Parametric Approaches (II): Extreme Value: formula sheet

Key formulas

Excess kurtosis

Excess kurtosis = kurtosis − 3

The normal distribution has kurtosis of 3, so excess kurtosis of 0. A positive value signals fat tails relative to the normal.

Normal VaR

VaR = μ_loss + z(α) × σ

For a loss distribution. z(0.99) ≈ 2.33 and z(0.999) ≈ 3.09. This is the benchmark that EVT improves on in the tails.

Tail index rule of thumb

Larger tail index ξ > 0 means a fatter tail

In EVT the shape parameter ξ (tail index) measures tail heaviness. ξ > 0 is heavy-tailed, ξ = 0 is exponential-type, ξ < 0 is a bounded tail. The normal falls in the ξ = 0 case.

GEV cumulative distribution (ξ ≠ 0)

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

x is the block maximum of losses. μ is location, σ > 0 is scale, ξ is the tail index.

Gumbel case (ξ = 0)

F(x) = exp{ −exp[−(x − μ) ÷ σ] }

Limit of the GEV as ξ → 0. Thin, exponentially decaying tail.

Tail index cases

ξ > 0: Fréchet; ξ = 0: Gumbel; ξ < 0: Weibull

Fréchet is the usual case for financial losses.

Quantile of block maxima (ξ ≠ 0)

x_p = μ − (σ ÷ ξ) × [1 − (−ln p)^(−ξ)]

p is the cumulative probability of the maximum. Obtained by inverting F(x) = p.

Gumbel quantile

x_p = μ − σ × ln(−ln p)

Use when ξ = 0.

Link to VaR for the original data

If a block has n observations and the VaR confidence is α, then p = α^n

The maximum of n observations is below x only if all n are. So P(max ≤ x) = F_loss(x)^n. Assumes independent, identically distributed losses.

Excess over threshold

Y = X − u, for X > u

POT models only these excesses, not the full loss.

GPD cumulative distribution

G(y) = 1 − (1 + ξy ÷ β)^(−1/ξ), for ξ ≠ 0, y ≥ 0, β > 0

For ξ < 0, y must also satisfy y ≤ −β ÷ ξ.

GPD with ξ = 0

G(y) = 1 − e^(−y ÷ β)

Exponential tail, the limit as ξ approaches 0.

Tail probability estimate

P(X > x) ≈ (Nu ÷ n) × (1 + ξ(x − u) ÷ β)^(−1/ξ), for x > u

Nu is the number of exceedances and n is the total sample size. Nu ÷ n estimates P(X > u).

Hill estimator

ξ̂ = (1 ÷ k) × Σ ln(X(i) ÷ X(k+1)), i = 1 to k

X(1) ≥ X(2) ≥ ... are ordered losses. It applies to heavy tails (ξ > 0). The choice of k is the threshold problem again.

Tail index and shape

α = 1 ÷ ξ, for ξ > 0

A smaller α means a heavier tail. The k-th moment exists only if α > k.

GPD tail estimator (POT)

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

Nu is the number of losses above u; n is the total sample size. Valid for ξ ≠ 0.

EVT VaR (ξ ≠ 0)

VaR(p) = u + (β ÷ ξ) × { [(n ÷ Nu) × (1 − p)]^(−ξ) − 1 }

p is the confidence level, e.g. 0.999. Use only when VaR(p) lies above u.

EVT VaR (ξ = 0)

VaR(p) = u − β × ln[(n ÷ Nu) × (1 − p)]

Exponential tail limit.

EVT expected shortfall

ES(p) = VaR(p) ÷ (1 − ξ) + (β − ξu) ÷ (1 − ξ), for ξ < 1

Equivalent to ES = [VaR + β − ξu] ÷ (1 − ξ). If ξ ≥ 1, ES is infinite.

Normal VaR for comparison

VaR(p) = μ + σ × zp

z0.99 = 2.326; z0.999 = 3.090 (loss with μ and σ for the loss distribution).

Normal expected shortfall

ES(p) = μ + σ × φ(zp) ÷ (1 − p)

φ is the standard normal density.

