### Signal or noise?

#### (Nov 25, 2016)

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):

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

which the CMI calls the APCI model.  $$m_x$$ is the central rate of mortality at age $$x$$ in year $$y$$ and $$\alpha_x$$, $$\beta_x$$, $$\kappa_y$$ and $$\gamma_{y-x}$$ are vectors of parameters to be estimated.  $$\bar y$$ is the average year, which is used to centre the time index around…

Tags: CMI, APCI, APC, Lee-Carter, Age-Period, smoothing

### Excel's Limits

#### (Jun 27, 2014)

We have written in the past about some of the reasons why we don't use Excel to fit our models.  However, we do use Excel for validation purposes - fitting models using two entirely separate tools is a good way of checking production code.  That said, there are some important limits to Excel, especially when it comes to fitting projection models.  Some of these limits are rather subtle, so it is important that an analyst is aware of all of Excel's limitations.

The first issue is that Excel's standard Solver feature won't work with more than 200 variables, i.e. parameters which have to be optimised in order to fit the model.  This is a problem for a number of important stochastic projection models, as shown in Table 1. …

Tags: Excel, Lee-Carter, APC, CBD

### 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…