Disaster risk and its implications for asset pricing Online appendix

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1 Disaster risk and its implications for asset pricing Online appendix Jerry Tsai University of Oxford Jessica A. Wachter University of Pennsylvania December 12, 2014 and NBER A The iid model This section derives some useful results for the iid model. The utility is given by V t = E t fc s, V s ) ds, t A.1) where fc, V ) = β 1 1 ψ C 1 1 ψ 1 γ)v ) 1 1 γ)v ) 1 1, A.2) and = 1 γ)/1 1 ). For ψ = 1, we assume ψ fc, V ) = β1 γ)v log C 1 ) log1 γ)v ). A.3) 1 γ Tsai: Department of Economics, Oxford University and Oxford-Man Institute of Quantitative Finance ; jerry.tsai@economics.ox.ac.uk; Wachter: Department of Finance, The Wharton School, University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104; Tel: 215) ; jwachter@wharton.upenn.edu. 1

2 Proposition 1. When consumption growth is iid, along the optimal consumption path, utility satisfies V t JC t ) = j C1 γ t 1 γ. where j > 0 and given by j = ) µ + 12 β ψ γ 1 1 ) σ 2 1 ψ λ E ν e 1 )). A.4) Proof of Proposition 1 Conjecture: JC t ) = j C1 γ t 1 γ, A.5) where = 1 γ)/1 1 ψ ). For convenience, let J t = JC t ). The HJB equation is: DJ t + fc t, J t ) = 0. Plug A.5) into A.2) we get: fc t, J t ) = βj t j 1 1 ], A.6) and by Ito s Lemma: DJ J = 1 J J C Cµ ) 2 J C 2 C2 σ 2 + λe ν JC e Z t ) JC) ]. A.7) The HJB can be rewritten as: 0 = 1 1 ) µ 12 ψ γ 1 1 ) σ ψ λ E ν e 1 + βj 1 β. A.8) 2

3 Solving this, we get j = ) µ + 12 β ψ γ 1 1 ) σ 2 1 ψ λ E ν e 1 )). Proposition 2. When consumption growth is iid, π t C γ t. That is, the state-price density is proportional to C γ t with a positive constant of proportionality. Proof. It follows from the form of fc t, V t ) that f C = β C 1 ψ t 1 1 ψ 1 γ)v t ) 1 1 C 1 ψ t C 1 γ) 1 1) t C γ t. where the second line follows from Proposition 1. Proposition 3. When consumption growth is iid, the riskfree rate is equal to r = β + 1 ψ µ 1 2 γ + γ ) σ 2 + λe ν e γzt 1) ) ] e 1 γ)zt 1). A.9) ψ Proof of Proposition 3 In the i.i.d case: f C C t, V t ) = βc γ t j 1 1 and ) f 1 V C t, V t ) = β. j 1 3

4 Therefore, π t = exp t 0 β + 1 ) ) ds β C γ j 1 t k 1 A.10) By Ito s Lemma: µ π = β + 1 ) γ C j 1 t µ + 1 C t 2 γγ + 1) C 2 Ct 2 t σ 2 A.11) Substituting j using A.4) in the main text and rearrange: µ π = β 1 ψ µ γ ) σ ) λe ν e 1 ]. ψ A.12) It also follows from no-arbitrage that r = µ π λe ν e γzt 1] = β + 1 ψ µ 1 γ + γ 2 ψ ) σ 2 + λe ν e γzt 1) ) e 1 )]. Proposition 4. When consumption growth is iid, and dividend is D t = C φ t dd t D t = µ D dt + φσ db t + e φzt 1) dn t. Let S t denote the price of the dividend claim,then the price-dividend ratio is given by S t = E t D t = t π s π t D s D t ds β µ D + 1 ψ µ 1 2 γ + γψ 2φγ ) σ 2 + λe ν 1 1 ) e 1 ) ] ) 1 e φ γ)zt 1). Proof of Proposition 4 Section A in the Appendix in the main text gives the general 4

