claim conditional on being in one of the rare states s4 ors5, to

insure themselves against this smoothing risk.

1.4.2 Aggregate Asset Price Moments

In Table 2, under our baseline calibration, we report the model implied first and second moments and autocorrelations for equity returns, the risk-free rate, the equity premium, price-dividend ratios, and the dividend yield as well as the Sharpe Ratio. Because we are able to solve the model exactly, we do not rely on simulation or estimation to produce moments from the model. This means the model moments reported in Table 2 and elsewhere are population moments and are computed without sampling error. Also, since we do not assume lognormal returns, all the returns, prices, and dividends reported in Table 2 are exact (not transformed on a log basis) and our model can produce both price-dividends and the dividend yield. Along with the population moments implied by the model, we report the corresponding estimates for these moments from annual data over the sample period 1930-2013. As shown in Table 2, the model produces values for all reported moments within the 95% confidence interval (and in many cases within one standard error) of the data estimates with the exceptions of the price-dividend and dividend yield volatilities and the first order autocorrelations of price-dividends. In particular, the model does a good job matching the equity premium and Sharpe Ratio while simultaneously producing both a low expected risk free rate and high enough volatility of the risk free rate to match the data. This feature is something that is typically difficult to generate in the class of representative agent asset pricing models without assuming unreasonable levels of risk aversion. To further understand how the model is generating these features, we can look at the model implied

equity and risk-free returns as well as expected returns to equity.                Rf s1 1.0380 s2 1.0235 s3 1.0057 s4 0.9419 s5 0.9179                               Et[Rm] s1 1.0828 s2 1.0904 s3 1.4763 s4 0.9451 s5 0.9335                               Et[Rm−Rf] s1 1.0448 s2 1.0669 s3 1.4706 s4 1.0032 s5 1.0156                               π s1 0.50 s2 0.28 s3 0.04 s4 0.13 s5 0.04               

The first thing to notice is that relative to periods of high consumption growth (s1) if the

economy is in a recession (s2) then demand for the risk-free asset increases as agents re-

balance their portfolios toward less risky securities. This pushes the price of the risk-free asset up and its net return down, as we see comparings1 and s2 in theRf vector. As ex-

pected, the inverse relationship shows up in expected equity returns as this rebalancing has the opposite effect on equity prices. If an exogenous shock elevates risk aversion (s3) then

these effects are only magnified because higher risk aversion induces even more portfolio rebalancing toward less risky securities driving the net return of the risk-free asset further down and expected equity returns even higher.

However, if there is an exogenous shock that depresses the EIS (s4 ands5) so that investors

have a stronger preference for smoothing consumption returns for both the risk free rate and expected equity returns fall. The reason is that all assets in the economy are vehicles for transferring consumption across future periods, even risky ones. Therefore, even though consumption growth is low, the investors prefer even smoother consumption and are will- ing to buy assets that will achieve this goal. Investor demand pushes prices of both the risk-free rate and equities higher and their preference for smoothing in these periods is so strong that they are willing to accept negative returns to ensure a smooth consumption

profile over future periods. Hence, investors end up paying a premium in the form of lower returns to transfer consumption into these rare states of depressed EIS.

As argued above, elevated risk aversion only amplifies the effect of a depressed EIS on dis- count factors so that if risk aversion is also elevated, demand for the risk-free asset is even higher, pushing its price up and return down. This can be seen comparing the gross risk-free return of 0.9419 ins4 (a net loss of about 5%) with the smaller gross return of 0.9179 ins5

(a net loss of about 8%). However, this accelerated increase in the price of the risk-free asset would make the price of the risky asset that pays off in state s5 relatively more attractive

than the one that pays off in state s4. Therefore, equity prices go up slightly in s5 relative

to s4 taking pressure off the risk-free rate, which is consistent with the expected equity

returns being lower in states5 than in states4.

