Stephen Richards
Articles written by Stephen Richards
The alias problem
Division of labour
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):
\[\log m_{x,y} = \alpha_x + \beta_x(y-\bar y) + \kappa_y + \gamma_{y-x}\qquad (1)\]
Getting to the root of time-series forecasting
When using a stochastic model for mortality forecasting, people can either use penalty functions or time-series methods . Each approach has its pros and cons, but time-series methods are the commonest. I demonstrated in an earlier posting how an ARIMA time-series model can be a better representation of a mortality index than a random walk with drift.
Pension freedom or trap for the unwary?
A momentary diversion
An important quantity in mathematical statistics is the moment of a distribution, i.e. the expected value of a given power of the observations. Moments can be either raw, centred about a particular value or standardised in some way. The simplest example is the mean of a distribution: this is the raw first moment, i.e. the expected value of each observation raised to the power 1:
Further reducing uncertainty
In a previous posting I looked at how using a well founded statistical model can improve the accuracy of estimated mortality rates. We saw how the relative uncertainty for the estimate of \(\log \mu_{75.5}\) could be reduced from 20.5% to 3.9% by using a simple two-parameter Gompertz model:
\(\log \mu_x = \alpha + \beta x\qquad (1)\)
Parameterising the CMI projection spreadsheet
The CMI is the part of the UK actuarial profession which collates mortality data from UK life offices and pension consultants. Amongst its many outputs is an Excel spreadsheet used for setting deterministic mortality forecasts. This spreadsheet is in widespread use throughout the UK at the time of writing, not least for the published reserves for most insurers and pension schemes.