5 no-arbitrage condition: µ π,t + µ S,t + D t S t + σ π,t σ S,t + λ t E ν e Z π+z S 1 ] = 0. A.13) Conjecture: S t D t = G, therefore S t = GD t and by Ito s Lemma: ds t S t = µ D dt + φσ db t + e φzt 1) dn t. A.14) In the iid case therefore π t C γ t, σ π = γσ, Z π = γz t. A.15) A.16) Substituting A.12) and A.14)-A.16) into A.13) implies β 1 ψ µ γ ) ψ σ 2 + µ D + G 1 γφσ ) λe ν e 1 ] + λe ν e φ γ)z t 1 ] = 0. Rearranging this verifies the conjecture and the dividend-price ratio is: G 1 = β µ D + 1 ψ µ 1 2 γ + γψ 2φγ ) σ 2 + λe ν 1 1 ) e 1 ) e φ γ)zt 1 )]. 5

6 B Data and results for time-varying disaster probabilities model We use annual data from 1948 to The aggregate market data come from CRSP. The market return is the gross return on the NYSE/AMEX/NASDAQ value-weighted index. Dividend growth is computed from the dividends on this index. The price-dividend ratio is price divided by the previous 12 months of dividends to remove the effect of seasonality in dividend payments in computing this dividend stream, we assume that dividends on the market are not reinvested). For the government bill rate, we use real returns on the 3-month Treasury Bill. We compute market returns, dividend growth, and government bill returns in real terms by adjusting for inflation using changes in the consumer price index also available from CRSP). We also compute consumption growth using real, per capital expenditures on non-durables and services for the U.S., available from the Bureau of Economic Analysis. Table A.1-A.6 reports results for the time varying disaster probability case. Parameters are as follows: using the 15% cutoff for disasters we set the average disaster probability λ = , and assume that there is a 40% probability of default on government bills in the case of a disaster. Discount rate β = 0.01 to match the low risk-free rate. Normal time consumption growth and volatility are set to match the postwar data, µ = , and σ = We set dividend growth µ D = 0.04 to get realistic price-dividend ratios and idiosyncratic volatility σ i = 0.05 to match the postwar dividend growth volatility. We set risk aversion γ = 3 and leverage φ = 3 to match the equity premium and return volatility. The mean-reversion κ λ = 0.12 to match persistence for price-dividend ratio. We look at six cases time additive utility EIS = 1/3), EIS = 1 and EIS = 2, for each of the three cases, we compare results with time-varying disaster probability, σ λ = , to ones with constant disaster probability, σ λ = Given other parameters and disaster distribution, the existence of the value function imposes a constraint on κ λ and σ λ. In particular, the disaster probability process cannot be highly persistent and too highly volatile at the same time. We choose κ λ and σ λ to best to match both the volatility of stock returns, the persistence of the price-dividend ratio, and the volatility of Treasury Bill returns 2 In order to solve the model using the general case, ψ can not be exactly one and σ λ can not be exactly 6

7 We simulate monthly data for 600,000 years to obtain population moments. We also simulate 100, year samples. In the tables, we report population values for each statistic, percentile values from the small-sample simulations, and percentile values for the subset of small-sample simulations that do not contain jumps. It is this subset of simulations that is the most interesting comparison for postwar data. C Production economy This section solves the two models with endogenous consumption choice are considered by Barro 2009). The first one allows for labor-leisure choice, and productivity is subject to disaster shocks, we will show that this case is equivalent to the iid endowment case. Then we will consider another case with capital accumulation. C.1 Recursive utility in discrete time Assume that continuation utility takes the following form: V t = 1 e β )C 1 1 ψ t + e β E t V 1 γ t+1 ] ) ] ψ 1 ψ 1 1 γ, C.1) with intertemporal marginal rate of substitution IMRS): ) 1 M t+1 = e β Ct+1 ψ C t V t+1 E t V 1 γ t+1 ] 1 1 γ ) 1 ψ γ. For these preferences, Epstein and Zin 1989) show that the intertemporal marginal rate of substitution IMRS) can be expressed as: ) M t+1 = e β Ct+1 ψ R 1 W,t+1 C, t C.2) zero, they are numbers very close to one and zero and 10 8, respectively). Since σ λ is not exactly zero, the autocorrelation of price-dividend ratios in Table A.4 and A.6 are not all equals one. 7

