ESTIMATING PENSIONS RISKS WITH SIMULATION METHODS

A5.4 RECOMMENDED READING

6.7 ESTIMATING PENSIONS RISKS WITH SIMULATION METHODS

Some final applications of simulation approaches are to the measurement of pensions risks. With pen- sions, the general method is to build a model that allows the pension fund to grow in line with pension- fund contributions and the (risky) returns made on past pension-fund investment. If the model is sophisticated, it would also allow for the effects of the pension-fund portfolio management strat- egy, which might also be dynamic, and for the possibility of interrupted contributions (e.g., due to the holder of the pension scheme being unemployed). When the holder retires, one of two things

might happen, depending on whether the pension scheme is a defined-benefit scheme or a defined- contribution scheme:

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With a defined-benefit (DB) scheme, the holder gets a predefined retirement income, usually specified in terms of a certain proportion of their final salary or the average of their last few years’

salary, plus possible add-ons such as inflation adjustment.²¹In these schemes, the pension-plan provider bears a considerable amount of risk, because the holder’s pension benefits are defined and yet their (and, where applicable, their employers’) contributions might not cover the provider’s pension liability. In such cases, the pension risk we are interested in is related to the probability and magnitude of the holder’s accumulated pension fund falling short of the amount needed to meet the costs of the holder’s defined pension benefits. Pension risk in this context is the risk of the pension provider’s assets falling short of its liabilities.

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With a defined-contribution (DC) scheme, the holder gets a pension based on the value of their accumulated fund and the way in which, and terms on which, the fund is converted to an annual retirement income.²²Usually, the accumulated fund would be converted into an annual retirement income by purchasing an annuity, in which case the retirement income depends not just on the value of the accumulated fund, but also on the going interest rate and the holder’s expected mortality.

With these schemes, the holder bears the pension risk, and the risk we are interested in is the risk of a low pension relative to some benchmark (e.g., such as final salary). Pensions risk in this context is the risk of the holder having a low pension.

We now consider each of these two types of scheme in turn.

6.7.1 Estimating Risks of Defined-benefit Pension Plans

To estimate the risks of a defined-benefit pension scheme, we must first clarify the precise features of the scheme concerned, and it is helpful to do so focusing on the pension provider’s assets and liabilities.

On the asset side, we need to make assumptions about the starting and ending dates of the scheme, the amounts contributed to the pension fund, the unemployment (or other contribution interruption) risk, and the way in which the pension fund is invested. To illustrate, we might assume that the pension-plan holder:

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Starts contributing at age 25 and aims to retire at age 65.

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Has a salary of $25,000 at age 25, contributes 20% of gross income to the fund, and expects salary to grow at 2% a year in real terms until retirement.

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Faces a constant unemployment risk of 5% a year, and contributes nothing to the scheme when unemployed.²³

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Does not benefit from any employer contributions to their pension fund.

21For more on DB schemes, see, e.g., Blake (2000, 2003) and Guptaet al.(2000).

22The reader who is interested in the mechanics and simulation of DC pension schemes might wish to explore the

‘pensionmetrics’ work of Blakeet al., and particularly Blakeet al.(2001a), where the pensionmetrics methodology and associated simulation issues are discussed in detail.

23These assumptions are unrealistically precise, but we need to make these or similar assumptions to estimate the risks involved, and these rather simple assumptions help to illustrate the approach. In practice, we might want to modify them in many ways: we might want to use a variety of alternative assumptions about start/end dates or contribution rates, real income growth rates, and so on. We might also wish to make real income growth stochastic, or allow a wage profile that peaks before retirement age and possibly depends on the holder’s profession, or we might allow unemployment risk to vary with age or profession (e.g., as in Blakeet al.(2001c)).

We also assume that the fund is invested entirely in one risky asset (e.g., equities), and that the return to this asset is normally distributed with annualised mean and standard deviation both equal to 0.1.²⁴

On the liability side, we can think of the pension provider as being obliged to purchase an annuity when the holder retires to give him/her a specified retirement income, and the provider’s liability is the cost of this annuity plus the cost of any add-ons such as guarantees against inflation. The provider’s liability therefore depends on the final salary (or the last few years’ average salary), the formula used to determine retirement income, the annuity rate, and the holder’s life expectancy conditional on reaching retirement. It follows, then, that in general the provider’s liability is subject to (at least) five (!) different sources of risk:

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Risk arising from uncertainty about the holder’s salary at or near retirement.

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Risk arising from uncertainty about the exact timing of retirement.

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Risk arising from uncertainty about employment, which creates uncertainty about the number of years the holder will contribute to the scheme.

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Risk arising from uncertainty about the annuity rate at retirement. This is important, because the annuity rate determines the cost of the annuity necessary to provide a given retirement income:

the lower the annuity rate, the greater the cost of the annuity, other things being equal.

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Risk arising from uncertainty about the holder’s life expectancy on retirement, and this is also important because of its effect on the cost of the annuity.

To make the analysis as transparent as possible, we now make the following illustrative assumptions:

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The pension is equal to the final salary times the proportion of years in which the holder has contributed to the fund. This implies that if the holder has worked and contributed throughout his/her working life, then he/she will get a pension equal to his/her final salary.

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The annuity rate on retirement is taken as 4%²⁵

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There are no add-ons, so the provider’s liability is only the cost of the annuity.

