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Johnson SU Distributions for Unequal Return-Tail Exponents

Article Quant Q&A · Author: Alex Craft
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Summary

The document asks how to model stock returns whose estimated left and right tail exponents differ, noting that a skewed Student-t distribution uses a shared tail exponent. It describes an analysis of daily prices for 250 stocks beginning in 1972, with tail exponents estimated using a peaks-over-threshold generalized Pareto approach. The author also compares rare-event value-at-risk estimates using Student-t tail parameters of 3 and 3.7, reporting relative differences for daily and monthly horizons in a hypothetical portfolio of ten stocks.

The response points to Johnson’s SU distribution as an alternative with different tail exponents and notes that it is available as a distributional choice in the rugarch R package’s GARCH specifications. The reported risk comparisons assume independent stocks, which the document acknowledges is unrealistic, and are described as approximate. It does not provide a derivation of the distribution’s tail behavior, a broader empirical comparison, or a full treatment of dependence and portfolio risk.

Key ideas

  • A skewed Student-t model uses the same tail exponent on both sides, which may not fit asymmetric tail estimates.
  • The document estimates return tails with a peaks-over-threshold generalized Pareto method applied to daily stock prices.
  • Johnson’s SU distribution is suggested as an alternative that can accommodate different tail exponents.
  • The reported value-at-risk comparisons rely on an independence assumption across stocks and are approximate.

Tags

EquitiesStatisticsRisk managementVolatility
Full text
# Extension of SkewStudentT with different left and right tail exponents?


# Extension of SkewStudentT with different left and right tail exponents?












I found that daily stock prices have different left ~3 and right ~3.7 tail exponents.

The $\text{SkewStudentT}(\mu,\sigma,\nu,\lambda)$ (Hansen and other variants) can't model that, it uses same tail exponent $\nu$ for both tails. And so, if we fix it as 3 - it will overestimate the probability of positive extreme events.

While the error in probability itself may be small, the extreme magnitude + exponentiation (of the log return) of the event magnifies the error and its impact on mean etc. (maybe it's better to fix it as 3.7 and underestimate the losses, the error should be smaller).

Is there an extension that allows different tail exponents something like $\text{SkewStudentT}(\mu,\sigma,\nu_l,\nu_r,\lambda)$?

P.S.

Data I used to estimate tail exponents: daily prices, 250 stocks starting 1972, using EVT POT GPD Deckers-Einmahl-de Haan tail estimator).

UPDATE:

Estimating VaR errors (relative): daily ~1.22, monthly ~1.24, for portfolio of 10 stocks and once in 10y event, tail exponent 3 vs 3.7.

Such calculation assumes independence of stocks, which is not true, so numbers are very approximate.

```
# Daily, typical daily log returns StudentT(0.001, 0.015)
p = 1-1/(365*10*10) # once in 10y for portfolio of 10 stocks
exp(
  quantile(StudentT(0.001, 0.015, 3), p) - 
  quantile(StudentT(0.001, 0.015, 3.7), p)
) # => 1.22

# Monthly, typical monthly log returns StudentT(0.01, 0.08)
p = 1-1/(12*10*10) # once in 10y for portfolio of 10 stocks
exp(
  quantile(StudentT(0.01, 0.08, 3), p) - 
  quantile(StudentT(0.01, 0.08, 3.7), p)
) # => 1.24
```
```

## Answer by Richard Hardy (score 2)

https://quant.stackexchange.com/a/83921

Johnson's $S_U$ distribution has different tail exponents. It is one of the distributional alternatives in the GARCH model specification in the `rugarch` package in R.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.