Fitting Extreme Value Models and Tail Risk Estimation

Understand Fitting Extreme Value Models and Tail Risk Estimation

Key rules to remember

GEV distribution function

G(x) = exp{ −[1 + ξ(x − μ)/σ]^(−1/ξ) } for ξ ≠ 0, where 1 + ξ(x − μ)/σ > 0

For ξ = 0 (Gumbel): G(x) = exp{ −exp[−(x − μ)/σ] }.

GPD distribution function

H(y) = 1 − (1 + ξy/σ)^(−1/ξ) for ξ ≠ 0; H(y) = 1 − exp(−y/σ) for ξ = 0

y = x − u ≥ 0, σ > 0. If ξ < 0, also y ≤ −σ/ξ.

GPD log-likelihood

l(ξ, σ) = −Nu ln σ − (1 + 1/ξ) Σ ln(1 + ξ yᵢ/σ), for ξ ≠ 0

yᵢ are the Nu excesses. For ξ = 0: l(σ) = −Nu ln σ − Σyᵢ/σ, so σ̂ = ȳ.

Tail probability (POT)

P(X > x) = (Nu ÷ n) × [1 + ξ(x − u)/σ]^(−1/ξ), for x > u

Nu = number of exceedances, n = total sample size.

High quantile (POT)

x_p = u + (σ/ξ) × { [(n ÷ Nu)(1 − p)]^(−ξ) − 1 }

Valid for p high enough that x_p > u. For ξ = 0: x_p = u + σ ln[Nu ÷ (n(1 − p))].

Return level (GEV, block maxima)

z_T = μ − (σ/ξ){ 1 − [−ln(1 − 1/T)]^(−ξ) } for ξ ≠ 0

T is the return period in blocks. Gumbel case: z_T = μ − σ ln[−ln(1 − 1/T)].

Mean excess function (GPD)

E[X − v | X > v] = [σ + ξ(v − u)] ÷ (1 − ξ), for v ≥ u and ξ < 1

Linear in v. A rising mean excess plot suggests ξ > 0.

Expected shortfall (GPD tail)

ES_p = x_p ÷ (1 − ξ) + (σ − ξu) ÷ (1 − ξ), for ξ < 1

The mean exists only if ξ < 1. State this condition.

How to solve Fitting Extreme Value Models and Tail Risk Estimation questions

1. 1Identify the method: block maxima (GEV, return levels) or threshold exceedances (GPD, quantiles, ES). Note which data are given.
2. 2Write down the model and its parameters. For POT, record u, n and Nu, and calculate Nu ÷ n.
3. 3For fitting, write the likelihood or log-likelihood from the density, differentiate, and set the derivatives to zero. State that the equations are solved numerically unless ξ = 0.
4. 4Check conditions on the estimates: σ > 0, the support condition 1 + ξ(x − u)/σ > 0, and ξ < 1 if you need a mean.
5. 5Choose the right formula: tail probability, quantile, return level, or expected shortfall. Write it in full before substituting.
6. 6Substitute carefully. Calculate the bracket first, then the power, then the final value. Keep four significant figures until the end.
7. 7State the result with units and its meaning, for example 'a loss exceeded once in 50 years on average'.
8. 8Comment briefly on reliability: sensitivity to the threshold, small Nu, and extrapolation beyond the data.

Quickest way: Plug-in method for POT quantiles and ES

Common mistakes in Fitting Extreme Value Models and Tail Risk Estimation

• Using 1 − p directly in the quantile formula without the factor n ÷ Nu.

  Students forget that the GPD describes only the excesses, not the whole distribution.

  Fix: Always form r = (n ÷ Nu)(1 − p). The factor n ÷ Nu converts a tail probability on the full data into one on the exceedances.

• Mixing up the GEV return level formula with the GPD quantile formula.

  Both involve powers of −ξ and look similar.

  Fix: Block maxima and return period T use the GEV with −ln(1 − 1/T). Threshold data use the GPD with Nu ÷ n. Decide the method first.

• Ignoring the ξ = 0 case and dividing by zero.

  The general formulas have ξ in the denominator.

  Fix: Check whether ξ is zero or the model is Gumbel or exponential. Use the logarithmic limiting forms.

• Quoting expected shortfall or mean excess when ξ ≥ 1.

  Students apply 1 ÷ (1 − ξ) without checking it.

  Fix: State that the mean exists only for ξ < 1. If ξ ≥ 1 the mean is infinite and ES is not defined.

• Treating a return period of T years as 'happens exactly every T years'.

  The wording suggests a regular cycle.

  Fix: Say the level is exceeded in any one block with probability 1/T, so on average once every T blocks.

• Choosing the threshold without justification, or calling the estimate exact.

  Students treat the fitted model as the truth.

  Fix: Mention the trade-off: a low threshold biases the fit, a high threshold leaves few exceedances and high variance. Support the choice with a mean excess plot or stability of ξ̂.

Worked examples