Log-normality assumption in consumption based asset pricing

Question

Consider a very basic discrete time representative consumer maximization problem with CRRA utility. There exist a risky asset with time $t$ price $p_t$ that pays time $t+1$ dividend $d_{t+1}$ , and a riskless asset with price $p_t^f$ that pays a constant payoff 1 at $t+1$. We assume that the dividends are a sequence of random variables that follow a Markov process. Assume further that the consumer has no other income streams (i.e. $y_t = 0 \ \forall t$). At time t consumer invests amount $\pi_t$ in the risky asset and amount $\pi_t^0$ in the riskless asset. Therefore, the maximization problem can be stated as

\begin{align*} & \underset{\{ c_t, \pi \}_0^\infty}{\text{max}} \ \ E_0 \sum_{t=0}^\infty \ \beta^t \ \frac{c_t^{1-\gamma} -1}{1-\gamma} \\ \\ \ s.t \ \ \ \ & c_t + \pi_t p_t + \pi_t^0 p_t^0 = (d_t+p_t) \pi_{t-1} + \pi_{t-1}^0 \\ & c_t \geq 0 \end{align*}

Say we want to find the equilibrium riskless rate and expected equity premium. In order to close the model, it is often assumed assumed (see e.g. Claus Munk's book Financial Asset Pricing Theory chapter 8.3) that the log-consumption growth and log-risky gross returns are jointly normally distributed. I.e

\begin{align*} & ln \ \Big(\dfrac{c_{t+1}}{c_t} \Big) \equiv \bar{g}_{t+1} \sim N(\mu_g, \sigma_g^2) \\ & ln R_{t+1} \equiv \bar{r}_{t+1} \sim N(\mu_r, \sigma_r^2) \ , \\ \end{align*}

where gross returns are defined as $$R_{t+1} \equiv \frac{p_{t+1} + d_{t+1}}{p_t} \ .$$

What I don't completely understand is where do thelog-normal distribution assumptions "come from". I know that since this is a representative agent economy, consumption of the agent must equal the aggregate dividend in the economy. But since we assumed that there is no income, $y_t = 0 \ \forall t$, the only exogenous dividend process in the economy is $d_t$ and therefore it should have the same distribution as the consumption growth. However, my impression is that when we say the risky rate has log-normal distribution this actually means the dividend process, since it is the 'random part' in the definition of returns (price $p_{t+1}$ is not exogenous but determined inside the model). To me it seems now that we have made two different assumptions about the same endowment process $d_t$. Where does the assumption for consumption come from or what does it stand for? How would the situation change if the consumer had some income stream $y_t > 0$?

Answer 1

The typical two-period Lagrangian is

$$\Lambda = \beta^t\cdot \Big(\frac{c_t^{1-\gamma} -1}{1-\gamma} + \lambda_t\cdot \big[(d_t+p_t) \pi_{t-1} + \pi_{t-1}^0- c_t - \pi_t p_t - \pi_t^0 p_t^0\big]\Big) \\ + \beta^{t+1}\cdot \Big(\frac{c_{t+1}^{1-\gamma} -1}{1-\gamma} + \lambda_{t+1}\cdot \big[(d_{t+1}+p_{t+1}) \pi_{t} + \pi_{t}^0- c_{t+1} - \pi_{t+1} p_{t+1} - \pi_{t+1}^0 p_{t+1}^0\big]\Big)$$

The first order conditions with respect to $c_t, \pi_t$ are

$$c_t^{-\gamma} = \lambda_t \implies ... \gamma\ln \frac {c_{t+1}}{c_t} = \ln \frac {\lambda_{t}}{\lambda_{t+1}} \tag{1}$$

$$-\beta^t\lambda_tp_t + \beta^{t+1}\lambda_{t+1}(d_{t+1}+p_{t+1})=0 \implies \frac {\lambda_{t}}{\lambda_{t+1}} = \beta \frac{p_{t+1} + d_{t+1}}{p_t} \tag{2}$$

and so, using also the definition of the gross return,

$$\ln \frac {\lambda_{t}}{\lambda_{t+1}} = \ln \beta + \ln R_{t+1} \tag{3}$$

Combining $(1)$ and $(3)$ we get

$$\ln \frac {c_{t+1}}{c_t} = \frac 1 {\gamma}\ln \beta + \frac 1 {\gamma}\ln R_{t+1} \tag{4}$$

So we see that at the optimal path, consumption growth is a direct affine function of the log-risk returns. This among other things implies that their correlation coefficient is equal to unity.

The normal distribution is closed under affine transformations (alternatively, under scaling and shifting), so if we assume that log-risky returns are normally distributed, then consumption growth is also normally distributed (with different mean and variance of course).

Note that although in general, the joint normality assumption is an additional one to be made when two normal random variables are not-independent, here, the fact that the one is an affine function of the other guarantees joint normality. By Cramer's condition for bivariate normality, it must be the case that all linear combinations of two normal random variables have a univariate normal distribution. In our case we have (generic notation) the random vavriable $Y$ and the random variable $X = a+bY$. Consider

$$\delta_1X + \delta_2 Y = \delta_1(a+bY) + \delta_2 Y = \delta_1a + (\delta_1b+\delta_2)Y$$

So for any $(\delta_1, \delta_2)$ (except the zero vector which is excluded a priori), $\delta_1X + \delta_2 Y$ follows a normal distribution if $Y$ does. So it is sufficient to assume that log-risk returns follow a normal distribution to obtain joint normality also.

