### Modelling improvements in experience data - II

#### (Jul 28, 2020)

In my previous blog I looked at the implied mortality improvements from time-varying traditional actuarial survival models.  In this blog we consider the implied improvements under the newer Hermite-spline model I proposed in Richards (2019).  This paper included an explicit attempt to model age-related mortality changes, as discussed in this blog.

However, the Hermite-spline family also offers an alternative approach to time-varying mortality, one that is analogous to the approach under traditional actuarial models.  We start by recapping the non-time-varying Hermite-spline model as follows for $$x\in [x_0, x_1]$$ and $$t=(x-x_0)/(x_1-x_0)$$:

$\log\mu_x = \alpha h_{00}(t) + m_0 h_{10}(t)… ### Modelling improvements in experience data - I #### (Jul 28, 2020) In the first of a pair of blogs we will look at how to allow for changes in mortality levels when calibrating models to experience analysis. We start with time-varying extensions of traditional parametric models proposed by actuaries, beginning of course with the Gompertz (1825) model: \[{\rm Gompertz}: \mu_{x,y} = e^{\alpha+\beta x + \delta(y-2000)}\qquad (1)$

where $$x$$ denotes exact age and $$y$$ denotes calendar time.  $$\alpha$$ is the intercept, i.e. (loosely) log(mortality) at age zero, while $$\beta$$ is the rate of increase in log(mortality) by age.  $$\delta$$ is the rate of change of mortality in time (more on this later), while $$-2000$$ is an offset to keep the parameters well-scaled. …

### Piquing interest in improvements

#### (Jul 22, 2020)

When underwriting a pension scheme for a bulk annuity or longevity swap, the first concern is understanding what mortality levels are, especially differentials amongst sub-groups.  The next concern is whether the recent mortality improvements in the pension scheme are in line with the pricing basis; if the scheme has experienced faster improvements, say, then this would be a valuable insight for pricing.

How do we check this when improvement bases are usually calibrated with population data, where the number of lives at a single age typically exceeds the number of lives in the entire pension scheme?  How can you reliably detect improvement rates in a portfolio's own experience data?  We can do this to a simple…

### Best practice in mortality work - regulatory comments

#### (Mar 10, 2020)

In a letter to the Chief Actuaries of UK insurance businesses, Malik (2019) highlighted two aspects of what the regulator regards as good practice in mortality work:

1. Considering trends in mortality in conjunction with seasonal variation, especially excess winter mortality, and
2. Considering how far into the future it is reasonable to project a cohort effect.

Seasonal variation in mortality occurs in all countries and affects insured portfolios as much as the wider population.  Seasonal variation also has a greater impact as age increases.  The point that Malik (2019) is making is that any estimate of recent trends will be impacted by whether the start and end points of the analysis were years of weaker or heavier…

### The cohort effects that never were

#### (May 20, 2019)

The analysis of cohort effects has long fascinated the actuarial community; these effects correspond to the observation that specific generations can have longevity characteristics different from those of the previous and the following ones. However, Richards (2008) conjectured that these cohort effects might be errors caused by sudden changes in fertility patterns.  Figure 1 shows the specific example of France, although the phenomenon is universal. The most significant fluctuations can be seen when birth rates fall dramatically during periods of war, such as World War I, and then spike afterwards.

Figure 1. Monthly births in France.  Source: Human Fertility Database.

To understand the impact of…

### Socio-economic differentials: convergence and divergence

#### (Jun 18, 2018)

Many western countries, including the UK, have recently experienced a slowdown in mortality improvements.  This might lead to the conclusion that the age of increasing life expectancies is over.  But is that the case for everyone?  Or are there some groups in the UK who are still experiencing mortality improvements?  The short answer is that mortality rates are still falling for the least deprived half of the population in England, while mortality improvements since 2011 have been virtually zero for the most deprived third.  This has important consequences for reserving for pensions and annuities, so let us explore in a bit more detail.  The findings in this blog are based on some early results of research…

### Fifty years of mortality improvements

#### (Jun 2, 2017)

In an earlier post we looked at the development of the distribution of age at death over time.  We saw how the peak adult age at death had continuously moved towards an ever-higher age.

Actuaries, of course, are very interested in the development of mortality rates over time.  Animation 1 shows the development of the crude observed force of mortality for males of retirement age in the UK since 1961. It shows that mortality rates have fallen at all ages, but much more so at ages below 80 than above.

Animation 1. Male mortality rates by age in England and Wales since 1961 (log scale). Click on the chart to restart the animation. Note that the mortality rates were calculated using post-2001 population estimates that had…

### Pensioners — the youth of today

#### (Sep 9, 2016)

This blog focuses on two particular features of mortality improvements: improvements around retirement age and improvements for the (very) old.  Figure 1 shows $$\mu_{x,1961}/\mu_{x,2012}$$, the ratios of the forces of mortality over the ages $$x = 50,\ldots,95$$, in 1961 and 2012 for four countries.  The figure was obtained as follows: data for ages 50-95 and years 1961-2012 were downloaded from the Human Mortality Database and the raw 2D mortality table was smoothed using 2D P-splines (see my earlier blog for details on 2D smoothing with P-splines).  The ratios plotted in Figure 1 were obtained from the resulting smooth mortality surface.

Figure 1.  Mortality ratios, $$\mu_{x, 1961}/\mu_{x, 2012}$$,…

### Reverse Gear

#### (Dec 8, 2015)

Against a background of long-term mortality improvements it is understandable to expect that societal change and developments in health care will be agents of progress. Recent research from Princeton Professor of Economics Anne Case and Nobel prize-winning economist Angus Deaton jolts such complacency in the starkest way. It reveals that since the late nineties, the all-cause mortality improvements experienced by white non-Hispanic Americans in midlife (ages 45 to 54) have not simply slowed, but slammed into reverse.

It quickly becomes clear that this research contains a telling illustration of basis risk. The researchers note that the scale of the effect had been missed since it played out so strongly…

### Reviewing forecasts

#### (Oct 19, 2015)

When making projections and forecasts, it can be instructive to compare them with what actually happened. In December 2002 the CMI published projections of mortality improvements that incorporated the so-called "cohort effect" (CMIB, 2002). These projections were in use by life offices and pension schemes in the United Kingdom from 2003 onwards. Since population mortality rates are now available for the ten-year period since then, we can compare actual improvements to the projections from 2003.  This is done in Figure 1.

Figure 1: Population mortality improvements 2003-2013 and corresponding CMI cohort projection bases. Source: Own calculations using national life tables from the ONS and the cohort…