Drawdown (economics)

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The Drawdown is the measure of the decline from a historical peak in some variable (typically the cumulative profit of a financial trading strategy).

Somewhat more formally, if X(t) is a random process [X(0) = 0, t \geq 0], the drawdown at any time, T, denoted D(T) is defined as

D(T)=Max \lbrack 0, Max_{t \in (0,T)} X(t)- X(T) \rbrack

The Maximum Drawdown (MDD) up to time T is the maximum of the Drawdown over the history of the variable. More formally,

MDD(T)=Max_{\tau\in (0,T)}\lbrack Max_{t \in (0,\tau)} X(t)- X(\tau) \rbrack

In finance, the use of the maximum drawdown as an indicator of risk is particularly popular in the world of Commodity trading advisors through the widespread use of three performance measures: the Calmar Ratio, the Sterling Ratio and the Burke Ratio. These measures can be considered as a modification of the Sharpe ratio in the sense that the numerator is always the excess of mean returns over the risk-free rate while the standard deviation of returns in the denominator is replaced by some function of the drawdown.

When X(T) is a standard Brownian motion, the expected behavior of the MDD as a function of time is known. A standard Brownian motion is represented as

X(t) = μt + σW(t),

where W(t) is a standard Wiener process. Then when μ > 0 the MDD grows logarithmically with time, μ = 0 the MDD grows as the square root of time and μ < 0 the MDD grows linearly with time.

[edit] References

  • M. Magdon-Ismail, A. Atiya, A. Pratap, Y. Abu-Mostafa, On the Maximum Drawdown of a Brownian Motion, Journal of Applied Probability, Volume 41, Number 1, pages 147-161, March, 2004.
  • M. Magdon-Ismail, A. Atiya, Maximum Drawdown, Risk Magazine, Volume 17, Number 10, pages 99-102, October, 2004.