An Analysis of the Maximum Drawdown
Risk Measure
Malik Magdon-Ismail (RPI)
May 6, 2004.
Joint work with:
Amir Atiya (Cairo University)
Amrit Pratap(Caltech)
Yaser Abu-Mostafa(Caltech)
Motivation
1
Example
Fund
Π1
Π2
Π3
µ(%)
25
30
25
σ(%)
10
10
12.5
max. DD(%)
−5
−7 1
2
1
−8 3
T (yrs)
1
1.5
2
Return over [0, T ]
Calmar Ratio =
max. DD over [0, T ]
Sterling Ratio =
Return over [0, T ]
max. DD over [0, T ] − 10%
Which fund is best?
2
The Complication
• µ and σ are annualized. The max. drawdown is computed over a
given time period.
• Problem: funds have statistics over different length time intervals.
How do we annualize M DD?
Why?
3
Common Practice
Compare funds over T = 3 yrs.
Artificial:
– Wasteful of useful data.
– 3 years data may not be available.
– Easily available data does not generally quote the MDD for 3 years.
4
√
T -Rule for Sharpe Ratio
µ(annualized)
Sharpe Ratio =
σ(annualized)
µτ
στ
)
over time periods of size τ
1
µ(annualized) = µτ ·
τ
s
σ(annualized) = στ ·
1
τ
µτ
Sharpe Ratio = √
στ τ
5
Similar scaling laws for Sterling-type Ratios?
Part I:
Analysis of the Maximum Drawdown.
Part II:
Application to Scaling Laws.
6
Part I:
Analysis of Maximum Drawdown
7
The Drawdown (DD)
The DD (current loss) is the loss from the peak to the current value.
X(t) is the cumulative return curve.
DD(T )
X(t)
t
T
DD(T ) = sup X(s) − X(T )
s∈[0,T ]
8
Distribution of DD(T )
• Since DD(T ) = max over [0, T ] − X(T ), the full distribution of DD(T )
can be obtained from the joint distribution of the max and the close.
• Joint distribution can be obtained in closed form for various stochastic
processes (see for example [Karatzas and Shreve, 1997])
9
The Maximum Drawdown (M DD)
The MDD is the maximum loss incurred from a peak to a bottom.
X(t)
M DD(T )
t
T
M DD(T ) = sup DD(t)
t∈[0,T ]
– DD is an extremum.
– M DD is an extremum of an extremum.
10
Sampling of Prior Work
– Previous work on the the MDD is mostly empirical or Monte-Carlo.
• [Acar and James; 1997]
• [Sornette; 2002]
• [Burghardt, Duncan and Liu; 2003]
• [Harding, Nakou and Nejjar; 2003]
• [Chekhlov, Uryasev and Zabarankin; 2003]
– The only analytical approach is for a Brownian motion with zero drift,
[Douady, Shiryaev and Yor; 2000].
11
Setup
X(t) is an (arithmetic) Brownian motion:
dX(t) = µdt + σdW (t)
0≤t≤T
µ = average return per unit time
(drift)
σ = std. dev. of the returns per unit time (volatility)
dW (t) = Wiener increment
(shocks)
Note: If the fund S(t) follows a geometric Brownian motion, then the
cumulative return sequence follows a Brownian motion.
We would like to study M DD(µ, σ, T ).
12
DD(t) is a Reflected Brownian Motion
Drawdown at time t is a stochastic process.
X(t)
DD(t)
t
t+∆
t
t+∆
DD(t) is reflecting at 0, i.e., DD(t) is a reflected Brownian motion.


−dX(t)
n
o
dDD(t) =

max 0, −dX(t)
DD(t) > 0
DD(t) = 0.
13
Maximum of a Reflected Brownian Motion
M DD(T ) = sup DD(t)
t∈[0,T ]
Absorbing Barrier
h
DD(t)
τ =stopping time
t
Reflecting Barrier
T
fτ (t, h|µ, σ)=stopping time density
[Dominé; 1996]
GH (h|µ, σ, T ) = P [M DD(µ, σ, T ) ≥ h] =
RT
dt fτ (t, h| − µ, σ)
0
[Magdon-Ismail, Atiya, Pratap and Abu-Mostafa; 2004]
14
Moments of GH
E[M DD(µ, σ, T )] =
Z ∞
0
dh GH (h|µ, σ, T )
15
Expected M DD(µ, σ, T )

2T
2

µ
2σ Q


p 2σ 2

µ








q √
πσ T
Theorem: E[M DD(µ, σ, T )] =
2









2

2
µ T

2σ

 µ Qn
2σ 2
Qp and Qn are “universal” functions.
– Only need to be computed
µ>0
µ=0
µ<0
once!
16
The Universal Qp and Qn
Bad news: Qp and Qn are integral series:

Z∞
 −u
Qp(x) = du 
e
0
x
−
3
cos2 (θn )
∞ sin (θn ) 1 − e
X
n=1
θn − cos(θn )sin(θn )

x

u
−
2
+ e− tanh(u) sinh(u) 1 − e cosh (u) 

θn are positive eigen solutions of tan(θn) = θun .
x
−
3 (θ ) 1−e cos2 (θn )
Z∞
sin
∞
n
X
u
Qn (x) = − du e
θn−cos(θn)sin(θn)
n=1
0
θn are positive eigen solutions of tan(θn) = − θun .
Good news: Only have to be computed once.
17
More Good News: Asymptotic Behavior
Most trading desks are interested in the long term.
Theorem:
 2
µ
σ
1


0.63519 + 2 log T + log σ

µ








T →∞ q π √
E[M DD] −→
σ T
2










µT + σ 2
µ
µ>0
µ=0
µ<0
Phase Transitions: Three different types of behavior:
√
log T, T , T
depending on the regime of µ.
18
Behavior of QM DD (x)
Comparison of QMDD(x) for Different µ
6
5
Q(x)
4
3
µ<0
2
µ=0
1
µ>0
0
0
1
2
3
4
5
x
http://www.cs.rpi.edu/∼magdon/data/Qfunctions.html
19
Some Useful Statistics
E[M DD]
σ
µ
σ
Sharpe ratio of expected performance
µT
E[M DD]
Calmar ratio of expected performance
Shrp =
Clmr =
Expected maximum drawdown per unit volatility
From now on, µ > 0.
20
M DD vs. Shrp
E(MDD) Per Unit σ Versus Sharpe Ratio (µ/σ)
2
T=3
E(MDD)/σ
1.75
1.5
1.25
T=2
1
T=1
0.75
0.5
0
E[M DD]
=
σ
1
2
Sharpe Ratio (µ/σ)
3
4
T
2
2Qp 2 Shrp T →∞ 0.63519 + 0.5 log T + log Shrp
Shrp
−→
Shrp
21
Calmar vs. Sharpe
Scaling of Calmar Ratio with Time
Calmar Ratio Versus the Sharpe Ratio
8
14
7
12
Shrp=1.5
5
Calmar Ratio
Calmar Ratio
6
4
Shrp=1
3
T=2
8
6
T=1
4
2
Shrp=0.5
1
0
0
T=3
10
2
Clmr
4
6
Time (Years)
8
2
10
0
0
1
2
Sharpe Ratio
3
4
T Shrp2
2
T
Shrp
T
→∞
2
−→
=
T
2
0.63519 + 0.5 log T + log Shrp
Qp 2 Shrp
Note: Clmr is monotonic in Shrp.
22
Part II:
Scaling Laws
23
Recap
Calmar(T ) =
Return over [0, T ]
µT
≈
= Clmr
M DD over [0, T ]
E[M DD]
Given two funds,
1 T1 .
Π1 : µ1, σ1, T1, M DD1, Clmr1 = MµDD
1
2 T2 .
Π2 : µ2, σ2, T2, M DD2, Clmr2 = MµDD
2
How to compare Clmr1 and Clmr2?
24
Normalized Calmar Ratio I
Normalize the ratios to a reference time τ , for example τ = 1 yr.
We know how to scale return over [0, T1],
µ1T1 −→ µ1τ = µ1T1 ·
τ
T1
We can scale M DD([0, T1]) → M DD([0, τ ]) using proportion,
M DD([0, τ ])
E[M DD([0, τ ])]
=
E[M DD([0, T1])]
M DD([0, T1])
25
Normalized Calmar Ratio II
Calmar 1(τ ) =
return([0, τ ])
M DD([0, τ ])
µ1T1
τ E[M DD([0, T1])]
·
=
·
M
DD([0,
T
])
T
τ ])]}
1 } | 1 E[M DD([0,
|
{z
{z
Calmar1(T1 )
γτ
Calmar 1(τ ) = γτ (T1, Shrp1) × Calmar1 (T1),
1 Q ( T1 Shrp2)
1
T1 p 2
γτ (T1, Shrp1) = 1
τ Shrp2)
Q
(
p
1
τ
2
26
Example – Revisited
Fund
Π1
Π2
Π3
µ(%)
25
30
25
σ(%)
10
10
12.5
max. DD(%)
−5
1
−7 2
1
−8 3
T (yrs)
1
1.5
2
Calmar Calmar
5
5
6
4.41
6
3.62
(normalized to τ = 1 yr.)
Π1 > Π2 > Π3
27
τ -Relative Strength
Fix a reference time τ .
βτ (Π1|Π2) =
Calmar 1(τ )
.
Calmar 2(τ )
– Relative normalized Calmar ratio of Π1 with respect to Π2.
– May depend on τ .
28
Relative Strength
Consider the long horizon, τ → ∞,
β(Π1|Π2) = lim βτ (Π1|Π2)
τ →∞
– The limit exists.
T1
1
2
Calmar1 (T1) T1 Qp( 2 Shrp1)
relative strength = β(Π1|Π2) =
×
Calmar2 (T2) 1 Qp( T2 Shrp2)
T2
2
2
Π1 Π2 ⇐⇒ β(Π1|Π2) ≥ 1
29
Example – Re-Revisited
Fund
Π1
Π2
Π3
