Calculations for DB pensions

I had thought that the calculations for DB pensions were really difficult. They are actually really simple. The tricky bit (where there is a need for actuaries) in in defining the life expectancy and in defining the probability of there being a qualifying spouse at the time of death, and the extra years of payments for that spouse.

You need four bits of info: age now, age when payments start, annual payments, life expectancy at age now.

Your life expectancy increases as you get older, and you can easily find online calculators that will tell you the life expectancy for a person of a given age. For yourself many calculators let you add your weight, whether you smoke, etc.. DB Pension Funds have to have one set of rules for everybody, so you can mimic them by not giving options for weight, smoker, gender etc..

The spouse’s pension is a typical add-on for a scheme. Depending on the scheme, you might have to be married at the time of ceasing contributing to the scheme, and still married to the same person at death, they have to outlive you (with sex equality etc. the spouse could be a man just as easily as it could be a woman, and could be older or younger), and the number of years they outlive might be wide ranging but I suspect on average is only 4 or 5. Historically the sex-equality thing was not the case: big final salary pensions were often held by men with younger wives, so the years of outliving could be 10-15. So let’s guess 90% have a spouse, 90% still have the same spouse at death (assuming it matters for that scheme), 50% of spouses outlive the member (i.e. no gender bias unlike was historically the case), they outlive by 5 years, and the pension is 50% of a full pension (my DB pays 33%, the NHS pays 50% or 37.5 or 33.75% depending on which scheme, but I’ve heard of 66%). 0.9×0.9×0.5x5x0.50=1.01 which means the benefit costs the fund about a year’s full pension, if my guesstimates are OK. So in all the models that follow you could simply allow for this by adding one year to life expectancy.

That’s the tricky bit out of the way. Now the simple maths:

This uses Net Present Values. When I use an annual factor of “RPI+3.3%” I seem to come very close to data that I can find online for various pensions (local government, NHS, etc) and a very close match to the transfer value that I was quoted a couple of years ago by my previous employer’s scheme. We will work all this out in today’s money because DB pensions increase with RPI, so we can ignore RPI and just focus on the 3.3%. NPV means that £1000 this year is worth £1000 this year, but £1000 next year is worth £9670 this year, and £1000 the year after that is worth 0.967×0.967×1000=£9350.89 this year.

If we do the maths for a transfer value we can see all the principles in action. Let’s say age now is 45, pension will start at 65, £1000 per year, and life expectancy of a 45 year old might be 88. The NPV of the first payment year at age 65 is £522.39, and each year the NPV of £1000 gets even lower. We sum all the NPVs from age 65 to age 88. That gives the transfer value of 8347.13 which is a multiplier of 8.34.

It’s worth saying now that if a scheme offers a very different transfer value then they have assumed very different criteria. The life expectancy is pretty easy to find (though a scheme might know that it’s members are unusually long lived, perhaps because of an unusual demographic). So the main reason for a very high transfer value is that they are assuming a very different NPV factor. If we take the same example but use an NPV factor of RPI+0% then each year is worth £1000 now, and the transfer value is £24,000 which is a multiplier of 24.

In a nutshell, this explains the unattractiveness of annuities and the appeal of pension flexibility. Annuities seem to be calculated assuming an annual factor of far less than RPI+3.3%, and so are rather unappealing.

If you have good reason to believe your life expectancy will be significantly longer (non-smoker, good weight, fit, good family history) then you might run the model with more years of payments and therefore attribute more value to staying in the DB scheme than the scheme itself has calculated. Similarly, your marital situation might cause you to value the spouse’s pension more than average or to disregard it.

We can do similar maths for the discounts for taking a pension early. Age now is 60, pension starts at 65 normally, £1000/year, NPV is RPI+3.3%, let’s say life expectancy of a 60 year old is 88. So the NPV for the £1000 at age 65 is £850.16, and summing all the years we see an NPV of 14,403.53. Now do the same again but start payments at age 60 and see what annual pension from age 60 to 88 has the same NPV of 14,403.53. The Excel “Data, What-if, Goal-seek” function is really handy for this. You will find that the pension should be 754.34 in today’s money, which is a reduction of 24.6% from the £1000. If you look online for reductions for schemes like the NHS and local government you can see very similar percentages.

You can also see that such a scheme starting at age 60 but allowing pensions to start at age 55 will have a smaller reduction: NPV at age 55 of paying £1000 from age 60 to 88 is 16232.97 and if paid from age 55 to 59 as well then to get the same NPV the annual pension is 775.82 which is a 22.4% reduction. Again, if you look online for reductions for schemes like the NHS and local government you can see very similar percentages.

There’s an argument for redefining the life expectancy for the older-starting pensioner. That would have the effect of reducing the reduction for starting early.

The commutation factor is simply based on the NPV. For a £1000 pension starting at age 65, NPV at age 65 is 16942.23 so the commutation factor is 16.94. For a £1000 pension starting at age 60, NPV at age 60 is £19094.12 so the commutation factor is 19.09.

Importantly, the NHS and local government schemes offer lower reduction percentages for the lump sum. We can see why. For a lump sum in 5 years time the NPV factor is 0.85016 so the reduction for the lump sum is only 15%.

In all this maths, the life expectancy and NPV percentage are crucial. Change the assumptions a little, and the maths changes. Given a transfer value, a commutation factor, etc. you can work back to the assumptions that must have been used. Just remember that different aged people have different life expectancy.