Compare and contrast: VaR v. CTE

(Mar 9, 2019)

Insurance reserving in many countries looks at extreme scenarios over a single year.  The idea is that the insurer faces an uncertain liability, as represented by a random variable, \(X\) say.  The question is then how to set a reserve for this liability, \(V_\alpha(X)\)?  In this blog we consider two common regulatory approaches to setting \(V_\alpha(X)\):

  1. Value-at-risk (VaR).  This is a quantile-based approach, i.e. \(V_\alpha(X)=Q_\alpha(X)\), where \(Q_\alpha(X)\) is the \(\alpha\)-quantile of \(X\), i.e. \(\Pr[X\leq Q_\alpha(X)]=\alpha\).  The quantile approach to reserving underlies Solvency II in the European Union.  The reserve \(V_\alpha(X)\) is thus set so that the liability \(X\)…

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Tags: conditional tail expectation, CTE, SST, quantile, percentile

Conditional tail expectations

(Oct 10, 2014)

In a recent posting I looked at the calculation of percentiles and quantiles, which underpin many calculations for ICA and Solvency II.  Simply put, an \(\alpha\)-quantile is the value which is not expected to be exceeded \(\alpha\times 100\)% of the time.  This value is denoted \(Q_{\alpha}\).  Mathematically, for a continuous random variable, \(X\), and a given probability level \(\alpha\) we have:

$$\Pr(X\leq Q_\alpha)=\alpha$$

Thus, ICA and Solvency II work is about 99.5%-quantiles or \(Q_{99.5\%}\). However, quantiles and percentiles are not universally used for determining regulatory capital.  In North America, for example, the conditional tail expectation (CTE) is widely used.  The CTE…

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Tags: conditional tail expectation, CTE, quantile, percentile, coherence, subadditivity

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