# emaxdrawdown

Compute expected maximum drawdown for Brownian motion

## Syntax

``ExpDrawdown = emaxdrawdown(Mu,Sigma,T)``

## Description

example

````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

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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

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

Data Types: `double`

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

Data Types: `double`

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

Data Types: `double`

## Output Arguments

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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.

 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.