Portfolio Optimization with
Drawdown Constraints
January 29, 2000
Alexei Chekhlov, TrendLogic Associates, Inc.
Stanislav Uryasev & Mikhail Zabarankin, University of
Florida, ISE
1
Introduction
• Losing client’s accounts is equivalent to death of business;
• Highly unlikely to hold an account which was in a drawdown for 2
years;
• Highly unlikely to be permitted to have a 50% drawdown;
• Shutdown condition: 20% drawdown;
• Warning condition: 15% drawdown;
• Longest time to get out of a drawdown - 1 year.
2

w( x , t )
- uncompounded portfolio value at time t;

x  ( x1, x2 ,, xm ) - set of unknown weights;



f ( x , t )  max w( x , ) w( x , t ) - drawdown function.
0 t
Three Measures of Risk:


•Maximum drawdown (MaxDD): M ( x )  max  f ( x , t )
•Average drawdown (AvDD):
•Conditional drawdown-at-risk
(CDaR):
0t T
T
 1

A( x )   f ( x, t ) dt
T0


1
 ( x) 
f
(
x
, t ) dt.

(1   )T t[ 0,T ], f ( x ,t ) ( x , )
3
Portfolio Equity & Underwater Curve
32,000,000
27,000,000
Equity/Drowdown
22,000,000
17,000,000
Net Equity
Draw Dow n
12,000,000
7,000,000
2,000,000
Nov-87
-3,000,000
Jun-88
Dec-88
Jul-89
Jan-90
Aug-90
Mar-91
Sep-91
Apr-92
Oct-92
Tim e
4
Portfolio Equity & Underwater Curve
67,000,000
57,000,000
Equity/Drawdown
47,000,000
37,000,000
Net Equity
Draw Dow n
27,000,000
17,000,000
7,000,000
May-93
-3,000,000
Sep-94
Feb-96
Jun-97
Nov-98
Mar-00
Tim e
5
Limiting the risk:
•
•
•
•

M( x )  1C

A( x )   2 C

 (x)   3C
MaxDD:
AvDD:
DVaR:



Combination: M( x )   1 C , A( x )   2 C ,   ( x )   3 C
for some 0  1, 2 , 3  1
6
Continuous Optimization Problems:
MaxDD:
AvDD:
CDaR:

max
R( x )


max
R( x )


max
R( x )

subject to

M ( x )   1 C
 
 xX
subject to

A( x )   2 C
 
 xX
subject to

  ( x )   3 C
 
 xX
“technological”
constraints:

X  x : xmin  xk  xmax ,  k  1, m
x
x

x

7
Discrete Optimization Problems:
MaxDD:
max
 
x
1
dC
AvDD:
 
