Defined contribution (DC) pension schemes are complicated financial products. However, they are ideal subjects for stochastic simulation methods1, and research on them has developed to the point where we now have the basis of a commercially feasible DC pension model2.
This article begins by explaining the structure of this model. We then work through an example and discuss how this approach can generate insights into the workings of a pension plan, how it can generate ‘hard’ answers to well-posed quantitative questions, and how it can be used to guide pension plan design.

PensionMetrics methodology
In a DC scheme, the individual contributes periodically to their pension fund. This is invested according to some asset allocation strategy, and tends to accumulate over time. When the individual retires, the fund is typically converted into an annuity at going annuity market rates, and the annuity provides the retirement income.
To model DC pension schemes ,we first identify the risk factors: unemployment risks, asset-return risks, interest-rate risks, and so forth. We then specify the exogenous processes (eg, unemployment, the asset return and interest rate processes) and the relationships involved (eg, typically expressed in terms of the ratio of pension to final salary3, and obtain our results from the distribution of simulated pension ratios.
The process is shown in flow chart form in figure 1. The top line shows the exogenous factors: the type of worker (male manual, etc), and the exogenous processes (asset-return, interest-rate processes, unemployment and salary-growth processes). The second line shows the two choice variables to be selected by the user, the contribution rate and the asset allocation strategy. The outputs from the first and second lines then feed into the calculation engine on the third line. This produces a simulated histogram of pension-ratio outcomes, which gives the results on the bottomline.
Illustrative simulation exercise: Suppose that a male managerial worker starts contributing to a pension fund at age 25, and invests 10% of his income each period into it. We now make a set of assumptions about the processes governing the risk factors, and carry out the simulation exercise4.
The resulting output is illustrated in figure 2, which shows the user interface of the basic PensionMetrics program developed by the authors. The input assumptions are listed on the right-hand side, and the output on the left and bottom. The latter consists of a chart plotting the pension ratio’s expected values and its 95% confidence interval for various possible retirement ages.
The figure shows that for a retirement age of 59, the pension ratio has an expected value of about 19.6% and a 95% confidence interval equal to 3.2%,_44.2%. In plain language, our individual’s best guess of his future replacement ratio is about 20%, and he can be 95% confident that he will get a ratio between 3.2% and 44.2%. His pension ratio is therefore highly uncertain, so his pension is very much at risk.
The model can be used to simulate how the replacement ratio might vary with any of the inputs: the contribution rate, the unemployment risk profile, the asset-allocation strategy, salary profile, etc, as well as the retirement age. If we carried out such an analysis, we would find that pension outcomes are proportional to the contribution rate (which follows from the structure of the model), are heavily affected by the retirement age (which is obvious) and asset allocation strategy (which most professionals would have expected, although we were surprised at how much the asset allocation strategy affected the outcome), and are not particularly sensitive to any of the other assumptions (which was a surprise to us).

Usefulness of stochastic simulation
This example highlights the practical usefulness of stochastic simulation:
q It generates qualitative insights about prospective pension outcomes. Perhaps the most striking of these is that DC schemes impose very large risks on plan holders. The model can generate an awareness of this exposure and give policyholders a chance to take corrective action (eg, increase contributions) before it is too late;
q The model provides quantitative answers to ‘what if’ questions: what would happen to pension outcomes if we change the asset-allocation strategy; if equity returns are lower than expected, and so on? Such exercises give us a sense of the robustness of our results to our assumptions, and enable us to identify those assumptions that matter;
q The model can be used to guide pension plan design. We can use the results to select an appropriate contribution rate or an asset allocation that suits the risk appetite of the client. This exercise also forces the user to confront reality and recognise the types of outcomes that particular choices entail: a user who wanted a low contribution rate would get results showing that their pension was likely to be very low, and so forth. Choices would then be informed – which is also useful to advisers and pension scheme managers who could demonstrate due diligence.
Stochastic simulation methods can also be applied to many other pensions problems. We can use them to examine: the impact of occupation and gender differences on pension outcomes; alternative post-retirement financial strategies (eg, how to invest post-retirement, etc); different types of pension (defined benefit versus DC, etc); the impact of tax changes; and pension fund shortfalls, which are a major concern with state and defined benefit schemes. Indeed, we would suggest that stochastic simulation should be the method of choice for almost any quantitative pension problem.
1 Some progress can be made if we resort to analytical approximations under assumptions of comonotonicity (see, eg, Kaas et alia, 2002). However, stochastic simulation methods are more flexible and can handle more difficult problems.
2 There are a variety of simulation-based approaches available (see, eg, Bodie (2002) and www.financialengines.com). Similar methods also appear in the actuarial literature under the heading of dynamic financial analysis (see, eg, Kaufman et alia, 2001).
3 Thus a pension ratio of 100% means that the pension income exactly matches the final salary, a ratio of 50% means that the pension income is 50% of final salary, and so on.
4 More details and a discussion of plausible alternatives are given in Blake et alia (2001).

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