# Normal vs leptokurtic distribution for financial returns

Financial returns have been shown to follow leptokutotic distributions, however volatility forecasting models like EWMA and DCC-GARCH assume conditionally (dependent on time) Normal distribution for returns.

Can anyone explain why? Especially when financial return distributions are clearly not normally distributed?

• Should the title say leptokurtic? I don't know what it or leptokutotic means, but when I Googled leptokutotic all of the results were for leptokurtic, and it seems to have something to do with the normal distribution. – Chill2Macht Jun 17 '17 at 20:34

EWMA

EWMA does not assume any distribution for the returns as it is a nonparametric forecasting method. There $$\sigma_t^2 = \lambda \sigma_{t-1}^2 + (1-\lambda) r_{t-1}^2$$ in the univariate case and $$\Sigma_t = \lambda \Sigma_{t-1} + (1-\lambda) \mathbf{r}_{t-1} \mathbf{r}_{t-1}'$$ in the multivariate case, for some lambda, e.g. $\lambda=0.96$. The unconditional variance of such process (if univariate) or its components (if multivariate) is $\infty$, which means the second moment is $\infty$ and thus the fourth moment is $\infty$ (because higher-order even moments do not exist if a lower-order even moment does not exist), making kurtosis ill-defined $\left( K=\frac{\mu_4}{(\sigma^2)^2}=\frac{\infty}{(\infty)^2} \right)$.

DCC-GARCH

DCC-GARCH requires an assumption on the conditional distribution of standardized returns. Similarly to the case of EWMA, when mesokurtic standardized returns are assumed, the actual returns due to GARCH (and DCC-GARCH) are leptokurtic (Bollerslev, 1986, p. 313).

Bollerslev, Tim. "Generalized autoregressive conditional heteroskedasticity." Journal of Econometrics 31.3 (1986): 307-327.

• in the riskmetrics document, ch5 I think, they have stated that for the EWMA model, a conditional normal distribution has been assumed. Secondly, they have also stated the formula assuming zero mean, does that mean they are actually considering residuals rather than real returns? – Hsk Jun 17 '17 at 18:02
• @Hsk, Hmm, if the mean is zero, then residuals are returns. Regarding the assumed normality, it does not play a role in EWMA, so they can assume whatever they like. But the key observation is that it is conditional normality and I suppose the assumption is on standardized returns (or standardized residuals), not on raw ones. Which makes the actual returns have ill-defined kurtosis. – Richard Hardy Jun 17 '17 at 18:08
• I am still confused about "why" assuming conditional normality is the norm when it's been shown that returns are clearly not normally distributed so I will wait for more answers. Also, in the original riskmetrics document conditional normality assumption is for raw returns – Hsk Jul 4 '17 at 7:51
• @Hsk, right, I had mixed up before: conditional normality is for returns while unconditional normality is for standardized residuals. It is because when conditioning the distribution of the returns on explanatory variables, only the standardized residuals remain random. And their distribution is not conditioned on anything. – Richard Hardy Jul 4 '17 at 9:59

I perhaps will just add a little pragmatism here (yet a view from a student, just keeping things simple) and try to explain using very popular language but formally you should refer to the answer provided by Richard Hardy.

For GARCH models we normally need to make certain assumptions on the conditional distribution of standardized returns as mentioned above. The most comfortable (for optimization) is obviously Gaussian (note most comfortable). There are numerous variations of ARCH/GARCH models which utilizing t-distributions (with various degrees of freedom, also skewed t-distr) for standardized returns, moreover from pragmatic financial (risk exposure management) view, which probably just adds uncertainty but supports your question, after GARCH "prefiltering" there is evidence that filtered returns are still heavy-tailed (speaking very simple language are best described by Generalized Pareto Distribution tails) for your reference:

McNeil, A. J., & Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of empirical finance, 7(3), 271-300.

However, I believe it is quite problematic (if not impossible) to specify (and optimize) a GARCH model with GPD tails, there will be numerous questions (note that extreme value theory has its big portion of criticism already, I am not mentioning GPD domain of attraction here), also what would you use for the body domain of your returns and etc.? So, from my limited practical experience GARCH + Normal distribution works quite well for most of the cases, if necessary you can go for t-distribution GARCH, but it should always depend why do need this, what for? If you want Value at Risk Estimation you can for McNeil & Frey (2000) pseudo-maximum likelihood approach and so on. Hope that helps.

I have written a proof that most financial returns cannot have a first moment and in subsequent papers replaced most of the math of financial modeling. This isn't well known and I keep getting desk rejected because "our readers would not be interested." Sorry, personal frustration. You can find my working papers at https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=1541471 .

The short answer as to why is to consider the Markowitzian model's intertemporal budget constraint on an asset by asset basis. That is the movement of future wealth by time. In a CAPM construction this would be $$\tilde{w}=R\bar{w}+\varepsilon.$$

This is one of three ways to approach this problem. The others are to solve the returns as the ratio of future value divided by present value minus one and to approximate a solution by finding the difference of the logarithms of the future value and the present value. Although obviously related, it turns out they have very different solutions.

For the above equation, Mann and Wald showed in 1943 that the estimator for R is the least squares estimator for all distributions of $\varepsilon$ centered on zero and with non-zero, defined variance. White showed that the sampling distribution of $R$ is the Cauchy distribution. Because the least squares estimator is a version of a sample mean and the Cauchy distribution has no population mean, this implies that for likelihood-based methods, no solution to CAPM models exists.