y N  x

 
1


max
y

dC N  x
x
subject to

subject to
   
max max { y j  x}  yi  x   1 C
1i  N 1 j i
 x min  xk  xmax ,  k  1, m


   
1 N
max
{
y
 N  1 j i j  x}  yi  x   2 C
 i 1
 x  x  x ,  k  1, m
k
max
 min
CDaR:
 
1


max
y

dC N  x
x
subject to



N





a  (1 1 ) N   max { y j  x}  yi  x      3 C
1 j i


i 1 

,
x min  xk  xmax ,  k  1, m

(g)+=max{0,g}.
8
Reward/Risk Ratios: MaxDD
ERatioMax
8.0
7.5
7.0
6.5
6.0
5.5
5.0
0.65
0.50
0.35
0.80
0.20
0.70
0.60
0.50
w eight BP
0.40
0.30
0.20
4.0
0.80
4.5
w eight US
9
Reward/Risk Ratios: AvDD
ERatioAv
30.00
29.50
29.00
28.50
28.00
0.20
0.50
0.80
0.70
0.60
0.50
0.40
0.80
0.30
27.50
0.20
w eight BP
w eight US
10
Table 1: MaxDD Solution
Risk
4.0% 5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 11.0% 12.0% 13.0% 14.0% 15.0% 16.0% 17.0%
Rate of
25.0% 36.3% 44.5% 51.4% 57.3% 63.0% 67.7% 71.7% 75.2% 78.0% 80.4% 81.9% 82.9% 83.0%
Return
0.200 0.397 0.740 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
AD
0.200 0.200 0.200 0.200 0.618 0.409 0.532 0.560 0.800 0.800 0.800 0.800 0.800 0.800
US
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.222 0.512 0.768 0.800 0.800 0.800
BP
0.251 0.592 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
CD
0.625 0.800 0.768 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
HG
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.629 0.800 0.800 0.800 0.800 0.800
DX
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.349 0.741 0.800 0.800 0.800
ED
0.200 0.200 0.200 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
EC
FXADJY 0.270 0.576 0.771 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
FXBPJY 0.200 0.200 0.200 0.200 0.200 0.200 0.533 0.800 0.800 0.800 0.800 0.800 0.800 0.800
FXEUBP 0.200 0.280 0.294 0.320 0.337 0.651 0.723 0.800 0.800 0.800 0.800 0.800 0.800 0.800
FXEUJY 0.200 0.200 0.405 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
FXEUSF 0.328 0.200 0.254 0.298 0.729 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
FXNZUS 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.266 0.800 0.800
FXUSSG 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.281 0.212 0.434 0.716 0.800 0.800 0.800
FXUSSE 0.200 0.800 0.800 0.652 0.732 0.704 0.598 0.350 0.200 0.200 0.200 0.800 0.800 0.800
0.200 0.200 0.387 0.581 0.519 0.503 0.541 0.800 0.800 0.800 0.800 0.800 0.800 0.800
FV
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.566 0.800 0.800
GC
0.200 0.228 0.336 0.250 0.372 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
JY
0.200 0.200 0.200 0.200 0.385 0.634 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
LFT
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.373 0.800 0.800 0.800 0.800 0.800 0.800
LGL
0.200 0.350 0.623 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
LBT
0.200 0.271 0.365 0.457 0.510 0.597 0.780 0.800 0.800 0.800 0.800 0.800 0.800 0.800
LAHD
0.200 0.300 0.418 0.450 0.437 0.800 0.800 0.800 0.770 0.800 0.800 0.800 0.800 0.800
MNN
0.200 0.200 0.366 0.393 0.519 0.520 0.633 0.751 0.800 0.800 0.800 0.800 0.800 0.800
SF
0.200 0.247 0.255 0.277 0.210 0.394 0.677 0.800 0.685 0.800 0.800 0.800 0.800 0.800
AAO
0.200 0.374 0.320 0.469 0.630 0.800 0.551 0.636 0.800 0.800 0.800 0.800 0.800 0.800
AXB
0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.200 0.396 0.800
SI
0.492 0.742 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
SJB
0.200 0.560 0.673 0.686 0.781 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
SNI
0.200 0.200 0.225 0.320 0.601 0.688 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
