Drawdown (economics): Difference between revisions

Revision as of 05:51, 15 January 2007

Latest revision as of 05:39, 28 September 2025

(93 intermediate revisions by 52 users not shown)

Line 1:

{{Short description|Measure of the decline from a historical peak}}

'''Drawdown''' is the measure of the decline from a historical peak in some variable (e.g., the variable of interest might be the cumulative profit since inception from the purchase of a share of common stock). Somewhat more formally, if X(t) is a random process [<math>X(0) = 0, t \geq 0</math>], the drawdown at any time, T, is defined as <math> - Min \lbrack 0, X(T) - Max_{t \in (0,T)} \lbrack X(t) \rbrack \rbrack </math>

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).<ref>{{Cite web|url=https://www.fidelity.com/learning-center/trading-investing/trading/what-is-drawdown-video|title=What Is A Drawdown? – Fidelity|website=www.fidelity.com|date=January 2001 |language=en-US|access-date=2019-08-04}}</ref>

Somewhat more formally, if <math display="inline">X(t), \; t \ge 0</math> is a [[stochastic process]] with <math display="inline">X(0) = 0</math>, the drawdown at time <math>T</math>, denoted <math display="inline">D(T)</math>,

In finance, the use of drawdown as an indicator of risk is particularly popular in the [[Commodity Trading Advisors - CTA]] world with the derivation 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 relatively to a risk-free rate while the standard deviation of returns in the denominator is replaced by other compilations of drawdown.

is defined as:<math display="block"> D(T) = \max_{t\in(0,T)}X(t)-X(T)

</math>The '''average drawdown''' (AvDD) up to time <math>T</math> is the time average of drawdowns that have occurred up to time <math>T</math>:<math display="block">\operatorname{AvDD}(T) = {1\over T}\int_0^T D(t) \, dt</math>The '''maximum drawdown''' (MDD) up to time <math>T</math>

is the maximum of the drawdown over the history of the variable. More formally, the MDD is defined as:<math display="block"> \operatorname{MDD}(T)=\max_{\tau\in (0,T)}D(\tau)=\max_{\tau\in (0,T)}\left[\max_{t \in (0,\tau)} X(t)- X(\tau) \right]</math>

==Pseudocode==

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 '''do'''

# peak will be the maximum value seen so far (0 to i), only get updated when higher NAV is seen

'''if''' (NAV[i] > peak) '''then'''

peak = NAV[i]

'''end if'''

DD[i] = 100.0 × (peak - NAV[i]) / 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[i] > MDD) '''then'''

MDD = DD[i]

'''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.{{citation needed|date=June 2015}}

:The '''Maximum Drawdown''', more commonly referred to as Max DD, is the worst (the maximum) peak to valley loss since the investment’s inception.{{citation needed|date=June 2015}}

In finance, the use of the maximum drawdown asis an indicator of risk is particularly popular in the [[Commodity Trading Advisors - CTA]] world withthrough the derivationuse of three performance measures: the [[Calmar Ratioratio]], the [[Sterling Ratioratio]] and the [[Burke Ratioratio]]. 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 relativelyover to athe risk-free rate while the standard deviation of returns in the denominator is replaced by othersome compilationsfunction 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.{{citation needed|date=June 2015}}

When <math>X</math> is [[Brownian motion#Mathematics|Brownian motion]] with drift, the expected behavior of the MDD as a function of

time is known. If <math>X</math> is represented as:<math display="block">X(t)=\mu t+ \sigma W(t)</math>Where <math>W(t)</math> is a standard [[Wiener process]], then there are three possible outcomes based on the behavior of the drift <math>\mu</math>:<ref>{{Cite journal|last1=Magdon-Ismail|first1=Malik|last2=Atiya|first2=Amir F.|last3=Pratap|first3=Amrit|last4=Abu-Mostafa|first4=Yaser S.|date=2004|title=On the Maximum Drawdown of a Brownian Motion|url=http://www.cs.rpi.edu/~magdon/ps/journal/drawdown_journal.pdf|journal=Journal of Applied Probability|volume=41|issue=1|pages=147–161|doi=10.1239/jap/1077134674|s2cid=122630605}}</ref>

* <math>\mu > 0</math> implies that the MDD grows logarithmically with time

* <math>\mu = 0</math> implies that the MDD grows as the square root of time

* <math>\mu < 0</math> 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 [[Mathematical optimization|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 programming|linear program]].<ref>{{Cite web|url=https://www.ise.ufl.edu/uryasev/files/2011/11/drawdown.pdf|title=Portfolio Optimization with Drawdown Constraints|last1=Chekhlov|first1=Alexei|last2=Uryasev|first2=Stanislav|date=2003|last3=Zabarankin|first3=Michael}}</ref><ref>{{Cite journal|last1=Chekhlov|first1=Alexei|last2=Uryasev|first2=Stanislav|last3=Zabarankin|first3=Michael|date=2005|title=Drawdown Measure in Portfolio Optimization|url=https://www.ise.ufl.edu/uryasev/files/2011/11/IJTAF_DrawDown_Paper.pdf|journal=International Journal of Theoretical and Applied Finance|volume=8|issue=1|pages=13–58|doi=10.1142/S0219024905002767}}</ref>

The authors start by proposing an auxiliary function <math>\Delta_{\alpha}(x)</math>, where <math>x\in\mathbb{R}^{p}</math> is a vector of portfolio returns, that is defined by:<math display="block">\Delta_\alpha(x) = \min_\zeta \left\{ \zeta + {1\over{(1-\alpha)T}}\int_0^T [D(x,t) - \zeta]_{+} \, dt \right\}</math>They call this the ''conditional drawdown-at-risk'' (CDaR); this is a nod to [[Expected shortfall|conditional value-at-risk]] (CVaR), which may also be [[Expected shortfall#Optimization of expected shortfall|optimized using linear programming]]. There are two limiting cases to be aware of:

* <math display="inline">\lim_{\alpha\rightarrow 0} \Delta_\alpha(x)</math> is the average drawdown

* <math display="inline">\lim_{\alpha\rightarrow 1} \Delta_\alpha(x)</math> is the maximum drawdown

==See also==

*[[Linear programming]]

*[[Risk measure]]

*[[Risk return ratio]]

== References ==

<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. [What 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.&nbsp;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.&nbsp;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 {{Webarchive|url=https://web.archive.org/web/20120227045357/http://alumnus.caltech.edu/~amir/mdd-risk.pdf |date=2012-02-27 }})

*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}}

[[Category:Business terms]]

[[Category:Financial risk]]

[[Category:Linear programming]]

[[Category:Financial models]]

[[Category:Financial risk modeling]]