Having established the structure of our model, we now program it, and one way to do so is provided by the function ‘dbpensionvar’. Leaving aside details, the programming strategy is to model the asset and liability sides separately, work out the terminal values of each under each simulation trial, and take the pension provider’s P/L to be the difference between the two. Once we have a set of simulated sample P/L values, we can then estimate VaR or ETL in the usual way. With the asset side, we build up the value of the pension fund over time, bearing in mind that the fund is equal to current contributions plus the last period’s fund value plus the return earned on last period’s fund value, and pension-fund contributions depend on whether the holder is working that period. We therefore need to take random drawings to determine the rate of return earned on previous pension-fund

24This assumption is also unrealistic in that it ignores the diversity of assets invested by pension funds, and ignores the possibility of dynamic investment strategies such as the lifestyling or threshold strategies followed by many pension funds (and which are explained, e.g., in Blakeet al.(2001a)).

25We therefore assume away uncertainty associated with both the annuity rate life expectancy. In practical applications we would certainly want to make the annuity rate stochastic, but would also want to ensure that the rate used was con- sistent with other contemporaneous rates. The correct way to treat the annuity rate is discussed in detail in Blakeet al.

(2001a).

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Loss () / profit () in ($k)

Frequency

VaR = 259.3

Figure 6.5 VaR of defined-benefit pension scheme.

Note: Obtained using the ‘dbpensionvar’ function with initial income $25k, income growth rate of 0.02, conditional life expectancy 80 years, contribution rate 0.15,µ=0.1,σ=0.1,p=0.05, annuity rate 0.04, 1000 trials, andcl=0.95.

investments and to determine whether the holder is working in the current period. On the liability side, we determine the final salary value and the number of contribution years, use these to determine the pension entitlement, and then apply the annuity rate and life expectancy to obtain the value of the pension annuity.

After all of this, we now choose our confidence level and run the program to estimate our VaR.

Given the assumptions made, our VaR at the 95% confidence level turns out to be 259,350. This is shown in Figure 6.5, which also shows the provider’s simulated L/P. Perhaps the most striking feature of this figure is the very wide dispersion in the L/P series: the provider can get outcomes varying from a loss of $400k to a profit of over $1.5m. The business of providing DB pensions is clearly a very risky one. The other point that stands out is that the VaR itself is quite high, and this in part is a reflection of the high risks involved. However, the high VaR is also partly illusory, because these prospective outcomes are 40 years’ off in the future, and should be discounted to obtain their net present values. If we then discount the future VaR figure at, say, a 5% annual discount rate, we obtain a ‘true’ VaR of about 36,310 — which is about 14% of the amount shown in the figure. When dealing with outcomes so far off into the future, it is therefore very important to discount future values and work with their present-value equivalents.

6.7.2 Estimating Risks of Defined-contribution Pension Plans

Defined-contribution pension schemes share much the same asset side as DB schemes; however, they differ in not having any distinct liability structure. Instead of matching assets against specified

0 0.5 1 1.5 2 2.5 3 3.5 4 0

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Pension ratio

Frequency

Pension ratio VaR = 0.501

Figure 6.6 VaR of defined-contribution pension scheme.

Note: Obtained using the ‘dcpensionvar’ function with initial income $25k, income growth rate of 0.02, conditional life expectancy 80 years, contribution rate 0.15,µ=0.1,σ =0.1,p=0.05, annuity rate 0.04, 1000 trials, andcl=0.95.

liabilities, they (usually) convert the assets available into an annuity to provide the pension.²⁶ This implies that the pension is determined by the size of the fund built up, and by the terms on which that fund can be annuitised.

To model a DC scheme, we would therefore have the same asset structure as the earlier DB scheme:

hence, we can assume that our pension-plan holder starts contributing at age 25, aims to retire at 65, has a starting salary of $25,000, and so on. Once he/she reaches retirement age, the accumulated fund is converted into an annuity at the going rate, and the pension obtained will also depend on the prevailing annuity rate and the holder’s life expectancy at that time.²⁷ To complete the model, we need assumptions about the annuity rate and the life expectancy on retirement, and we may as well make the same assumptions about these as we did before. However, with DC schemes, the notion of pensions risk refers to the value of the pension itself, not to a possible pension-fund shortfall, and it is convenient to express this risk in terms of the pension divided by the final salary.

We now program the model, and a program is provided by the IMRM function ‘dcpensionvar’.

The programming strategy is to model the terminal value of the pension fund under each simulation

26The practice of annuitising funds on retirement is however not necessarily the best way to convert the fund into a pension.

These issues are explored in more detail in Blakeet al.(2001b) and Milevsky (1998).

27As viewed from the time when the scheme is first set up, this means that the pension is subject to a number of sources of risk — leaving aside retirement age and contribution rate risks, there are also risks arising from the returns earned on pension-fund investments, risks arising from the possibility of unemployment and interrupted contributions, and risks arising from uncertainty about the annuity rate and life expectancy that will prevail at the time of retirement. Furthermore, these risks are borne entirely by the plan holder, and not by the pension-plan provider, who essentially bears no risk.

trial, annuitise each trial fund value at the going rate, and then divide these by the final salary to obtain a set of normalised pension values. The distribution of these pension values then gives us an indication of our pension risks.

If we carry out these calculations, we find that the pension ratio has a sample mean of 0.983 and a sample standard deviation of 0.406 — which should indicate that DC schemes can be very risky, even without looking at any VaR analysis. The pension VaR — the likely worst pension outcome at the relevant (and in this case, 95%) confidence level — is 0.501, and this indicates that there is a 95%

chance of a pension ratio higher than 0.501, and a 5% chance of a lower pension ratio.