Keating and Shadwick (2002) proposed Omega (referred to as Gamma in their original paper) as a way to capture all of the higher moments of the returns distribution.
an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns
L is the loss threshold that can be specified as zero, return from a benchmark index, or an absolute rate of return - any specified level
one of: simple, interp, binomial, blackscholes
one of: point (in time), or full (distribution of Omega)
risk free rate, as a single number
TRUE/FALSE whether to ouput the standard errors of the estimates of the risk measures, default FALSE.
Control parameters for the computation of standard errors. Should be done using the RPESE.control function.
any other passthru parameters
Mathematically, Omega is: integral[L to b](1 - F(r))dr / integral[a to L](F(r))dr
where the cumulative distribution F is defined on the interval (a,b). L is the loss threshold that can be specified as zero, return from a benchmark index, or an absolute rate of return - any specified level. When comparing alternatives using Omega, L should be common.
Input data can be transformed prior to calculation, which may be useful for introducing risk aversion.
This function returns a vector of Omega, useful for plotting. The steeper, the less risky. Above it's mean, a steeply sloped Omega also implies a very limited potential for further gain.
Omega has a value of 1 at the mean of the distribution.
Omega is sub-additive. The ratio is dimensionless.
Kazemi, Schneeweis, and Gupta (2003), in "Omega as a Performance Measure" show that Omega can be written as: Omega(L) = C(L)/P(L) where C(L) is essentially the price of a European call option written on the investment and P(L) is essentially the price of a European put option written on the investment. The maturity for both options is one period (e.g., one month) and L is the strike price of both options.
The numerator and the denominator can be expressed as: exp(-Rf=0) * E[max(x - L, 0)] exp(-Rf=0) * E[max(L - x, 0)] with exp(-Rf=0) calculating the present values of the two, where rf is the per-period riskless rate.
The first three methods implemented here focus on that observation. The first method takes the simplification described above. The second uses the Black-Scholes option pricing as implemented in fOptions. The third uses the binomial pricing model from fOptions. The second and third methods are not implemented here.
The fourth method, "interp", creates a linear interpolation of the cdf of
returns, calculates Omega as a vector, and finally interpolates a function
for Omega as a function of L. This method requires library Hmisc,
which can be found on CRAN.
Keating, J. and Shadwick, W.F. The Omega Function. working paper. Finance Development Center, London. 2002. Kazemi, Schneeweis, and Gupta. Omega as a Performance Measure. 2003.
data(edhec)
Omega(edhec)
#> Convertible Arbitrage CTA Global Distressed Securities
#> Omega (L = 0%) 2.848491 1.618552 2.756588
#> Emerging Markets Equity Market Neutral Event Driven
#> Omega (L = 0%) 1.752959 4.291785 2.630127
#> Fixed Income Arbitrage Global Macro Long/Short Equity
#> Omega (L = 0%) 3.369045 2.89794 2.314433
#> Merger Arbitrage Relative Value Short Selling Funds of Funds
#> Omega (L = 0%) 3.955367 3.662014 0.9247907 2.185667
# CRAN (questionably(ahem) requires these methods to not run if you don't have Suggests loaded)
if(requireNamespace("Hmisc", quietly = TRUE)){
Omega(edhec[,13],method="interp",output="point")
Omega(edhec[,13],method="interp",output="full")
} # end spurious CRAN check
#> Funds of Funds
#> -0.0705 292.0000000
#> -0.0705 292.0000000
#> -0.0618 218.7500000
#> -0.0616 166.4285714
#> -0.06 132.1818182
#> -0.0272 102.4117647
#> -0.0269 81.0400000
#> -0.0266 67.9411765
#> -0.0264 58.9318182
#> -0.0262 51.3214286
#> -0.0252 45.7101449
#> -0.0222 41.3614458
#> -0.0205 37.8673469
#> -0.0202 34.9824561
#> -0.0192 32.5496183
#> -0.0176 30.4630872
#> -0.0163 28.6488095
#> -0.0156 27.0531915
#> -0.0149 25.6363636
#> -0.0148 24.3679654
