Stephen Richards

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

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

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

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

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

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):

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

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.

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:

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:

\(\log \mu_x = \alpha + \beta x\qquad (1)\)