! Portfolio with a DrawDown Risk Measure. (PortDrawDnR.lng)
In a sequence of periods, the drawdown of a portfolio in period t
is the difference: (highest previous portfolio value) - (current value).
One measure of portfolio risk is the sum of the k largest drawdowns,
over a period of time, the K Median drawdown.
The objective in this model is:
Maximize (Ending value)/ KMedianDrawDown.;
! Keywords: Draw down, Fractional programming, Hyperbolic programming, Portfolio,
Ratio objective, Risk measurement;
SETS:
SCENE: R, VP, VPMX, DD, XSDD;
ASSET: X, USE;
SXA( SCENE, ASSET): VE, CG;
ENDSETS
DATA:
!Here are several cases;
! Case original Markowitz example, 1943 to 1954;
!CHarry VP0 = 1; !CHarry KMED = 2; !CHarry ALPHA = 0.01; !CHarry ASSET =
SP500 ATT GMT USX;
! Which assets to use/consider;
!CHarry USE =
0 1 1 1;
!CHarry VE =
1.258997 1.3 1.225 1.149
1.197526 1.103 1.29 1.26
1.364361 1.216 1.216 1.419
0.919287 0.954 0.728 0.922
1.05708 0.929 1.144 1.169
1.055012 1.056 1.107 0.965
1.187925 1.038 1.321 1.133
1.31713 1.089 1.305 1.732
1.240164 1.09 1.195 1.021
1.183675 1.083 1.39 1.131
0.990108 1.035 0.928 1.006
1.526236 1.176 1.715 1.908;
! Case Start75: Data from 1975 to 2021;
!Start75; VP0 = 1!Start75; KMED = 5! 1 2 3 4 5 6 7 8 9 10 11 12 13 14;
!CStart75; ASSET =
SP500 Chevron CocaCola DupontDow PandG IBM Weyerhauser Ford GE ExxonMobil Deere Boeing Merck MMM;
! Which assets to use/consider;
!CStart75; USE =
0 1 1 1 1 1 1 1 1 1 1 1 1 1;
! Data from 1STOCK1, 1975 to 2021;
!CStart75; VE =
1.3846 1.3202 1.5519 1.6659 1.092 1.3348 1.3562 1.3184 1.382 1.3733 1.217 1.5476 1.0433 1.2033
1.2420 1.3957 0.9605 0.9468 1.052 1.2447 1.2525 1.3977 1.206 1.2085 1.2193 1.8359 0.9838 1.0203
0.9222 0.9482 0.9430 0.6167 0.9172 0.9798 0.5887 0.9299 0.8966 0.8974 0.7826 1.257 0.8147 0.8565
1.0641 1.2058 1.1779 0.9299 1.0349 1.0914 0.8995 0.9208 0.9449 1.0208 1.399 2.5378 1.2185 1.3015
1.1869 1.2027 0.7863 1.2915 0.8354 0.8626 1.2893 0.7596 1.0743 1.1221 1.13 1.0639 1.0684 0.796
1.3276 1.765 0.9674 1 0.9276 1.0544 1.0748 0.625 1.2099 1.4626 1.2268 1.3074 1.173 1.1741
0.9467 0.8618 1.0412 0.8171 1.167 0.8379 0.8498 0.8375 0.9367 0.7752 0.7396 0.5099 1 0.9237
1.2122 0.7464 1.4964 0.9857 1.4712 1.6923 1.2414 2.3209 1.6536 0.952 0.831 1.5056 0.9985 1.3761
1.2313 1.082 1.0288 1.2899 0.9619 1.2675 0.9375 1.635 1.2358 1.1218 1.3051 1.2915 1.0679 1.1
1.0596 0.9025 1.1659 0.824 1.0022 1.0092 0.863 1.0767 0.9659 1.3483 0.7727 1.2943 1.0401 0.953
1.3224 1.22 1.3547 1.4909 1.2237 1.2629 1.0558 1.2712 1.2848 1.225 0.9664 1.3841 1.4574 1.1415
1.1906 1.1902 1.3402 1.4268 1.095 0.7717 1.2276 1.4547 1.1821 1.2721 0.7957 0.9785 1.8084 1.2953
1.0569 0.8733 1.0099 1.5385 1.1178 0.9625 1.0265 1.34 1.0262 1.0873 1.5191 0.7237 1.2795 1.1075
1.1664 1.1546 1.1705 0.975 1.019 1.0552 0.9726 1.34 1.0142 1.1541 1.3813 1.6385 1.0931 0.9631
