Optimal solutions for optimization in practice 77
We consider a subuniverse of our global stocks universe of 250 randomly
drawn stocks (with complete history) over the last 4 years, measured weekly.
The only constraint we apply here is:
^002 wi 002..The measurement of drawdowns allows certain flexibility with regard to the hori-
zon over which it is calculated. Our analysis starts with the following definitions:
DRR()ττMax[, ( ()/ 01 Maxt∈(,) 0 τ ()t)]
(3.10)where τ (0, T ).
The maximum drawdown (MDD) as the worst of all drawdowns then is:
MDD()TRRMaxττ∈∈(, ) 00 Tt[(()/τ Max (,)() )]t 1
(3.11)or — using the results of calculations based on the first definition
MDD()TDMaxτ∈(, ) 0 T (τ)
(3.12)While these definitions allow drawdowns over horizons up to T , we limited
this horizon subsequently to any period within a calendar month and then any
period within 4 weeks, so:
Dm( )Max[, ( 01 Mintmm∈(,)seRt Rm()/ ( )s)]
(3.13)where m s and m e are the beginning and end of a month m (1 ... 35), respec-
tively, and
Dw 14 () 40 Max[,(Mintww∈(,)RtRw()/() )] 1
(3.14)where w (1 ... 152) is a week of the cumulative return series.
In both cases, the maximum drawdown would be calculated similar to
Equation (3.12).
Finally , the “ drawdown ” could be calculated over a fixed period of 4 weeks,
which is effectively the 4 weeks return, i.e.,
D w^2 () (^441 Rw )/()Rw (3.15)^Please note that in this case, the “ drawdown ” is any 4 weeks return, i.e., in
contrast to the other drawdown definitions, definition (3.15) results in nega-
tive drawdown figures for negative returns.
In our attempt to find a suitable loss aversion coefficient that would allow
us to meet the drawdown target, we focused on the last definition (3.15).
However, analysis for other drawdown definitions is available from the authors
as well. We used loss aversion coefficients ranging from 0.0 to 5.5 in 0.5 steps
(i.e., 12 backtests). The resulting returns for a loss aversion coefficient of 0 and
3.5 based on definition (3.15) are shown in the appendix. On the basis of these