Scaling Stock Returns to Address Non-Normality
Summary
The document asks whether stock returns can be rescaled to look more normally distributed and describes an attempt to reproduce a published method on S&P 500 returns. The approach clips each return at a multiple of an evolving volatility estimate, updates that estimate using an exponentially weighted squared clipped return, and divides the next return by the updated volatility. The author reports that the result around Black Monday is much more extreme than in the paper, raising questions about the implementation or method.
The replies offer alternatives and context. One recommends scaling returns by realized variance or volatility from a GARCH model, citing empirical studies that find lower kurtosis after volatility adjustment. Another suggests an inverse hyperbolic sine transformation associated with the Johnson SU distribution. A further reply attributes non-normality partly to regime changes and mentions modeling distinct distributional components. These are pointers and explanations rather than a resolved diagnosis of the replication discrepancy; the discussion does not establish that any transformation makes returns truly normal or appropriate for every application.
Key ideas
- Volatility changes can contribute to the heavy tails observed in raw returns.
- Scaling returns by realized volatility or a GARCH estimate can make their distribution closer to normal.
- The attempted recursive scaling clips returns before updating volatility, so implementation details affect the transformed series.
- The Johnson SU distribution motivates an inverse hyperbolic sine transformation as another route to approximate normality.
- Regime changes and distributional asymmetry may also explain why returns depart from normality.
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Full text
# Stocks' returns distribution
# Stocks' returns distribution
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 by Kevin (score 4)
https://quant.stackexchange.com/a/69196
##### 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 by jaehyukchoi49 (score 2)
https://quant.stackexchange.com/a/69221
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 by majeed simaan (score 1)
https://quant.stackexchange.com/a/69192
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.