1.32 1.4809 1.7311 1.2201 1.6149 0.7723 1.0995 0.8639 1.4413 1.1364 1.2813 1.4691 1.342 1.2843
0.9658 1.072 1.2039 0.6655 1.2331 1.2005 0.7919 0.6103 0.8895 1.035 0.7622 1.1463 1.1597 1.0769
1.3095 0.9501 1.7258 1.1316 1.0837 0.7876 1.2571 1.0563 1.3333 1.1763 1.024 1.0523 1.8526 1.1108
1.076 1.0072 1.0436 1.0651 1.1425 0.566 1.3409 1.5244 1.1176 1.0041 0.9115 0.8403 0.7815 1.0564
1.1017 1.2536 1.0675 0.9913 1.0629 1.1216 1.2102 1.5044 1.2266 1.0327 1.6914 1.0779 0.7925 1.0807
1.0119 1.0244 1.1541 1.185 1.0877 1.3009 0.8403 0.8643 0.9726 0.9723 0.8953 1.0867 1.1091 0.9816
1.3802 1.1737 1.4417 1.0446 1.3387 1.2432 1.1533 1.0359 1.4118 1.3116 1.5962 1.6676 1.7213 1.2436
1.2306 1.2411 1.4175 1.1157 1.2967 1.658 1.0954 1.1169 1.3733 1.2174 1.1489 1.3589 1.2133 1.2505
1.3367 1.1846 1.2672 1.2951 1.4832 1.3812 1.0356 1.5058 1.4842 1.2487 1.4383 0.919 1.3312 0.9887
1.2873 1.0771 1.0047 0.8959 1.1441 1.7622 1.0357 1.2085 1.3901 1.1951 0.5644 0.6667 1.3915 0.8667
1.2111 1.0445 0.8694 1.4694 1.1999 1.1702 1.4133 0.9084 1.5172 1.1017 1.3194 1.2701 0.911 1.3761
0.9089 0.9747 1.0461 0.8223 0.7159 0.7879 0.7067 0.4396 0.9293 1.0791 1.0562 1.5928 1.3935 1.2312
0.8802 1.0613 0.7737 0.9223 1.0088 1.4231 1.0656 0.6707 0.8361 0.9041 0.953 0.5876 0.628 0.981
0.7773 0.7419 0.9298 0.8792 1.0861 0.6407 0.9099 0.5916 0.6075 0.8891 1.0502 0.8507 0.9628 1.0431
1.2872 1.2995 1.1576 1.3997 1.1622 1.1959 1.3005 1.7204 1.2723 1.1734 1.4188 1.2774 0.8161 1.3792
1.1082 1.2156 0.8205 1.191 1.1029 1.0637 1.0503 0.915 1.1782 1.2502 1.1437 1.2285 0.6957 0.9652
1.0479 1.0811 0.9681 0.8851 1.0508 0.8338 0.9869 0.5273 0.9603 1.0958 0.9155 1.3568 0.9897 0.9443
1.1574 1.2952 1.197 0.9105 1.1104 1.1819 1.065 0.9728 1.0616 1.3643 1.3958 1.2648 1.3706 1.0055
1.0546 1.2693 1.2719 0.988 1.1424 1.1127 1.0437 0.8961 0.9962 1.2226 1.959 0.9845 1.3328 1.082
0.6278 0.7926 0.7377 0.3828 0.842 0.7785 0.4151 0.3403 0.437 0.8521 0.4115 0.4879 0.5231 0.6824
1.2711 1.0408 1.2591 1.831 0.9808 1.5554 1.4093 4.3668 0.934 0.8542 1.4115 1.2686 1.202 1.4367
1.1487 1.1852 1.1539 1.2356 1.061 1.1212 1.0512 1.679 1.2089 1.0723 1.5354 1.2056 0.9863 1.0439
1.0207 1.166 1.0639 0.8424 1.037 1.2529 0.9863 0.6409 0.9792 1.1592 0.9314 1.124 1.0461 0.947
1.1588 1.0164 1.0362 1.1241 1.0177 1.0417 1.4901 1.2035 1.172 1.0211 1.1173 1.0274 1.0859 1.1361
1.3243 1.1551 1.1396 1.3733 1.1991 0.9792 1.1348 1.1915 1.3354 1.1693 1.0568 1.8112 1.2225 1.5105
1.1381 0.8981 1.022 1.0273 1.1189 0.8554 1.1368 1.0045 0.9015 0.9135 0.9687 0.9523 1.1347 1.1716
1.0131 0.8019 1.0175 1.1287 0.8718 0.8578 0.8353 0.909 1.2327 0.8432 0.8621 1.1124 0.9301 0.9167
1.1193 1.3084 0.9651 1.1115 1.0588 1.2061 1.0037 0.8609 1.0144 1.1579 1.351 1.0767 1.1145 1.1854
1.2194 1.0636 1.1066 1.2447 1.0928 0.9243 1.1718 1.0297 0.5522 0.9267 1.5189 1.8943 0.9558 1.3181
