Portfolio optimization with “ Threshold Accepting ” : a practical guide 207
Quantiles
A quantile of the cumulative distribution function (CDF) of a sample r is
defined as:
QqCDF^1 () min{|qrrqCDF()}where q may range from 0% to 100% (we drop the % sign in subscripts).
So, in words, the q th quantile is a number Qq such that q of the observations
are smaller, and (100% q ) are larger than Qq. When we estimate quantiles,
several numbers may satisfy this definition ( Hyndman & Fan, 1996 ). A simple
estimator is the order statistics of portfolio returns []rr[] [ ] 12 ...r[ ]nS^ , i.e., a step
function. If k is the smallest integer not less than qn S , then the q th quantile is
the max( k , 1)th order statistic. This is consistent with the convention in many
statistical packages like Matlab or R that Q 0 is the minimum of the sample
01rdrd r (^) d
0
1
When rd is fixed
The illustration pictures the return distributions of two portfolios. If we aim to find a portfolio
that minimizes a function of the returns below rd, there should be little debate about which
portfolio to choose.
When rd equals some quantile
We would certainly prefer the more variable distribution on the right; a decision rule based
on a moment centered around rd would, however, choose the dominated distribution on the left.
Figure 9.2 Setting the desired return r d.