Reading: Return and Standard Deviation

In the preceding sections, we have worked with the assumption that returns are normally distributed. As it drastically simplifies the analysis, we will continue to work with this assumption later on. However, it is important to understand and remember that this is a simplification. In reality, returns are often not normally distributed.

In what follows, we briefly want to mention two of the many potential violations of the assumption of normally distributed returns:

• Discrete returns cannot be normally distributed
• "Fat tails."

Normality of Discrete returns

If a return is normally distributed, it can, in theory, go from minus infinity (\(-\infty\)) to plus infinity (\(+\infty\)), though such extreme values have, of course, an extremely low probability. Still, the normal distribution does not end at -100%. And there lies the problem. Therefore, whenever we assume normal returns, we implicitly assume that a return in question can be smaller than -100%.

When we work with discrete (simple) returns, this can be problematic. If the return (R) is smaller than -100%, we would end up with a negative asset value (P₁):

\(P_1 = P_0 \times (1+R) \)

\( if \ R < -100 \% \implies P_1 < 0 \)

Since the assets that we are considering here have limited liability, this outcome does not make too much sense. Investors cannot lose more than their full investment. Therefore, the smallest possible discrete return is -100%! Consequently, discrete returns, by definition, cannot truly be normally distributed!

To solve this problem, we should work with continuously compounded returns. As we have seen in the module "Time Value of Money" the future value of an asset with continuous compounding is:

\( P_1 = P_0 \times e^{r \times T}\),

where r is the continuously compounded return: r = ln(1+R).

This expression cannot take on negative values, regardless of the return we enter. For example, if \(P_0\) is 100 and the continuously compounded return \(r\) is -150%, \(P_1\) is 22.31 in 1 year (\(t=1\)):

\( P_1 = P_0\times e^{r \times T} = 100\times e^{-1.5}=22.31\).

With continuous compounding, security returns can therefore not take on negative values, which is easier to square with reality. Many academics therefore assume that continuously compounded returns are normally distributed, not discretely compounded returns. 

Fat tails

The logical follow-up question then is whether continuously compounded returns are, indeed, normally distribution. Empirically, we often observe that there are more return observation around the mean (in the center of the distribution) and in the far tails of the distribution. The latter characteristic is generally referred to as "fat tails."

This has very important implications: When we assume normal returns, we generally underestimate the probability of very large positive and/or negative outcomes! In particular in the context of risk management, where we are interested in exactly these outcomes, the normal distribution is therefore a dangerous tool.

For the purpose our investigation, namely the development of the basics of portfolio theory and the relation between risk and return, the assumption of normally distributed returns is an acceptable approximation (remember the initial graph with the return distribution of U.S. stocks). Therefore, in what follows, we keep working with this assumption.