0.9559 0.869 1.032 0.7509 1.0004 0.7409 0.62 0.6125 0.4338 0.8153 0.9531 1.0936 1.3579 0.8095
1.311 1.1077 1.169 2.7766 1.3588 1.1792 1.3815 1.2157 1.5298 1.0233 1.1615 1.0101 1.1903 0.9259
1.1576 0.7008 0.9908 1.1076 1.114 0.9391 1.1103 0.9452 0.9677 0.5907 1.5529 0.6571 0.8994 0.9908
1.2689 1.3896 1.0797 1.1376 1.1757 1.0906 1.2282 2.3629 1.0934 1.4845 1.2744 0.9405 1.3092 1.0162 ;
ENDDATA
! Constants:
VP0 = portfolio value at start = initial investment,
VE( t, j) = growth factor over period t of asset j,
CG( t, j) = compounded growth of asset j, = CG( t-1,j)* VE( t, j)
KMED = k median index = how many of the the largest drawdowns to consider,
! Variables:
X( J) = amount invested in ASSET J at 1st period, following buy-and-hold strategy,
VP( t) = portfolio value at end of period t,
VPMX( t) = maximum portfolio value seen up to period t,
DD( t) = drawdown measured at end of period t, (decrease from earlier max),
THETAKM = k median of drawdowns,
XSDD( t) = Excess drawdown beyond THETAKM in period t,
SUMKHI = sum of K highest drawdowns.
;
! Compute some constants;
NSCENE = @SIZE( SCENE); ! Number periods;
! Compute compounded growth by asset;
@FOR( ASSET( J):
CG( 1, J) = VE( 1, J); ! Asset values, end of period 1;
@FOR( SCENE( t) | t #GT# 1: ! Remaining periods;
CG( t, J) = CG( t-1, J)* VE( t, J);
);
);
! Maximize (end of horizon portfolio value) / RiskMeasure;
[OBJ] MAX = VP( NSCENE)/ SUMKHI;
! Budget constraint at beginning;
[BUDGET] @SUM( ASSET( J) | USE( J) : X( J)) = VP0;
! Portfolio value at end of each period;
@FOR( SCENE( t) :
VP( t) = @SUM( ASSET( J) | USE( J) : CG( t, J) * X( J));
);
! Enforce max portfolio value seen up to period t;
VPMX( 1) = VP( 1);
DD( 1) = 0; ! Compute drawdown;
@FOR( SCENE( t) | t #GT# 1:
VPMX( t) >= VPMX( t-1); ! Max value of portfolio up to t;
VPMX( t) >= VP( t);
DD( t) >= VPMX( t) - VP( t); ! Max loss of value up to t;
);
! Sum K-Median calculations, where
THETAKM = value of kth highest drawdown.
Excess drawdown beyond the k median ;
@FOR( SCENE( t):
XSDD( t) >= DD( t) - THETAKM;
);
! Compute sum of k largest;
SUMKHI >= KMED * THETAKM + @SUM( SCENE( t): XSDD( t));
! Argument:
Assume objective (maximize) is a decreasing function of SUMKHI.
I.e., high SUMKHI is bad.
Claim: At optimum, THETAKM = kth highest value of DD( t).
Proof:
a) Suppose THETAKM is too small, so number DD( t) > THETAKM is r > KMED.
Increasing THETAKM by epsilon will
decrease SUMKHI by r*epsilon - KMED * epsilon, so solution could not be optimal.,
b) Suppose THETAKM is too large, so number DD( t) > THETAKM is r < KMED.
Decreasing THETAKM by epsilon will
decrease SUMKHI by KMED * epsilon - r*epsilon, so solution could not be optimal.,
! Ref. Uryasev et al. discussion of CVaR;
END
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