8 where = 1 γ)/1 1 ψ ), and R W,t+1 is the gross return on an asset that delivers aggregate consumption as its dividend. C.2 Leisure-labor choice This is the first case in Barro 2009). In this model, output is given by Y t = A t L α t, C.3) where A t is productivity and L t is the quantity of labor employed. Here, assumptions on the endowment process are replaced by assumptions on productivity, so that A t is given by: log A t+1 A t = µ + ɛ t if there is no disaster at t + 1 Z t+1 if there is a disaster at t + 1 C.4) where ɛ t+1 is an iid standard normal shock and disaster occurs with probability 1 e λ. We modify C.1) by replacing C t with period utility that allows for preferences over leisure: U t = C t 1 L t ) χ. C.5) As we show in Section 6.1, a constant share α) of the consumption comes from labor income, and the rest 1 α) comes from dividend. Therefore, consumption/output growth and dividend growth both equal to technology growth: log C t+1 C t = log D t+1 D t = log Y t+1 Y t = log A t+1 A t. 8

9 That is, log consumption growth process is the same as log productivity process 3 : 0 with prob. e λ log C t+1 = log C t + µ + ɛ t+1 + Z t+1 with prob. 1 e λ. C.6) Period utility depends on both consumption and leisure in this model, as stated by C.5). In the iid case, however, labor share L t, is constant over time, it follows that the IMRS only depends on consumption growth and return on consumption wealth. Conjecture the consumption-to-wealth ratio is constant, that is, C t /W t = cw for all t. The budget constraint for the representative agent: W t+1 = R W,t+1 W t C t ), can be written as R W,t+1 = 1 C t+1. C.7) 1 cw C t And we can rewrite the IMRS by plugging C.7) into C.2): ) γ ) 1 M t+1 = e β Ct+1 1. C.8) C t 1 cw The Euler equation implies that the consumption-wealth ratio is determined by ] W t+1 C t+1 E t M t+1 = W t 1, C t+1 C t C t or, ] C t+1 1 E t M t+1 = 1 C t cw cw 1, 3 Notice that this specification is almost the same as the one in the continuous time endowment economy. The only difference is the interpretation of µ. In normal times, log consumption growth is normally distributed with mean µ in this discrete time model, and it is µ 1 2 σ2 ) t in the continuous time model. 9

10 Substituting C.8) into the IMRS and rearrange: ) ] 1 γ 1 cw) = E t e β Ct+1 C t { = exp β + 1 γ)µ + 1 } e 2 1 γ)2 σ 2 λ + 1 e λ )E ν e. Therefore, log 1 cw) = β ) µ ψ 2 1 γ)2 σ log e λ + 1 e λ )E ν e, C.9) which verifies the conjecture. We are interested in the value when the time interval goes to zero. Since λ scale with the time interval, in the limit, λ is close to zero and: 1 e λ λ, moreover, for x close to zero, log1 + x) x x=0x = x. Therefore the last term in C.9), log e λ + 1 e λ )E ν e log 1 λ + λeν e λeν e 1 ]. In the limit, Equation C.9) can be written as: log 1 cw) β ) µ ψ 2 1 γ)2 σ λe ν e 1 ]. C.10) Next we can solve for the risk-free rates and returns on consumption claim in the economy, 10

11 and we will focus on the limit when time interval approaches zero: log R f t+1 = log E t M t+1 ] = β + γµ 1 2 γ2 σ 2 log e λ + 1 e λ )E ν e γz t + 1) log1 cw) = β + 1 ψ µ 1 2 σ2 γ 2 1 ) 1 γ) 2 log e λ + 1 e λ )E ν e γz t 1 + log e λ + 1 e λ )E ν e β + 1 ψ µ 1 2 σ2 γ 2 1 ) 1 γ) 2 λe ν e γz t 1 ] + 1 λe ν e 1 ] = β + 1 ψ µ 1 2 σ2 γ + γ ψ 1 ψ ) + λe ν e γzt 1 ) + 1 e 1 )]. C.11) Comparing with the continuous time version, there is an extra ψ σ2 in the risk-free rate, and this is because the mean of log consumption growth is µ instead of µ 1 2 σ2. The return on consumption claim is given by C.7): log E t R W,t+1 ] = µ 1 2 σ2 + log e λ + 1 e λ )E ν e Z t log1 cw) = β + 1 ψ µ σ2 1 ) 1 γ2 ) + log e λ + 1 e λ )E ν e Z t 1 log e λ + 1 e λ )E ν e β + 1 ψ µ σ2 1 ) e 1 γ2 Z ) + λe t ν 1 ) 1 e 1 )]. C.12) Next we can calculate the risk premium for the consumption claim. Using C.11) and C.12): log E t R W ] log R f = γσ 2 λe ν e γz t 1 ) e Zt 1 )]. C.13) Notice that the equity premium on consumption claim in the continuous time limit is the same as the one we obtain in the iid endowment economy Section 3.1). 11

