Apr 09

Asset returns: like a stegosaurus or a T-Rex?

by David Harper, CFA, FRM, CIPM

FRM | Quant |

The first 2007 FRM reading is from Linda Allen's excellent book Understanding Market, Credit, and Operational Risk. (Last year, we told customers to start here. Now it tops the list. Coincidence...?)

This first reading contains a big theme: asset returns are not normal (i.e., normally distributed). 

Value at risk (VaR) is mostly approached in one of three ways:

  • Delta-normal
  • Historical simulation
  • Monte Carlo (forward simulation)

Two of the three (delta-normal and Monte Carlo) depend on an assumption about the random (stochastic) distribution of returns over time. We assume returns are random but we impose a distributional assumption on the randomness. The randomness is "illustrated by" or "characterized by" a probability distribution.

New learners sometimes see the formulas and let probability distributions scare them. They don't deserve to scare you. Distributions just draw a line-plot of the arc of what could happen. Over to the left, small odds. In the middle, more likely. Over to the right, less likely again. They give shape to randomness; they map the contours of risk.

It would be really great (read: easy) if asset returns were normally distributed. Like the bell curve. A normal distribution reminds me of the stegosaurus (not really, it's the best I could do). As dinosaurs go, the stegosaurs, being a herbivore, is relatively normal:

 

In my imagination, the stegosaurus is symmetrical, skinny-tailed, and stable. That's how asset returns would be if they were normal. But they totally aren't. Asset returns are more like the T-Rex: skewed, fat-tailed, and unstable.

Skewed refers to the empirical observation that, as Allen says, "declines are more severe than increases." Skewness is the third moment about the mean (The first moment is the mean, a location parameter. The second moment is the variance, a scale parameter). If the distribution is not skewed, it is symmetrical, and the coefficient of skewness will be zero (0).

Fat-tailed is important for the exam. You must know that asset returns tend to be fat-tailed. Fat-tailed returns have a critical implication on delta-normal VaR: actual extreme losses are more likely than predicted by a normal distribution. Tail density is related to peakedness: higher peaks implies skinnier tails because there is more density in the middle; lower peaks implies more density in the "fat" tails. Peakedness is also called kurtosis. Therefore, kurtosis is a measure of tail density (fat or skinny).

So, briefly, kurtosis is the "fourth moment" measure of peakedness (and, by implication, tail density). A normal distribution produces a coefficient of kurtosis equal to three (3). So, we say "kurtosis = 3 for a normal distribution" But note also, we sometimes instead say "excess kurtosis = 0 for a normal distribution." (as in, "the excess above or below three").

Finally, unstable refers the the unfortunate reality that, even if we manage to create good parameters (i.e., for our distribution), the parameters stubbornly change over time. In fact, they tend to change dramatically. As Allen says, "large moves will always occur 'out of the blue.'

She asks "can normality be salvaged?" In other words, can we do something to the normal distribution (e.g., make it conditional, transform it) to make it useful. Her answer is...no.

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