Valuing a promise

A defined benefit scheme is a stack of promises: so much a year, to this member, from that date, for as long as they live. There is no pot with anyone's name on it. One collective fund backs the whole list, and the payments stretch decades into the future. Before you can ask whether the fund is big enough, you have to answer a harder question first: what is a promise to pay money in ten or thirty years' time worth today?

The tool for that question is discounting, and nearly everything else in this course is an application of it.

One payment

Start with the simplest promise a scheme could hold: £5,000, payable in exactly ten years. You do not need £5,000 today, because money invested in the meantime grows. If you can earn 5% a year, ten years of growth multiplies your money by 1.05^10, about 1.6289. Run that backwards and you need only:

£5,000 ÷ 1.05^10 = £3,069.57

Check it forwards: £3,069.57 in the fund today, earning 5% a year, grows to the full £5,000 just as the payment falls due.

That £3,069.57 is the present value of the promise, and a present value is what a pension liability is. When a scheme reports liabilities of £500m, nobody owes £500m this afternoon. The figure is thousands of future payments, each discounted back like this, added up.

The rate is the whole game

Rerun the same sum at 3%:

£5,000 ÷ 1.03^10 = £3,720.47

Nothing about the promise changed: same member, same £5,000, same date. Yet the liability rose by about 21%, purely because the discount rate fell two percentage points. That is the single most important mechanism in UK pensions: when yields fall, liabilities rise. The next lesson covers what happened when this mechanism ran one way for twenty years and then reversed.

Where does the rate come from? For funding purposes it is normally anchored to gilt yields, the return on UK government bonds. The logic: a pension promise is about as certain as a payment gets, so you price it off the most certain investments available. Funding discount rates are influenced by market interest rates and the scheme’s investment strategy. If market yields rise, the discount rate used to value the liabilities will often rise too, reducing their present value.

Streams and the annuity factor

Real pensions are streams, not single payments. Say a scheme owes a member £1,000 at the end of each of the next five years, discounting at 4%. You could discount all five payments separately and add them up. The shortcut the profession runs on is the annuity factor:

(1 - 1.04^-5) ÷ 0.04 = 4.4518

£1,000 × 4.4518 = £4,451.82

Read the factor as a price tag: £1 a year for five years costs £4.45 today. A real pension is the same object stretched over decades, with one refinement: each payment is also weighted by the probability the member is alive to collect it, which bakes a mortality table (the assumed death rates at each age) into the factor.

Prudent or best estimate?

A promise does not have one price, because assumptions are choices. Two terms to know:

• Best estimate: assumptions intended to be realistic, with no deliberate margin for caution or optimism.
• Prudent: assumptions deliberately tilted towards caution, such as a lower discount rate or longer lifetimes. Prudence pushes the liability number up, and where that opens a shortfall it asks more money of the employer sooner, which buys members security.

Statutory funding valuations are required to be prudent; transfer values (the sum a member can move to another scheme instead of the promise) are calculated on best estimate. Same promises, different prices. The split is deliberate: a transfer value is paid out of the collective fund, so a leaver paid on the prudent basis would take out more than the promise is expected to cost, at the expense of the members who stay.

Rate and term together

The rate is only half of it. The other half is how far away the payment is, because discounting compounds. Here is the same £5,000 promise priced at four rates, first ten years out and then forty, the far end of a real scheme's payment stream:

Read down the columns, not across the rows. At ten years, taking the rate from 5% to 2% raises the liability by about a third. At forty years it more than triples it. The same yield fall does far more damage to a promise that is decades away.

These numbers leave out mortality and inflation. Real valuations layer both back in, but the core never changes: every liability is a discounted payment, or a sum of them.

Layer mortality back in

A valuation model puts one row per projected payment on a sheet, with the discount factor visible in its own column, so a reviewer can trace any number back to an input. Below is that sheet, for a pensioner aged 60 drawing £1,000 a year for five years, the same stream as the annuity factor above, with one column added for the probability the member is alive to collect each payment, built from each year's mortality rate, written q on the sheet.

Now do one by hand: