Aug 27

Normal & standard normal distribution

by David Harper, CFA, FRM, CIPM

FRM |

bellcurve_clock

Learning objective

  • Describe the key properties of the normal distribution, the standard normal distribution

Why is the normal so common?

Two reasons, convenience and dazzling talent:

  • Convenience: The normal is economical because it only requires two parameters (mean and variance). The standard normal is even more economical: it requires zero parameters (mean = 0, variance = 1)
  • Dazzling talent: The central limit theorem (CLT) says that sampling distribution of sample means tends to be normal (i.e., converges toward a normally shaped distributed) regardless of the shape of the underlying distribution. This is profound: regardless of the distribution, the sample mean converges to normality.

Properties of normal

Key properties of the normal include:

  • Symmetrical around its mean value (symmetry = 0. Third moment = 0)
  • Peaks at mean but descends rapidly to tails
  • ~68% at 1s, ~95% at 2s, ~99.7% at 3s
  • Only requires (fully described by) two parameters, mean and variance
  • A linear combination (function) of two normally distributed random variables is itself normally distributed
  • Skew = 0, Kurtosis = 3 (excess kurtosis = 0)

Key role in (parametric) delta normal value at risk (VaR)

In the FRM, we tend to assume a VaR confidence of 95% or 99%, Since VaR is about losses not gains, this is a one-tailed probability:

normal_table

Are 95% and 99% confidence magic numbers?

Not even. We can lower the confidence to consider less extreme losses; there is nothing wrong with looking at 80% VaR. It is likely more a shareholder perspective than a solvency perspective.

And we can certainly go higher. Here are the Basel II confidences:

  • Credit risk (IRB): 99.9% CaR one-year holding period. But not normal!
  • Market risk (IMA): 99% VaR ten-day holding period. Probably normal!
  • Operational risk (AMA): 99.9% CaR one-year holding period. Not normal (LDA compounds frequency and severity: will not be normal)

Where else do we see it?

The Black-Scholes assumes Brownian motion. Under this model of the stock price process, price levels (and wealth relatives; i.e., future price/today’s price) are lognormally distributed which means the period returns are roughly normally distributed (Greek Phi is often used to denote a normal distribution):

gmb_weiner

Comments

  1. Be the first to leave a comment!

Leave a Comment