### Getting to the root of time-series forecasting

#### (Oct 3, 2016)

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. In this posting we will examine the structure of an ARIMA model and how one might go around selecting and fitting it.

Assume we have an index at time $$t$$, $$\kappa_t$$, and an error term, $$\epsilon_t$$ ($$\kappa_t$$ could be the mortality index in the Lee-Carter model, for example). For mortality applications the simplest non-trivial forecasting model is the…

### Parameterising the CMI projection spreadsheet

#### (May 18, 2016)

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.

Following Willets (1999), the basic unit of the CMI spreadsheet is the mortality-improvement rate:

$1 - \frac{q_{x,t}}{q_{x,t-1}}\qquad(1)$

where $$q_{x,t}$$ is the probability of death aged $$x$$ in year $$t$$, assuming a life is alive at the start of the year.

### Volatility v. Trend Risk

#### (Oct 8, 2010)

The year 1992 was important in the development of forecasting methods: Ronald Lee and Lawrence Carter published their highly influential paper on forecasting US mortality.  The problem is difficult: given matrices of deaths and exposures (rows indexed by age and columns by year) can we forecast future death rates?  Lee and Carter designed a model specifically to solve this problem:

log μx,yαx + βxκy        (1)

where αx measures the average mortality at age xκy measures the effect of year y; this year effect is modulated by an age dependent coefficient, βx.  Lee and Carter used US data up to 1989 and here I've followed them by using data on US males aged 60-90 between 1933-1989,…

#### (Jul 12, 2010)

One of the most written-about models for stochastic mortality projections is that from Lee & Carter (1992).  As Iain described in an earlier post, the genius of the Lee-Carter model lies in reducing a two-dimensional forecasting problem (age and time) to a simpler one-dimensional problem (time only).

A little-appreciated fact is that there are two ways of approaching the time-series projection of future mortality rates.  A simple method is to treat the future mortality index as a simple random walk with drift.  This makes the strong simplifying assumption that the mortality trend changes at a constant rate (apart from the random noise).  Figure 1 shows an example projection for males in England &…