12 C.3 Capital accumulation In the second case of Barro 2009), output is given by: Y t = AK t, where K t is the capital stock. Productivity A is constant in this model while capital evolves according to: K t+1 = K t + I t δ t+1 K t, where δ t+1 is depreciation. Depreciation has a normal time and disaster component: 0 if there is no disaster at t δ t+1 = δ + 1 e Zt if there is a disaster at t. First, conjecture a constant investment-to-capital ratio ζ = I t /K t, consumption growth equals capital growth, thus investment growth: C t+1 C t = A ζ)k t+1 A ζ)k t = K t+1 K t = 1 + ζ δ t+1. C.14) Similar to the previous case, we can conjecture and verify that the consumption-wealth ratio is constant: 1 cw) = E t ) ] 1 γ e β Ct+1, C t with C.14), the right hand side is a constant. In this case, return on capital equals return on wealth: R W,t+1 = R S t+1 = 1 + A δ t+1, 12

13 and the Euler equation can be written as: e β E t Ct+1 C t ) ψ R S t+1 ) ] = 1. C.15) We can rewrite C.15) as: e β E ν 1 + ζ δ t+1 ) ψ 1 + A δt+1 ) ] = 1, or β + log E ν 1 + ζ δ t+1 ) ψ 1 + A δt+1 ) = 0. C.16) We are interested in the limit when time interval goes to zero: E ν 1 + ζ δ t+1 ) ψ 1 + A δt+1 ) ] ] = e λ 1 + ζ δ) ψ 1 + A δ) + 1 e λ )E ν 1 + ζ δ + e Zt 1) ψ 1 + A δ + e Z t 1) = 1 + ζ δ) ψ 1 + A δ) e λ + 1 e λ e Zt + ζ δ) ψ e Z t + A δ) )E ν 1 + ζ δ) ψ 1 + A δ) 1 + ζ δ) ψ 1 + A δ) e 1 Zt + ζ δ) ψ e Z t + A δ) λ + λe ν 1 + ζ δ) ψ 1 + A δ) = 1 + ζ δ) ψ 1 + A δ) e 1 Zt + ζ δ) ψ e Z t + A δ) + λe ν ζ δ) ψ 1 + A δ) Therefore: ) log 1 + ζ δ) ψ 1 + A δ) e 1 Zt + ζ δ) ψ e Z t + A δ) + λe ν ζ δ) ψ 1 + A δ) ] ψ ζ δ) + A δ) + λe e Zt + ζ δ) ψ e Z t + A δ) ν ζ δ) ψ 1 + A δ) ψ ζ δ) + A δ) + λe ν e 1 ]. 13

14 We can then solve for the investment-capital ratio ζ using C.16): β ψ ζ δ) + A δ) + λe ν e 1 ] = 0, and 1 ψ ζ = ψ 1 ) λe ν e 1 ] + δ + ψ A δ β). γ 1 C.17) Next we can also solve for the risk-free rate using the Euler equation: 1 = e β E ν Ct+1 C t ) ψ R S t+1 ) 1 R f ] = e β R f E ν 1 + ζ δ t+1 ) ψ 1 + A δt+1 ) 1 ]. Similar to the above calculation, the Euler condition becomes: β + r f ψ ζ δ) + 1) A δ) + λe ν e γz t 1 ] = 0, where log R f = log1 + r f ) r f. Plugging in ζ using C.17): r f = A δ + λe ν e γz t e Zt 1) ], or R f = 1 + A δ + λe ν e γz t e Zt 1) ]. Recall that the expected return on equity is a constant: ER S ] = 1 + A δ + λe ν e Z t 1 ], hence the equity premium for consumption claim is: ER S ] R f = λe ν e γz t 1)e Zt 1) ]. 14

