### From magical thinking to statistical thinking

#### (Dec 4, 2019)

The Institute and Faculty of Actuaries in the UK has recently added mortality projection to its syllabus, so this year I have been teaching the subject for the first time to students at Heriot-Watt University.

As an exercise, I asked the students to imagine they were an actuary back in 1980, and to first fit a Lee-Carter model to data from the Human Mortality Database for males in England and Wales, 1940-1980.  Then the students had to project the fitted mortality surface forward 25 years, to 2005.  This is easily done using some R programs written by Iain Currie for our recent book, Macdonald, Richards and Currie (2018). The programs $$\tt Lee\_Carter.r$$ and $$\tt Forecast\_LC.r$$ referred to there are freely…

### Mortality by the book

#### (Dec 22, 2017)

Our book, Modelling Mortality with Actuarial Applications, will appear in Spring 2018.  I wrote the second of the three parts, where I describe the modelling and forecasting of aggregate mortality data, such as provided by the Office for National Statistics, the Human Mortality Database or indeed by any insurer whose own data is suitable.  I have divided my contribution into four chapters. In the first chapter I deal with one-dimensional data, for example, deaths by age for a given year.  The Gompertz model is used to introduce the regression-based approach; estimation is initially by least squares, but by the end of the chapter I use generalized linear models (GLMs) with both Poisson and binomial errors.

Tags: GLMs, mortality projections, R

### Reviewing forecasts

#### (Oct 19, 2015)

When making projections and forecasts, it can be instructive to compare them with what actually happened. In December 2002 the CMI published projections of mortality improvements that incorporated the so-called "cohort effect" (CMIB, 2002). These projections were in use by life offices and pension schemes in the United Kingdom from 2003 onwards. Since population mortality rates are now available for the ten-year period since then, we can compare actual improvements to the projections from 2003.  This is done in Figure 1.

Figure 1: Population mortality improvements 2003-2013 and corresponding CMI cohort projection bases. Source: Own calculations using national life tables from the ONS and the cohort…

### Simulating the Future

#### (Jan 13, 2015)

This blog has two aims: first, to describe how we go about simulation in the Projections Toolkit; second, to emphasize the important role a model has in determining the width of the confidence interval of the forecast.

We use US male mortality data for years 1970 to 2009 downloaded from the Human Mortality Database. Figure 1 shows the observed log mortality. Unlike UK mortality (which shows accelerating improvements in log mortality over the same period) the US improvement is perfectly well described by a straight line. We fit the simplest of models: $$y_j = a + b x_j + \epsilon_j$$, where $$x_j$$ is year $$j$$, $$y_j = \log(d_j/e_j)$$ with $$d_j$$ the observed number of deaths in year $$j$$ and $$e_j$$ the corresponding…

### Demography's dark matter: measuring cohort effects

#### (May 19, 2014)

My last blog generated quite a bit of interest so I thought I'd write again on cohorts. It's easy to (a) demonstrate the existence of a cohort effect and to (b) fit models with cohort terms, but not so easy to (c) interpret or forecast the fitted cohort coefficients. In this blog I'll fit the following three models:

 LC $$\log \mu_{i,j} = \alpha_i + \beta_i \kappa_j$$ LCC $$\log \mu_{i,j} = \alpha_i + \beta_i \kappa_j + \gamma_{j-i}$$ APC $$\log \mu_{i,j} = \alpha_i + \kappa_j + \gamma_{j-i}$$

where $$\mu_{i,j}$$ is the force of mortality at age $$i$$ in year $$j$$. These are the Lee-Carter model (LC), the Lee-Carter model with an added cohort effect (LCC) (a special case of the Renshaw-Haberman model) and the Age-Period-Cohort…

### Forecasting with cohorts for a mature closed portfolio

#### (Mar 27, 2014)

At a previous seminar I discussed forecasting with the age-period-cohort (APC) model:

$$\log \mu_{i,j} = \alpha_i + \kappa_j + \gamma_{j-i}$$

where $$\mu_{i,j}$$ is the force of mortality at age $$i$$ in year $$j$$; the parameters $$\alpha_i$$ , $$\kappa_j$$ and $$\gamma_{j-i}$$ are usually thought of as the age, period and cohort parameters respectively. These parameters are not identifiable from the formula above, so we need a set of constraints to fix unique estimates.  Suppose we number the cohorts from 1 (oldest) to $$n_c$$ (youngest).  One set of constraints is as follows:

$$\sum \kappa_j = 0,\; \sum \gamma_c = 0,\; \sum c \gamma_c = 0.$$

Forecasting the mortality table now depends on forecasting…

### The best available approximation to the truth

#### (Mar 9, 2013)

In my role as guest editor of the British Actuarial Journal, I wrote an editorial piece about how actuaries can assess the suitability (or otherwise) of models for projecting mortality rates.  The editorial can be downloaded on the right; it covers various aspects of mortality-projection models including:

Tags: mortality projections, BAJ

### Benchmarking VaR for longevity trend risk

#### (Mar 1, 2013)

I recently wrote about an objective approach to setting the value-at-risk capital for longevity trend risk.  This approach is documented in Richards, Currie & Ritchie (2012), which was recently presented to a meeting of actuaries in Edinburgh.  One of the topics which came up during the discussion was how the answers from the value-at-risk (VaR) method squared with how life offices actually change their projection bases in practice.  In particular, commentators were interested in what might be regarded as a "real world" example of a sudden change in projection basis, and how this might compare with the results from the VaR framework.

As it happens, we do have an historical example to call upon, and furthermore…

### Hitting the target, but missing the point

#### (Jan 31, 2013)

Targeting methods are popular in some areas for mortality forecasting. One well known current example is the CMI's model for forecasting mortality. The CMI model allows the user to fix the long-term mortality-improvement rate; Stephen has written previously about some of the undesirable consequences of this, while Gavin has written about dealing with its lack of an uncertainty measure. However, other consequences of a targeting assumption are less well known, so we will explore them here. We present a simple example which illustrates the impact that an assumption about the future - which after all is what a target is - can have on a forecast.

To illustrate, we use male mortality data, i.e. death counts and…

### Canonical correlation

#### (Jul 28, 2012)

At our seminar earlier this year I looked at the validity of assumptions underpinning some stochastic projection models for mortality.  I looked at the assumption of parameter independence in forecasting, and examined whether this assumption was borne out by the data.  It transpires that the assumption of independence is a workable assumption for some models, but not for others.  This has important consequences in a Solvency II context - an internal model must be shown to have assumptions grounded in fact.

To illustrate, we contrast two models which share a number of features. First, the Lee-Carter model:

and second the age-period-cohort (APC) model:

Forecasting in these models depends on an assumption…