ReIns (version 1.0.7)

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)

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.

Value

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

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

# NOT RUN {
# 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]))
# }