**As actuaries we often have to derive mortality bases from portfolio experience data, but how certain are we of the results? Stephen Richards presents a method of quantifying this risk**

Actuaries commonly derive mortality bases from portfolio experience data for valuation and risk quantification purposes. However, this is also required for pricing block transfers, such as longevity swaps, reinsurance treaties and bulk annuities.

Actuaries commonly derive mortality bases from portfolio experience data for valuation and risk quantification purposes. However, this is also required for pricing block transfers, such as longevity swaps, reinsurance treaties and bulk annuities.

In each case it is useful to know two things:

(i) what uncertainty surrounds the mortality basis, and (ii) what financial impact this uncertainty has. Both of these questions come under the heading of mis-estimation risk.

As an example, a UK pension scheme is considering a longevity swap. The scheme and insurer have agreed a basis for future mortality improvements, but both parties have to decide on a basis for current mortality rates. Furthermore, both parties want to understand the mis-estimation risk surrounding the basis, and thus the potential financial impact. The scheme has *n *= 14,802 living pensioners and also has 2,265 records for past deaths observed over the period 2007-2012.

The two parties have slightly different rationales in wanting to understand the mis-estimation risk. The scheme wants to know the financial impact to judge if it is worth paying the insurer's premium to remove the risk. In contrast, the insurer wants to know if its pricing margin covers the risk of mis-estimation based on the scheme's recent experience. In particular, the insurer (or reinsurer) will have to hold regulatory capital for mis-estimation risk if the longevity swap is agreed.

**Modelling current mortality**

There are many ways to analyse mortality, but one of the better approaches is to use survival models for individual lives. This involves a parametric

model for the force of mortality, which makes the best use of all available information. The model fitted here is the time-varying version of the Makeham-Perks law:

where

where, for example, male is the change in mortality from being male and male is an indicator variable taking the value 1 when life *i* is male and 0 otherwise. The other parameters and indicator variables are defined similarly. The model is fitted to the scheme's data and the resulting parameter estimates are shown in Table 1.

**Correlations and concentration of risk**

The parameter estimates in Table 1 are shown with their standard errors. In a sense these standard errors are the beginning of understanding mis-estimation, as they tell us the degree of confidence we can have in each parameter estimate. For example, the estimate of the age parameter is 0.148 and an approximate 95% confidence interval for the true underlying value is (0.138, 0.158). At a superficial level, therefore, one might think that the standard errors are all we would need to assess mis-estimation. However, with all statistical models there are usually correlations between the parameters. Some of these correlations can be quite material, as shown in Table 2, and they must be taken into account when assessing mis-estimation risk.

The other aspect of mis-estimation risk is that it doesn't affect all lives equally, and that not all lives are of equal financial impact. For example, the large-pension cases account for the top 10% of lives, but they account for 39.8% of the total scheme pension. Table 1 shows that such cases have markedly lower mortality, but the standard error shows that there is relatively greater uncertainty over just how much lower. Furthermore, Table 2 shows that there is a correlation of -19% between the parameters for large pension cases and males, so it is not sufficient to stress any one parameter in isolation.

**Quantifying the risk**

If parameters are correlated to varying degrees, how can we perform a mis-estimation assessment? We cannot simply stress each parameter by a multiple of its standard error, as this ignores correlations. This is illustrated in Figure 1 (below) for a simple Gompertz model with

Our solution is to use the whole variance-covariance matrix to generate consistent alternative parameter groups. This not only allows for the uncertainty over the parameters themselves, but it also allows for their correlations. There is also the question of how to allow for the fact that individual liabilities are affected to different extents. Our solution is to value the entire portfolio life-by-life with each alternative parameter set. We repeat this m times to generate a set, S, of alternative portfolio valuations. S describes the financial impact of parameter risk and parameter correlations, while allowing for all individual characteristics and concentrations of liability. The percentiles of S can be used to investigate the financial effect of mis-estimation risk, say by comparing the excess of a given percentile to the median.

**Results**

For the pension scheme in question, we generated m =10,000 sets of alternative parameter values with the covariance matrix. In each case we valued the in-force liabilities with each parameter set. The 99.5th percentile of S was 3.97% higher than the median (the median of S was very close to the mean). This compares loosely to a typical insurer pricing margin of around 4%-5%.

It is also possible to express mis-estimation results as a percentage of a standard table using the equivalent-annuity calculation. For this portfolio the equivalent best-estimate percentages of S2PA were 88.5% for males and 87.2% for females. Using the appropriate percentiles of S we can use the mis-estimation assessment to find a 95% confidence interval for these percentages. For males we get (78.7%, 99.5%) and for females we have (79.3%, 96.1%). The width of these intervals reflects the modest size of the scheme and the concentration of risk in a relatively small subset of lives. A larger portfolio would probably have a narrower confidence interval.

There are many potential risk factors that affect a demographic risk like mortality and the effect of these risk factors can be estimated using a parametric statistical model. The parameters in such a model have both uncertainty around their estimates and correlations with each other. Using the variance-covariance matrix for the estimated parameters, the mis-estimation risk for a portfolio can be straightforwardly assessed using the portfolio's own experience data.