Solving asset pricing models with stochastic volatility

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Solving asset pricing models with stochastic volatility Oliver de Groot 2014-71 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Solving asset pricing models with stochastic volatility (cid:3) Oliver de Groot y Federal Reserve Board August 27, 2014 Abstract This paper provides a closed-form solution for the price-dividend ratio in a standardassetpricingmodelwithstochasticvolatility. Thesolutionisusefulinallowing comparisons among numerical methods used to approximate the non-trivial closedform. Keywords: Endowment model, Price-dividend ratio, Closed-form solution JEL classi(cid:133)cations: C61, C62, G12 1 Introduction Thepurposeofthispaperistoobtainanexactexpressionfortheprice-dividendratiofora simple asset pricing model with stochastic volatility. Stochastic volatility has become an important feature of macroeconomic models that seek to jointly explain stylized business cycle and asset pricing facts. Since closed-form solutions elude richer macroeconomic models, various numerical methods have been proposed to provide an approximated solution. The contribution of this paper is to present a simple stochastic volatility model in which an exact solution exists, which may serve as a benchmark from which to compare alternative numerical approximation methods. I would like to thank E. de Groot, M. Gonzalez-Astudillo and J. Roberts for useful discussions and (cid:3) insightful comments. A¢ liation: Federal Reserve Board, Washington D.C., Email: oliver.v.degroot@frb.gov. The views y expressedinthispaperarethoseoftheauthoranddonotnecessarilyre(cid:135)ectthoseoftheFederalReserve Board. 1

Burnside (1998) provided an exact solution for the Lucas (1978) asset pricing model withGaussian,autoregressivedividendgrowthshocksandtimeseparableconstantrelative risk aversion (CRRA) preferences.1 Tsionas (2003) extended Burnside(cid:146)s solution to an arbitrary shock distribution while Chen, Cosimano, and Himonas (2008) and Collard, FØve, and Ghattassi (2006) extended it to the case with non-time separable preferences through habits in consumption. In each case, the solutions provide a useful benchmark against which to test numerical solution algorithms. This paper follows in that tradition. It extends the Burnside model by adding stochastic volatility to the dividend growth process. Since Bansal and Yaron (2004) showed the importance of stochastic volatility to account for stylized asset pricing facts, the use of stochastic volatility has become a widespread addition to macro-(cid:133)nance models. Stochastic volatility is attractive because it generates heteroskedastic aggregate (cid:135)uctuations, a basic property of many time series (such as consumption) and adds extra (cid:135)exibility in accounting for asset-pricing patterns. Due to the increasing importance of stochastic volatility, which naturally adds additional non-linearity into the solution of models, a growing literature has been testing how di⁄erent numerical solution methods that solve equilibrium models with stochastic volatility perform. Caldara, FernÆndez-Villaverde, Rubio-Ram(cid:237)rez, and Yao (2012), for example, compare perturbation methods (of second and third order), Chebyshev polynomials and value function iteration in a real business cycle model with stochastic volatility. In this paper, I show the exact solution for the price-dividend ratio of a simple asset pricing model as a non-trivial function of the model(cid:146)s two state variables, the current dividend growth rate and the current volatility of the dividend growth process. The solution hasthefollowingproperties: First, theprice-dividendratioincreaseswhenthevolatilityof dividend growth increases as well as when the volatility of the stochastic volatility process increases. Second, the sensitivity of the price-dividend ratio to a change in the volatility state is increasing in the persistence of the stochastic volatility process. In addition, I derive an expression for the unconditional mean of the price-dividend process that is also increasing in the volatility and persistence of the stochastic volatility process. Finally, I provide parameter conditions under which the price-dividend ratio and its unconditional mean are (cid:133)nite. The rest of the paper is structured as follows. Section 2 presents the basic assetpricing model with stochastic volatility and section 3 presents and discusses the results. Section 4 concludes. The appendix provides a detailed derivation of the key results of the paper as well as discussing a variant of the basic model. 1AnearlycontributionbyLabadie(1989)alsoprovidedthesolutioninaslightlymoregeneralcontext. 2

2 The asset pricing model There is a representative agent who maximizes the expected discounted stream of utility c1 (cid:13) E (cid:12)t t(cid:0) ; (1) 0 1 t=0 1 (cid:13) (cid:0) P subject to the budget constraint c +s p (d +p )s ; (2) t t+1 t t t t (cid:20) where E is mathematical expectations operator conditional on the time t information t set, c is consumption and s denotes units of an asset whose price at date t is p with t t t dividends, d . The discount factor is (cid:12) (0;1) and the coe¢ cient of relative risk aversion t 2 is (cid:13) > 0 and (cid:13) = 1. The growth rate of dividends, denoted x log(d =d ), is assumed t t t 1 6 (cid:17) (cid:0) to follow a Gaussian AR(1) process x = x+(cid:26)(x x)+p(cid:17) " ; (3) t t 1 t t (cid:0) (cid:0) wherexisthesteadystategrowthrateofdividends, (cid:26) [0;1)isthepersistenceparameter 2 and " is a sequence of i.i.d. innovations from the standard normal distribution. The t innovations to x are scaled by p(cid:17) . (cid:17) is therefore the conditional variance of dividend t t t growth and is time varying. In particular, it follows an AR(1) process (cid:17) = (cid:17) +(cid:26) (cid:17) (cid:17) +!" ; (4) t (cid:17) t 1 (cid:17);t (cid:0) (cid:0) (cid:0) (cid:1) where (cid:17) is its steady state, (cid:26) [0;1) is the persistence of the stochastic volatility process, (cid:17) 2 ! is a scalar and " is a sequence of i.i.d. innovations from the standard normal distrib- (cid:17);t ution.2 The (cid:133)rst-order equilibrium condition of the agent(cid:146)s maximization problem, equations (1)-(2), is c (cid:13)p = E (cid:12)c (cid:13) (p +d ): (cid:0)t t t (cid:0)t+1 t+1 t+1 Market clearing, s = 1, implies that c = d , and, in de(cid:133)ning the price-dividend ratio as t t t 2This formulation of the stochastic volatility process ensures a closed-form expression for the pricedividendratiobutcouldtechnicallycausethestandarddeviationofdividendgrowthtobecomenegative. However, under reasonable calibrations of the process, this happens rarely. Bansal and Yaron (2004) use the same process and choose the following parameter values based on a monthly frequency: (cid:17) = 1:232 10 3; (cid:26) =0:987; and ! =0:04658 10 3. Simulating this process 105 times for 840 quarters (cid:2) (cid:0) (cid:17) (cid:2) (cid:0) results in the process turning negative in 0:13% of the simulations. A discussion of the model solution using an appropriately truncated normal distribution is provided in Appendix A.2. 3

