# Central limit theorem and normality assumption of asset return distribution

Can central theorem justify normality assumption of assets return distribution? And if it can why the empirical evidence show this assumption, which many finance models are based on, is a far cry from reality?

No. I just published a paper on this. If return is defined as $$r_t=\frac{p_{t+1}q_{t+1}}{p_tq_t},$$ and since returns are not data while prices and volumes are, then it follows that the distribution of returns depends entirely upon the distribution of prices and the distribution of quantities. For example, in bankruptcy, $q_{t+1}=0$.

As it is a very long paper, there is no single distribution or even family of distributions involved in returns. Under the standard stated assumptions in Markowitzian style models, the distribution of returns would integrate out to be $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$ This assumption follows from the fact that there are many buyers and sellers and that stocks are sold in a double auction so the winner's curse does not obtain. If the winner's curse were present, as in antique auctions, then the distribution would be the ratio of two Gumbel distributions.

Regardless, with some interesting special exceptions, returns on equity securities cannot have a mean. Since there cannot be a mean return, then it follows that $\beta$ as defined in models such as the CAPM cannot exist. An assumption in all of these finance models has been that the parameters are known with probability one, but if you drop that assumption, you will find that no estimator exists that could converge to the population parameter.

Ratio distributions are well known and taught as standard graduate work in statistics. The most common method to find a ratio distribution if the underlying are continuous densities is that if $Z=\frac{Y}{X}$, then the density of $Z$ is $$g(z)=\int_{-\infty}^\infty|x|f(x,zx)\mathrm{d}x.$$

With this you can derive just about any form of return except single period discount bonds and cash-for-stock mergers as well as accounting ratios and growth rates.

As a rule of thumb, the central limit theorem is strongly violated for any financial return data, as well as quite a bit of macroeconomic data. Further, again as a rule of thumb, no non-Bayesian estimator exists for financial data. I will be presenting that along with a replacement for Black-Scholes at a conference in Albuquerque in a few weeks. The short form of this would go as follows:

For an equation such as $$x_{t+1}=\beta{x}_t+\epsilon_{t+1},\beta>1$$ only non-parametric regression methods such as Theil's regression or quantile regression could work. Bayesian methods are never less risky than Frequentist methods, though the converse is not true. If we assume that the Frequentist tool chosen has the same sampling density as the posterior density if the prior density were flat and nuisance parameters were marginalized out, then the Frequentist density would still be inadmissible.

The issue is that the Frequentist sampling distribution is symmetric and would include regions where $\beta\le{1}$. If that region has a density of $K$ and the region greater than unity has area $1-K$ then it follows that the Bayesian posterior density with a prior where $\Pr(\beta\le{1})=0$ will have zero mass in the region less than or equal to one, but whose density would be $$\frac{1-K}{K}$$ times greater than the equivalent non-Bayesian component.

Because admissibility can be defined in term of stochastic dominance, it follows that the Bayesian estimator always stochastically dominates a Frequentist estimator. As such, no admissible non-Bayesian method is available for most finance problems and most macroeconomic problems where growth is involved.

• Would you please give a link to the paper - Thanks – vonjd Feb 20 '18 at 21:51
• Is it this one? file.scirp.org/pdf/JMF_2017083015172459.pdf – vonjd Feb 20 '18 at 21:54
• Yes. You will be able to pick up the black-scholes paper in the proceedings of the SWFA Conference. – Dave Harris Feb 21 '18 at 4:07

Normality is just an assumption which can be used to inform a model. No model is ever right, but some might be useful. Also, the assumption of normality doesn't really depend on the actual process being normal. Rather, it just requires that an investor's decisions are driven by mean and variance. While the underlying process of asset price returns is almost never normal - per se - we usually assume normality because:

1. Returns are approximately normal;
2. When an outcome is produced by many small effects acting additively and independently, its distribution will be close to normal; and,
3. It is much harder to do economics without normality

An investor who assumes joint normality need not be interested in anything else because the normal distribution is maximum entropy distribution which is completely described by two parameters: mean $\mu$ and variance $\sigma^2$. Therefore, the assumption of normality imposes the minimal prior structural constraint beyond these moments. An investor who only cares about mean and variance who doesn't use normality is taking into account additional assumptions which may or may not be correct or robust.

There are theories which define investor preferences for moments beyond the first two, but not much evidence of which I am aware that supports such a position. See: The possible preferences of investors for higher than first 2 moments of return distribution?

Approximate normality of logarithmic returns is classically demonstrated through the Lindeberg–Lévy central limit limit theorem (CLT). Normality also arises due to a variation of the CLT in which the sample of means of any distribution will tend toward normal. So, when you begin to look at a large number of assets and returns, the distributions will just tend towards normality.

If however you decide not to assume normality, you should probably have a compelling reason since it is much more difficult to do economics without this assumption.

For example, variance minimization techniques, such as those defined under Modern Portfolio Theory (MPT), are implicitly predicated on the assumption of joint normality. Even though there will be a set of portfolio weights which minimizes variance regardless of the underlying distributions, correlation (which also does not assume normality) is only a complete measure of association if the joint multivariate distribution is normal; i.e., covariance is only an exhaustive measure of co-movement if the joint distributions are themselves normal. We can see this is true because the joint distribution of X and Y is defined by joint normality:

${\frac {1}{2\pi \sigma _{X}\sigma _{Y}{\sqrt {1-\rho ^{2}}}}}\iint _{X\,Y}\exp \left[-{\frac {1}{2(1-\rho ^{2})}}\left({\frac {X^{2}}{\sigma _{X}^{2}}}+{\frac {Y^{2}}{\sigma _{Y}^{2}}}-{\frac {2\rho XY}{\sigma _{X}\sigma _{Y}}}\right)\right]\,\mathrm {d} X\,\mathrm {d} Y$

Which through a proof can be show to produce:

$\sigma _{X+Y}={\sqrt {\sigma _{X}^{2}+\sigma _{Y}^{2}+2\rho \sigma _{X}\sigma _{Y}}},$

If now, we define $\omega_i \sigma^2_i=\sigma_X$, and $\omega_j \sigma^2_j=\sigma_Y$, then we get back the equation which is used as the basis of mean variance optimization of a two asset portfolio:

$\mathbb{E}[\sigma _{p}^{2}]=\omega_{i}^{2}\sigma _{i}^{2}+\omega_{j}^{2}\sigma _{j}^{2}+2\omega_{i}\omega_{j}\sigma _{i}\sigma _{j}\rho _{ij}$

So while the portfolio covariance matrix can always be computed, to the extent that underlying assets have returns which are not normal the optimization is likely to be spurious.