Distribution of simple returns vs logreturns

Question

I understand that stock prices are conditionally modeled using a log normal distribution by the relationship

$ y_t/y_{t−1}∼logN(μ_{daily},σ^2_{daily})$

$y_t∼logN(log(y_{t-1})+μ_{daily},σ^2_{daily}))$

which implies

$log(y_t)∼N(log(y_{t−1})+μ_{daily},σ^2_{daily})$ then $\frac{y_t−y_{t−1}}{y_{t−1}}≈log(y_t)−log(y_{t−1})∼N(μ_{daily},σ^2_{daily})$ ....equation(1)

According to above equation(1), returns can be approximated with a normal distribution (for now, we shall IGNORE about FAT TAIL and other problems).

But The simple return is defined by −1 < $R_t$ < ∞,since the price is always > 0

i.e. the minimum value of a simple return is capped at -100% (when $y_t$ becomes zero and hence $\frac{y_t−y_{t−1}}{y_{t−1}}$ becomes -1)

This means the probability density function of the simple return $R_t$ can never be symmetrical which contradicts above equation (1) as normal distribution is symmetrical.

Please guide me on what I missing here. It is clear I am logically wrong somewhere, but I am unable to figure out

Answer 1

Let us start with a definition.

Definition A statistic is any function of the data.

Returns are calculated as $$\text{Return}=\frac{\text{Future Value}}{\text{Present Value}}-1=R_t.$$

$$\text{Present Value}=\text{Price}_t\times\text{Quantity}_t=p_tq_t.$$ $$\text{Future Value}=\text{Price}_{t+1}\times\text{Quantity}_{t+1}=p_{t+1}q_{t+1}.$$

So $$R(p_t,p_{t+1},q_t,q_{t+1})=\frac{p_{t+1}q_{t+1}}{p_tq_t}$$

So returns are a function of prices and quantities. As such, you have to derive the distribution and not assume the distribution. It is a statistic just like Student's t-distribution or the F distribution. Fortunately, the solution to this problem has been known since 1941 in the field of statistics.

If you assume $q_t=q_{t+1}$, $p_t>0$ and $p_{t+1}>0,$ then you get a truncated distribution. You are correct, the normal distribution cannot be the distribution of returns. You haven't missed anything.

Using auction theory and assuming no effects from dividends, such as liquidating dividends; no mergers; bankruptcy or effects from liquidity costs then the distribution of equity returns must be $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}.$$

Because mergers, bankruptcy, dividends, and liquidity costs exist, the actual distribution is a complicated mixture distribution. Likewise, other auction structures, such as English style auctions yield a different distribution and different payout structures, such as those with bonds yield a different distribution.

In finance, you need to translate the center of location to the equilibrium, so that $(0,0)$ is at $p_t^*,p^*_{t+1}$.

The general method to calculate a ratio distribution is as follows.

If $Z=\frac{Y}{X}$, then the cumulative distribution function of $z$ is $D(z)=\Pr(Z\le{z}).$

The density function of $Z$, when the variables have support on the entire real number line, ends up being $$p(z)=\int_{-\infty}^\infty|x|f(x,zx)\mathrm{d}x.$$

The only possible way returns on equity securities could follow a normal or log-normal distribution is if you assume prices and quantities do not exist. Of course, that would make them certificates of deposit. If you remove government insurance, unsurprisingly, they can be modeled with a normal or log-normal distribution because bankruptcy implies a state where no payout may happen.

Curtiss, J. H. (1941). On the distribution of the quotient of two chance variables. Annals of Mathematical Statistics, 12:409-421.

Harris, D. E. (2017). The distribution of returns. The Journal of Mathematical Finance, 7(3):769-804.

Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association, 60(309):193-204.

Answer 2

You're missing that normality and log-normality of returns and prices are simplifying assumptions. Inspecting skew and kurtosis, it's even more obvious that they're assumptions.

Your example that return going to -100, but exceeding +100 defying the normality assumption is a triviality, particularly given we've already established returns aren't perfectly normal. For the vast majority of circumstances, it's good enough.