! Solve an Omega ratio portfolio problem as a nonlinear program
Maximize (E(R)-tau)/E(max(0,tau - R)),
where the random variable R is the portfolio return, and
tau is a give target, such as the risk free rate of return.
The Omega ratio is sometimes suggested as an alternative to
the Sharpe ratio, when the distribution of R is very asymmetric;
! Keywords: Investment, LINGO, Omega ratio, Portfolio, Risk, Scenario, Sharpe ratio, Stocks;
SETS:
STOCK: X;
OBS: R, DwnSide;
OxS( OBS, STOCK): GF;
ENDSETS
DATA:
Tau = 1.02;
! These data were retrieved from the site https://www.1stock1.com/;
STOCK =
3M Chevron CocaCola DupontDow PandG IBM Weyerhauser Ford GE ExxonMobil Deere Boeing Merck;
OBS, GF =
1975 1.2033 1.3202 1.5519 1.6659 1.092 1.3348 1.3562 1.3184 1.382 1.3733 1.217 1.5476 1.0433
1976 1.0203 1.3957 0.9605 0.9468 1.052 1.2447 1.2525 1.3977 1.206 1.2085 1.2193 1.8359 0.9838
1977 0.8565 0.9482 0.943 0.6167 0.9172 0.9798 0.5887 0.9299 0.8966 0.8974 0.7826 1.257 0.8147
1978 1.3015 1.2058 1.1779 0.9299 1.0349 1.0914 0.8995 0.9208 0.9449 1.0208 1.399 2.5378 1.2185
1979 0.796 1.2027 0.7863 1.2915 0.8354 0.8626 1.2893 0.7596 1.0743 1.1221 1.13 1.0639 1.0684
1980 1.1741 1.765 0.9674 1 0.9276 1.0544 1.0748 0.625 1.2099 1.4626 1.2268 1.3074 1.173
1981 0.9237 0.8618 1.0412 0.8171 1.167 0.8379 0.8498 0.8375 0.9367 0.7752 0.7396 0.5099 1
1982 1.3761 0.7464 1.4964 0.9857 1.4712 1.6923 1.2414 2.3209 1.6536 0.952 0.831 1.5056 0.9985
1983 1.1 1.082 1.0288 1.2899 0.9619 1.2675 0.9375 1.635 1.2358 1.1218 1.3051 1.2915 1.0679
1984 0.953 0.9025 1.1659 0.824 1.0022 1.0092 0.863 1.0767 0.9659 1.3483 0.7727 1.2943 1.0401
1985 1.1415 1.22 1.3547 1.4909 1.2237 1.2629 1.0558 1.2712 1.2848 1.225 0.9664 1.3841 1.4574
1986 1.2953 1.1902 1.3402 1.4268 1.095 0.7717 1.2276 1.4547 1.1821 1.2721 0.7957 0.9785 1.8084
1987 1.1075 0.8733 1.0099 1.5385 1.1178 0.9625 1.0265 1.34 1.0262 1.0873 1.5191 0.7237 1.2795
1988 0.9631 1.1546 1.1705 0.975 1.019 1.0552 0.9726 1.34 1.0142 1.1541 1.3813 1.6385 1.0931
1989 1.2843 1.4809 1.7311 1.2201 1.6149 0.7723 1.0995 0.8639 1.4413 1.1364 1.2813 1.4691 1.342
1990 1.0769 1.072 1.2039 0.6655 1.2331 1.2005 0.7919 0.6103 0.8895 1.035 0.7622 1.1463 1.1597
1991 1.1108 0.9501 1.7258 1.1316 1.0837 0.7876 1.2571 1.0563 1.3333 1.1763 1.024 1.0523 1.8526
1992 1.0564 1.0072 1.0436 1.0651 1.1425 0.566 1.3409 1.5244 1.1176 1.0041 0.9115 0.8403 0.7815
1993 1.0807 1.2536 1.0675 0.9913 1.0629 1.1216 1.2102 1.5044 1.2266 1.0327 1.6914 1.0779 0.7925
1994 0.9816 1.0244 1.1541 1.185 1.0877 1.3009 0.8403 0.8643 0.9726 0.9723 0.8953 1.0867 1.1091
1995 1.2436 1.1737 1.4417 1.0446 1.3387 1.2432 1.1533 1.0359 1.4118 1.3116 1.5962 1.6676 1.7213
1996 1.2505 1.2411 1.4175 1.1157 1.2967 1.658 1.0954 1.1169 1.3733 1.2174 1.1489 1.3589 1.2133
1997 0.9887 1.1846 1.2672 1.2951 1.4832 1.3812 1.0356 1.5058 1.4842 1.2487 1.4383 0.919 1.3312
1998 0.8667 1.0771 1.0047 0.8959 1.1441 1.7622 1.0357 1.2085 1.3901 1.1951 0.5644 0.6667 1.3915
1999 1.3761 1.0445 0.8694 1.4694 1.1999 1.1702 1.4133 0.9084 1.5172 1.1017 1.3194 1.2701 0.911
2000 1.2312 0.9747 1.0461 0.8223 0.7159 0.7879 0.7067 0.4396 0.9293 1.0791 1.0562 1.5928 1.3935
