Asset Pricing in Production Economies

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1 Urban J. Jermann 1998 Presented By: Farhang Farazmand October 16, 2007

2 Motivation Can we try to explain the asset pricing puzzles and the macroeconomic business cycles, in one framework.

3 Motivation: Equity Premium Puzzle Using standard power utility function, aggregate consumption growth is too smooth and not sufficiently correlated with the magnitude of the equity sharpe ratio for reasonable values of the risk aversion coefficient.

4 Motivation: Equity Premium Puzzle Using standard power utility function, aggregate consumption growth is too smooth and not sufficiently correlated with the magnitude of the equity sharpe ratio for reasonable values of the risk aversion coefficient. E t (Rt+1 e ) σ t (Rt+1 e ) = σ t(m t+1 ) E t (M t+1 ) ρ t(m t+1, Rt+1) e M t+1 = δ ( ) γ Ct+1 C t

5 Motivation: Risk Free Rate Puzzle Do a log-linearization Er f t+1 = ln δ + γe( c t+1 ) γ2 2 σ2 ( c t+1 )

6 Motivation: Risk Free Rate Puzzle Do a log-linearization Er f t+1 = ln δ + γe( c t+1 ) γ2 2 σ2 ( c t+1 ) Again given the arguments stated earlier, the model has difficulty explaining the low historical value of the risk free rate.

7 Business Cycle Aggregate output and employment fluctuate over time. Understanding these phenomena has been the task of real-business-cycle(rbc) models.

8 So Far in Asset Pricing We have seen. Long-Run risk models that depart from the assumption of i.i.d. growth rates. Habit-formation models that assume habit persistence. In fact in both cases we saw that the power-utility assumption was relaxed.

9 Extending the RBC-Model In this study we look at an RBC-Model that incorporates internal habit and adjustment costs. We will see that both are needed to arrive at the desired results.

10 Extending the RBC-Model Why? Because in an RBC framework consumption is endogenous and as always agents will try to obtain a smooth consumption-path. So unless restrictions are imposed, an RBC-Model with internal habit will fair miserably at explaining the high equity premium and the low risk-free rate.

11 Extending the RBC-Model Why? Because in an RBC framework consumption is endogenous and as always agents will try to obtain a smooth consumption-path. So unless restrictions are imposed, an RBC-Model with internal habit will fair miserably at explaining the high equity premium and the low risk-free rate.

12 The Model One-sector RBC-Model with a single consumption-investment good. Representative agent has preferences captured by a power utility function with internal habit. Introduce capital adjustment costs.

13 Notation N: quantity of labour. L: Leisure. Λ: marginal utility. I: Investment. A: Stochastic productivity level. W: Wage. K: Capital stock. F( ): Output function. X: deterministic trend in labour augmenting technical change. φ( ) captures the idea of costly capital adjustment(concave).. a t : Vector of financial assets held at time t. V a t : Vector of asset prices. D: Dividends.

14 The Agent Maximises the value of the firm { n β k Λ t+k E t (A t+k F (K t+k, X t+k N t+k ) W t+k N t+k I t+k ) Λ t k=0 ( ) It K t+1 = (1 δ)k t + φ K t K t It s an all equity firm that does not raise new capital by issuing new shares D t = A t F (K t, X t N t ) W t N t I t }

15 The Agent and expected lifetime utility of consumption In equilibrium max E t n β k u (C t+k ) k=0 s.t.w t N t + a t (V t a + Da t ) = C t + a t+1 V t a 1 = N t + L t A t F (K t, X t N t ) = C t + I t

16 Solution The model is solved by employing the technique of log-linearizing first-order conditions and assuming that the returns and the marginal utility are conditionally log-normal. Thus given V t (s t ) = βk E t [Λ t+k (s t+k )D t+k (s t+k )] Λ t (s t ) and the approximations, we obtain E(Rt,t+1(s 1 t )) = 1 ( ) 1 exp βγ t 2 (var(e tλ t+1 λ t )) var(λ t+1 E t λ t+1 )

17 Calibration Parameters are chosen or picked to match long-run model behaviour. These parameters include the depreciation rate, labour share, parameter of risk aversion(5). Parameters governing habit persistence, capital adjustment costs, the discount factor are estimated using GMM. We note that the moments matched are: standard deviation of consumption growth divided by standard devation of output growth, standard deviation of investment growth divided by the standard deviation of output growth, the mean risk-free rate and the equity premium.

18 Notice The data they are using is quarterly and spans 200 periods. When trying to match moments in a GMM framework we hope that the sample moments are close to the true values. Also note that the model does not provide an expression for the equity premium puzzle. As such an SMM procedure must be built into the estimation system.

19 Results As we will see he model(benchmark) seems to do a good job of capturing the standard deviations, the risk-free rate and equity premium. Furthermore, it is the combination of habit persistence and costly capital adjustment that is of importance.

20 Results The following table summarizes the results

21 Results Consider the premium on a strip that pays D k at time t + k. R D k t,t+1 R t,t+1 = exp( cov t (λ t+1, E t+1 d t+k )) exp( cov t (λ t+1, E t+1 λ t+k λ t+1 )) In a model with adjustment costs dividends are pro-cyclical, i.e. stocks pay dividends when marginal utility is low and are therefore more risky than a (perpetual) bond.

22 Issues Two problems persist The premium for long-term bonds is too high in the model. The risk-free rate is too volatile. They show that they can increase equity premium compared to bon-premium but at the expense of obtaining a much too volatile dividend process.

23 Conclusion and Discussion The model surely provides us with a way to address the issue of explaining the asset pricing puzzles while capturing some features of the business-cycle. However, the estimation procedure focused on coming close to explaining the low risk-free rate, the equity premium and the standard deviation ratios for the business cycle variables. This tells us that the model is flexible enough to capture these moments but is the model any good at forecasting. Furthermore, as the next presentation will point out this model implies counter-cyclical employment??

24 Conclusion and Discussion More needs to be done to the existing model to actually capture the fluctuations in output and employment(next paper.) We have seen that habit persistence per se is not enough and some sort of rigidity needs to be added. So what if we consider a model with n consumer and m investment goods and impose limited access to certain markets depending on the type of the consumer. One could also entertain the notion of external habit with a constant risk-free rate as in Cochrane and Campbell.

25 Conclusion and Discussion Also what if we consider Keynesian business-cycle models that model fluctuations based on sluggish nominal price and wage adjustments.

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