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Stephen Richards

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Articles written by Stephen Richards

Conditional tail expectations

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:

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

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

Don't cut corners

An important class of mortality-projection models is the Cairns-Blake-Dowd (CBD) family.
Tags: cohort effect, Cairns-Blake-Dowd, M6 model, M7 model

Quantiles and percentiles

Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. They are generally described as q-quantiles, where q specifies the number of intervals which are separated by q−1 points.
Tags: quantile, percentile, Solvency II, Excel, R language

Creative thinking around longevity risk

The U.K. has been a hotbed of innovation when dealing with the longevity risk found in pension schemes.
Tags: longevity risk, regulation, longevity swaps

Excel's limits

We have written in the past about some of the reasons why we don't use Excel to fit our models.  However, we do use Excel for validation purposes — fitting models using two entirely separate tools is a good way of checking production code.  That said, there are some important limits to Excel, especially when it comes to fitting projection models.
Tags: Excel, Lee-Carter, APC, Cairns-Blake-Dowd

(Un)Fit for purpose

Academics lay great store by anonymous peer review and in openly publishing their results.  There are good reasons for this — anonymous peer review allows expert third parties (usually two) to challenge assumptions without fear of retribution, while open publishing allows others to test things and find their limitations. 
Tags: Lee-Carter, Renshaw-Haberman, model risk

A second pension-scheme revolution

In his book Unseen Revolution, Peter Drucker drew attention to the structural changes in economic ownership which were silently ushered in with the growth of corporate pension schemes.
Tags: pension schemes, bulk annuities, buy-out

Spotting quality issues with limited data

In an earlier posting I showed how to use the Kaplan-Meier function to identify subtle data problems.  However, what can you do when you don't have the detailed information to build a full survival curve?
Tags: data validation, survival rates, standard table

S2 mortality tables

The CMI has released the long-awaited S2 series of mortality tables based on pension-scheme data.  These are the first new tables since the CMI changed its status (the S2 series is only available to paying subscribers, unlike prior CMI tables). 
Tags: S2 Series, S1 Series, CMI, mortality improvements

The perils of parameter interpretation

With some notable exceptions, such as the Kaplan-Meier estimator, most mortality models contain parameters. In a statistical model these parameters need to be estimated, and it is a natural thing for people to want to place interpretations on those parameter estimates. However, this can be tricky, as parameters in a multi-parameter model are dependent on each other.

Tags: Lee-Carter, parameterisation