y p =d , the (cid:133)rst-order equilibrium condition becomes t t t (cid:17) d 1 (cid:13) t+1 (cid:0) y = E (cid:12) (y +1): (5) t t t+1 d (cid:18) t (cid:19) Iterating forward and making using of x , we are left with t y = (cid:12)iE exp (1 (cid:13)) i x : (6) t 1 i=1 t (cid:0) j=1 t+j (cid:16) (cid:17) P P 3 The model solution Equation (6) shows that, in this asset pricing model, the price-dividend ratio at time t is simply a function of expected future dividend growth. Finding an exact solution for y means (cid:133)nding a closed-form expression for E exp (1 (cid:13)) i x for i = 1;2;:::in t t (cid:0) j=1 t+j terms of the current state, x and (cid:17) . In the case w(cid:16)ithout stochastic v(cid:17)olatility Burnside t t P (1998)derivedsuchasolution. Thetheorembelowshowsanexactsolutionwith stochastic volatility. Theorem 1 The solution to equation (6) is y = (cid:12)iexp A x+B (x x)+C (cid:17) +D ((cid:17) (cid:17))+F !2 (7) t 1 i=1 i i t (cid:0) i i t (cid:0) i P (cid:0) (cid:1) where A (1 (cid:13))i; B 1 (cid:13) (cid:26)(1 (cid:26)i) i (cid:17) (cid:0) i (cid:17) 1 (cid:0) (cid:26) (cid:0) 2 (cid:0) C 1 1 (cid:13) i 2(cid:26)1(cid:16)(cid:26)i + (cid:17) (cid:26)21 (cid:26)2i ; i (cid:17) 2 1 (cid:0) (cid:26) (cid:0) 1 (cid:0) (cid:26) 1 (cid:0) (cid:26)2 (cid:0) (cid:0) (cid:0) D i (cid:17) 1 2 1 1 (cid:0) (cid:0) (cid:13) (cid:26) 2 (cid:18) (cid:26) (cid:16) (cid:17) 1 1 (cid:0) (cid:0) (cid:26) (cid:26) i (cid:17) (cid:17) (cid:17) (cid:0) (cid:16) 2(cid:26)i (cid:17) (cid:26) 1 (cid:0) 1 (cid:0) ((cid:26) (cid:26) (cid:0)(cid:17) (cid:0)(cid:17) 1 1 (cid:26) (cid:26) )i +(cid:26)i (cid:17) (cid:26)2 (cid:17)1 (cid:0) 1 (cid:0) ((cid:26) (cid:26) (cid:0)(cid:17) (cid:0)(cid:17) 1 1 (cid:26) (cid:26) 2 2 )i (cid:19) ; (cid:16) (cid:17) i(cid:30)2 1 +(cid:30)2 2 1 1 (cid:0) (cid:26) (cid:26) 2 (cid:17) 2 i +(cid:30)2 3 1 1 (cid:0) (cid:26) (cid:26) 2 2 i +(cid:30)2 4 1 1 (cid:0) (cid:26) (cid:26) 4 4 i F i (cid:17) 1 4 (cid:16) 1 1 (cid:0) (cid:0) (cid:13) (cid:26) (cid:17) 4 0 B B +2(cid:30) + 2 (cid:30) 2 3 (cid:30) 1 1 (cid:0) 1 (cid:30) ( 2 (cid:26) (cid:26) 1 1 (cid:17) (cid:0) (cid:0) (cid:26) (cid:26) ) (cid:26) (cid:26) i i (cid:17) (cid:17) + (cid:0) + 2 (cid:17) 2 (cid:30) (cid:30) 2 1 (cid:30) (cid:30) 4 3 1 1 (cid:0) 1 1 (cid:0) (cid:0) (cid:0) ( (cid:26) (cid:26) (cid:26) (cid:26) i (cid:17) (cid:26) + (cid:26) 2 2 ) 2 i (cid:30) + 1 (cid:30) 2 (cid:0) 4 (cid:30) 1 1 3 (cid:0) (cid:0) (cid:30) (cid:26) (cid:26) 4 2 2 i 1 1 (cid:0) (cid:26) (cid:26) 3 3 i 1 C C ; B (cid:0) (cid:17) (cid:0) (cid:17) (cid:0) C @ A and where 1 (cid:26) (cid:26) +(cid:26) (1 (cid:26))2 2(cid:26)2 (cid:26)4 (cid:30) ; (cid:30) (cid:0) (cid:17) (cid:17) (cid:0) ; (cid:30) (cid:26)i 1; and (cid:30) : 1 (cid:17) 1 (cid:26) 2 (cid:17) (cid:26) (cid:26)2 1 (cid:26) (cid:26) (cid:26) 3 (cid:17) (cid:26) (cid:26) (cid:0) 4 (cid:17) (cid:0) (cid:26) (cid:26)2 (cid:0) (cid:17) (cid:17) (cid:0) (cid:0) (cid:0) (cid:1)(cid:17) (cid:17) (cid:0) (cid:17) (cid:0) (cid:18) (cid:17) (cid:0) (cid:19) (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1) Proof. See Appendix A.1. In Burnside (1998), the solution without stochastic volatility is y = (cid:12)iexp(A x+B (x x)+C (cid:17)); t 1 i=1 i i t (cid:0) i P 4

therefore, it is the term D ((cid:17) (cid:17))+F !2 inside the exponential function in equation (7) i t i (cid:0) that is novel. It is straightforward to show (see equation (13) and (15) in Appendix A.1) that both D > 0 and F > 0.3 It follows that @yt > 0 and @yt > 0: A rise in the i i @((cid:17) (cid:17)) @!2 t(cid:0) volatility of dividend growth unambiguously increases the price-dividend ratio as does a riseinthevolatilityofthestochasticvolatilityprocessitself. Sincetheagentisriskaverse, greater uncertainty reduces the agent(cid:146)s demand for the asset, reducing the price. It also follows that @@yt=@(xt x) > 0 and @@yt=@(xt x) > 0: The price-dividend ratio responds j (cid:0) j j (cid:0) j @((cid:17) (cid:17)) @!2 t(cid:0) more to movements in the dividend growth rate in a high volatility state than in a low volatility state as well as in an environment with greater stochastic volatility. The insight from this result is that the heteroskedasticity (inherent in the exogenous dividend growth process) will be more pronounced in the endogenous price-dividend ratio. Equations (13) and (15) also show clearly that @Di; @Fi > 0: A rise in the persistence of the stochastic @(cid:26) @(cid:26) (cid:17) (cid:17) volatility process increases the sensitivity of the price-dividend ratio to both changes in dividend growth and volatility. Since the price-dividend ratio is the sum of an in(cid:133)nite sequence, it is not clear from equation (7) whether the price-dividend ratio is (cid:133)nite. The following theorem states the parameter conditions under which the price-dividend ratio is (cid:133)nite. Theorem 2 The series in equation (7) converges if and only if 1 1 (cid:13) 2 1 1 2 (cid:12)exp (1 (cid:13))x+ (cid:0) (cid:17) + !2 < 1: (8) (cid:0) 2 1 (cid:26) 2 1 (cid:26) (cid:18) (cid:0) (cid:19) (cid:18) (cid:0) (cid:17)(cid:19) !! Proof. See Appendix A.3. In Burnside (1998), the convergence criterion is 1 1 (cid:13) 2 (cid:12)exp (1 (cid:13))x+ (cid:0) (cid:17) < 1; (cid:0) 2 1 (cid:26) ! (cid:18) (cid:0) (cid:19) and thus less demanding that the condition in Theorem 2, conditional on the same parameters for (cid:12);(cid:13);x;(cid:26) and (cid:17). To get a better understanding of the restriction the condition in Theorem 2 places on the parameters of the stochastic volatility process, I following Schmitt-GrohØ and Uribe (2004) and Bansal and Yaron (2004) in parameterizing the asset pricing model as follows: (cid:12) = 0:95; x = 0:0179; and (cid:17) = 6:084 10 5. In addition, I consider three (cid:0) (cid:2) di⁄erent parameterizations of the pair ((cid:26);(cid:13)) using (cid:26) = 0:137;0:9 and (cid:13) = 2:5;11 . f(cid:0) g f g I ignore the high persistence, high risk aversion combination since the price-dividend 3The exception is logarithmic preferences ((cid:13) = 1) in which case A = B = C = D = F = 0 and i i i i i the price-dividend ratio becomes constant. With logarithmic preferences B = 0 because the wealth i and subsitution e⁄ects of a change in the dividend-growth rate exactly o⁄set eachother. Since the price-dividend ratio remains constant in response to dividend growth movements, it follows that the price-dividend ratio is also invariant to changes in the volatility of those movements. 5

