Stephen Richards Profile Picture

Stephen Richards

Managing Director
Contact Stephen Richards

Articles written by Stephen Richards

Age rating

Back in the days before personal computers, actuaries relied solely on published tables for their calculations. These were not just the mortality tables, but monetary functions of these tables known as commutation factors. My old student tables from 1980 list commutation and other factors at discount rates of 4%, 6% and 8% (the latter rate seems almost comically high by current standards).

Tags: age rating, Makeham-Beard

How much data do you need?

There are two common scenarios when an actuary has to come up with a mortality basis for pensioners or annuitants.
Tags: credibility, basis risk, concentration risk

Twin peaks

If you are over forty, the title of this blog will call to mind an iconic, sometimes disturbing, television series of the same name from 1990.  If you clicked on the link expecting murder, surreal horror and an undercurrent of sleaze, however, then this posting is as far away from all that as you are ever likely to get.
Tags: value-at-risk, bimodal distribution, Solvency II, model risk

Fifty years of mortality improvements

In an earlier post we looked at the development of the distribution of age at death over time. We saw how the peak adult age at death had continuously moved towards an ever-higher age.
Tags: mortality improvements

Changing patterns of mortality

In an earlier post we introduced the idea of the so-called curve of deaths, which is simply the distribution of age at death.  This is intimately bound up with survival models and the idea of future lifetime as a random variable.

On the (funding) level

When I read Allan Martin's earlier blog on how pension-scheme reserves routinely fail to include expenses, I was so surprised I had to ask him if it was really true.  As a former life-insurance actuary, any reserve which didn't include an allowance for expenses simply wasn't a complete assessment of the liability in my view. 
Tags: pension schemes, expenses, market consistency

The alias problem

A problem that can crop up during mortality modelling is that of aliasing, specifically extrinsic aliasing.  The situation can be illustrated by an example of the sort of data available for a pension scheme.
Tags: extrinsic aliasing

Division of labour

At this time of year insurers have commenced their annual valuation of liabilities, part of which involves setting a mortality basis.  When doing so it is common for actuaries to separate the basis into two components.
Tags: valuation, Solvency II, mis-estimation risk, trend risk

Season's Greetings to all our readers!

\[y = \frac{\log_e\left(\frac{x}{m}-sa\right)}{r^2}\]

\[\Rightarrow yr^2 = \log_e\left(\frac{x}{m}-sa\right)\]

\[\Rightarrow e^{yr^2} = \frac{x}{m}-sa\]

\[\Rightarrow me^{yr^2} = x-msa\]

\[\Rightarrow me^{rry} = x-mas\]

Signal or noise?

Each year since 2009 the CMI in the UK has released a spreadsheet tool for actuaries to use for mortality projections. I have written about this tool a number of times, including how one might go about setting the long-term rate. The CMI now wants to change how the spreadsheet is calibrated and has proposed the following model in CMI (2016a):

\[\log m_{x,y} = \alpha_x + \beta_x(y-\bar y) + \kappa_y + \gamma_{y-x}\qquad (1)\]

Tags: CMI, APCI, APC, Lee-Carter, Age-Period, smoothing