Summary: wish to relate the MDD to the drift, we cannot assume that the drift is zero.
Portfolio optimisation using the drawdown has also been considered in
Chekhlov, Uryasev & Zabarankin (2003).
The expected maximum drawdown
Assume that the value of a portfolio follows a Brownian motion:
where time is measured in years, and µ is the average return per unit time,
is the standard deviation of the returns per unit time and dW is the usual
Wiener increment. This model assumes that profits are not reinvested. If
profits are reinvested, then a geometric Brownian motion is the appropri-
For such a case, equivalent formulas can be obtained by taking a log trans-
formation: if x = log s, then x follows a Brownian motion with µ = µ^ 1/2
^2 and = ^. (The MDD in this case is defined with respect to the per-
centage drawdown rather than the absolute drawdown.) If the portfolio
value follows a more complicated process, then the results for the Brown-
ian motion can be used as a benchmark.
Using results on the first passage time of a reflected Brownian motion,
we find that the expected MDD has drastically different behaviour accord-
ing to whether the portfolio is profitable, breaking even or losing money.
This `phase shift' in the behaviour is highlighted by the asymptotic (T )