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Population Frailty Models in the context of COVID-19

As actuaries, we are all aware of how important selection effects are: from the well-known fact that insured individuals have lower mortality rates than the general population, to adverse selection leading to higher claim rates in companies with lax underwriting practices. Unfortunately, quantifying these selection effects is rarely straightforward, since they are normally the consequence of a complex combination of many underlying factors. However, sometimes it is possible to use a simple model to estimate the effect of a single underlying component.

For life and pensions actuaries, how to account for the long-term effects of Covid on population mortality will probably be a key topic of discussion in years to come, with some early articles already being published. For example, Issue 13 of the IFoA Longevity Bulletin contains several interesting articles analysing the potential long-term effects of Covid, from the direct effects of long Covid on survivors to the consequences of the economic recessions caused by widespread lockdowns. Working Paper 139 by the CMI Covid-19 Working Party, published last October, dealt with how assumptions about future mortality or morbidity may be affected by the pandemic.

One possible effect of Covid comes from the selection caused by deaths disproportionately affecting those with other underlying conditions. That is, it is possible that the post-Covid elderly population will be, on average, healthier than it would have been otherwise. It was thinking about these issues that reminded me of the so-called frailty models: a type of mortality model where we allow for different mortality curves for each individual within a population. Although conceptually simple, these models can be very powerful indeed when analysing this kind of event. This article does not delve into mathematical detail, but further information can be found here. However, a run through of the core concepts is worthwhile to better illustrate the use of these models.

The model

The basic idea behind the model is that the force of mortality at age x, m(x), follows a very simple law for each individual. For example, the well-known Gompertz law says that:



This law can be summarised saying that the force of mortality grows exponentially with age.

This model can be extended to account for different levels of mortality for different individuals. For example, we can introduce an extra parameter z, so that:

In[μ(x,z)]=Inz+a+b∗x → μ(x,z)=z∗μ(x)

The force of mortality for any specific individual is proportional to the mortality of a “baseline” individual with z = 1.

The wider population will consist of many individuals, each of them with their own (random) frailty, measured by z. We can assume that these random values of z follow a certain probability distribution, and that would allow us to find the population-level force of mortality from the individual level m(x, z).

A key feature of this model is that, for a cohort of people of the same age, the population level mortality will change with the average value of z in the population. At young ages, when few people in the cohort have died, the mean of z will remain roughly constant and mortality for the population will grow exponentially. However, as the cohort ages and only the healthiest individuals survive (those with a low value of z), the average value of z will start decreasing. This dampens the exponential growth of mortality with age, resulting in a slowdown at very high ages (which we observe in many developed countries with large elderly populations).

Our scenarios

Of course, there are many aspects to Covid other than mortality selection, but I will not delve into them here. Instead, I will use a simple example to illustrate the type of issue that can be explored using frailty models. What I will do is show how a frailty model can help produce estimates of future mortality after an extreme event introduces some type of selection in the surviving population (say, an epidemic that mostly affects people with a specific profile).

The model will be based on a portfolio of 10,000 annuitants, all of them of the same exact age. I will assume that some extreme event causes, say, 5% of them to die before the first annuity payment is due. I will then project future survival for the remaining 95% under three scenarios: the dead are chosen at random (scenario 1), they are the 5% of annuitants with the highest frailty (scenario 2), and they are the strongest 5% (scenario 3). Assuming the annuity pays £1 in advance every month we can calculate the cashflow out under each scenario and therefore estimate the effect of the selection on the liabilities of this portfolio.

The first step of this analysis is using the constant force of mortality approximation to calculate probabilities of the likes of 1/12qx(z) for each individual from our formula m(x, z) = zexp(a + bx). We assume that the z’s for our annuitants follow a gamma distribution with shape parameter k = 5.7 and scale q = 1/5.7. We set the two parameters in the Gompertz law to a = -10 and b = 0.1. These are in line with those used in the article linked above, where a model like this was fitted to 1980 England and Wales mortality for males of ages between 40 and 90.

With this choice of parameters, the mortality of an individual of average frailty (z = 1) will be 0.018 at age 60, and their maximum attainable age 100. Of course, actual mortality and maximum lifespan will be frailty-dependent, with stronger individuals able to survive well beyond that age.

We use R to simulate values of z for this population. Once we have the mortality for each individual, we assume 500 of our 10,000 annuitants (i.e. the 5% referred to above) die before the first annuity payment. To recap, the three scenarios we are going to analyse are:

Lastly, and because it is widely accepted that adding a stochastic component is guaranteed to improve any model at any time, we use our q-rates to simulate possible future scenarios for the (random) number of individuals who will survive to a certain month in the future in each of the three scenarios. This is so that we can plot a range of possible outcomes for each of them, instead of simply a central prediction, and get a better idea of how different the outcomes for each scenario can be.

Modelling results

Figure 1: Expected Payout over projection period

Figure 1 shows the expected pay out for each month during the next 40 years. For each month, we calculate 100 possible values of the cashflow out in each scenario, and for each of them plot both the mean and 10% and 90% percentiles of the expected outflow. We see that the behaviour is exactly what we would have expected: Scenario 2, where the frailest in the population died before the annuity payments started, results in higher payments being made than the other two cases. When the strongest in the population die, scenario 3, we get the lowest level of payments. Scenario 1 is in the middle of the other two.

A very interesting feature is that the fluctuations due to the random number of actual deaths observed that we tried to estimate by calculating the cashflows stochastically is not enough to mask the effects of the selection introduced by the extreme mortality event: that is, we see clear differences between the three scenarios, with only a slight overlap between 1 and 2 and 2 and 3, and a reasonably large gap between scenarios 1 and 3. For a portfolio of this size, therefore, the selection effect we have introduced is sufficiently powerful to create a material difference in the value of future liabilities.

Figure 2: Scenario Comparison

Figure 2 illustrates how these differences change throughout the life of the portfolio. Here we plot the difference in expected cash outflow between each pair of scenarios. We see that there is a difference not only in the value of the outflows, but also the timing at which the change in expected outflows is the largest.

For example, when the frailest policyholders die before the annuity starts (red and black lines, showing the difference in between the scenarios where the frailest policyholders die compared to the scenario with random deaths and the scenario where strongest policyholders die respectively), we see cashflows that are most different early on (since we do not have frail survivors that will die early into their annuity), but the difference wears off relatively fast (since the long term survivors will always be the strongest annuitants).

However, the difference in outflows when the strongest died in the extreme event (blue line, showing the difference between the scenario where the strongest policyholders die and the scenario with random deaths) lasts longer in time. This is because most annuitants who would have survived to a very high age died at time zero, leading to a difference that persists up to the highest ages in our model.

The magnitude of the differences reaches a peak of between approximately 100 and 300 pounds depending on the pair of scenarios chosen. If we compare this with the roughly 3,500 expected outflow we have around year 20, we see that frailty alone can change the value of the expected cashflows by around 2.5% to 8.5%. This is a very noticeable difference, and one that the annuity provider should only ignore at their own risk!


The conclusions we get from these results are not unexpected:

  1. Selection effects at time zero can have a fairly large impact on the value of liabilities later in the life of the portfolio.
  2. Even a very conceptually simple model can give us a lot of insight into possible future scenarios when used appropriately.
  3. Setting assumptions about future mortality is always a difficult task, but especially so when an extreme event happens that changes the profile of the population under consideration (and may therefore affect their long-term mortality).

I hope these three points have been clearly illustrated throughout the article, and that readers now feel more familiar with oft-overlooked frailty models.

Cristian Redondo Loures

July 2021