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In many theories of financial mathematics it is assumed that asset return rates are normally distributed (e.g. VaR models) or lognormally distributed (e.g. Black-Scholes model). In practice, asset return rates are neither normally distributed nor lognormally distributed. It is easy to be checked by downloading time series of selected asset prices, computing selected return rates and running statistical tests like Kolmogorov–Smirnov test or Shapiro–Wilk test. There is also academic discussion on this topic:

Kjersti Aas, "To log or not to log: The distribution of asset returns"

Abstract:

"In the context of the measurement of market risk, the random variable is taken as the rate of return of a financial asset. One may define the return in different ways, the two most common are arithmetic and geometric returns. The distinction between these two types of returns is not well understood. They are frequently assumed to be approximately equal. Moreover they both are assumed to be normally distributed. In this paper we explain the difference. We show that both types cannot be normally distributed, and that the difference grows larger as the volatility of the financial asset increases and the time resolution decreases."

Question:

Why can we assume that asset return rates are normally distributed (or lognormally distributed) if empirical and statistical tests show that it is not true? How can it be justified?

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In the colloquial sense of the word "justified," it is not justified. I will describe why it is justified mathematically and under what circumstances and in what case it is not justified.

Let me begin with the simplest of equations $$\tilde{w}=R\bar{w}+\epsilon,\epsilon\sim\mathcal{N}(0,\sigma^2).$$ Let us assume that this equation is an element of our problem. From a static model, it maps to $w_{t+1}=Rw_t+\epsilon_{t+1}.$ Through Donsker's scale invariance, it can then be shown that this, in turn, could be mapped to a continuous time model, but we won't do that here as it won't add any value to the discussion.

Using either Ito or Stratonovich calculus, we can correctly solve a wide variety of problems, though both methods of calculus assume that all parameters are known. It is a very important assumption because the above equation has no solution within Frequentist axioms and remaining consistent with mean-variance finance.

To understand why, if $R$ is unknown, then Mann and Wald have shown that the maximum likelihood estimator is the ordinary least squares estimator for any $\epsilon$ drawn from any distribution centered on zero with a defined, fixed variance. However, note that if $R<1$, then capital will go to zero. If $R=1$, then it follows that $R$ is essentially currency and bears no interest so nobody would "invest" in it, though they may hold money for a variety of other reasons. It must be the case that $R>1$.

Expecting a positive return is not a surprising thing. The estimator for $R$ is the least squares estimator in all circumstances, so it is both the best estimator in Fisher's Likelihood-based method of statistics and Pearson and Neyman's Frequentist method of statistics. So far, so good.

The question then becomes, "what is the sampling distribution of $\hat{R}$?"

That is the rub. White in 1958 was able to show that the limiting distribution is the Cauchy distribution, which has neither a mean nor, as a consequence, a variance. Any use of least squares has zero power to find the parameter.

In other words, if mean-variance models are true, there cannot exist a test to measure it with positive power as the distribution-free methods available are Thiel's polynomial regression and quantile regression. Both are median based.

So, if you assume normality, the models are valid if all assumptions are met, including models with a normal distribution. Mathematically, the models are valid but inapplicable to a world where the parameters are not known with certainty.

I have proposed a new stochastic calculus that first-order stochastically dominates Ito methods, but it is in peer review right now. I will try and remember to come back and post if it is published. I dropped the assumption in Ito calculus that the parameters were known and proposed both a Bayesian and a Frequentist stochastic calculus.

If the parameters are unknown, then it is possible to derive the distribution of returns. That is because if $$r_t=\frac{p_{t+1}q_{t+1}}{p_tq_t}-1,$$ then $r$ is a function of prices and quantities, which are data. The definition of a statistic is any function of data. As such, returns are not data; they are statistics. Their distribution should be derived.

As $r$ is the product of the ratios of prices and the ratio of quantities, then $r$ is the sum of the price ratios times the existential states quantities could finish in. Note also that we just ignored dividends and liquidity costs. That is ill-advised but would make this a very long, long, post.

The existential states are bankruptcy where $q_{t+1}=0$, cash-for-stock mergers where $q_{t+1}=w$, stock-for-stock mergers where $q^f_{t+1}=kq^j_{t+1},$ and the going concern state where $q_{t+1}=mq_t$ where $m$ corrects for splits and stock dividends.

The remainder concerns the ratio of prices. The distribution of prices can be derived by combining auction theory with the terms and conditions of the contract or asset. As such, antiques should have a different return than stocks, which should have a different return than bonds.

From auction theory, in equilibrium, we know that in a double auction there is no winner's curse so the optimal solution is for each bidder to bid their expectation. The sampling distribution of very many expectations is the normal distribution. For argument purposes here I am ignoring thin markets because the answer comes out the same, but it takes another forty pages of proofs.

If we restrict ourselves to the case where $q_t=q_{t+1}$ and impose an equilibrium assumption, which is overly restrictive, but again, it is a length of discourse issue, then prices are the ratio of two normal distributions that are truncated at -100%.

If one treats the equilibrium prices as (0,0) by translating them as $p_t-p_t^*,\forall{t}$, then by well-known theorems, the distribution of returns will converge to the Cauchy distribution, though truncated.

For the going concern case, the distribution of returns, ignoring dividends and not correcting for liquidity costs, must be $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}-1.$$

If you want to test it, I would suggest downloading Carnival Cruise Lines daily prices. Construct daily returns correcting for weekends. Construct the Bayesian posterior predictive distribution and you will find it nearly perfectly overlaps the kernel density estimate.

The problem with using the normal distribution is that the Cauchy distribution has no first or higher moments that are defined. The consequence of this is that estimates of $\beta$ are completely without power and have perfect asymptotic relative inefficiency when compared to any valid median estimator.

With respect to the log-normal distribution, everything that is listed above still holds. Because log-normal models can be derived from normal models, nothing is different. For example, you can derive Black-Scholes from the Capital Asset Pricing Model. That is because, while the normal distribution assumes additive errors, they can be converted to multiplicative errors with a model change by noting the relationship between difference equations and models using exponential constructions.

This counter-intuitive observation does depend on knowledge of the parameters. When they are unknown, the concave nature of the logarithm will create a different result.

See

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

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

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

Marsaglia, G. (1965) Ratios of Normal Variables and Ratios of Sums of Uniform Variables. Journal of the American Statistical Association, 60, 193-204.

Marsaglia, G. (2006) Ratios of Normal Variables. Journal of Statistical Software, 16, 1-10.

Mann, H. and Wald, A. (1943) On the Statistical Treatment of Linear Stochastic Difference Equations. Econometrica, 11, 173-200.

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

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  • $\begingroup$ Thanks for your input! +1 $\endgroup$ – B_B May 23 at 6:50
  • $\begingroup$ +1, I also have begun reading your "The Distribution of Returns". Its one of the clearest and well written articles I've seen in a long time. Only read 6 pages so far but will finish! $\endgroup$ – Attack68 May 23 at 7:58
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It is justified in that you obtain some better information than you have without it.

For example, you can make the assumption and acknowledge some inaccuracy in your model. The inaccuracy of pricing a swaption based on the assumption of normality (or lognormality) leaves you in a far better position than having no assumption, nor any model to begin with.

To correct the inaccuracy vol skews are introduced which assert a different level of kurtosis to the distribution than that of normality (or lognormality). So you are in a more knowledgable position again.

You may never find absolute truth but you may very well find some truth that gains a profit or allows you a framework for risk management of a market-making portfolio.

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  • $\begingroup$ Thanks for your input! +1 $\endgroup$ – B_B May 23 at 6:50

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