Risks for the Long Run:Estimation and Inference

Ravi Bansal, Dana Kiku, and Amir YaronDraft 2007

Presented for Fall 2009 NYU Asset Pricing Seminar by Jason LevineSeminar by Jason Levine

IntroductionIntroduction• Empirically evaluate how well a long-run risk model can explain

asset returnsasset returns• Asset market data should be informative about investors’

behavior• Exploit the dynamics of aggregate consumption growth and p y gg g p g

Euler restrictions to solve for unobserved return on claim to consumption stream

• LRR Model sources of riskShort run risks in consumption– Short-run risks in consumption

– Long-run risk in consumption– Fluctuations in consumption uncertainty – consumption volatility risk

• Derive expressions for the IRMS (intertemporal marginal rate of p ( p gsubstitution) in terms of the risk sources for many combinations of parameters

Summary of ResultsSummary of Results• LRR component is highly persistent

LRR f f h i k i• LRR component accounts for most of the risk premium• Assets with large mean returns (value and small assets) are more

sensitive to innovations in the LRR variable and news about economic uncertaintyy

• Annual data leads to RA> 15 and IES <1• Adjusting for time averaging can lead to a more reasonable RA

near 10 and IES near 2Ti i d fi it l ff t h t ff t• Time averaging and finite sample effects have strong effects. Decision intervals should match up with sample frequency.

• The model proposed is not rejected • LRR model prices time series and cross sectional variations wellLRR model prices time series and cross sectional variations well

(even in some variations where RA and IES parameters are not in the proper range)

Some Supporting ResearchSome Supporting Research• Asset market data contains information about

investors’ behavior– Cochrane and Hansen (1992)

• Development of long-run risks modelDevelopment of long run risks model– Bansal and Yaron (2004) – can account for risk-free rate,

equity premium, and volatility puzzles• Exploit Asset Pricing Euler Equations through GMMExploit Asset Pricing Euler Equations through GMM

– Hansen and Singleton (1982)• LRR Model

K d P t (1978)– Kreps and Porteus (1978)– Epstein and Zin (1989)– Weil (1989)

OutlineOutline

• ModelModel• Data• Empirical Analysis• Empirical Analysis• Simulation Comparison

SMM b d ti t• SMM based estimates• Implication of using the market return in

th i i k lthe pricing kernel• Concluding Remarks

ModelModel• Variables

– 0 < δ < 1 reflects agent’s time preferences– γ = coefficient of risk aversion– ψ = elasticity of intertemporal substitution

Model (2)Model (2)

UtilityUtility

Budget Constraint

Consumption DynamicsConsumption Dynamics

• ∆ct+1 = growth rate of log consumption• xt = consistent component that captures long-run

i k i i hrisks in consumption growth• ρ=persistence in the conditional mean of

consumption growthp g• time varying volatility in consumption leads to time-

varying risk premia (same as Bansal and Yaron)

Euler EquationEuler Equation

• m = log of IMRSm = log of IMRS• r = log of gross return on asset j

Long Run Risk Model’s IMRSLong Run Risk Model s IMRS

• r t 1 is the continuous return on therc,t+1 is the continuous return on the consumption asset

IMRSIMRS

• Barred variables are means

Log-price Consumption RatioLog price Consumption Ratio

Solution for ASolution for A

• A1 >0 IES parameter > 1• A1 >0 IES parameter > 1• For the price-consumption ratio to respond negatively to an

increase in economic uncertainty, IES > 1 assuming risk aversion >1>1

• Can solve numerically:

Return to IMRSReturn to IMRS• Plug back ing

This is correct other than the approximation error from the linearization around the theoretical value of the average price-consumption ratio. Empirically, this error is small.

Risk premium for an assetRisk premium for an asset

• This is based on the IMRS expressionE h β i ith t t th ith f i k i ζ f• Each βi,j is with respect to the ith source of risk in ζt+1 for asset j

• λi is the ith entry in the vector of market prices of risks Λλi is the ith entry in the vector of market prices of risks Λ

Special Case note: IES=1Special Case note: IES 1

• The case where the IES=1 comesThe case where the IES=1 comes naturally as a limit of the general formulas already discussedformulas already discussed.

• The IES can be estimated as a free parameterparameter.

Recovering the State VariablesRecovering the State Variables• Long-run risk component, xt, can be identified by g p , t, y

regressing consumption growth on the risk-free rate and market price-dividend ratio

2 i i il d ti id l• σ2t is similar, regress squared consumption residual

on the same variables• This identifies short-run consumption risk ηThis identifies short run consumption risk η

Recovering the State Variables (2)Recovering the State Variables (2)

• The shocks can be recovered by fitting independent y g pAR(1) dynamics

• All components needed to construct the IMRS are fully bl hrecoverable as shown.

