Английская Википедия:Drawdown (economics)
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).[1]
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>, is defined as:<math display="block"> D(T) = \max\left[\max_{t\in(0,T)}X(t)-X(T),0 \right ] \equiv \left[ \max_{t\in(0,T)}X(t)-X(T) \right ]_{+} </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.Шаблон:Citation needed
- The Maximum Drawdown, more commonly referred to as Max DD, is the worst (the maximum) peak to valley loss since the investment’s inception.Шаблон:Citation needed
In finance, the use of the maximum drawdown is an indicator of risk through the 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.Шаблон:Citation needed
When <math>X</math> is 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>:[2]
- <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 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.[3][4]
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 conditional value-at-risk (CVaR), which may also be 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
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. 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 Шаблон:Webarchive)
- 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.
