## A model point

The current issue of *The Actuary* magazine carries an article on the selection of model points. Model points were widely used by actuaries in the 1980s and 1990s, when computing power was insufficient to perform complex policy calculations on every policy in a reasonable time-frame. The idea is to select a much smaller number of sample policies, whose behaviour in aggregate mimics that of the portfolio overall.

There are several ways of selecting policies as model points, or even creating them from scratch: sometimes actuaries would create model points which had no counterpart in the portfolio. However, computing power has come a long way since the times when model points were necessary. By way of illustration, in a paper last year Iain Currie and I simulated a portfolio of over 200,000 lives in run-off 10,000 times in an hour. The hardware involved is neither complicated nor difficult to source. Alternatively a portfolio of 2 million lives could have been simulated in run-off 1,000 times. Simulation of individual lifetimes using survival models is particularly fast.

So model points are no longer as necessary as they once were. Time and effort can now be saved with the elimination of the model-point construction process. This brings other benefits, too, including the removal of risk that the model points do not adequately represent the portfolio, and not having to convince auditors and regulators that your model points do what they are supposed to.

But what advantages does whole-portfolio simulation bring? In our 2009 paper, Iain and I explored how the number of lives and the concentration of their benefits impacted on the overall portfolio risk. We were able to quantify how a larger portfolio could justify smaller valuation margins than a smaller one. Table 1 below shows an illustration for two annuity portfolios, where we assume that interest rates and mortality rates are precisely known and the only uncertainty is who dies when. This is variously known as stochastic risk, idiosyncratic risk or binomial risk. Rather obviously, you cannot measure this risk if you have reduced your portfolio to a smaller number of model points!

Table 1. Extra capital required as percentage of median discounted value to be 99.5% sure of covering stochastic risk (idiosyncratic risk) only. Source: Richards and Currie (2009).

Portfolio | Number of lives | Capital margin |

Small | 15,429 | 1.07% |

Large | 207,190 | 0.50% |

Table 1 shows the practical impact of the law of large numbers, namely that a portfolio with a large number of lives experiences less volatility than a smaller portfolio. This assumes that all other things are equal, such as age structure and distribution of benefit amounts. However, the benefit of whole-portfolio simulation is that these things are all automatically taken into account. As the process is very quick, we can add risks stepwise to see their additional impact. For example, using a stochastic projection model we can add trend risk, which gives the results in Table 2:

Table 2. Extra capital required as percentage of median discounted value to be 99.5% sure of covering stochastic risk (idiosyncratic risk) and trend risk. Source: Richards and Currie (2009).

Portfolio | Number of lives | Capital margin |

Small | 15,429 | 3.50% |

Large | 207,190 | 3.12% |

Table 2 shows a number of new insights. As expected, the extra capital requirement has increased because there is now more risk. The figures in Table 2 are much larger than those in Table 1, demonstrating that trend risk is the dominant of the two risks for both portfolios. Furthermore, the advantage of size for the large portfolio has narrowed from 0.57% (=1.07%-0.50%) to 0.38% (=3.50%-3.12%). Since whole-portfolio simulation is nowadays very quick, actuaries are able to explore aspects of risk — and risk interactions — which are not possible using model points.

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