The MDD is defined as the maximum loss incurred from peak to bottom during a specified period of time. It is often preferred over some of the other risk measures because of the tight relationship between large drawdowns and fund redemptions. Also, a large drawdown can even indicate the start of a deterioration of an otherwise successful trading system, for example due to a market regime switch. Overall, the MDD is a very important risk measure. To be able to use it more insightfully, its analytical properties have to be understood. As a step towards this direction, we have presented in this article some analytic results that we have developed. We hope more and more results will come out from the research community analyzing this important measure

Abu-Mostafa, Yaser

Atiya, Amir

Magdon-Ismail, Malik

Pratap, Amrit

English

Caltech Authors - Main

The Maximum Drawdown of the Brownian Motion 
Malik Magdon-Ismail 
Dept Computer Science 
Rensselaer Polytechnic Inst it ut e 
Room 207, Ldly Bldg 
110 8th Street 
Troy, NY 12180 
magdon@cs.rpi.edu 
Amir Atiya 
Dept Computer Engineering 
Cairo University 
Giza, Egypt 
amiratiya@link.net 
Amrit Pratap 
Dept Computer Science 
Caltech 136-93 
Pasadena, CA 91125 
Yaser Abu-Mostafa 
Dept Electrical Engineering 
Caltech 136-93 
Pasadena, CA 91125 
The MDD is defined as the maximum loss incurred from peak to bottom during a 
specified period of time. Mathematically it is given by 
where X ( t )  represents the equity curve of the trading system or fund. The maximum 
drawdown MDD is the most widespread risk measure among money managers and 
hedge funds. It is often preferred over some of the other risk measures because of 
the tight relationship between large drawdowns and fund redemptions. Also, a large 
drawdown can even indicate the start of a deterioration of an otherwise successful 
trading system, for example due to a market regime switch. 
Despite the importance of such a problem, there has been a lack of theoretical results 
covering that problem. In this article, we develop new analytic formulas that relate a 
0-7803-7654-4/03/$I7.00 02003 IEEE 243 CIFEr'03 HONG KONG 
trading strategy’s return, standard deviation of returns, and MDD. A widely followed 
measure of risk adjusted return based on MDD is the Sterling Ratio. It is defined as 
Return over [0, T ]  
M D D  over [O,T] 
Sterling (T )  = 
It basically measures how much return to expect for every unit of downside risk, akin 
to the widely used Sharpe ratio. In an environment where markets exhibit frequent 
crashes and portfolios encounter sustained periods of losses the Sterling ratio has 
emerged as one of the leading performance measures used in the industry. In this 
article we also develop formulas that relate the Sterling ratio with the Sharpe ratio. 
The only currently available way to analyze MDD is to measure it on a previous 
historical period. The problem withi this approach is that this represents only a one- 
sample-point estimate. Our developed formulas will give the expected MDD in a 
specific time period for a given return and standard deviation. As such, they will 
provide some guidance as to what h4DD to expect and also whether a trading system 
with a given observed MDD is considered a  favorable^' or “unfavorable” system. 
Assume a trading system follows a Brownian motion: 
where p is the average return per unit time, 0 is the standard deviation of the returns 
per unit time for the trading system, and dw is the usual Wiener increment. This 
model assumes that profits are not reinvested using the trading model. If profits are 
reinvested, then the Geometric Brownian motion is the appropriate model. For such 
a case equivalent formulas can similarly be derived by taking a log transformation. 
(The MDD in that case is defined ils the percentage loss rather than absolute loss.) 
There could potentially be trading systems that have different or more coinplicated 
process models. However, the benefit from considering a standard Brownian motion 
is that the resulting formulas could provide a baseline benchmark, against which we 
can compare any given system. 
A starting step in our derivations has been to show that the drawdown process is 
equivalent to a Brownian motion with drift -p  and one absorbing barrier and one 
reflecting barrier. That called for the first passage time distribution for such a process. 