For Pearson and Neyman Frequentists a solution does exist, but it abandons mean-variance finance. Obviously the method of least squares is gone, but Theil's regression still survives; unfortunately, what you would end up with is median-interquartile range finance without a covariance structure. The only way to solve mean-variance finance in Frequentist methods is to abandon it.

Fortunately, the method White used to show the limiting distribution has a Bayesian interpretation. He multiplied the square root of Fisher information by the likelihood, which in Bayesian terms is the same as multiplying the Jeffrey's prior by the likelihood. While this gives you the posterior distribution of a mean, if $\tilde{x}$ is drawn from a Cauchy distribution of $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x-\mu)^2},$$ then $\bar{x}$ is drawn from $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(\bar{x}-\mu)^2},$$ which is precisely the same distribution. So the posterior of $R$ should be the Cauchy distribution and the likelihood should be the Cauchy distribution.

The relationship between the likelihood and the density is that if $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x-\mu)^2},\forall\{\mu,\sigma\}\in\Theta$$ is your likelihood then $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x-\mu)^2},\forall{x}\in\chi$$ is your density.

If you do not use regression, but instead take the ratio then the answer is less clear. It depends upon the number of actors buying and selling, the rules in the auction, whether or not you are in equilibrium, the independence or dependence of errors, the terminal state of the asset, liquidity constraints, and other constraints in the system such as information asymmetry.

For the logarithmic case, the distribution is the hyperbolic secant distribution in the simple case. As the hyperbolic secant distribution lacks anything resembling a covariance structure, you cannot have a CAPM or Fama-French style model.

The easiest way to think about this is to consider the ratio case of $p_{t+1}/p_t$. Returns are not data; prices are data. Returns are the transformation of data. If prices have a random component then returns must follow a ratio distribution. In the Markowitz case, the distribution of the shock to prices must be normal. If you view this in the error space rather than price space so that in equilibrium errors are located at (0,0) and derive the distribution, the returns are the ratio of two independent, normal variates. This implies that returns follow a Cauchy distribution. For firms not merging out of existence or going bankrupt, this is very close to the empirical truth once you truncate the distribution at -100%. I argue the difference is from the intertemporal budget constraint, but that is to be argued by others later.

EDIT The reason that conditional normality is assumed, even though the original GARCH paper states that stocks strongly violate the assumptions of GARCH, is that one way to get leptokurtic distributions is to assume a mixture of normals where the mean is moving around. The problem with this is that White's proof above shows that those models go to the Cauchy distribution as time becomes arbitrarily large. The perception is that if the field can remain inside normality in some manner, then it can use a math that is well understood.

In a separate paper, I show that there does not exist a non-Bayesian estimator that is admissible and so that Frequentist decision theory is excluded. This excludes almost all of macroeconomics and financial economics. The short form of the argument goes like this:

1. The distributions involved lack statistics sufficient for the parameter
2. The Bayesian likelihood function is always minimally sufficient
3. It follows that for purposes of projection, but not necessarily inference, that non-Bayesian methods will lose information and their sampling distributions will be noisier than is necessary.
4. The Bayesian estimator will stochastically dominate the non-Bayesian estimator
5. Standard tools such as Ito calculus or optimal control theory won't be available.

It is very sensible to try to preserve normality because without a distribution that is at least in the exponential family, the tool loss will be fantastic. If you go back and read the initial papers following Markowitz, you will see that they were non-rigorous. The first paper by Markowitz is non-rigorous. This is problematic because no one notices this. A normal or normal-like distribution is essential for Markowitz to be valid. Because of this, a variety of tools have been developed on an ad hoc basis to try and preserve some form of normality so that a covariance matrix can be preserved.

If a covariance matrix cannot exist, and it cannot exist, then so much vanishes. Diversification is no longer inherently prudent. There is no such thing as "systematic" risk versus "idiosyncratic" risk. While it may exist, no one knows what those phrases mean now. Based on my work on option pricing, idiosyncratic risk has to be priced. OLS, GLS, FGLS and so forth are useless. The list goes on. It is very reasonable to try and preserve some relationship either to normality or at least to the exponential family of distributions.

I wont' work, but it does make sense.

• With all due respect, I do not think this answers the actual question. – Richard Hardy Jun 17 '17 at 18:05
• Perhaps not directly, but it at least seems to show that there is research in finance which doesn't assume everything is normally distributed. (And one of the OP's premises seems to be that it is always assumed that everything is normally distributed in finance.) So maybe with a caveat/note at the beginning that the answer doesn't directly address the question asked, but rather one of the premises underlying the question, would be helpful. I think it is a useful contribution to the thread. – Chill2Macht Jun 17 '17 at 19:15
• @Chill2Macht, While the text might be interesting and valuable on its own, I think it misses the point of the question. The OP does not doubt the returns are nonnormally distributed. His/her question is on why particular models assume normality. I do not think this answer addresses that. But you may disagree. – Richard Hardy Jun 17 '17 at 19:21
• @Chill2Macht The vast majority of current research in Finance does not use normal distributions for returns, this is not something that is debated at all and the OP has no doubts about this. The question was, "if we know returns aren't normal, why do we use GARCH?". A possible answer is "GARCH can reproduce distributions with fatter tails than normal" (because of the time-varying volatility, essentially). This answer doesn't mention GARCH at all. – Chris Haug Jun 17 '17 at 19:40