The Chi-square distribution

• The chi-square distribution is a family of distributions, depending on degrees of freedom:
• d.f. = n â€“ 1

Example (FRM Exam)

Assume you have empirical data showing historical returns (v) for a given financial variable
(e.g.: Forex rate), how could you perform a quick test of the validity of the power law
Prob(v > x) = K * x-a where x is large, as a good model of the tail of the distribution?

1. Plot the probability of v exceeding x standard deviations against x
2. Plot the probability of v exceeding x standard deviations against Log of x
3. Plot the Log of the probability of v exceeding x standard deviations against x
4. Plot the Log of the probability of v exceeding x standard deviations against the Log of x

Solution

D.
The mathematical relationship in the question can be rewritten (by taking the logs on both sides): Log(Prob(v > x)) = Log(K) â€“ aLog(x), i.e. the plot of the Log of the probability of v exceeding x standard deviations against the log of x should be a straight (decreasing) line if the relationship strictly holds.
The intercept is an estimate of Log of K and the slope of the line yields the parameter a.