Stephen Richards
Articles written by Stephen Richards
Seasonal mortality and age
The Hermite model of mortality
In Richards (2012) I compared seventeen different parametric models for modelling the mortality of a portfolio of UK annuitants. The best-fitting model, i.e. the one with the lowest AIC, was the Makeham-Beard model:
\[\mu_x = \frac{e^\epsilon+e^{\alpha+\beta x}}{1+e^{\alpha+\rho+\beta x}}\qquad(1)\]
Mortality down under
Is your mortality model frail enough?
Mortality at post-retirement ages has three apparent stages:
A broadly Gompertzian pattern up to age 90 (say), i.e. the mortality hazard is essentially linear on a logarithmic scale.
The rate of increase in mortality slows down, the so-called "late-life mortality deceleration".
Compare and contrast: VaR v. CTE
New year, new insights
Happy New Year to all our readers!
Hedging or betting?
Last week I presented at Longevity 14 in Amsterdam. A recurring topic at this conference series is index-based approaches to managing longevity risk. Indeed, this topic crops up so reliably, one could call it a hardy perennial.
'D' is for deficiency
Valuing liabilities with survival models
Regular readers of this blog will know that we are strong advocates of the benefits of modelling mortality in continuous time via survival models. What is less widely appreciated is that a great many financial liabilities can be valued with just two curves, each entirely determined by the force of mortality, \(\mu_{x+t}\), and a discount function, \(v^t\).
Testing the tests
Examining residuals is a key aspect of testing a model's fit. In two previous blogs I first introduced two competing definitions of a residual for a grouped count, while later I showed how deviance residuals were superior to the older-style Pearson residuals. If a model is correct, then the deviance residuals by age should look like random N(0,1) variables.