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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(t), \; t \ge 0 is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
with X(0) = 0, the drawdown at time T, denoted D(T), is defined as: D(T) = \max\left max_X(t)-X(T),0 \right \equiv \left \max_X(t)-X(T) \right The average drawdown (AvDD) up to time T is the time average of drawdowns that have occurred up to time T:\operatorname(T) = \int_0^T D(t) \, dtThe maximum drawdown (MDD) up to time T is the maximum of the drawdown over the history of the variable. More formally, the MDD is defined as: \operatorname(T)=\max_D(\tau)=\max_\left max_ X(t)- X(\tau) \right/math>


Pseudocode

The following
pseudocode In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...
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 do # peak will be the maximum value seen so far (0 to i), only get updated when higher NAV is seen if (NAV > peak) then peak = NAV end if DD = 100.0 × (peak - NAV / peak # Same idea as peak variable, MDD keeps track of the maximum drawdown so far. Only get updated when higher DD is seen. if (DD > MDD) then MDD = DD end if end for


Trading definitions

There are two main definitions of a drawdown:


1. How low it goes (the magnitude)

:Put 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) in the value of an investment. :The Maximum Drawdown, more commonly referred to as 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 advisor A commodity trading advisor (CTA) is US financial regulatory term for an individual or organization who is retained by a fund or individual client to provide advice and services related to trading in futures contracts, commodity options and/or ...
s through the widespread use of three performance measures: the Calmar ratio, the Sterling ratio and the
Burke ratio Burke is an Anglo-Norman Irish surname, deriving from the ancient Anglo-Norman and Hiberno-Norman noble dynasty, the House of Burgh. In Ireland, the descendants of William de Burgh (–1206) had the surname ''de Burgh'' which was gaelicised in ...
. 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 In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
, then there are three possible outcomes based on the behavior of the drift \mu: * \mu > 0 implies that the MDD grows logarithmically with time * \mu = 0 implies that the MDD grows as the square root of time * \mu < 0 implies that the MDD grows linearly with time


Banking or other finance definitions


Credit offered

Where an amount of credit is offered, a drawdown against the line of credit results in a debt (which may have associated interest terms if the debt is not cleared according to an agreement.)


Funds offered

Where funds are made available, such as for a specific purpose, drawdowns occur if the funds – or a portion of the funds – are released when conditions are met.


Optimization of drawdown

A passing glance at the mathematical definition of drawdown suggests significant difficulty in using an optimization framework to minimize the quantity, subject to other constraints; this is due to the non-convex nature of the problem. However, there is a way to turn the drawdown minimization problem into a
linear program Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is ...
. The authors start by proposing an auxiliary function \Delta_(x), where x\in\mathbb^ is a vector of portfolio returns, that is defined by:\Delta_\alpha(x) = \min_\zeta \left\They call this the ''conditional drawdown-at-risk'' (CDaR); this is a nod to conditional value-at-risk (CVaR), which may also be optimized using linear programming. There are two limiting cases to be aware of: * \lim_ \Delta_\alpha(x) is the average drawdown * \lim_ \Delta_\alpha(x) is the maximum drawdown


See also

*
Linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
* Risk measure * Risk return ratio


References


Further reading

*Burghardt, G., Duncan, R. and L. Liu, "Understanding Drawdowns", working paper, Carr Futures (September 4), 2003 *Eckholdt, H., "Risk Management: Using SAS to Model Portfolio Drawdown, Recovery and Value at Risk" (February), 2004. hat journal was this in?*Goldberg, L.R. and O. Mahmoud, "On a Convex Measure of Drawdown Risk", working paper, Center for Risk Management Research, UC Berkeley, 2014. (https://ssrn.com/abstract=2430918) *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. *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. *Magdon-Ismail, M. and A. Atiya, "Maximum Drawdown", ''Risk Magazine'' (October), 2004. (http://alumnus.caltech.edu/~amir/mdd-risk.pdf ) *Steiner, Andreas, "Ambiguity in Calculating and Interpreting Maximum Drawdown," working paper (December), 2010. *Wilkins, K., C. Morales and L. Roman, "Maximum Drawdown Distributions with Volatility Persistence", working paper, 2005. {{Financial risk Business terms Financial risk Linear programming Financial models Financial risk modeling