Upper tail dependence coefficient

λU = lim (u→1) P(Y > F_Y⁻¹(u) | X > F_X⁻¹(u))

λU = 0 means asymptotic independence; λU > 0 means extremes cluster.

Lower tail dependence coefficient

λL = lim (u→0) P(Y ≤ F_Y⁻¹(u) | X ≤ F_X⁻¹(u))

Relevant for joint crashes in losses.

Sklar's theorem

F(x, y) = C(F_X(x), F_Y(y))

Any joint distribution splits into a copula C and its marginals.

Tail properties of common copulas

Gaussian: λ = 0 (correlation < 1); Student t: λ > 0; Clayton: λL > 0; Gumbel: λU > 0

Gaussian copula understates joint extremes.

GPD tail shape

ξ > 0 heavy tail; ξ = 0 exponential tail; ξ < 0 bounded tail

Used for each margin; ξ is the hardest parameter to estimate.

Quick revision

• EVT models only the tail, not the whole distribution.
• Block maxima uses the largest loss per period and fits a GEV distribution.
• POT uses all losses above a high threshold and fits a GPD.
• GEV has three types: Fréchet (heavy tail), Gumbel (light tail), Weibull (bounded tail).
• Tail index ξ > 0 means a fat, heavy tail; most financial losses show this.
• POT usually uses data more efficiently than block maxima.
• Threshold choice: too low gives bias, too high gives high variance.
• Expected shortfall is larger than VaR at the same confidence level.
• Normal-based VaR tends to understate losses far into the tail.
• Estimates beyond the observed data are extrapolations and carry high uncertainty.
• Extremes in different assets often occur together, so tail dependence matters.
• Always state the method, the tail index and the interpretation in your reasoning.

Common mistakes

• Saying the normal distribution overestimates extreme losses. Fix: Fat tails mean more extreme outcomes than the normal predicts. Normal VaR at high confidence is too low.
• Thinking EVT fits the entire return distribution. Fix: EVT is a tail-only method. It deliberately ignores the centre of the distribution.
• Treating the location and scale as the mean and standard deviation. Fix: Say location is a centring parameter and scale is a spread parameter for the maxima. They are not the mean or standard deviation, except by coincidence.
• Mixing up the sign of ξ: calling ξ > 0 Weibull or thin-tailed. Fix: Remember: positive ξ = fat tail = Fréchet. Zero = Gumbel. Negative = Weibull with a finite upper limit.
• Reversing the threshold trade-off, saying a higher threshold lowers variance. Fix: A higher threshold means fewer data points, so variance rises. It cuts bias, not variance. Lower threshold: low variance, high bias.
• Mixing up GEV and GPD. Fix: GEV models block maxima. GPD models excesses over a threshold. Ask what the data are: maxima per block or exceedances.
• Using 1 − p alone inside the bracket instead of (n ÷ Nu) × (1 − p). Fix: Always compute n ÷ Nu first and multiply by 1 − p.
• Using a positive exponent, q^(+ξ), or forgetting the minus sign. Fix: The exponent is −ξ. Since q < 1, q^(−ξ) > 1, so VaR rises above u.
• Treating correlation as a measure of tail dependence. Fix: Remember correlation reflects the whole distribution and linear dependence; tail dependence is a separate, limiting conditional probability.
• Saying the Gaussian copula has tail dependence when correlation is high. Fix: For correlation below 1 the Gaussian copula has zero tail dependence, however high the correlation.

Exam tips

• Expect questions that give a kurtosis, QQ plot description or exceedance count and ask what it implies. Link it to fat tails and understated normal VaR.
• Memorise the contrast: normal fits the centre and extrapolates, EVT fits only the tail.
• Know the two EVT approaches by name: GEV for block maxima, Generalized Pareto for peaks over threshold.
• Watch for overclaiming options such as exact, guaranteed or assumption-free. These are usually wrong.
• On a phone, scan options for the direction word first: understate or overstate.
• Expect questions that give ξ and ask you to name the type. Learn the signs cold: positive Fréchet, zero Gumbel, negative Weibull.
• Read whether a quantile is for the block maximum or the single-period loss. Examiners test the p = α^n conversion.
• Know that the block size trade-off is bias versus variance, and that block maxima use data less efficiently than peaks over threshold.