## Simulation and survival

In an earlier post we discussed how a survival model was directly equivalent to assuming future lifetime was a random variable.  One consequence of this is that survival models make it quick and simple to simulate a policyholder's future lifetime for the purposes of ICAs and Solvency II.

The survival curve is the proportion of lives surviving to each age, i.e. tpx in actuarial parlance.  Below is a sample survival curve in red for a life aged x, showing how to read off the probability of survival to age x+t:

For simulation purposes we simply reverse this procedure: we generate a pseudo-random number uniformly distributed over the interval (0, 1), place it on the vertical axis and look up the age at death x+t.

A huge advantage of survival models in simulation lies in the following simple formula for the survival probability:

where Hx(t) is known as the integrated hazard function.  For many survival models Hx(t) has a closed-form expression, and a list for the common actuarial laws of mortality is given in Richards (2008).  The general formula above can then be re-arranged as follows for simulation purposes:

Depending on the choice of mortality law, this formula can be further manipulated to get a direct, closed-form expression for the simulated age at death, x+t.  Using survival models can therefore give a quick and simple way of  simulating the exact age at death from a single U(0,1) variate.  This is particularly efficient when simulating an entire portfolio in run-off, life-by-life, since it means no model points need to be selected.

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Stephen Richards is the Managing Director of Longevitas
##### Run-off simulations in Longevitas

Longevitas does full portfolio run-off simulations on a life-by-life basis. The output enables the measurement of trend risk, idiosyncratic risk and concentration risk, and you can examine the variation in time lived, cash paid and the value of cashflows paid.  The risks associated with model-point selection are side-stepped by simulating every single life.

Longevitas can also perturb a nominated parameter between each portfolio simulation.  This is done with respect to the estimated standard error for the parameter concerned, with the rest of the model being fitted around the perturbed value.  This "perturbation run-off" allows the measurement of the impact of parameter risk on time lived, cash paid and the value of cashflows.