Information Matrix
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Posts feedHedging 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
Smooth Models Meet Lumpy Data
Most of the survival models used by actuaries are smooth or piecewise smooth; think of a Gompertz model for the hazard rate, or constant hazard rates at individual ages. When we need a cumulative quantity, we use an integral, as in the cumulative hazard function, \(\Lambda_x(t)\):
\[ \Lambda_x(t) = \int_0^t \mu_{x+s} \, ds. \qquad (1) \]
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, \(v^t\).
More than one kind of information
This collection of blogs is called Information Matrix, and it is named after an important quantity in statistics. If we are fitting a parametric model of the hazard rate, with log-likelihood:
\[ \ell( \alpha_1, \ldots, \alpha_n ) \]
as a function of \(n\) parameters \(\alpha_1, \ldots, \alpha_n\), then the information matrix is the matrix of second-order partial derivatives of \(\ell\). That is, the matrix \({\cal I}\) with \(ij\)th component:
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.
Socio-economic differentials: convergence and divergence
Getting animated about longevity
Less is More: when weakness is a strength
A mathematical model that obtains extensive and useful results from the fewest and weakest assumptions possible is a compelling example of the art. A survival model is a case in point. The only material assumption we make is the existence of a hazard rate, \(\mu_{x+t}\), a function of age \(x+t\) such that the probability of death in a short time \(dt\) after age \(x+t\), denoted by \({}_{dt}q_{x+t}\), is:
\[{}_{dt}q_{x+t} = \mu_{x+t}dt + o(dt)\qquad (1)\]