We utilize the formulas for that first passage distribution, given by [l] and [2]. We 
analyzed both the continuous time case and the discrete case, and established that 
the continuous case can be obtained as the limiting value of the discrete time case. 
According to our derivations, as it turned out, the cases of p > 0, p = 0, and p < 0 
are qualitatively different and result in formulas that are quite different. The formulas 
for the expected MDD are as follows: 
CIFEr’03 HONG KONG 244 
The functions Q, and Qp are integrals of series expansions that can be evaluated 
numerically. They are given by the following integrals: 
where 0, is the solution of the eigenvalue problem 
We note that care has to be excercized when numerically evaluating them (especially 
Q,) because the series converge rather slowly. However, the interesting thing here is 
that Q, and Qp do not depend on any constant or quantity such as p, a, or T .  As 
such, they are universal functions that can be evaluated once and for all. Then, they 
can be tabulated and used in a lookup manner (rather than evaluated numerically 
every time they are used). Figures 1 and 2 show plots for Qp and Qn respectively. 
From the formulas, one can deduce a relationship between the MDD and the well- 
known Sharpe ratio. The Sharpe ratio is given by p/a where p and LT are measured 
in units of one year. One can deduce that for the case p > 0: 
E ( M D D )  = -Qp((Sharpe)2T) 0 
Sharpe 
Now comes the question, how does the drawdown behave asymptotically? Most 
trading desks are interested in long term performance results. That is, what system 
can survive over the long period ahead, and provide superior return with a level of 
drawdown that is acceptable and does not threaten its survival. We have derived 
formulas for the drawdown for T -+ 00 (T is the investment horizon). They are 
given by the following theorem: 
245 CIFEr'03 HONG KONG 
Theorem: 
The limiting expectation as T + 03 is given by. 
2(--0.5416 + 0.5log(T) + log:) if p > 0 
1.2552afi i f p = O  
= {I,,, i f p < O  E(MDD)IT-w 
One can see that the drawdown can scale in three different ways (log(T), 0, or T )  
depending on the regime of p. It is as if the problem undergoes a “phase shift” as p 
goes from negative to  positive. It cain therefore be used as a sort of hypothesis test: is 
the return positive, even or negative? This observation also supports the importance 
of the drawdown as an indicator of the effectiveness of a trading strategy. 
We introduced earlier in the article the Sterling Ratio. Mathematically it is defined 
as 
P 
E ( M D D ( T ) )  
Sterling(T) = 
where M D D ( T )  represents the maximum drawdown in an interval of length T .  Fo- 
cusing our attention to the more interesting case of p > 0, we get 
(Sharp e) 
Sterling(T) = 
Q p  ( (Sha r~e )~T)  
Considering also the asymptotic case T 00, 
(Sharpe)2 
-0.5416 + 0.5log(T) + log(Sharpe) Sterling(T - CO) = 
These formulas indicate a direct :relationship between the Sterling Ratio and the 
Sharpe Ratio. A deviation observed from historical data indicates a tendency to 
either have or avoid excessive successive loss periods. This feature depends on the 
trading system, the types of markets (for example the trendiness or mean reversion 
tendency of the markets), and the degree of diversification. 
Overall, the MDD is a very important risk measure. To be able to use it more insight- 
fully, its analytical properties have to be understood. As a step towards this direction, 
we have presented in this article some analytic results that we have developed. We 
hope more and more results will come out from the research community analyzing 
this important measure. 
CIFEr’03 HONG KONG 246 
Comparison of Q,,(x) for Different p 
6 -  
5- 
4- 
X 
Figure 1: Behavior of the Q functions for different p. 
References: 
[l] M. Domine,“First passage time distribution of a Wiener process with drift con- 
cerning two elastic barriers”, Journal of Applied Probability, 33: 164-175, 1996. 
[2} Blasi, “On a random walk between a reflecting and an absorbing barrier”, annals 
of Probability, 4(4):695-696, 1976. 
247 CIFEr‘03 HONG KONG 


The maximum drawdown of the Brownian motion

Domine,“First passage time distribution of a Wiener process with drift concerning two elastic barriers”,

https://authors.library.caltech.edu/10808/1/MAGcife03.pdf

The maximum drawdown of the Brownian motion

Abstract

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