# CTE: Conditional Tail Expectation In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

 CTE R Documentation

## 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.

### 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.

### Value

Vector with the CTE corresponding to each element of p.

### Author(s)

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

### 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

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]))

## End(Not run)


ReIns documentation built on Nov. 3, 2023, 5:08 p.m.