ReIns (version 1.0.10)

CTE: Conditional Tail Expectation

Description

Compute Conditional Tail Expectation (CTE) \(CTE_{1-p}\) of the fitted spliced distribution.

Usage

CTE(p, splicefit)
ES(p, splicefit)

Value

Vector with the CTE corresponding to each element of \(p\).

Arguments

p

The probability associated with the CTE (we estimate \(CTE_{1-p}\)).

splicefit

A SpliceFit object, e.g. output from SpliceFitPareto, SpliceFiticPareto or SpliceFitGPD.

Author

Tom Reynkens with R code from Roel Verbelen for the mixed Erlang quantiles.

Details

The Conditional Tail Expectation is defined as $$CTE_{1-p} = E(X | X>Q(1-p)) = E(X | X>VaR_{1-p}) = VaR_{1-p} + \Pi(VaR_{1-p})/p,$$ where \(\Pi(u)=E((X-u)_+)\) is the premium of the excess-loss insurance with retention u.

If the CDF is continuous in \(p\), we have \(CTE_{1-p}=TVaR_{1-p}= 1/p \int_0^p VaR_{1-s} ds\) with \(TVaR\) the Tail Value-at-Risk.

See Reynkens et al. (2017) and Section 4.6 of Albrecher et al. (2017) for more details.

The ES function is the same function as CTE but is deprecated.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Reynkens, T., Verbelen, R., Beirlant, J. and Antonio, K. (2017). "Modelling Censored Losses Using Splicing: a Global Fit Strategy With Mixed Erlang and Extreme Value Distributions". Insurance: Mathematics and Economics, 77, 65--77.

Verbelen, R., Gong, L., Antonio, K., Badescu, A. and Lin, S. (2015). "Fitting Mixtures of Erlangs to Censored and Truncated Data Using the EM Algorithm." Astin Bulletin, 45, 729--758

See Also

qSplice, ExcessSplice, SpliceFit, SpliceFitPareto, SpliceFiticPareto, SpliceFitGPD

Examples

if (FALSE) {

# Pareto random sample
X <- rpareto(1000, shape = 2)

# Splice ME and Pareto
splicefit <- SpliceFitPareto(X, 0.6)

p <- seq(0.01, 0.99, 0.01)
# Plot of CTE
plot(p, CTE(p, splicefit), type="l", xlab="p", ylab=bquote(CTE[1-p]))
}