The dynamics just described generate volatility across states in the risk-free rate and ex- pected returns and therefore, in the equity premium. As expected, ins3 investors demand

a very high risk premium to hold equities because low demand for risky assets in periods of elevated risk aversion drives prices down and investors must be compensated for this risk in the form of higher returns. In states with depressed EIS, we see that investors demand less of a risk premium than in the other three states even though the economy is in a recession due to a stronger preference for consumption smoothing, which increases demand for both risk-free and risky assets. Overall, these effects generate variation in the equity premium but with reasonable average levels for risk aversion because these states of irregular preferences happen very rarely in the model. The steady state probabilities in theπ vector show that the model spends almost 80% of the time in states where investors have regular preferences. We only require rare and temporary periods of irregular preferences to generate the model’s equity premium of 6.21.

Price Dividend Volatility

Although the model does a good job in matching the first moments of price-dividend ra- tios and dividend yields, the most obvious area where the model struggles is in generating enough volatility in these variables. Counterfactually low price-dividend volatility is an issue that the long-run risks models of Bansal and Yaron (2004) and Bansal et al. (2012) also struggle with, so despite this weakness our model is in good company. Those models also use recursive preference specifications but very different consumption and dividend dy- namics than our model assumes.

There are a few things we might do to improve the model’s ability to generate price-dividend volatility. First, our model has a very small state space and expanding the state space for consumption and dividend growth could potentially add additional variation that the cur- rent model is unable to capture. Second, the dividend growth process we have specified (leveraged consumption growth) is too auto-correlated and too strongly cross correlated with consumption growth (it is equal to 1 by construction) relative to dividends data. This strong correlation between dividend growth and consumption growth results in the model producing an unconditional contemporaneous correlation of 0.46 between excess returns and consumption growth, which is much too high relative to the low correlation found in the data. However, even though we do not assume separate processes for consumption and dividends, our model implied value of 0.46 is less than the value of 1.0 produced by the standard time-separable model and close to the value of 0.47 produced by Campbell and Cochrane (1999) from simulations at an annual frequency.23 The reason this correlation is not 1.0 in our model is that some of the variation in returns is being explained by varia- tion in preference parameters that are not directly tied to changes in consumption growth (transitioning between states 2-5 in the model). This counterfactual result, as Cochrane and Hansen (1992) point out, is a major factor in the empirical failures of the consumption-based

23_{The models of Barberis et al. (2001) and Bansal and Yaron (2004) produce a contemporaneous correlation}

asset pricing model. Relaxing this unrealistic restriction in future iterations of the model would introduce more dividend volatility into the model and could improve the model’s fit.

Alternative Calibrations

Given the model’s ability to match key asset pricing moments, it is useful to look at a few alternative calibrations to reveal what features of the model are responsible for this success. In Table 1 we presented several different alternative calibrations to our baseline calibration. One point of debate between proponents of either long-run risk models following Bansal and Yaron (2004) or habits models following Campbell and Cochrane (1999) is whether or not consumption growth is independently and identically distributed (iid). The long-run risks model assumes a predictable, long run component in consumption growth, while the habits model assumes consumption is a random walk (ρC = 0). In our baseline calibration, we do

not assume consumption is a random walk, we calibrate ρC to the data sample we have.

However, in Table 1 we specify an alternative calibration (4) of the model withρC = 0 that

is otherwise nearly identical to our baseline calibration with the exception that regular level of risk aversion is calibrated to be 7.5 and the risk aversion scaling parameterkelev = 4.93.

We report the model fit in Table 3. As shown in the table, the model performs just as well under the assumption of iid consumption growth with the exception that the volatility of the risk-free rate is too high. The takeaway from this exercise is that the particular nature of the consumption growth dynamics being assumed as either iid or having a predictable component is not crucial for our model to fit the data.

Calibrations (1)-(3) in Table 1 are special cases of the baseline calibration that maintain the exact same calibration as the baseline but shut down variation in the EIS parameter, risk aversion or both. Calibration (1) sets b = 0, which shuts down time-variation in the risk aversion parameter and only allows the EIS parameter to be time-varying in order to highlight the importance of variation in the EIS. Comparing this calibration to the baseline,