Figure 1: Regions of convergence in the parameter space r = 0.139 and g = 2.5 r = 0.139 and g = 11 r = 0.85 and g = 2.5 0.998 0.998 0.998 converge diverge 0.996 0.996 0.996 0.994 0.994 0.994 h h h r r r 0.992 0.992 0.992 Bansal & Yaron 0.99 parameters 0.99 0.99 0.988 0.988 0.988 0 5 0 5 0 5 w w w 4 4 4 x 10 x 10 x 10 Note: Red crosses mark the parameter space for which the condition in Theorem 2 holds, blue circles the parameter space for which the condition is violated and the price-dividend ratio is no longer (cid:133)nite. The black square denotes parameters values (cid:26) (cid:17) = 0:987; and ! = 0:0465 10 (cid:0) 3 used by Bansal and (cid:2) Yaron (2004). Remaining parameters are (cid:12) = 0:95; x = 0:0179 and (cid:17) = 6:084 10 (cid:0) 5: (cid:2) ratio is never (cid:133)nite in this case. Figure 1 shows the (cid:26) ;! pairs (the two parameters (cid:17) describing the stochastic volatility process) for which the condition for a (cid:133)nite price- (cid:0) (cid:1) dividendratio(inTheorem2)holds. Theplotsshowthatwhenboththepersistenceofthe endowment growth process and risk aversion are low (the left panel), then the conditions on the stochastic volatility process to ensure that the price-dividend ratio is (cid:133)nite are relatively weak. Bansal and Yaron (2004) choose parameter values of (cid:26) = 0:987 and (cid:17) ! = 0:0465 10 3 (indicated in the (cid:133)gure), signi(cid:133)cantly inside the convergent parameter (cid:0) (cid:2) space. However, as either the level of risk aversion (middle panel) or the persistence of thedividendgrowthprocess(rightpanel)increases, theparameterspaceforthestochastic volatility process consistent with a (cid:133)nite price-dividend process shrinks considerably. The same condition as in Theorem 2 also ensures that the unconditional mean of the price-dividend ratio is (cid:133)nite, as stated in the next theorem. Theorem 3 The mean of the price-dividend ratio is A x+C (cid:17) +F !2 i i i E(y ) = (cid:12)iexp ; t 1 i=1 0 +1 B i 2(cid:17) + !2 (cid:13)2 i;1 2(cid:13) i;1 (cid:13) i;2 + (cid:13)2 i;2 1 P 21 (cid:0) (cid:26)2 2 1 (cid:0) (cid:26)2 (cid:17) (cid:0) 1 (cid:0) (cid:26) (cid:17) (cid:26)2 1 (cid:0) (cid:26)4 @ (cid:16) (cid:17) A 6

where B2 (cid:26) B2 (cid:26)2 (cid:13) i (cid:17) +D , (cid:13) i ; i;1 (cid:17) 2 (cid:26) (cid:26)2 i i;2 (cid:17) 2 (cid:26) (cid:26)2 (cid:18) (cid:17) (cid:0) (cid:19) (cid:17) (cid:0) and is (cid:133)nite if and only if the condition in Theorem 2 holds. Proof. See Appendix A.4. The unconditional mean price-dividend ratio is increasing in both the volatility, ! and the persistence (cid:26) of the price-dividend ratio (as is made clear by the quadratic expression (cid:17) in (19) in Appendix A.4). 4 Conclusion Thispaperprovidesanexactexpressionfortheprice-dividendratioinanendowmentasset pricing model with CRRA preferences, Gaussian autoregressive shocks and stochastic volatility. Thesolutionprovidesausefulbenchmarkagainstwhichtotesttheperformance of alternative numerical solution algorithms which one may wish to use to solve more elaborate macro-(cid:133)nance models with stochastic volatility. Sincethestructureofthemodelwithstochasticvolatilitysharesmanyoftheproperties of the basic Burnside asset pricing model, it should be possible to derive an exact solution for this stochastic volatility model with the addition of multivariate and higher order autoregressive processes as in Burnside (1998) or with habits in consumption as in Chen, Cosimano, and Himonas (2008) and Collard, FØve, and Ghattassi (2006). This would be a fruitful direction for future research. References Bansal, R. and A. Yaron (2004). Risks for the long run: A potential resolution of asset pricing puzzles. The Journal of Finance 59(4), 1481(cid:150)1509. Burnside, C. (1998). Solving asset pricing models with gaussian shocks. Journal of Economic Dynamics and Control 22(3), 329 (cid:150)340. Caldara, D., J. FernÆndez-Villaverde, J. F. Rubio-Ram(cid:237)rez, and W. Yao (2012). Computing DSGE models with recursive preferences and stochastic volatility. Review of Economic Dynamics 15(2), 188 (cid:150)206. Chen, Y., T. Cosimano, and A. Himonas (2008). Solving an asset pricing model with hybrid internal and external habits, and autocorrelated gaussian shocks. Annals of Finance 4(3), 305(cid:150)344. Collard, F., P. FØve, and I. Ghattassi (2006). A note on the exact solution of asset pricing models with habit persistence. Macroeconomic Dynamics 10, 273(cid:150)283. 7

Labadie, P. (1989). Stochastic in(cid:135)ation and the equity premium. Journal of Monetary Economics 24(2), 277 (cid:150)298. Lucas, Robert E., J. (1978). Asset prices in an exchange economy. Econometrica 46(6), pp. 1429(cid:150)1445. Schmitt-GrohØ, S. and M. Uribe (2004). Solving dynamic general equilibrium models using a second-order approximation to the policy function. Journal of Economic Dynamics and Control 28(4), 755(cid:150)775. Tsionas, E. G. (2003). Exact solution of asset pricing models with arbitrary shock distributions. Journal of Economic Dynamics and Control 27(5), 843 (cid:150)851. A Appendix A.1 Solution: Proof of Theorem 1 The ultimate aim is to rewrite the expression E exp (1 (cid:13)) i x for i = 1;2;::: (9) t (cid:0) j=1 t+j (cid:16) (cid:17) P in terms of the time t state variables, x and (cid:17) . Iterating forward the dividend growth t t process, equation (3), so that x is in terms of x gives t+j t x t+j = x+(cid:26)j(x t (cid:0) x)+ j k=1 (cid:26)j (cid:0) k p(cid:17) t+k " t+k : P Substituting this into (9) gives E t exp (1 (cid:0) (cid:13)) i j=1 x+(cid:26)j(x t (cid:0) x)+ j k=1 (cid:26)j (cid:0) k p(cid:17) t+k " t+k : (cid:16) (cid:16) (cid:17)(cid:17) P P Collecting terms for x, (x x) and each " gives t t+j (cid:0) i (x+(cid:26)j(x x)) E exp (1 (cid:13)) j=1 t (cid:0) : t (cid:0) + i jP=1 i k (cid:0) = j 1 +1(cid:26)k (cid:0) 1 p(cid:17) t+j " t+j !! (cid:16) (cid:17) P P Using the standard results of geometric progressions gives (1 (cid:13))ix+(1 (cid:13))(cid:26)1 (cid:26)i (x x) E exp (cid:0) (cid:0) 1 (cid:0) (cid:26) t (cid:0) : t +(1 1 (cid:0) (cid:13) (cid:26) ) i j=1 (1 (cid:0) (cid:26)i (cid:0) j+1 (cid:0) )p(cid:17) t+j " t+j ! (cid:0) P Since the (cid:133)rst row in the previous expression is only in terms of x and (x x), the t (cid:0) expectations operator can be moved, leaving 8

