### Mortalityrating and GDPR

#### (Apr 18, 2018)

Previously our mortalityrating.com service processed a simple file format that included postcode, gender and date of birth alongside pension amount and commencement date for individuals in an occupational pension scheme. This combination of attributes when taken together is often capable of identifying "natural persons" in the language of the upcoming EU General Data Protection Regulation (GDPR). Some might choose to mitigate risk by deleting scheme data as soon as ratings complete. However, an alternative approach would be to perform ratings without requiring a combination of attributes that may be personally identifiable. How could such a thing be acheved?

An important observation is that a postcode…

### Stopping the clock on the Poisson process

#### (Apr 12, 2018)

"The true nature of the Poisson distribution will become apparent only in connection with the theory of stochastic processes$$\ldots$$"

Feller (1950)

In a previous blog, we showed how survival data lead inexorably toward a Poisson-like likelihood. This explains the common assumption that if we observe $$D_x$$ deaths among $$n$$ individuals, given $$E_x^c$$ person-years exposed-to-risk, and we assume a constant hazard rate $$\mu$$, then $$D_x$$ is a Poisson random variable with parameter $$E_x^c\mu$$. But then $$\Pr[D_x>n]>0$$. That is, an impossible event has non-zero probability, even if it is negligibly small. What is going on?

Physicists are ever alert to the tiniest difference between…

### Thymus of the essence?

#### (Apr 6, 2018)

We've considered cancer and its relationship to aging on a number of previous occasions. Studies published in the British Journal of Cancer in 2011 and 2018 concluded that around 40% of cases are attributable to known modifiable lifestyle and environmental factors, which is a substantial minority. Whilst risk for specific cancer subtypes will be more or less amenable to lifestyle and environment interventions than this, it is beyond doubt that the longer we live, the higher our risk of cancer becomes. But why?

A 2014 analysis of analysis of age and cancer risk proposed the view that "For most adults, age is coincidentally associated with preventable chronic conditions, avoidable exposures, and modifiable…

### Lump sum or annuity?

#### (Mar 28, 2018)

People are often faced with a decision whether to live off their savings or buy an annuity.  Normally such decisions are made around the retirement ages of 60-65.  However, an interesting counter-example has just been provided by eighteen-year-old Charlie Lagarde, the winner of a lottery in Canada.  She had to decide between taking a C$1million lump sum or an annuity of C$1,000 each week for life.  She opted for the latter, forswearing an opportunity to become an instant millionaire.  But was it the right decision?

We obtained the most recent mortality tables for the population of Quebec and performed some simple calculations of the discounted present value of C\$1,000 per week.  The results are shown in Table…

Tags: annuity, life expectancy

### The Curse of Cause of Death Models

#### (Mar 22, 2018)

Stephen's earlier blog explained the origin of the very useful result relating the life-table survival probability $${}_tp_x$$ and the hazard rate $$\mu_{x+t}$$, namely:

${}_tp_x = \exp \left( - \int_0^t \mu_{x+s} \, ds \right). \qquad (1)$

To complete the picture, we add the assumption that the future lifetime of a person now aged $$x$$ is a random variable, denoted by $$T_x$$, and the connection with expression (1) which is:

${}_tp_x = \Pr[ T_x > t ]. \qquad (2)$

The 'package' of random lifetime $$T_x$$, hazard rate $$\mu_{x+t}$$, survival function $${}_tp_x$$, and expression (1) tying everything together, sums up the mathematics of a survival model.  For the statistics, we have observations…

### Analysis of VaR-iance

#### (Mar 13, 2018)

In recent years we have published a number of papers on stochastic mortality models.  A particular focus has been on the application of such models to longevity trend risk in a one-year, value-at-risk (VaR) framework for Solvency II.  However, while a small group of models has been common to each paper, there have been changes in the calculation basis, most obviously where updated data have been used.  Sometimes these changes stemmed from more data being available, but, as Richard Willets covered in his blog, the ONS also restated the population estimates following the 2011 census.  This makes it tricky to compare results between papers. We therefore thought it would be instructive to do a step-by-step analysis…

### Constraints: a lot of fuss about nothing?

#### (Mar 5, 2018)

Our paper, "A stochastic implementation of the APCI model for mortality projections", was presented at the Institute and Faculty of Actuaries in October 2017. There was quite a discussion of the role of constraints in the fitting and forecasting of models of mortality. This got me wondering if constraints weren't in fact a red herring. This blog is a short introduction to the results of my investigation into the role, or indeed the non-role, of constraints in modelling and forecasting mortality.

We illustrate our argument with data on UK males from ages 40-104 and for years 1961-2015 from the Office for National Statistics (ONS). We have the number of deaths, $$d_{x,y}$$, age $$x$$ last birthday in year $$y$$…

### Introducing the Product Integral

#### (Feb 26, 2018)

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

${}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)$

Stephen covered the derivation of this in a previous blog, but I want to look more closely at the right-hand side of equation (1).  In particular, we can find an entirely different representation of $${}_tp_x$$ as a product integral, which leads to many insights in survival models.

Recall how the integral in equation (1) is constructed.  Choose a partition of the interval $$[0,t]$$, that is some sequence $$\Delta_1,\Delta_2,\ldots,\Delta_n$$ of non-overlapping sub-intervals that exactly cover the interval.  Define…

### Fathoming the changes to the Lee-Carter model

#### (Feb 19, 2018)

Ancient Greek philosophers had a paradox called "The Ship of Theseus"; if pieces of a ship are replaced over time as they wear out until every one of the original components is gone, is it still the same ship?  At this point you could be forgiven for thinking (a) that this couldn't possibly be further removed from mortality modelling, and (b) that I had consumed something a lot more potent than tea at breakfast.  However, this philosophical parable is relevant to the granddaddy of all stochastic projection models: the one proposed by Lee & Carter (1992).

In their original paper Lee & Carter (1992) proposed the following model:

$\log m_{x,y} = \alpha_x+\beta_x\kappa_y+\epsilon_y$

where $$m_{x,y}$$…

Tags: Lee-Carter, P-splines, ARIMA

### Solid Progress

#### (Feb 8, 2018)

When we previously discussed the progress of immunotherapy within cancer treatment, some of the most exciting results were in the field of leukaemia and melanoma, with progress in other solid cancers lagging somewhat behind. In the UK, solid tumour-forming cancers account for the overwhelming majority of cancer mortality for both females and males, so similar progress in these areas could have a transformative impact on the prognosis for sufferers.

The CAR-T approaches so successful with blood cancers are undergoing active trials in the UK and have already been approved as a treatment by the US FDA. Such techniques extract, modify and reintroduce the patient's immune cells after engineering the ability…