Information Matrix

Pricing block transactions is a high-stakes business.  An insurer writing a bulk annuity has one chance to assess the price to charge for taking on pension liabilities.  There is a lot to consider, but at least there is data to work with: for the economic assumptions like interest rates and inflation, the insurer has market prices.  For the mortality basis, the insurer usually gets several years of mortality-experience data from the pensi

One interesting aspect of maximum-likelihood estimation is the common behaviour of estimators, regardless of the nature of the data and model.  Recall that the maximum-likelihood estimate, \(\hat\theta\), is the value of a parameter \(\theta\) that maximises the likelihood function, \(L(\theta)\), or the log-likelihood function, \(\ell(\theta)=\log L(\theta)\).  By way of example, consider the following three single-parameter distributions:

In a previous posting I looked at how using a well founded statistical model can improve the accuracy of estimated mortality rates. We saw how the relative uncertainty for the estimate of \(\log \mu_{75.5}\) could be reduced from 20.5% to 3.9% by using a simple two-parameter Gompertz model:

\(\log \mu_x = \alpha + \beta x\qquad (1)\)