exp(A i x+B i (x t (cid:0) x))E t exp (cid:18) i j=1 1 (cid:0) (cid:26)i (cid:0) j+1 p(cid:17) t+j " t+j ; (10) (cid:16) (cid:17) where P (cid:0) (cid:1) 1 (cid:13) A (1 (cid:13))i; B (cid:18)(cid:26) 1 (cid:26)i and (cid:18) (cid:0) : i i (cid:17) (cid:0) (cid:17) (cid:0) (cid:17) 1 (cid:26) (cid:18) (cid:0) (cid:19) (cid:0) (cid:1) At this stage it is instructive to rewrite the expression with the expectations operator in (10) as an integral of probabilistic outcomes "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i"t+1 (cid:1)(cid:1)(cid:1) "t+i exp (cid:16) (cid:18) i j=1 1 (cid:0) (cid:26)i (cid:0) j+1 p(cid:17) t+j " t+j (cid:17) " d t F +1 (cid:1)(cid:1)(cid:1)" d t F +i" d (cid:17); G t+1 (cid:1)(cid:1)(cid:1)" d (cid:17); G t+i ; R R R R P (cid:0) (cid:1) where F and G are the density functions for the i.i.d. random variables " and " , respec- (cid:17) tively. Since the " innovations are independent, we can rewrite the problem as "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i i j=1 "t+j exp (cid:18) 1 (cid:0) (cid:26)i (cid:0) j+1 p(cid:17) t+j " t+j " d t F +j!" d (cid:17); G t+1 (cid:1)(cid:1)(cid:1)" d (cid:17); G t+i ; R R Q R (cid:0) (cid:0) (cid:1) (cid:1) Usinga standardresult forrandomvariables, namelythat if z N (0;1) andk is a scalar, (cid:24) then E(exp(kz)) = exp k2 , we get 2 (cid:16) (cid:17) (cid:18)2 i exp 1 (cid:26)i j+1 2 (cid:17) dG dG ; "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i(cid:18) j=1 (cid:18) 2 (cid:0) (cid:0) t+j (cid:19)(cid:19) "(cid:17);t+1 (cid:1)(cid:1)(cid:1)"(cid:17);t+i R R Q (cid:0) (cid:1) or (cid:18)2 exp i 1 (cid:26)i j+1 2 (cid:17) dG dG : (11) "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i (cid:18) 2 j=1 (cid:0) (cid:0) t+j (cid:19) "(cid:17);t+1 (cid:1)(cid:1)(cid:1)"(cid:17);t+i R R P (cid:0) (cid:1) If we assumed (cid:17) = (cid:17) for all i = 1;2;::: the expectations operator would disappear from t+i the above expression and with a little further manipulation we would recover the solution inBurnside(1998). Instead, withstochasticvolatilitythereismoreworktodo. Iterating forward the stochastic volatility process, equation (4), so that (cid:17) is in terms of (cid:17) gives t+j t (cid:17) = (cid:17) +(cid:26)j ((cid:17) (cid:17))+ j (cid:26)j k!" : t+j (cid:17) t (cid:0) k=1 (cid:17)(cid:0) (cid:17);t+k P Substituting this expression into (11) gives (cid:18)2 exp i 1 (cid:26)i j+1 2 (cid:17) +(cid:26)j ((cid:17) (cid:17))+ j (cid:26)j k!" dG dG : "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i (cid:18) 2 j=1 (cid:0) (cid:0) (cid:16) (cid:17) t (cid:0) k=1 (cid:17)(cid:0) (cid:17);t+k (cid:17) (cid:19) "(cid:17);t+1 (cid:1)(cid:1)(cid:1)"(cid:17);t+i R R P (cid:0) (cid:1) P 9

Collecting terms for (cid:17), ((cid:17) (cid:17)) and each " gives t (cid:17);t+j (cid:0) i (1 (cid:26)i j+1) 2 (cid:17) (cid:18)2 j=1 (cid:0) (cid:0) "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i exp0 2 0 +! i + i j=P i 1 ( j 1 +1 (cid:0) 1 (cid:26)i (cid:0) j (cid:26) + i 1) j 2 + (cid:26) 2 j (cid:17) ( k (cid:17) t 2 (cid:0) (cid:26)k (cid:17)) 1 " 11 " d (cid:17); G t+1 (cid:1)(cid:1)(cid:1)" d (cid:17); G t+i : R R B B B B j=1P k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:17);t+j C C C C @ @ (cid:16) (cid:17) AA P P (cid:0) (cid:1) Since the (cid:133)rst two rows in the previous expression are only in terms of (cid:17) and ((cid:17) (cid:17)), the t (cid:0) integral can be moved, leaving exp(C (cid:17) +D ((cid:17) (cid:17))) (12) i i t (cid:0) (cid:18)2! exp i i j+1 1 (cid:26)i j+2 k 2 (cid:26)k 1 " dG dG ; (cid:2) "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i (cid:18) 2 j=1 (cid:16) k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:17) (cid:17);t+j (cid:19) "(cid:17);t+1 (cid:1)(cid:1)(cid:1)"(cid:17);t+i R R P P (cid:0) (cid:1) where (cid:18)2 (cid:18)2 C i 1 (cid:26)i j+1 2 and D i 1 (cid:26)i j+1 2 (cid:26)j: (13) i (cid:17) 2 j=1 (cid:0) (cid:0) i (cid:17) 2 j=1 (cid:0) (cid:0) (cid:17) Notice that D 0, P@Di (cid:0) 0 and @Di (cid:1) 0. Expanding t P he qu (cid:0) adratic term (cid:1) s in C and D i (cid:21) @(cid:26) (cid:20) @(cid:26) (cid:17) (cid:21) i i gives C = (cid:18)2 i 1 2(cid:26)i(cid:26) (j 1) +(cid:26)2i(cid:26) 2(j 1) i 2 j=1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) D i = (cid:18) 2 2 i j=1 (cid:26) (cid:17)P (cid:26) (cid:17) j (cid:0) 1 (cid:0) (cid:0) 2(cid:26) (cid:17) (cid:26)i (cid:26) (cid:17) (cid:26) (cid:0) 1 j (cid:0) 1 +(cid:26) (cid:17) (cid:26)2i (cid:26) (cid:17)(cid:1) (cid:26) (cid:0) 2 j (cid:0) 1 ; (cid:16) (cid:17) and using the standarPd results of geometric(cid:0)progre(cid:1)ssions gives(cid:0) (cid:1) C = (cid:18)2 i 2(cid:26)1 (cid:26)i +(cid:26)21 (cid:26)2i i 2 (cid:0) 1 (cid:0) (cid:26) 1 (cid:0) (cid:26)2 (cid:0) (cid:0) D i = (cid:18) 2 2 (cid:18) (cid:26) (cid:17)1 1 (cid:0) (cid:0) (cid:26) (cid:26) (cid:17) i (cid:17) (cid:0) (cid:16) 2(cid:26)i (cid:17) (cid:26) 1 (cid:0) 1 (cid:0) ((cid:26) (cid:26) (cid:0)(cid:17) (cid:0)(cid:17) 1 1 (cid:26) (cid:26) )i +(cid:26)i (cid:17) (cid:26)2 (cid:17)1 (cid:0) 1 (cid:0) ((cid:26) (cid:26) (cid:0)(cid:17) (cid:0)(cid:17) 1 1 (cid:26) (cid:26) 2 2 )i (cid:19) The (cid:133)nal expression left to evaluate is the integral expression in (12), (cid:18)2! exp i i j+1 1 (cid:26)i j+2 k 2 (cid:26)k 1 " dG dG ; (14) "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i (cid:18) 2 j=1 (cid:16) k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:17) (cid:17);t+j (cid:19) "(cid:17);t+1 (cid:1)(cid:1)(cid:1)"(cid:17);t+i R R P P (cid:0) (cid:1) which becomes exp(F !2) where i (cid:18)4 2 F i i j+1 1 (cid:26)i j+2 k 2 (cid:26)k 1 : (15) i (cid:17) 4 j=1 k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:16) (cid:17) P P (cid:0) (cid:1) Notice that F 0, @Fi 0 and @Fi 0. The above expression is another geometric i (cid:21) @(cid:26) (cid:20) @(cid:26) (cid:17) (cid:21) progression (albeit a more tedious one). Expand to give (cid:18)4 2 F i = 4 i j=1 k i (cid:0) = j 1 +1 (cid:26)k (cid:17)(cid:0) 1 (cid:0) 2(cid:26)i (cid:0) j+1 (cid:26) (cid:17) (cid:26) (cid:0) 1 k (cid:0) 1 +(cid:26)2(i (cid:0) j+1) (cid:26) (cid:17) (cid:26) (cid:0) 2 k (cid:0) 1 : (cid:16) (cid:16) (cid:17)(cid:17) P P (cid:0) (cid:1) (cid:0) (cid:1) 10

