Quadratic probing

The drawdown is the measure of the decline from a historical peak in some variable (typically the cumulative profit or total open equity of a financial trading strategy).

Somewhat more formally, if $X=(X(t),t\ge 0)$ is a random process with X(0) = 0, the drawdown at time T, denoted $D(T)$, is defined as

$D(T)=\max \{0,{\max }_{t\in (0,T)}X(t)-X(T)\}$

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

$\text{MDD}(T)={\max }_{\tau \in (0,T)}[{\max }_{t\in (0,\tau )}X(t)-X(\tau )]$

The following pseudocode computes the Drawdown ("DD") and Max Drawdown ("MDD") of the variable "NAV", the Net Asset Value of an investment. Drawdown and Max Drawdown are calculated as percentages:

MDD = 0
peak = -99999
for i = 1 to N step 1
  if (NAV[i] > peak) # peak will be the maximum value seen so far (0 to i)
    peak = NAV[i]
  endif
  DD[i] = 100.0 * (peak - NAV[i]) / peak
  if (DD[i] > MDD) # Same idea as peak variable, MDD keeps track of the maximum drawdown so far.
    MDD = DD[i]
  endif
endfor

Finance Definitions

There are two main definitions of a drawdown:

1. How low it goes (the magnitude)

Putting it plainly, a drawdown is the “pain” period experienced by an investor between a peak (new highs) and subsequent valley (a low point before moving higher).

Next, the Maximum Drawdown, or more commonly referred to as Max DD. This is pretty much self explanatory, as the Max DD is the worst (the maximum) peak to valley loss since the investment’s inception.

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.

2. How long it lasts (the duration)

The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs.

The Max Drawdown Duration is the worst (the maximum/longest) amount of time an investment has seen between peaks (equity highs).

Many assume Max DD Duration is the length of time between new highs during which the Max DD (magnitude) occurred. But that isn’t always the case. The Max DD duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss (and usually is, because the program needs a long time to recover from the largest loss), but it doesn’t have to be.

When X is Brownian motion with drift, the expected behavior of the MDD as a function of time is known. If X is represented as $X(t)=\mu t+\sigma W(t),$ where $W(t)$ is a standard Wiener process, then: when $\mu >0$ the MDD grows logarithmically with time; when $\mu =0$ the MDD grows as the square root of time; and when $\mu <0$ the MDD grows linearly with time (Magdon-Ismail et al. 2004).

Further reading

• Burghardt, G., Duncan, R. and L. Liu, "Understanding Drawdowns", working paper, Carr Futures (September 4), 2003 (http://www.intelligenthedgefundinvesting.com/pubs/rb-bdl.pdf)
• Eckholdt, H., "Risk Management: Using SAS to Model Portfolio Drawdown, Recovery and Value at Risk" (February), 2004. (http://www.intelligenthedgefundinvesting.com/pubs/rb-he.pdf)
• Grossman, S. J. and Z. Zhou, "Optimal Investment Strategies for Controlling Drawdowns", Mathematical Finance 3, pp. 241–276, 1993.
• Hamelink, F. and M. Hoesli, "The Maximum Drawdown as a Risk Measure: The Role of Real Estate in the Optimal Portfolio Revisited", working paper (June 24), 2003. (http://www.intelligenthedgefundinvesting.com/pubs/rb-fhmh.pdf)
• Hayes, B. T., "Maximum Drawdowns of Hedge Funds with Serial Correlation", Journal of Alternative Investments (vol 8, no 4) (Spring), pp. 26–38, 2006.
• Kim, Daehwan, "Relevance of Maximum Drawdown in the Investment Fund Selection Problem when Utility is Nonadditive", working paper (July), 2010. (http://www.intelligenthedgefundinvesting.com/pubs/dk_rmd.pdf)
• Steiner, Andreas, "Ambiguity in Calculating and Interpreting Maximum Drawdown," working paper (December), 2010. (http://www.intelligenthedgefundinvesting.com/pubs/as_acm.pdf)
• Wilkins, K., C. Morales and L. Roman, "Maximum Drawdown Distributions with Volatility Persistence", working paper, 2005. (http://www.intelligenthedgefundinvesting.com/pubs/rb-kwcmlr.pdf)
• Magdon-Ismail, M. and A. Atiya, "Maximum Drawdown", Risk Magazine (October), 2004. (http://alumnus.caltech.edu/~amir/mdd-risk.pdf)
• Magdon-Ismail, M., A. Atiya, A. Pratap, and Y. Abu-Mostafa, "On the Maximum Drawdown of the Brownian Motion", Journal of Applied Probability (vol 41, no 1), 2004. (http://alumnus.caltech.edu/~amir/drawdown-jrnl.pdf)

References

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