### The cohort effects that never were

#### (May 20, 2019)

The analysis of cohort effects has long fascinated the actuarial community; these effects correspond to the observation that specific generations can have longevity characteristics different from those of the previous and the following ones. However, Richards (2008) conjectured that these cohort effects might be errors caused by sudden changes in fertility patterns.  Figure 1 shows the specific example of France, although the phenomenon is universal. The most significant fluctuations can be seen when birth rates fall dramatically during periods of war, such as World War I, and then spike afterwards.

Figure 1. Monthly births in France.  Source: Human Fertility Database.

To understand the impact of…

### 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…

### Don't cut corners

#### (Oct 7, 2014)

An important class of mortality-projection models is the Cairns-Blake-Dowd (CBD) family. These models are described in a landmark paper by Cairns et al (2009).  Three of the most important of the CBD models are M5, M6 and M7, as defined below for age $$x$$ and calendar year $$y$$:

 M5 $$\log \mu_{x,y} = \kappa_{0,y} + \kappa_{1,y}S(x)$$ M6 $$\log \mu_{x,y} = \kappa_{0,y} + \kappa_{1,y}S(x) + \gamma_{y-x}$$ M7 $$\log \mu_{x,y} = \kappa_{0,y} + \kappa_{1,y}S(x) + \gamma_{y-x} + \kappa_{2,y}Q(x)$$

where:

\eqalign{S(x) &= \left(x - \bar x\right)\\ Q(x) &= \left(x - \bar x\right)^2-\hat\sigma^2\\ \hat\sigma^2 &= \displaystyle\frac{1}{n_x}\sum_{i=1}^{n_x} (x_i-\bar x)^2}

The original…

### 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…

### 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…

### Order, order!

#### (Apr 6, 2011)

Mortality improvements can be analysed in a number of ways.  A common desire is to want to separate mortality improvements into components for period and cohort.  However, this is much trickier than it seems, as we shall show here.  In particular, the order in which calculations are performed can be very important.

Figure 1 shows the mortality improvements derived from a two-dimensional smoothed model, identical to the ones presented by Richards, Kirkby and Currie (2006).  The vast majority - 97.3% - of the improvements are positive, and there is a pronounced diagonal pattern reflecting the tendency for mortality improvements to track year of birth, or cohort.  Indeed, the diagonal cohort patterns are…

### A rose by any other name

#### (Oct 25, 2010)

How important are the labels we give to things?  In a seminal paper Richard Willets brought a particular mortality phenomenon to the attention of the UK actuarial profession:

"The 'cohort effect' has been a wave of rapid improvements, rippling upwards through mortality rates in the UK.  For the past four decades, people born between 1925 and 1945 have benefited from faster mortality improvements than those born in adjacent generations."

Willets, R. C. (1999) Mortality in the next millennium,
Staple Inn Actuarial Society, London.

The cohort effect for UK mortality has been a major issue for pension liabilities.  It has reduced mortality rates and thus pushed up the life expectancy for people…