Using (for the penultimate time) the results of geometric progressions gives F = (cid:18)4 i 1 (cid:0) (cid:26)i (cid:17)(cid:0) j+1 2(cid:26)i j+1 1 (cid:0) (cid:26) (cid:17) (cid:26) (cid:0) 1 i (cid:0) j+1 +(cid:26)2(i j+1) 1 (cid:0) (cid:26) (cid:17) (cid:26) (cid:0) 2 i (cid:0) j+1 2 : i 4 j=1 1 (cid:26) (cid:0) (cid:0) 1 (cid:26) (cid:26) 1 (cid:0) 1 (cid:26) (cid:26) 2 (cid:17) (cid:0) (cid:17) (cid:1)(cid:0) (cid:0) (cid:17) (cid:1)(cid:0) ! (cid:0) (cid:0) (cid:0) P It is useful to reverse the indexation for j = 1;:::;i by rewriting i+j 1 = j, in which (cid:0) case (cid:18)4 1 (cid:26)j 1 (cid:26) (cid:26) 1 j 1 (cid:26) (cid:26) 2 j 2 F = i (cid:0) (cid:17) 2(cid:26)j (cid:0) (cid:17) (cid:0) +(cid:26)2j (cid:0) (cid:17) (cid:0) : i 4 j=1 1 (cid:26) (cid:0) 1 (cid:26) (cid:26) 1 1 (cid:26) (cid:26) 2 (cid:17) (cid:0) (cid:17) (cid:0)(cid:1) (cid:0) (cid:17) (cid:0)(cid:1) ! (cid:0) (cid:0) (cid:0) P Further manipulation gives (cid:18)4 F = i (cid:30) +(cid:30) (cid:26)j 1 +(cid:30) (cid:26)j 1 +(cid:30) (cid:26)2(j 1) 2 : i 4 j=1 1 2 (cid:17)(cid:0) 3 (cid:0) 4 (cid:0) P (cid:0) (cid:1) where 1 (cid:26) (cid:26) +(cid:26) (1 (cid:26))2 2(cid:26)2 (cid:26)4 (cid:30) ; (cid:30) (cid:0) (cid:17) (cid:17) (cid:0) ; (cid:30) (cid:26)i 1; and (cid:30) : 1 (cid:17) 1 (cid:26) 2 (cid:17) (cid:26) (cid:26)2 1 (cid:26) (cid:26) (cid:26) 3 (cid:17) (cid:26) (cid:26) (cid:0) 4 (cid:17) (cid:0) (cid:26) (cid:26)2 (cid:0) (cid:17) (cid:17) (cid:0) (cid:0) (cid:0) (cid:1)(cid:17) (cid:17) (cid:0) (cid:17) (cid:0) (cid:18) (cid:17) (cid:0) (cid:19) (cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1) Multiplying out the quadratic term gives (cid:18)4 (cid:30)2 1 +(cid:30)2 2 (cid:26)2 (cid:17) (j (cid:0) 1) +(cid:30)2 3 (cid:26)2(j (cid:0) 1) +(cid:30)2 4 (cid:26)4(j (cid:0) 1) F = i +2(cid:30) (cid:30) (cid:26)j 1 +2(cid:30) (cid:30) (cid:26)j 1 +2(cid:30) (cid:30) (cid:26)2(j 1) : i 4 j=10 1 2 (cid:17)(cid:0) 1 3 (cid:0) 1 4 (cid:0) 1 P B +2(cid:30) 2 (cid:30) 3 (cid:26) (cid:17) (cid:26) j (cid:0) 1 +2(cid:30) 2 (cid:30) 4 (cid:26) (cid:17) (cid:26)2 j (cid:0) 1 +2(cid:30) 3 (cid:30) 4 (cid:26)3(j (cid:0) 1) C @ A (cid:0) (cid:1) (cid:0) (cid:1) Using (for the (cid:133)nal time) the results of geometric progressions gives i(cid:30)2 1 +(cid:30)2 2 1 1 (cid:0) (cid:26) (cid:26) 2 (cid:17) 2 i +(cid:30)2 3 1 1 (cid:0) (cid:26) (cid:26) 2 2 i +(cid:30)2 4 1 1 (cid:0) (cid:26) (cid:26) 4 4 i F i = (cid:18) 4 4 0 B B +2(cid:30) + 2 (cid:30) 2 3 (cid:30) 1 1 (cid:0) 1 (cid:30) ( 2 (cid:26) (cid:26) 1 1 (cid:17) (cid:0) (cid:0) (cid:26) (cid:26) ) (cid:26) (cid:26) i i (cid:17) (cid:17) + (cid:0) + 2 (cid:17) 2 (cid:30) (cid:30) 2 1 (cid:30) (cid:30) 4 3 1 1 (cid:0) 1 1 (cid:0) (cid:0) (cid:0) ( (cid:26) (cid:26) (cid:26) (cid:26) i (cid:17) (cid:26) + (cid:26) 2 2 ) 2 i (cid:30) + 1 (cid:30) 2 (cid:0) 4 (cid:30) 1 1 3 (cid:0) (cid:0) (cid:30) (cid:26) (cid:26) 4 2 2 i 1 1 (cid:0) (cid:26) (cid:26) 3 3 i 1 C C : B (cid:0) (cid:17) (cid:0) (cid:17) (cid:0) C @ A This completes the proof. (cid:4) A.2 Ruling out negative volatility with a truncated normal Drawing the " innovations from the standard normal distribution creates the technical (cid:17) possibility that we get negative values for (cid:17) . One candidate solution might be to draw t from a truncated standard normal distribution which, with appropriate truncation, can guarantee non-negative values for (cid:17) . To (cid:133)nd the natural truncation point, we can t calculate the value of (cid:17) (without loss of generality, we set (cid:17) = (cid:17)) following a sequence t+i t 11

