Pareto Models for Risk Management

Gabaix (2009) claimed that similar results can be obtained when exponents are different, unfortunately, this yields only asymptotic power tails, which will be discussed in this chapter.

\(\overline{F}_{u'}\) is a truncated Pareto distribution, with density equals to \(f(x)/(1-F(u'))\). This property can be observed directly using Eq. (8), where both \(\alpha \) and \(\lambda \) remain unchanged.

Note that this property is quite intuitive, since the GPD distribution appears as a limit for exceeding distributions, and limit in asymptotic results are always fixed points: the Gaussian family is stable by addition (and appears in the Central Limit Theorem) while Fréchet distribution is max-stable (and appears in the first theorem in extreme value theory).

Historically, extremes were studied through block-maximum—yearly maximum, or maximum of a subgroup of observations. Following Fisher and Tippett (1928), up to some affine transformation, the limiting distribution of the maximum over n i.i.d observations is either Weibull (observations with a bounded support), Gumbel (infinite support, but light tails, like the exponential distribution) or Fréchet (unbounded, with heavy tails, like Pareto distribution). Pickands (1975) and Balkema and de Haan (1974) obtained further that not only the only possible limiting conditional excess distribution is GPD, but also that the distribution of the maximum on subsamples (of same size) should be Fréchet distributed, with the same tail index \(\gamma \), if \(\gamma >0\). For instance in the USA, if the distribution of maximum income per county is Fréchet with parameter \(\gamma \) (and if county had identical sizes), then the conditional excess distribution function of incomes above a high threshold is a GPD distribution with the same tail index \(\gamma \).

The quantile function U is defined as \(U(x)=F^{-1}(1-1/x)\).

Using the expansion \((1+y^a)^b \approx 1+b y^a\), for small \(y^a\), in (22) yields (20).

Albrecher et al. (2017, Sect. 4.6) give an approximation, based on \((1+\delta -\delta y^\tau )^{-\alpha }\approx 1-\alpha \delta +\alpha \delta y^\tau \), which can be very poor. Thus, we do not recommend to use it.

Even if Hill estimator can be can be seen as a Maximum Likehood estimator, for some properly chosen distribution.

Given a sample \(\lbrace x_1,\ldots ,x_n \rbrace \), let \(\lbrace x_{1:n},\ldots ,x_{n:n} \rbrace \) denote the ordered version, with \(x_{1:n}=\min \lbrace x_1,\ldots ,x_n\rbrace \), \(x_{n:n}=\max \lbrace x_1,\ldots ,x_n\rbrace \) and \(x_{1:n}\le \ldots x_{n-1:n}\le x_{n:n} \).

The study of the limiting distribution of the maximum of a sample of size n made us introduce a normalizing sequence \(a_n\). Here, a continuous version is considered—with U(t) instead of U(n)—and the sequence \(a_n\) becomes the auxiliary function a(t).

See the R packages ReIns or TopIncomes.

It is the danishuni dataset in the CASdatasets package, available from http://​cas.​uqam.​ca/​.

Available from https://​www.​nasdaq.​com/​market-activity/​commodities/​bz%3Anmx.