DataData• Summary

– Time period – 1930 – 2002– Long span should give more reliable statistical inference– Annual frequency

P tf li• Portfolios– Market– Riskless asset (3-month t-bill less 1-year expected inflation to

adjust to real terms)adjust to real terms)– Size based (sorted in deciles)– Book-to-market (value versus growth) (sorted in deciles)– Portfolio CompositionsPortfolio Compositions

• Value-weighted monthly returns• per-share price and dividend series (Campbell and Shiller (1988),

Bansal, Dittmar, and Lundblad (2005), and Hansen, Heaton, and Li (2005))Li (2005))

Data SummaryData Summary

Consumption DataConsumption Data

• Seasonally adjusted per-capita data onSeasonally adjusted per capita data on real consumption and GCP from the NIPA tables on the Bureau of Economic Analysis website

• Aggregate consumption = consumer gg g pexpenditures on non-durables and services

• Growth rates = first difference of log series

Empirical ResultsEmpirical Results

Evidence on Consumption Growth d U iand Uncertainty

Consumption growth is highly predictable as indicated by the R2.

Returns and BetasReturns and Betas

• Use full set of 10 deciles for both value/growth and sizeand size

• These three betas explain 84% of the cross-sectional variation in mean returnssectional variation in mean returns.

consumption, long-run growth, consumption uncertainty, standard CCAPM

Euler Equation Estimate EvidenceEuler Equation Estimate Evidence

• Main results of the paperMain results of the paper• Attempt to estimate Risk Aversion and IES• Discount rate (delta) = 0 987Discount rate (delta) = 0.987• Risk aversion around 15, IES <0.5

– Not great but generates only small pricing errorsNot great, but generates only small pricing errors, cannot reject model since p-values are high

• Notice how the Small and Value portfolios are pmispriced in one direction, and Large/Growth in the other direction

Robustness TestRobustness Test• To check robustness, use a richer set of predictive p

variables– two-year moving average of lagged consumption growth– log of consumption-to-GDP ratiog p– aggregate market price-dividend ratio– short interest rate– default premiumdefault premium

• results– adjusted R2 of 37%

persistence parameter ρa =0 67 is large– persistence parameter ρa =0.67 is large– confirms the low frequency dynamics of consumption growth

Decision-Interval and Time A iAveraging

• Sample Frequency is assumed to match up with ti d i i f t b th b tconsumption decision frequency, may not be the best

assumption• Time averaging to annual data led to a downward bias

i th IESin the IES• Bias corrected values are more consistent with

economic implications of recursive-preferences based modelsmodels

– Risk aversion ≈ 10– IES ≈ 2

• Estimation is redone assuming agents make monthly• Estimation is redone assuming agents make monthly decisions, but sampling is annual

– Monthly consumption is replaced by a 12-month rolling average of consumptionaverage of consumption

Simulation: Consumption, Di id d d A RDividends, and Asset Returns

– d = dividendsdi id d th– μ = mean dividend growth

– Φ = leverage on persistent consumption– η = i.i.d. consumption shock– u = dividend news (allow for cross-sectional correlation)

Simulation DataSimulation Data

• 876 months (73 annual observations)876 months (73 annual observations)• 500 paths

i k i 10• risk aversion = 10• IES = 2

Table infoTable info

• Panels VIII A and IX A show parametersPanels VIII A and IX A show parameters• Panel VIII B shows a good match on the

dynamics of consumption growthdynamics of consumption growth• Panel IX B mostly good

– For Small and Value portfolios modelFor Small and Value portfolios, model volatility is smaller than data (exception made because the extreme volatility driven b f d t i t )by a few data points)

• Table X shows small pricing errors

Finite-Sample Biases and Time A i Eff E i iAveraging Effects on Estimation

• Implicationsp– Medians of risk aversion and IES are close to those estimated

in table IV• the RA is large, and right-skewed• IES < 1• Despite IES=2 and RA=10 in simulations

– Model not rejected, J-stat close to that estimated from data– Table XI comes close to table IV, providing additional support– The biases in IES and RA come from average and finite

sample effectsP V l i f l l l– Pop-Values is from a long annual sample

• better values for RA and IES• misspecification detected due to large J-stat and small P-value

SMM based estimatesSMM based estimates

• SMM procedure of Duffie and Singleton (1993)SMM procedure of Duffie and Singleton (1993) can account for time averaging effects

• The pricing errors are small and near zero p gsince the standard errors are large

• Results in good IES and RA valuesg

Implication of incorrectly using the k i h i i k lmarket return in the pricing kernel

• Use simulated data to estimate the model with market-return based pricing kernel

• Risk aversion is too low, partly due to large volatility of pricing kernel

Concluding remarksConcluding remarks• Develops methods for estimating the LRR p g

model of Bansal and Yaron (2004)• LRR model works well to capture time-series

and cross-sectional variation in returnsand cross-sectional variation in returns• Drawback is the how sensitive IES and Risk

Aversion are to time averaging and finite l bi Bi id l tisample biases. Biases provide an explanation

of large RA and low IES of previous papers.• The paper confirms that long-run risks are e pape co s t at o g u s s a e

more important than short-run risks for pricing assets.