Stocks' returns distribution

Question

I understand that stocks' returns are not normally distributed. However, is there any method that we can rescale the stocks' returns so they look more like normal distributions?

I managed to find a paper talking about this: L. C. G. Rogers: Sense, nonsense and the S&P500, Decisions in Economics and Finance (2018) 41:447–461

https://link.springer.com/content/pdf/10.1007/s10203-018-0230-3.pdf

where the author rescaled the SP500 returns as follows:

As you can see, he managed to scale the returns on the extreme periods (for example Black Monday 1987) to look more like normal distributions.

I tried to replicate this method using Python using the same parameters as in the paper with K = 4, Beta is 0.025. N was not specified but I chose N to be 100.

SP500['returns'] = np.log(SP500['Adj Close']/SP500['Adj Close'].shift(1))
SP500['returns_sq'] = np.square(SP500['returns'])
SP500.loc[:, 'vol'] = 0
SP500.loc[:,'Vol_rescaled_returns'] = 0
K = 4
Beta = 0.025
SP500.loc[101,'vol'] = np.sqrt(SP500.loc[1:101,'returns_sq'].mean())  

for i in range(101,len(SP500)-1):
  Y = max(-K*SP500.loc[i,'vol'],min(K*SP500.loc[i,'vol'],SP500.loc[i,'returns']))
  SP500.loc[i+1,'vol'] = np.sqrt(Beta*(Y**2) + (1-Beta)*(SP500.loc[i,'vol']**2))
  SP500.loc[i+1,'Vol_rescaled_returns'] = SP500.loc[i+1,'returns'] / SP500.loc[i+1,'vol']

However, my result is different from the paper, as shown below with significant negative returns on Black Monday around -16 while on the paper it's -6. Is there anything wrong with my code above? I have checked a few times but it seems quite straightforward or is there a problem with this method? Thanks a lot!

Answer 1

Standardised log-returns are approximately normally distributed, $\frac{r_t}{\sqrt{\text{RV}_t}}\sim N(0,1)$.

As @noob2 says, heteroscedasticity is a big reason why (log) returns aren't normally distributed. If you correct for this fact (divide by the square root of realised variance, the sum of squared returns), then returns look quite close to being normal. Instead of realised variance, you could even use a simple GARCH model if you don’t want to use high frequency data.

To see some empirical evidence, check out these two papers:

• Andersen, Bollerslev, Diebold and Ebens (2001, JFE) who study the returns of the stocks in the Dow Jones and
• Andersen, Bollerslev, Diebold and Labys (2001, JASA) who study FX rates.

From the JFE paper: You see that $r_t$ has fat tails (kurtosis about 5). After scaling by realised variance, $r_t/v_t$, the kurtosis is much closer to the three of a normal distribution. You can see the resemblance to the normal distribution from the plot below.

Answer 2

There are several distributions that generalize normal distribution with skewness and kurtosis, and some of them are known to model the stock return better than a normal distribution.

For example, Johnson's SU distribution is a good candidate. See the QQ plot (Figure 4) in Choi et al. (2019). Basically, the SU distribution is the sinh transformation of the normal variate. So, the arcsinh transformation (or rescale as you put it) of the stock return very closely follows a normal distribution.

Reference:

• Choi, J., Liu, C., & Seo, B. K. (2019). Hyperbolic normal stochastic volatility model. Journal of Futures Markets, 39(2), 186–204. https://doi.org/10.1002/fut.21967

Answer 3

It is important to understand what causes non-normality. Consistent with noob2's comment, it could be attributed to a change in regimes. These regimes cause excess skewness and kurtosis, two properties that are inconsistent with normality. In this regard, one can impose a structural model on the data generating function. One potential is a combination of elliptical and non-elliptical components. Under this specification, one can perform a decomposition that filters the elliptical component from the returns. It all depends on the application. For instance, this paper illustrates this decomposition for portfolio selection.