µ(%)
25
30
25
σ(%)
10
10
12.5
max. DD(%)
−5
1
−7 2
1
−8 3
T (yrs)
1
1.5
2
β
1.00
0.97
0.64
(relative strengths w.r.t Π1.)
Π1 > Π2 > Π3
30
Properties of Relative Strength
Complete: β(Π1|Π2) = β(Π 1|Π )
2 1
Π1 Π2 or Π2 Π1
Transitivity: β(Π1|Π3) = β(Π1|Π2)β(Π2 |Π3)
Π1 Π2 and Π2 Π3 =⇒ Π1 Π3
β(Π |Π )
Independent of Reference instrument: β(Π1|Π2) = β(Π1|Π3)
2 3
β(Π1|Π3) ≥ β(Π2|Π3) =⇒ Π1 Π2
i.e., the relative strength defines a total order.
31
Real Data∗
Fund
S&P 500
F T SE100
N ASDAQ
DCM
N LT
OIC
T GF
µ(%)
10.04
7.01
11.20
15.65
3.35
17.19
8.48
σ(%)
15.48
16.66
24.38
5.78
16.03
4.52
9.83
T (yrs)
24.25
19.83
19.42
3.08
3.08
1.16
4.58
M DD
46.28
48.52
75.04
3.11
25.40
0.42
8.11
Calmar
5.261
2.865
2.899
15.50
0.4062
47.48
4.789
E[MDD]
44.56
55.54
77.87
4.770
31.35
2.493
15.84
Calmar
0.6104
0.4395
0.4402
6.541
0.2202
42.31
1.752
β
1
0.5003
0.5407
27.76
0.1331
212.0
3.589
DCM =Diamond Capital Management;
N LT =Non-Linear Technologies;
OIC=Olsen Investment Corporation;
T GF =Tradewinds Global Fund.
– Normalized Calmar ratio is to τ = 1 yr.
– Relative strength index β is computed w.r.t. S&P 500.
∗ International
Advisory Services Group http://iasg.pertrac2000.com/mainframe.asp
32
Discussion
1. Studied MDD for a Brownian motion.
– geometric Brownian?
2. We now have scaling laws for MDD and Sterling-type ratios.
3. Portfolio opt. to maximize Calmar?
Since Clmr is monotonic in Shrp, optimization of
Shrp implies optimization of Clmr.
Advertisement:
[Magdon-Ismail, Atiya, Pratap, Abu-Mostafa; 2004]
[Magdon-Ismail, Atiya; 2004 submitted]
http://www.cs.rpi.edu/∼magdon
33
Thank You!
Advertisement:
[Magdon-Ismail, Atiya, Pratap, Abu-Mostafa; 2004]
[Magdon-Ismail, Atiya; 2004 submitted]
http://www.cs.rpi.edu/∼magdon
34
References
[1] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, 1997.
[2] E. Acar and S. James, Maximum loss and maximum drawdown in financial markets. Int.
Conference on Forecasting Financial Markets, London, UK, 1997.
[3] D. Sornette, Why Do Stock Markets Crash: Critical Events in Complex Financial Systems.
Princeton University Press, 2002.
[4] Burghardt, G., Duncan, R. and L. Liu, Deciphering drawdown. Risk magazine, Risk management for investors, September, S16-S20, 2003.
[5] D. Harding, G. Nakou, and A. Nejjar, The pros and cons of drawdown as a statistical
measure for risk in investments. AIMA Journal, pp. 16-17, April 2003.
[6] Chekhlov, A., Uryasev, S., and M. Zabarankin, Drawdown Measure in Portfolio Optimization. Research Report 2003-15. ISE Dept., University of Florida, September 2003.
[7] R. Douady, A. Shiryaev, and M. Yor, On the probability characteristics of downfalls in a
standard Brownian motion. SIAM, Theory Probability Appl., Vol 44, pp. 29-38, 2000.
[8] M. Dominé First passage time distribution of a Wiener process with drift concerning two
elastic barries. Journal of Applied Probability, DD:164–175, 1996.
[9] M. Magdon-Ismail, A. Atiya, A. Pratap and Y. Abu-Mostafa, On the Maximum Drawdown
of a Brownian Motion. Journal of Applied Probability, Volume 41, Number 1, 2004.
35