of lowest-possible realizations of " , namely "min to give (cid:17) (cid:17) (cid:17)min = (cid:17) +(cid:26)i 1!"min + +!"min: t+i (cid:17)(cid:0) (cid:17) (cid:1)(cid:1)(cid:1) (cid:17) The non-negativity constraint requires lim(cid:17)min > 0, in which case t+i i !1 1 (cid:26)i (cid:17) 1 (cid:26) (cid:17) + lim (cid:0) (cid:17) !"min > 0 or "min > (cid:0) (cid:17) : i 1 (cid:26) (cid:17) (cid:17) (cid:0) ! !1 (cid:0) (cid:17) (cid:0) (cid:1) This expression implies that for a small ! relative to a large (cid:17) (and low persistence, (cid:26) ), the probability of (cid:17) becoming negative can be extremely small and of no practical (cid:17) t concern. Bansal and Yaron (2004) use the following parameterization for the stochastic volatility process: (cid:17) = 1:232 10 3; (cid:26) = 0:987; and ! = 0:04658 10 3. In this case (cid:0) (cid:17) (cid:0) (cid:2) (cid:2) "min = 0:344. However, drawing from this distribution would also lower the volatility (cid:17) (cid:0) of the process that Bansal and Yaron targeted since 2"min(cid:30) "min var "trunc = 1+ (cid:17) (cid:17) < 1; (cid:17) 1 2(cid:8) "min (cid:0) (cid:0) (cid:17) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) where the trunc superscript denotes that it is the truncated random variable and the 1 on the right-hand side of the expression is the variance of the non-truncated standard normal. To consider how the model solution would be altered by the additional truncation, reconsider (14), reproduced here (cid:18)2! exp i i j+1 1 (cid:26)i j+2 k 2 (cid:26)k 1 " dG dG ; "(cid:17);t+1 (cid:1)(cid:1)(cid:1) "(cid:17);t+i (cid:18) 2 j=1 (cid:16) k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:17) (cid:17);t+j (cid:19) "(cid:17);t+1 (cid:1)(cid:1)(cid:1)"(cid:17);t+i R R P P (cid:0) (cid:1) and rewrite it as (cid:18)2! i exp i j+1 1 (cid:26)i j+2 k 2 (cid:26)k 1 " dG : j=1 "(cid:17);t+j (cid:18) 2 (cid:16) k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:17) (cid:17);t+j (cid:19) "(cid:17);t+j Q R P (cid:0) (cid:1) In general, the moment generating function of a stochastic variable X with distribution G is M (t) = Eexp((cid:28)X) = 1 exp((cid:28)X)dG(" (cid:17) ); (cid:28) R: 2 (cid:0)1 Rewriting (14) using the moment generaRting function becomes (cid:18)2! i M i j+1 1 (cid:26)i j+2 k 2 (cid:26)k 1 : (16) j=1 2 k (cid:0) =1 (cid:0) (cid:0) (cid:0) (cid:17)(cid:0) (cid:18) (cid:19) (cid:16) (cid:17) Q P (cid:0) (cid:1) 12

Using the results of geometric progressions gives 1 (cid:26)i j+2 (cid:26)2(i j+2) 1 (cid:26) (cid:26)2 2 (cid:0) + (cid:0) 2 + (cid:26)i j+1; 1 (cid:26) (cid:0) (cid:26) (cid:26) (cid:26)2 (cid:26)2 (cid:0) 1 (cid:26) (cid:0) (cid:26) (cid:26) (cid:26)2 (cid:26) (cid:17)(cid:0) (cid:18) (cid:0) (cid:17) (cid:0) (cid:17) (cid:0) (cid:17) (cid:19) (cid:18) (cid:0) (cid:17) (cid:0) (cid:17) (cid:0) (cid:17)(cid:19) for the summation in expression (16). The term F !2 in equation (7) from the main text i is therefore replaced with the following expression: (cid:18)2! 1 2(cid:26)i (cid:0) j+2 + (cid:26)2(i (cid:0) j+2) i logM 1 (cid:0) (cid:26) (cid:17) (cid:0) (cid:26) (cid:0) (cid:26) (cid:17) (cid:26)2 (cid:0) (cid:26)2 (cid:17) : j=1 0 2 0 1 2 (cid:26) + (cid:26)2 (cid:26)i j+1 11 P (cid:0) 1 (cid:0) (cid:26) (cid:17) (cid:0) (cid:26) (cid:0) (cid:26) (cid:17) (cid:26)2 (cid:0) (cid:26) (cid:17) (cid:17)(cid:0) @ @ (cid:16) (cid:17) AA With " drawn from a symmetrically truncated standard normal distribution with "min = (cid:17) (cid:17) (cid:17)(1 (cid:26) ) (cid:0) (cid:17) , the moment generating function is given by (cid:0) ! (cid:28)2 (cid:8) "min (cid:28) (cid:8) "min (cid:28) exp (cid:0) (cid:17) (cid:0) (cid:0) (cid:17) (cid:0) : 2 1 2(cid:8) "min (cid:18) (cid:19) (cid:0) (cid:0) (cid:1) (cid:17) (cid:0) (cid:1)! (cid:0) (cid:1) Inthelimit, (cid:17)(1 (cid:0) (cid:26) (cid:17) ) , themomentgeneratingfunctionwouldbeexp (cid:28)2 , recovering ! ! 1 2 the solution in the main text. (cid:16) (cid:17) A.3 Convergence: Proof of Theorem 2 The aim is to show that the in(cid:133)nite summation (cid:12)iexp A x+B (x x)+C (cid:17) +D ((cid:17) (cid:17))+F !2 ; 1 i=1 i i t (cid:0) i i t (cid:0) i P (cid:0) (cid:1) convergences to a (cid:133)nite number. First, I de(cid:133)ne z = (cid:12)iexp A x+B (x x)+C (cid:17) +D ((cid:17) (cid:17))+F !2 ; i i i t i i t i (cid:0) (cid:0) so that the price-dividend(cid:0)ratio given by (cid:1) y = z : t 1 i=1 i P To test convergence, it is su¢ cient to show that lim zi+1 < 1. It follows that i zi !1(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) z (A A )x+(B B )(x x) i+1 i+1 i i+1 i t = (cid:12)exp (cid:0) (cid:0) (cid:0) ; (cid:12) z i (cid:12) +(C i+1 (cid:0) C i )(cid:17) +(D i+1 (cid:0) D i )((cid:17) t (cid:0) (cid:17))+(F i+1 (cid:0) F i )!2 ! (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 13

