May 29

The assumptions behind normal value at risk (VaR) are all wrong, except for one

by David Harper, CFA, FRM, CIPM

FRM | Risk |

Normal parametric value at risk (VaR) depends on the following assumptions (there are other non-parametric VaR approaches, namely Monte Carlo simulation and historical simulation; further, there are parametric VaR approaches that do not assume a normal distribution):

* Stationarity: the (shape of the) probability distribution is constant over time

* Random walk: tomorrow's outcome is independent of today's outcome

* Non-negative: requirement: assets cannot have negative value

* Time consistent: what is true for a single period is true for multiple periods; e.g., assumptions about a single week can be extended to a year

* Normal: expected returns follow a normal distribution

 

Only one of these assumptions is ultimately reliable: assets don't typically have negative value. The other assumptions are routinely violated in financial markets:

Actual returns are not necessarily independent and identically distributed (i.i.d.)

The i.i.d. requirement subsumes stationarity and the independence of the random walk. Stationarity refers to a stable distribution over time. In the case of a normal distribution, it means the mean and variance are the same from period to period. (Note, when the variance is finite and constant, this is called homoscedasticity). But we know actual returns aren't so well-behaved. Linda Allen says volatility time-varies; Jorion agrees and adds that conditional mean is often also time-varying.

We can't always extend volatility over n-periods

Note that Linda Allen recommends continuous compounding because it easily extends over multiple periods, unlike the simple percentage. Continuous compounding enable time consistency. However, volatility clustering and the tendency of mean reversion both wreck the natural extension of volatility according to the square root rule.

Actual returns are not normal

This is the most devastating violation. Normal distributions are great at describing the middle, the central tendency. But a normal distribution is not well-suited to outliers. Unfortunately, in risk, that's what we care about: low frequency, high severity (LFHS) events:

 

 We know that actual financial returns tend to be fat-tailed. A normal distribution has kurtosis (the so-called fourth moment) of three. A fat-tailed distribution has excess kurtosis: kurtosis greater than three. This is called leptokurtic (less than three is called playtykurtic).

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