TY
0.200 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800
DGB
11
Table 2: AvDD Solution
Risk
Rate of
Return
AD
US
BP
CD
HG
DX
ED
EC
FXADJY
FXBPJY
FXEUBP
FXEUJY
FXEUSF
FXNZUS
FXUSSG
FXUSSE
FV
GC
JY
LFT
LGL
LBT
LAHD
MNN
SF
AAO
AXB
SI
SJB
SNI
TY
DGB
0.8%
0.9%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
25.1% 30.9% 35.6% 54.5% 67.4% 76.7% 82.7% 83.0%
0.298
0.200
0.200
0.256
0.416
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.398
0.200
0.200
0.246
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.200
0.229
0.200
0.200
0.384
0.200
0.200
0.665
0.457
0.200
0.200
0.309
0.481
0.200
0.204
0.200
0.200
0.200
0.200
0.342
0.557
0.200
0.200
0.513
0.200
0.200
0.292
0.200
0.200
0.389
0.200
0.200
0.200
0.402
0.200
0.200
0.547
0.292
0.200
0.800
0.572
0.200
0.200
0.373
0.597
0.200
0.297
0.200
0.200
0.200
0.200
0.590
0.622
0.200
0.200
0.744
0.200
0.200
0.378
0.200
0.200
0.522
0.200
0.200
0.200
0.462
0.200
0.200
0.669
0.331
0.200
0.800
0.800
0.200
0.200
0.800
0.800
0.200
0.323
0.200
0.331
0.501
0.578
0.800
0.800
0.200
0.476
0.800
0.525
0.200
0.800
0.200
0.200
0.800
0.337
0.200
0.539
0.800
0.620
0.200
0.800
0.657
0.200
0.800
0.800
0.200
0.200
0.800
0.800
0.200
0.378
0.615
0.602
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.200
0.800
0.200
0.425
0.800
0.665
0.679
0.800
0.800
0.800
0.200
0.800
0.800
0.200
0.800
0.800
0.405
0.666
0.800
0.800
0.200
0.725
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.200
0.800
0.313
0.773
0.800
0.800
0.800
0.800
0.800
0.800
0.200
0.800
0.800
0.721
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.467
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.546
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
0.800
12
E ffic ie n t F ro n tie r: M a x D D
0.9
0.8
0.7
0.6
R (x )
Figure 1: MaxDD
Efficient Frontier
0.5
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
M (x )
E ffic ie n t F ro n tie r: A v D D
0 .9
0 .8 5
0 .8
0 .7 5
0 .6
0 .5 5
R ( x)
Figure 2: AvDD
Efficient Frontier
0 .7
0 .6 5
0 .5
0 .4 5
0 .4
0 .3 5
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
0
0
0 .0 0 5
0 .0 1
0 .0 1 5
0 .0 2
0 .0 2 5
0 .0 3
0 .0 3 5
0 .0 4
A (x )
13
E ffi c i e n t F r o n ti e r
90%
80%
70%
Figure 3: Efficient
Frontier as a function
of MaxDD
R(x )
60%
50%
Max DD
40%
A v DD
30%
20%
10%
0%
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
M (x )
E ffi c i e n t F r o n ti e r
90%
80%
70%
60%
R(x )
Figure 4: Efficient
Frontier as a function
of AvDD
50%
Max DD
A v DD
40%
30%
20%
10%
0%
0 .0 %
0 .5 %
1 .0 %
1 .5 %
2 .0 %
A (x )
2 .5 %
3 .0 %
3 .5 %
4 .0 %
14
Reward/Risk
8
7
MaxDDRatio
Figure 5: MaxDDRatio
as a function of MaxDD
6
5
MaxDD
4
AvDD
3
2
1
0
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
M(x)
R e w a r d / R i sk
40
35
Figure 6: AvDDRatio
as a function of AvDD
AvDDRa tio
30
25
Max DD
20
A v DD
15
10
5
0
0 .0 %
0 .5 %
1 .0 %
1 .5 %
2 .0 %
2 .5 %
3 .0 %
3 .5 %
4 .0 %
A (x )
15
Underwater Curves: MaxDD and AvDD:
Draw dow n Curves: MaxDD & AvDD
0
Jul-98
Nov-98
Feb-99
M ay-99
Aug-99
Dec-99
-500,000
-1,000,000
M ax DD
Av DD
-1,500,000
-2,000,000
-2,500,000
16
Conclusions
• Introduced a one-parameter family of risk measures based on a notion
of a drawdown (underwater) curve;
• Mapped Portfolio Allocation problem into linear programming
problems to be solved using efficient computer solvers;
• Solved a particular real-life example on the basis of historical equity
curves;
• CDaR-generated solutions are more stable for practical weights’
allocation.
17