which, when de(cid:133)ning X X X becomes i i+1 i (cid:17) (cid:0) z i+1 = f (cid:12)exp Ax+B (x x)+C (cid:17) +D ((cid:17) (cid:17))+F !2 ; z i t (cid:0) i i t (cid:0) i (cid:12) i (cid:12) (cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) e e e e e where (cid:12) (cid:12) A 1 (cid:13); B (1 (cid:13))(cid:26)i+1 i (cid:17) (cid:0) (cid:17) (cid:0) C (cid:18)2 1 2(cid:26)i+1 +(cid:26)2(i+1) ; D (cid:18)2 (cid:26)i+1 2(cid:26) (cid:17) (cid:26) (cid:26)i(1 e i (cid:26) (cid:17) ) 2 (cid:26)i 1 (cid:0) e (cid:26) + (cid:26) (cid:17) (cid:26)2 (cid:26)2i(1 (cid:26)2) (cid:26)i 1 (cid:26) ; i (cid:17) 2 (cid:17) (cid:0) (cid:26) (cid:17)(cid:0) (cid:26) (cid:0)e (cid:0) (cid:0)(cid:17) (cid:0) (cid:17) (cid:26) (cid:17)(cid:0) (cid:26)2(cid:1) (cid:0) (cid:0) (cid:17) (cid:0) (cid:17) (cid:16)(cid:16) (cid:0) (cid:30)2 (cid:0)+(cid:30)2(cid:26)2 (cid:1) i (cid:1)+(cid:30)2(cid:26)2i +(cid:0)(cid:30)2(cid:26)4i (cid:0) (cid:1)(cid:1) (cid:17)(cid:17) 1 2 (cid:17) 3 4 e and F (cid:18)4 +2(cid:30) (cid:30) (cid:26)i +2(cid:30) (cid:30) (cid:26)i +2(cid:30) (cid:30) (cid:26)2i : i (cid:17) 4 0 1 2 (cid:17) 1 3 1 4 1 +2(cid:30) (cid:30) (cid:26) (cid:26) i +2(cid:30) (cid:30) (cid:26) (cid:26)2 i +2(cid:30) (cid:30) (cid:26)3i B 2 3 (cid:17) 2 4 (cid:17) 3 4 C e @ A (cid:0) (cid:1) (cid:0) (cid:1) Taking the limit of these terms gives limA = 1 (cid:13); limB = 0 i i (cid:0) i !1 !12 limC = 1 1 (cid:13) ; e i 2 1 (cid:0) (cid:26) e i (cid:0) !1 (cid:16) (cid:17) 2 i l ! im 1 D i = 0; aned i l ! im 1 F i = (cid:18) 2(1 (cid:18) (cid:0) 2 (cid:26) (cid:17) ) (cid:19) : e e It then follows that 2 z 1 1 (cid:13) 2 (cid:18)2 lim i+1 = (cid:12)exp (1 (cid:13))x+ (cid:0) (cid:17) + !2 : i !1(cid:12) z i (cid:12) 0 (cid:0) 2 (cid:18) 1 (cid:0) (cid:26) (cid:19) 2 1 (cid:0) (cid:26) (cid:17) ! 1 (cid:12) (cid:12) (cid:12) (cid:12) @ (cid:0) (cid:1) A (cid:12) (cid:12) A.4 Mean price-dividend ratio: Proof of Theorem 3 In order to calculate the unconditional mean, it is necessary to appropriately capture the autocorrelation created by the " innovations in the dividend growth process. Iterating (cid:17) backwardthestochasticvolatilityprocess,equation(4), sothat(cid:17) isintermsofasequence t of past " realizations gives (cid:17) (cid:17) (cid:17) = (cid:26)k (cid:17) (cid:17) +! k (cid:26)s 1" : (17) t (cid:0) (cid:17) t (cid:0) k (cid:0) s=1 (cid:17)(cid:0) (cid:17);t+1 (cid:0) s (cid:0) (cid:1) P Taking the limit gives lim(cid:17) (cid:17) = ! (cid:26)s 1" ; k t (cid:0) 1 s=1 (cid:17)(cid:0) (cid:17);t+1 (cid:0) s !1 in which case P (cid:17) (cid:17) = ! (cid:26)s 1" : t+1 (cid:0) j (cid:0) 1 s=1 (cid:17)(cid:0) (cid:17);t+2 (cid:0) j (cid:0) s P 14

Similarly, x can be written as t x t (cid:0) x = (cid:26)k(x t (cid:0) k (cid:0) x)+ k j=1 (cid:26)j (cid:0) 1 p(cid:17) t+1 (cid:0) j " t+1 (cid:0) j ; P and k limx t (cid:0) x = 1 j=1 (cid:26)j (cid:0) 1 p(cid:17) t+1 (cid:0) j " t+1 (cid:0) j : !1 Substituting in for equation (17) givesP x x = (cid:26)j 1 (cid:17) +! (cid:26)s 1" " : (18) t (cid:0) 1 j=1 (cid:0) 1 s=1 (cid:17)(cid:0) (cid:17);t+2 (cid:0) j (cid:0) s t+1 (cid:0) j (cid:16)q (cid:17) P P The unconditional mean of y is t E(y ) = (cid:12)iexp A x+C (cid:17) +F !2 Eexp(B (x x)+D ((cid:17) (cid:17))); t 1 i=1 i i i i t (cid:0) i t (cid:0) P (cid:0) (cid:1) which means we need only evaluate the expectations term Eexp(B (x x)+D ((cid:17) (cid:17))): i t i t (cid:0) (cid:0) To do this, (cid:133)rst substitute using equation (18), which gives Eexp B (cid:26)j 1 (cid:17) +! (cid:26)s 1" " +D ! (cid:26)j 1" : i 1 j=1 (cid:0) 1 s=1 (cid:17)(cid:0) (cid:17);t+2 j s t+1 j i 1 j=1 (cid:17)(cid:0) (cid:17);t+1 j (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:16) (cid:16)q (cid:17) (cid:17) (cid:16) (cid:17)(cid:17) P P P At this stage it is instructive to rewrite the expectations operator as an integral of probabilistic outcomes B (cid:26)j 1 (cid:17) +! (cid:26)s 1" " i 1 j=1 (cid:0) 1 s=1 (cid:17)(cid:0) (cid:17);t+2 j s t+1 j exp (cid:0) (cid:0) (cid:0) dF dF dG dG ; "(cid:17);t (cid:1) " (cid:1) (cid:17) (cid:1) ;t "t (cid:1) " (cid:1) t (cid:1) 0 P +(cid:16)Dq i ! 1 j= P 1 (cid:26)j (cid:17)(cid:0) 1" (cid:17);t+1 j (cid:17) 1 "t (cid:1)(cid:1)(cid:1)"t (cid:0)1 "(cid:17);t (cid:1)(cid:1)(cid:1)"(cid:17);t (cid:0)1 R R(cid:0)1R (cid:0)R1 (cid:0) @ (cid:16) (cid:17) A P Rearranging the above expression gives exp B (cid:26)j 1 (cid:17) +! (cid:26)s 1" " dF dF "(cid:17) R ;t (cid:1) " (cid:1) (cid:17) (cid:1) ;t R(cid:0)1 Q 1 j=1 " R t (cid:1) " (cid:1) t (cid:1) (cid:0)R1 (cid:16) i (cid:0) (cid:16)q P 1 s=1 (cid:17)(cid:0) (cid:17);t+2 (cid:0) j (cid:0) s (cid:17) t+1 (cid:0) j (cid:17) "t (cid:1)(cid:1)(cid:1)"t (cid:0)1 ! exp D ! (cid:26)j 1" dG dG : (cid:2) i 1 j=1 (cid:17)(cid:0) (cid:17);t+1 (cid:0) j "(cid:17);t (cid:1)(cid:1)(cid:1)"(cid:17);t (cid:16) (cid:16) (cid:17)(cid:17) (cid:0)1 P Using the same standard result as before for Gaussian shocks gives B2 exp i (cid:26)2(j 1) (cid:17) +! (cid:26)s 1" (cid:1)(cid:1)(cid:1) 1 j=1 2 (cid:0) 1 s=1 (cid:17)(cid:0) (cid:17);t+2 (cid:0) j (cid:0) s "(cid:17);t "(cid:17);t (cid:18) (cid:19) R R(cid:0)1Q (cid:0) P (cid:1) exp D ! (cid:26)j 1" dG dG ; (cid:2) i 1 j=1 (cid:17)(cid:0) (cid:17);t+1 (cid:0) j "(cid:17);t (cid:1)(cid:1)(cid:1)"(cid:17);t (cid:16) (cid:16) (cid:17)(cid:17) (cid:0)1 P 15

