208 Optimizing Optimization
(an estimator for the worst-case return), and Q 100 is the maximum. Of course,
for quantiles in the tails this requires that we have a reasonably large number
of observations.
VaR is the loss (negative return) only to be exceeded with a given, usually
small, probability at the end of the investment period. Thus, VaR is a quan-
tile of the distribution of returns for this period. In our notation, VaR for a
probability of 1% can be written as Q 1. Quantiles may also be used as reward
measures; we could, for example, maximize a higher quantile (e.g., the 90th).
Drawdown
The functions described so far are functions of the distribution of final wealth,
but we may also observe the time path of portfolio wealth. Let v be a time
series of portfolio values, with observations at t 0, 1, 2, ... , T. Then, the
drawdown Dt of this series at time t is defined as:
Dttvvtmax
(9.6)where vtmax is the running maximum, i.e., vvsttsmaxmax{ | ∈[ , ]}0.
The symbol D stands for the whole vector of length T 1; subscripts indi-
cate a scalar value, for instance the drawdown’s maximum, or the drawdown
at a particular point in time. Other functions may be computed to capture the
information in the drawdown vector, e.g., the mean time underwater (i.e., the
average time elapsed between two consecutive values in D that are sufficiently
close to 0), or the correlation between a portfolio’s drawdown and the draw-
down of an alternative asset like an index.
Typical examples used in risk – reward ratios may be the drawdown’s mean,
its maximum, or its standard deviation, which can be computed by:
DDDDDDDmeanmaxstd mean
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1TTt
ttTt
ttT∑
∑
max( )()respectively. The definition in Equation (9.6) gives D in currency terms (i.e.,
the absolute drawdown). Usually, a relative drawdown is preferred, obtained
by using the logarithm of v or by dividing every Dt by vtmax.