15 The equity premium here again is the same as the one in the iid endowment economy model, except for not having the volatility term. References Barro, Robert J., 2009, Rare disasters, asset prices, and welfare costs, American Economic Review 99, Epstein, Larry, and Stan Zin, 1989, Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica 57,

16 Table A.1: Power utility and time-varying disaster probability No-Disaster Simulations All Simulations Data Population ER b ] σr b ) ER e R b ] σr e ) Sharpe ratio expep d σp d) AR1p d) E c] σ c) E d] σ d) Notes: Parameters are as follows: average disaster probability λ = , discount rate β = 0.01, risk aversion γ = 3, normal time consumption growth µ c = , consumption growth volatility σ c = , dividend growth µ d = 0.04, idiosyncratic volatility σ i = 0.05, leverage φ = 3, and mean-reversion κ λ = 0.12, and volatility σ λ =

17 Table A.2: Power utility and constant disaster probability No-Disaster Simulations All Simulations Data Population ER b ] σr b ) ER e R b ] σr e ) Sharpe ratio expep d σp d) AR1p d) E c] σ c) E d] σ d) Notes: Parameters are as follows: average disaster probability λ = , discount rate β = 0.01, risk aversion γ = 3, normal time consumption growth µ c = , consumption growth volatility σ c = , dividend growth µ d = 0.04, idiosyncratic volatility σ i = 0.05, leverage φ = 3, and mean-reversion κ λ = 0.12, and volatility σ λ =

18 Table A.3: EIS=1 and time-varying disaster probability No-Disaster Simulations All Simulations Data Population ER b ] σr b ) ER e R b ] σr e ) Sharpe ratio expep d σp d) AR1p d) E c] σ c) E d] σ d) Notes: Parameters are as follows: average disaster probability λ = , discount rate β = 0.01, risk aversion γ = 3, normal time consumption growth µ c = , consumption growth volatility σ c = , dividend growth µ d = 0.04, idiosyncratic volatility σ i = 0.05, leverage φ = 3, and mean-reversion κ λ = 0.12, and volatility σ λ =

19 Table A.4: The equity premium EIS=1 and constant disaster probability No-Disaster Simulations All Simulations Data Population ER b ] σr b ) ER e R b ] σr e ) Sharpe ratio expep d σp d) AR1p d) E c] σ c) E d] σ d) Notes: Parameters are as follows: average disaster probability λ = , discount rate β = 0.01, risk aversion γ = 3, normal time consumption growth µ c = , consumption growth volatility σ c = , dividend growth µ d = 0.04, idiosyncratic volatility σ i = 0.05, leverage φ = 3, and mean-reversion κ λ = 0.12, and volatility σ λ =

20 Table A.5: EIS=2 and time-varying disaster probability. No-Disaster Simulations All Simulations Data Population ER b ] σr b ) ER e R b ] σr e ) Sharpe ratio expep d σp d) AR1p d) E c] σ c) E d] σ d) Notes: Parameters are as follows: average disaster probability λ = , discount rate β = 0.01, risk aversion γ = 3, normal time consumption growth µ c = , consumption growth volatility σ c = , dividend growth µ d = 0.04, idiosyncratic volatility σ i = 0.05, leverage φ = 3, and mean-reversion κ λ = 0.12, and volatility σ λ =

21 Table A.6: EIS=2 and constant disaster probability No-Disaster Simulations All Simulations Data Population ER b ] σr b ) ER e R b ] σr e ) Sharpe ratio expep d σp d) AR1p d) E c] σ c) E d] σ d) Notes: Parameters are as follows: average disaster probability λ = , discount rate β = 0.01, risk aversion γ = 3, normal time consumption growth µ c = , consumption growth volatility σ c = , dividend growth µ d = 0.04, idiosyncratic volatility σ i = 0.05, leverage φ = 3, and mean-reversion κ λ = 0.12, and volatility σ λ =