Information Matrix

In earlier blogs I discussed two techniques for handling outliers in mortality forecasting models:

• Chen & Liu (1993) for ARIMA models for univariate time series, and

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

I recently came across an online article by W. S. Warren, the deputy editor of Science Advances.  In the article Warren outlines some easy ways for submitting authors to improve their paper's chances of being accepted for journal publication.

In the course of a recent investigation, with my colleagues Dr Oytun Haçarız and Professor Torsten Kleinow, a key parameter was the mortality rate of persons suffering from Hypertrophic Cardiomyopathy (HCM), an inherited heart disorder characterized by thickening of the left ventricular muscle wall.  It is quite rare, so precision is not to be expected, and indeed an annual mortality rate of 1% \((q_x=0.01)\), independent of age \(x\), is widely cited.  I

Interest rates and gilt yields are critical drivers of pension-scheme reserving and bulk-annuity pricing.  However, many UK pension schemes self-insure when it comes to economic risks, with Liability Driven Investment (LDI) a common approach.  This makes the turmoil in the UK Gilts market in Autumn 2022 of particular interest.  Daily movements of 10-20 standard deviations arose as the

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 mortality forecasting work we often deal with downward trends.  It is often tempting to jump to the assumption of a linear trend, in part because this makes for easier mathematics.  However, real-world phenomena are rarely purely linear, and the late Iain Currie advocated linear adjustment as means of judging linear-seeming patterns.  This involves calculating a line between the first and last points, and deducting the line value at ea

In my previous blog I showed how univariate stochastic mortality models, like the Lee-Carter and APC models, can be robustified to cope with data affected by the covid-19 pandemic.  Such robustification is necessary because outliers, such as the 2020 experience, bias parameter estimates and affect value-at-risk (VaR) capital requirements.  Kleinow & Richards (2016) showed how one-year VaR-style capital requirements are heavily de