2001 0.981 1.0613 0.7737 0.9223 1.0088 1.4231 1.0656 0.6707 0.8361 0.9041 0.953 0.5876 0.628
2002 1.0431 0.7419 0.9298 0.8792 1.0861 0.6407 0.9099 0.5916 0.6075 0.8891 1.0502 0.8507 0.9628
2003 1.3792 1.2995 1.1576 1.3997 1.1622 1.1959 1.3005 1.7204 1.2723 1.1734 1.4188 1.2774 0.8161
2004 0.9652 1.2156 0.8205 1.191 1.1029 1.0637 1.0503 0.915 1.1782 1.2502 1.1437 1.2285 0.6957
2005 0.9443 1.0811 0.9681 0.8851 1.0508 0.8338 0.9869 0.5273 0.9603 1.0958 0.9155 1.3568 0.9897
2006 1.0055 1.2952 1.197 0.9105 1.1104 1.1819 1.065 0.9728 1.0616 1.3643 1.3958 1.2648 1.3706
2007 1.082 1.2693 1.2719 0.988 1.1424 1.1127 1.0437 0.8961 0.9962 1.2226 1.959 0.9845 1.3328
2008 0.6824 0.7926 0.7377 0.3828 0.842 0.7785 0.4151 0.3403 0.437 0.8521 0.4115 0.4879 0.5231
2009 1.4367 1.0408 1.2591 1.831 0.9808 1.5554 1.4093 4.3668 0.934 0.8542 1.4115 1.2686 1.202
2010 1.0439 1.1852 1.1539 1.2356 1.061 1.1212 1.0512 1.679 1.2089 1.0723 1.5354 1.2056 0.9863
2011 0.947 1.166 1.0639 0.8424 1.037 1.2529 0.9863 0.6409 0.9792 1.1592 0.9314 1.124 1.0461
2012 1.1361 1.0164 1.0362 1.1241 1.0177 1.0417 1.4901 1.2035 1.172 1.0211 1.1173 1.0274 1.0859
2013 1.5105 1.1551 1.1396 1.3733 1.1991 0.9792 1.1348 1.1915 1.3354 1.1693 1.0568 1.8112 1.2225
2014 1.1716 0.8981 1.022 1.0273 1.1189 0.8554 1.1368 1.0045 0.9015 0.9135 0.9687 0.9523 1.1347
2015 0.9167 0.8019 1.0175 1.1287 0.8718 0.8578 0.8353 0.909 1.2327 0.8432 0.8621 1.1124 0.9301
2016 1.1854 1.3084 0.9651 1.1115 1.0588 1.2061 1.0037 0.8609 1.0144 1.1579 1.351 1.0767 1.1145
2017 1.3181 1.0636 1.1066 1.2447 1.0928 0.9243 1.1718 1.0297 0.5522 0.9267 1.5189 1.8943 0.9558
2018 0.8095 0.869 1.032 0.7509 1.0004 0.7409 0.62 0.6125 0.4338 0.8153 0.9531 1.0936 1.3579
2019 0.9259 1.1077 1.169 2.7766 1.3588 1.1792 1.3815 1.2157 1.5298 1.0233 1.1615 1.0101 1.1903
2020 0.9908 0.7008 0.9908 1.1076 1.114 0.9391 1.1103 0.9452 0.9677 0.5907 1.5529 0.6571 0.8994
2021 1.0162 1.3896 1.0797 1.136 1.1757 1.0906 1.2282 2.3629 0.9015 1.4845 1.2744 0.9405 1.3092
2022 0.6751 1.5295 1.0743 0.8496 0.9265 1.0541 0.7528 0.5589 1.2327 1.8026 1.2504 0.9462 1.4477
2023 0.9116 0.831 0.9264 1.1209 0.9669 1.1608 1.1216 1.0482 1.0144 0.9064 0.9326 1.3684 0.9826 ;
ENDDATA
SUBMODEL RatioNLP:
! The Omega ratio portfolio problem solved as a nonlinear program;
! The fractions invested must sum to 1;
@SUM( STOCK( s): X(s)) = 1;
! For each scenario, compute return from portfolio;
@FOR( OBS( i):
R( i) = @SUM( STOCK( s): GF( i, s)* X( s));
! Compute downside below target for scenario i;
DwnSide( i) >= Tau - R( i);
);
! Compute numerator of objective ratio;
X2 = ( @SUM( OBS( i): R( i))/ @SIZE( OBS)) - Tau;
! Compute denominator of objective ratio;
X1 = @SUM( OBS( i): DwnSide( i))/ @SIZE( OBS);
MAX = RATIO;
RATIO = X2/ X1;
ENDSUBMODEL
CALC:
@SET( 'OROUTE',1); ! Buffer size for routing output to window;
@SET( 'TERSEO',2); ! Output level (0:verb, 1:terse, 2:only errors, 3:none);
@SOLVE( RatioNLP);
@WRITE( ' Optimal Omega ratio= ', RATIO, @NEWLINE( 1));
@WRITE( ' Stock Fraction', @NEWLINE( 1));
@FOR( STOCK( s) | X( s) #GT# 0:
@WRITE( @FORMAT( STOCK( s),'12s'), ' ', @FORMAT( X( s), '8.3f'), @NEWLINE( 1));
);
ENDCALC
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