which can be rewritten as B2 exp i (cid:26)2(j 1) (cid:17) +! (cid:26)s 1" +D ! (cid:26)j 1" dG dG ; "(cid:17) R ;t (cid:1) " (cid:1) (cid:17) (cid:1) ;t R(cid:0)1 (cid:18) 2 P 1 j=1 (cid:0) (cid:0) P 1 s=1 (cid:17)(cid:0) (cid:17);t+2 (cid:0) j (cid:0) s (cid:1) i (cid:16) P 1 j=1 (cid:17)(cid:0) (cid:17);t+1 (cid:0) j (cid:17) (cid:19) "(cid:17);t (cid:1)(cid:1)(cid:1)"(cid:17);t (cid:0)1 Removing the constants term from the integral gives B2! H exp i (cid:26)2(j 1) (cid:26)s 1" +D ! (cid:26)j 1" dG dG ; i "(cid:17) R ;t (cid:1) " (cid:1) (cid:17) (cid:1) ;t R(cid:0)1 (cid:18) 2 P 1 j=1 (cid:0) (cid:0)P 1 s=1 (cid:17)(cid:0) (cid:17);t+2 (cid:0) j (cid:0) s (cid:1) i (cid:16) P 1 j=1 (cid:17)(cid:0) (cid:17);t+1 (cid:0) j (cid:17) (cid:19) "(cid:17);t (cid:1)(cid:1)(cid:1)"(cid:17);t (cid:0)1 where 1 B2(cid:17) H exp i : i (cid:17) 21 (cid:26)2 (cid:18) (cid:0) (cid:19) Focussing on the integral term, the above expression is rearranged in order to bring together " innovations with the same time subscript: (cid:17) B2! "(cid:17) R ;t (cid:1) " (cid:1) (cid:17) (cid:1) ;t R(cid:0)1 exp (cid:18) P 1 j=1 (cid:18) 2 i (cid:26)j (cid:17)(cid:0) 1 (cid:16) P j s=1 (cid:0) (cid:26) (cid:0)(cid:17) 1(cid:26)2 (cid:1) s (cid:0) 1 (cid:17) +D i !(cid:26)j (cid:17)(cid:0) 1 (cid:19) " (cid:17);t+1 (cid:0) j (cid:19) d "(cid:17) G ;t (cid:1)(cid:1)(cid:1)"(cid:17) d ;t G (cid:0)1 : Again, using the results of standard normals and geometric series gives !2 B2 1 (cid:26) 1(cid:26)2 j 2 exp i (cid:26)j 1 (cid:0) (cid:0)(cid:17) +D (cid:26)j 1 : 0 2 1 j=1 2 (cid:17)(cid:0) 1 (cid:26) 1(cid:26)2 i (cid:17)(cid:0) 1 (cid:0) (cid:0) (cid:0)(cid:17) (cid:1) ! ! P @ A This can be rewritten as !2 exp (cid:13) (cid:26)j 1 (cid:13) (cid:26)2(j 1) 2 ; (19) 2 1 j=1 1 (cid:17)(cid:0) (cid:0) 2 (cid:0) (cid:18) (cid:19) P (cid:0) (cid:1) where B2 (cid:26) B2 (cid:26)2 (cid:13) i (cid:17) +D and (cid:13) i : i;1 (cid:17) 2 (cid:26) (cid:26)2 i i;2 (cid:17) 2 (cid:26) (cid:26)2 (cid:18) (cid:17) (cid:0) (cid:19) (cid:17) (cid:0) Multiplying out the quadratic term in expression (19) gives !2 exp 2 1 j=1 (cid:13)2 i;1 (cid:26)2 (cid:17) (j (cid:0) 1) (cid:0) 2(cid:13) i;1 (cid:13) i;2 (cid:26) (cid:17) (cid:26)2 j (cid:0) 1 +(cid:13)2 2 (cid:26)4(j (cid:0) 1) ; (cid:18) (cid:19) (cid:16) (cid:17) P (cid:0) (cid:1) And using the standard results of geometric series gives !2 (cid:13)2 2(cid:13) (cid:13) (cid:13)2 exp 1 1 2 + 2 : 2 1 (cid:26)2 (cid:0) 1 (cid:26) (cid:26)2 1 (cid:26)4 (cid:18) (cid:18) (cid:0) (cid:17) (cid:0) (cid:17) (cid:0) (cid:19)(cid:19) Thus, the unconditional mean price-dividend ratio is A x+C (cid:17) +F !2 i i i E(y ) = (cid:12)iexp : t 1 i=1 0 +1 B i 2(cid:17) + !2 (cid:13)2 i;1 2(cid:13) i;1 (cid:13) i;2 + (cid:13)2 i;2 1 P 21 (cid:0) (cid:26)2 2 1 (cid:0) (cid:26)2 (cid:17) (cid:0) 1 (cid:0) (cid:26) (cid:17) (cid:26)2 1 (cid:0) (cid:26)4 @ (cid:16) (cid:17) A 16

Next, it is necessary to show that the condition for convergence of the in(cid:133)nite summation in the expression above is the same as the condition stated in Theorem 2. Let A x+C (cid:17) +F !2 i i i z = (cid:12)iexp ; i 0 +1 B i 2(cid:17) + !2 (cid:13)2 i;1 2(cid:13) i;1 (cid:13) i;2 + (cid:13)2 i;2 1 21 (cid:0) (cid:26)2 2 1 (cid:0) (cid:26)2 (cid:17) (cid:0) 1 (cid:0) (cid:26) (cid:17) (cid:26)2 1 (cid:0) (cid:26)4 @ (cid:16) (cid:17) A so that E(y ) = z . Then t 1 i=1 i P z i+1 = (cid:12)exp Ax+C i (cid:17) +F i !2 + 2(1 (cid:0) (cid:17) (cid:26)2) B i 2 +1 (cid:0) B i 2 + ! 2 2 : z 0 1 (cid:13)2 (cid:13)2 2 (cid:13) (cid:13) (cid:13) (cid:13) + 1 (cid:13)2 (cid:13)2 1 (cid:12) (cid:12) i (cid:12) (cid:12) (cid:2) 1 (cid:0) (cid:26)2 (cid:17) i+1;1 (cid:0)e i;1 e(cid:0) 1 (cid:0) (cid:26) (cid:17)e(cid:26)2 i+1;1 i+1;2(cid:0) (cid:0) i;1 i;2 (cid:1) 1 (cid:0) (cid:26)4 i+1;2 (cid:0) i;2 (cid:12) (cid:12) @ (cid:16)(cid:16) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:17)(cid:17) A (cid:12) (cid:12) TheparametersA;C ;andF arethesameasinSectionA.3. SinceSectionA.3alsoshows i i that limB = limD = 0, it follows naturally (or after much tedious manipulation4) that i i i i e e e this re!s1ult also!im1plies that e e lim B2 B2 = lim (D D ) = 0; i i+1 (cid:0) i i i+1 (cid:0) i !1 !1 (cid:0) (cid:1) lim (cid:13)2 (cid:13)2 = lim (cid:13) (cid:13) (cid:13) (cid:13) = lim (cid:13)2 (cid:13)2 = 0: i i+1;1 (cid:0) i;1 i i+1;1 i+1;2 (cid:0) i;1 i;2 i i+1;2 (cid:0) i;2 !1 !1 !1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 4Available upon request. 17