emaxdrawdown

Compute expected maximum drawdown for Brownian motion

Syntax

ExpDrawdown = emaxdrawdown(Mu,Sigma,T)

Description

ExpDrawdown = emaxdrawdown(Mu,Sigma,T) computes the expected maximum drawdown for a Brownian motion for each time period in T using the following equation:

$$dX\left(t\right)=\mu dt+\sigma dW\left(t\right).$$

If the Brownian motion is geometric with the stochastic differential equation

$$dS\left(t\right)={\mu }_{0}S\left(t\right)dt+{\sigma }_{0}S\left(t\right)dW\left(t\right)$$

then use Ito's lemma with X(t) = log(S(t)) such that

$$\begin{array}{c}\mu ={\mu }_0-0.5{\sigma }_0{}^2,\\ \sigma ={\sigma }_0\end{array}$$

converts it to the form used here.

Examples

Compute Expected Maximum Drawdown

This example shows how to use log-return moments of a fund to compute the expected maximum drawdown (EMaxDD) and then compare it with the realized maximum drawdown (MaxDD).

load FundMarketCash
logReturns = log(TestData(2:end,:) ./ TestData(1:end - 1,:));
Mu = mean(logReturns(:,1));
Sigma = std(logReturns(:,1),1);
T = size(logReturns,1);

MaxDD = maxdrawdown(TestData(:,1),'geometric')

MaxDD = 
0.1813

EMaxDD = emaxdrawdown(Mu, Sigma, T)

EMaxDD = 
0.1545

The drawdown observed in this time period is above the expected maximum drawdown. There is no contradiction here. The expected maximum drawdown is not an upper bound on the maximum losses from a peak, but an estimate of their average, based on a geometric Brownian motion assumption.

Input Arguments

Mu — Drift term of a Brownian motion with drift

numeric

Drift term of a Brownian motion with drift., specified as a scalar numeric.

Data Types: double

Sigma — Diffusion term of a Brownian motion with drift

numeric

Diffusion term of a Brownian motion with drift, specified as a scalar numeric.

Data Types: double

T — A time period of interest

numeric | vector

A time period of interest, specified as a scalar numeric or vector.

Data Types: double

Output Arguments

ExpDrawdown — Expected maximum drawdown

numeric

Expected maximum drawdown, returned as a numeric. ExpDrawdown is computed using an interpolation method. Values are accurate to a fraction of a basis point. Maximum drawdown is nonnegative since it is the change from a peak to a trough.

Note

To compare the actual results from maxdrawdown with the expected results of emaxdrawdown, set the Format input argument of maxdrawdown to either of the nondefault values ('arithmetic' or 'geometric'). These are the only two formats that emaxdrawdown supports.

References

[1] Malik, M. I., Amir F. Atiya, Amrit Pratap, and Yaser S. Abu-Mostafa. “On the Maximum Drawdown of a Brownian Motion.” Journal of Applied Probability. Vol. 41, Number 1, March 2004, pp. 147–161.

Version History

Introduced in R2006b