Answer 2

I recently produced a paper deriving the distribution of returns for all asset and liability classes. The log-normal return only appears in two cases. The first is with single period discount bonds, the second with cash-for-stock mergers. It comes from an assumption, I believe originally by Boness to eliminate the problem in Markowitz of infinitely negative prices. While it was logically derived, it has a critical assumption that makes it generally untrue.

Most finance models assume that the parameters are known with probability one. You do not need to estimate $\mu$ with $\bar{x}$ because it is assumed to be known. On the surface, this is not a problem because this is the general methodology of null hypothesis based methods. You assert a null is true and hence the parameters are known and a test is made against this null.

The difficulty happens when the parameters are not known. It turns out the proof collapses without that assumption, in general. The same is true for Black-Scholes. I am presenting a paper at the SWFA conference this spring where I argue that if the assumptions of the Black-Scholes formula are literally true, then there cannot exist an estimator that converges to the population parameter. Everyone just assumed the formula under perfect knowledge equaled the parameter estimator. No one ever actually checked its properties. In their initial paper, Black and Scholes empirically tested their formula and they reported that it didn't work. Once you drop the assumption that the parameters are known, the math comes out differently. Different enough to not be able to think about it the same way.

Let us consider a case of an NYSE traded equity security. It is traded in a double auction so the winner's curse does not obtain. Because of this, the rational behavior is to create a limit order whose price is equal to $\mathbb{E}(p_t),\forall{t}$. There are many buyers and sellers so the limit book should be statically normal, or at least it will become so as the number of buyers and sellers go to infinity. So $p_t$ is statically normal about $p_t^*$, the equilibrium price.

Of course, we have ignored the distribution of $(q_t,q_{t+1})$. If you ignore splits and stock dividends, then it either continues to exist or it does not. So you have to create a mixture distribution for stock-for-stock returns, cash-for-stock returns, and bankruptcy. We will ignore these cases for simplicity, although doing so precludes the ability to solve an option pricing model.

So, if we restrict ourselves to $r_t=\frac{p_{t+1}}{p_t}$ and assume away all dividends, then our returns will be the ratio of two normals about the equilibrium. I am excluding dividends because they create a mess and I am excluding cases such as the 2008 financial crisis because you get a weird result that would consume page after page after page of text.

Now simplify our derivation, if we translate our data from $(p_t^*,p_{t+1}^*)$ to $(0,0)$ and define $\mu=\frac{p_{t+1}^*}{p_t^*}$ we can easily see the distribution. In the absence of a limitation on liabilities or an intertemporal budget constraint, by well-known theorem, the density of returns must be the Cauchy distribution, which has neither a mean nor a variance. When you translate everything back to price space, the density becomes $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$

Since there is no mean, you cannot take expectations, perform a t or F test, use any form of least squares. Of course, this would be different if it were an antique instead.

If it were an antique at an auction the winner's curse obtains. The high bidder wins the bid and the limiting density of high bids is the Gumbel distribution. So you would solve the same problem but as the ratio of two Gumbel distributions instead of two normal distributions.

The problem is not actually this simple. The limitation of liability truncates all underlying distributions. The intertemporal budget constraint skews all underlying distributions. There is a different distribution for dividends, mergers for cash, mergers for stock or property, bankruptcy, and a truncated Cauchy distribution for going concerns as above. There are six types of distributions present for equity securities in a mixture.

Different markets with different rules and different existential states create different distributions. An antique vase has the case where it is dropped and shatters. It also has the case of wear and tear or some other change in intrinsic quality. Finally, it also has the case that if enough similar vases are destroyed that the center of location moves.

Finally, because of truncation and the lack of a sufficient statistic for the parameters, there does not exist a computable and admissible non-Bayesian estimator.

You can find a derivation of the ratio of two normal variates and an explanation at http://mathworld.wolfram.com/NormalRatioDistribution.html

You can also find what appears to be the first paper on the topic at

Curtiss, J.H. (1941) On the Distribution of the Quotient of Two Chance Variables. Annals of Mathematical Statistics, 12, 409-421.

There is also a follow-up paper at

Gurland, J. (1948) Inversion Formulae for the Distribution of Ratios. The Annals of Mathematical Statistics, 19, 228-237

For the autoregressive form for Likelihoodist and Frequentist methods at

White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197,

and its generalization by Rao at

Rao, M.M. (1961) Consistency and Limit Distributions of Estimators of Parameters in Explosive Stochastic Difference Equations. The Annals of Mathematical Statistics, 32, 195-218

My paper takes these four and other papers, such as a paper by Koopman and one by Jaynes, to construct the distributions if the true parameters are unknown. It observes that the above White paper has a Bayesian interpretation and permits a Bayesian solution even though no non-Bayesian solution exists.

Do note that $\log(R)$ has a finite mean and variance, but no covariance structure. The distribution is the hyperbolic secant distribution. This is also by a well-known result in statistics. It cannot really be a hyperbolic secant distribution because of the side cases such as bankruptcy, mergers and dividends. The existential cases are additive, but the log implies multiplicative errors.

You can find an article on the hyperbolic secant distribution at

Ding, P. (2014) Three Occurrences of the Hyperbolic-Secant Distribution. The American Statistician, 68, 32-35

My article is at

Harris, D. (2017) The Distribution of Returns. Journal of Mathematical Finance, 7, 769-804

Before you read mine, you should read the above four papers first. It also wouldn't hurt to read E.T. Jaynes tome as well. It is, unfortunately, a polemical work, but it is rigorous nonetheless. His book is:

Jaynes, E.T. (2003) Probability Theory: The Language of Science. Cambridge University Press, Cambridge, 205-207