Troubleshooting Expected Shortfall for Stable Return Distributions
Summary
The document concerns parametric expected shortfall (ES) when returns are modeled with a four-parameter Paretian stable distribution. It describes a question about implementing a closed-form ES method attributed to Stoyanov, with VaR defined for returns as a negative quantile and the calculation focused on the left tail. The author reports using numerical integration in R but failing to reproduce the paper’s target results except in the symmetric case.
A table compares the author’s ES estimates with target values across different stability parameters, alongside VaR and integration diagnostics. The reported integral errors are small, yet the mismatch grows substantially across the displayed parameter range. The document does not include the implementation or a resolution, so it cannot establish whether the issue comes from parameterization, tail conventions, or a coding error. It is a troubleshooting prompt rather than a validated procedure.
Key ideas
- The question concerns parametric ES under a Paretian stable model with four distribution parameters.
- The author defines risk from returns and integrates over the left tail.
- Numerical integration appears accurate by the reported error estimates, but the ES outputs diverge from the paper’s targets.
- The symmetric case is reported as an exception where the results match.
- No implementation details or definitive explanation for the discrepancy are provided.
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Full text
# Expected shortfall of stable distribution by Stoyanov # Expected shortfall of stable distribution by Stoyanov I've been working on calculating parametric ES assuming the returns follow Paretian stable law. Given the four parameters - $\alpha, \beta,\sigma,\mu$- Stoyanov introduces closed form solution of the problem (assuming returns as the r.v., not losses - hence VaR is the negative quantile and we are integrating over the left tail, as I understand it. - see below. I have been struggling for two days with implementing this in `R`, but I cannot get the results they publish at the end of the paper - expect for a special symmetric case ($\beta=0$) Any ideas where I could possibly go wrong? Here are my results: I use `integrate` in `R`for the integral and it seems to give reasonable values of the integral with absolute error around `1e-04`. Any help would be greatly appreciated. ``` > dat alpha myES target relError integral myVaR absErrorIntegral subdivisions 1 1.04 160.837 157.204 0.023 -0.159 6.119 0.000 7 2 1.07 93.020 86.393 0.077 -0.166 5.755 0.000 6 3 1.10 66.317 58.302 0.137 -0.174 5.430 0.000 5 4 1.13 52.234 43.331 0.205 -0.184 5.140 0.000 4 5 1.16 43.650 34.082 0.281 -0.194 4.878 0.000 4 6 1.19 37.961 27.831 0.364 -0.205 4.642 0.000 4 7 1.22 33.996 23.344 0.456 -0.217 4.428 0.000 4 8 1.25 31.154 19.979 0.559 -0.231 4.234 0.000 3 9 1.28 29.094 17.369 0.675 -0.246 4.057 0.000 3 10 1.31 27.617 15.292 0.806 -0.264 3.895 0.000 3 11 1.34 26.599 13.604 0.955 -0.283 3.746 0.000 3 12 1.37 25.965 12.200 1.128 -0.305 3.610 0.000 2 13 1.40 25.672 11.030 1.327 -0.331 3.486 0.000 2 14 1.43 25.703 10.030 1.563 -0.360 3.371 0.000 2 15 1.46 26.058 9.170 1.842 -0.395 3.266 0.000 2 16 1.49 26.758 8.430 2.174 -0.436 3.169 0.000 2 17 1.52 27.840 7.771 2.583 -0.486 3.080 0.000 2 18 1.55 29.360 7.200 3.078 -0.546 2.998 0.000 2 19 1.58 31.399 6.680 3.701 -0.619 2.923 0.000 2 20 1.61 34.067 6.229 4.469 -0.710 2.854 0.000 2 21 1.64 37.506 5.817 5.448 -0.824 2.791 0.000 2 22 1.67 41.917 5.445 6.698 -0.967 2.733 0.000 2 23 1.70 47.576 5.100 8.329 -1.148 2.679 0.000 1 24 1.73 54.877 4.790 10.457 -1.383 2.630 0.000 2 25 1.76 64.422 4.500 13.316 -1.690 2.585 0.000 2 26 1.79 77.145 4.250 17.152 -2.103 2.544 0.000 2 27 1.82 94.616 4.010 22.595 -2.673 2.505 0.000 2 28 1.85 119.654 3.790 30.571 -3.497 2.470 0.000 3 29 1.88 157.937 3.580 43.116 -4.766 2.437 0.000 3 30 1.91 222.734 3.390 64.703 -6.928 2.406 0.000 3 31 1.94 353.958 3.220 108.925 -11.330 2.378 0.001 3 32 1.97 751.596 3.050 245.425 -24.725 2.351 0.001 4 ```
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