#> -0.0142 23.2244094
#> -0.0141 22.1870504
#> -0.014 21.1677632
#> -0.0138 20.2447130
#> -0.0133 19.3472222
#> -0.0132 18.5333333
#> -0.0127 17.7909739
#> -0.0126 17.1103753
#> -0.0122 16.4835391
#> -0.0119 15.9038462
#> -0.0116 15.3657658
#> -0.0108 14.8646362
#> -0.0104 14.3964968
#> -0.0099 13.9579580
#> -0.0095 13.5254958
#> -0.0093 13.1204819
#> -0.0089 12.7401774
#> -0.0083 12.3822115
#> -0.0082 12.0445205
#> -0.0079 11.7252986
#> -0.0077 11.4229576
#> -0.0074 11.1360947
#> -0.0072 10.8634652
#> -0.0071 10.5935252
#> -0.007 10.3370593
#> -0.0069 10.0930041
#> -0.0068 9.8347758
#> -0.0063 9.5823928
#> -0.0062 9.3436599
#> -0.0059 9.1174033
#> -0.0054 8.9025845
#> -0.0049 8.6982813
#> -0.0044 8.4978593
#> -0.004 8.3015873
#> -0.0037 8.0993789
#> -0.0036 7.9028757
#> -0.0034 7.7165971
#> -0.0033 7.5268440
#> -0.0031 7.3471753
#> -0.0028 7.1767442
#> -0.0027 7.0147982
#> -0.0025 6.8606664
#> -0.0022 6.7105263
#> -0.0021 6.5673931
#> -0.0019 6.4307452
#> -0.0018 6.3001133
#> -0.0015 6.1750731
#> -0.0012 6.0552408
#> -0.001 5.9402678
#> -9e-04 5.8298368
#> -7e-04 5.7214863
#> -6e-04 5.6173149
#> -5e-04 5.5170628
#> -4e-04 5.4204916
#> -3e-04 5.3273827
#> -2e-04 5.2375350
#> 1e-04 5.1507634
#> 4e-04 5.0668967
#> 6e-04 4.9842296
#> 8e-04 4.9042821
#> 9e-04 4.8269089
#> 0.0013 4.7492223
#> 0.0015 4.6740551
#> 0.0017 4.6012745
#> 0.0018 4.5295293
#> 0.0019 4.4600217
#> 0.0021 4.3914975
#> 0.0022 4.3217750
#> 0.0024 4.2542817
#> 0.0025 4.1878812
#> 0.0026 4.1225744
#> 0.0028 4.0593093
#> 0.003 3.9970658
#> 0.0031 3.9367270
#> 0.0032 3.8781984
#> 0.0033 3.8213918
#> 0.0034 3.7654259
#> 0.0035 3.7110747
#> 0.0037 3.6567667
#> 0.0039 3.6032993
#> 0.004 3.5506689
#> 0.0041 3.4988710
#> 0.0043 3.4478998
#> 0.0046 3.3983834
#> 0.005 3.3502545
#> 0.0051 3.3034502
#> 0.0052 3.2573330
#> 0.0053 3.2124601
#> 0.0057 3.1687769
#> 0.0058 3.1262322
#> 0.0059 3.0847777
#> 0.006 3.0443678
#> 0.0064 3.0049595
#> 0.0066 2.9660413
#> 0.0067 2.9280718
#> 0.0068 2.8901225
#> 0.0069 2.8526635
#> 0.007 2.8161148
#> 0.0071 2.7800304
#> 0.0072 2.7448078
#> 0.0073 2.7100251
#> 0.0075 2.6756812
#> 0.0076 2.6417745
#> 0.0077 2.6086611
#> 0.0078 2.5763109
#> 0.0079 2.5446952
#> 0.008 2.5131231
#> 0.0082 2.4819423
#> 0.0083 2.4511505
#> 0.0085 2.4210526
#> 0.0086 2.3916232
#> 0.0088 2.3628380
#> 0.0089 2.3346740
#> 0.009 2.3065521
#> 0.0091 2.2787633
#> 0.0092 2.2515708
#> 0.0093 2.2249538
#> 0.0094 2.1988924
#> 0.0095 2.1731204
#> 0.0096 2.1476366
#> 0.0097 2.1222037
#> 0.0099 2.0970671
#> 0.0104 2.0724500
#> 0.0106 2.0481144
#> 0.0108 2.0242739
#> 0.0109 2.0007015
#> 0.0111 1.9776011
#> 0.0113 1.9547561
#> 0.0114 1.9321647
#> 0.0119 1.9100179
#> 0.0121 1.8881128
#> 0.0125 1.8664478
#> 0.0126 1.8452018
#> 0.0127 1.8243615
#> 0.013 1.8039143
#> 0.0131 1.7838480
#> 0.0133 1.7641509
#> 0.0134 1.7444866
#> 0.0136 1.7251828
#> 0.0137 1.7062287
#> 0.0138 1.6876139
#> 0.0139 1.6693285
#> 0.014 1.6513629
#> 0.0142 1.6337079
#> 0.0145 1.6163546
#> 0.0147 1.5991557
#> 0.0148 1.5822471
#> 0.0152 1.5656208
#> 0.0153 1.5492689
#> 0.0156 1.5331842
#> 0.0157 1.5173592
#> 0.016 1.5017871
#> 0.0163 1.4863393
#> 0.0164 1.4711357
#> 0.0169 1.4561698
#> 0.0171 1.4410888
#> 0.0172 1.4262456
#> 0.0175 1.4116341
#> 0.0182 1.3972484
#> 0.0185 1.3830826
#> 0.0189 1.3691311
#> 0.0191 1.3553887
#> 0.0199 1.3418501
#> 0.0202 1.3285103
#> 0.0203 1.3153646
#> 0.0204 1.3024081
#> 0.0205 1.2895408
#> 0.0206 1.2768569
#> 0.0209 1.2643523
#> 0.0213 1.2520227
#> 0.0216 1.2398639
#> 0.0217 1.2278722
#> 0.0219 1.2160436
#> 0.022 1.2043745
#> 0.0222 1.1928613
#> 0.0223 1.1815005
#> 0.0225 1.1702888
#> 0.0233 1.1592229
#> 0.0244 1.1482996
#> 0.0256 1.1375160
#> 0.0267 1.1268689
#> 0.0274 1.1163556
#> 0.0275 1.1059732
#> 0.0282 1.0957189
#> 0.0286 1.0855903
#> 0.0303 1.0755846
#> 0.0311 1.0656994
#> 0.0312 1.0559323
#> 0.0313 1.0462808
#> 0.0317 1.0367428
#> 0.0334 1.0273160
#> 0.034 1.0179981
#> 0.0373 1.0087872
#> 0.0384 0.9996811
#> 0.04 0.9906778
#> 0.0435 0.9817755
#> 0.0483 0.9729721
#> 0.0622 0.9642659
#> 0.0666 0.9556551