Expected Present Value: pricing a promise

Strip a term assurance, a pure endowment or an annuity down and you find promises of one shape: an amount of money, paid at a future date, if a named person is then alive, or then dead. Price the shape and you can price the products. The tool is the expected present value, EPV, and for one promise it is three numbers multiplied together:

EPV = amount × probability of payment × discount factor

That line values the promise inside everything from a £10-a-month term policy to a multi-billion-pound buy-in. The only thing that changes is how many of those promises you add up.

The ingredients

From the Acttuary Standard Table (built in the first lesson of this module), for a life aged 60:

• Survivors: l₆₀ = 10,000 and l₆₅ = 9,654.8, so ₅p₆₀ = 0.96548.
• Deaths in each of the next five years: d₆₀ = 50.0, d₆₁ = 59.7, d₆₂ = 69.2, d₆₃ = 78.6, d₆₄ = 87.7.

And discount factors at 4% a year, where v = 1 ÷ 1.04:

Each factor answers: what is £1 then worth now, if money earns 4% in between? £1 due in five years is worth 82.2p today, because 82.2p invested at 4% grows back into £1.

The simplest promise: a pure endowment

A pure endowment pays a fixed sum on a future date if the policyholder is then alive: no death benefit, nothing along the way. Take one: £10,000 at age 65, sold to a life now aged 60.

EPV = 10,000 × 0.96548 × 0.821927 = £7,935.54, the fair price of that promise.

The price sits below £10,000 for two separate reasons, and it pays to keep them separate. The survival haircut removes 3.452%, because some policyholders will not live to collect. The discounting haircut removes a further 17.8%, because money five years away is cheaper than money today.

Fair here has a precise meaning. Collect £7,935.54 from each of 10,000 60-year-olds: £79.36m. Invest it at 4% for five years and it grows to £96.55m. The table expects 9,654.8 survivors, each owed £10,000: £96.55m, so the fund empties to the last pound. Per policyholder: £7,935.54 × 1.04⁵ = £9,654.80, which is exactly £10,000 × 0.96548, the expected payout per life who signed up.

Five promises stapled together: term assurance

Now the classic protection product. A 5-year term assurance for £100,000 on our 60-year-old pays out if death occurs within the term, at the end of the year of death in the standard simplification. That is five promises: £100,000 at the end of year one if death comes in year one, at the end of year two if in year two, and so on. Each gets the same three-number treatment, using the probability of dying in that particular year, dx ÷ l₆₀:

EPV = £3,040.62, the fair single premium for £100,000 of five-year cover on this basis. It is not the price a customer would be quoted: real premiums add expenses, profit and the cost of holding capital on top, and the next lesson puts them there. Add the printed column and you get £3,040.63, a penny more, because five rows were each rounded to the nearest penny before being added. The total comes from the unrounded figures, £3,040.6209. Round at the end, not on the way, and say so when a checker asks why your total does not match their addition.

Two checks are worth building in. The probability column sums to 0.03452, which is exactly 1 minus 0.96548: dying within the term and surviving it are the only two exits, so the column must total the complement of ₅p₆₀. And 0.03452 × £100,000 = £3,452 is what the cover would cost with no discounting at all; the £411 gap is the interest the premium is assumed to earn between coming in and the claims going out.

Run it, then bend it

Both prices in one short loop. Press Run, then change i and run again: 0.03 first, then 0.05.

At 3% both prices rise: the endowment to about £8,328, the term assurance to about £3,136. At 5% both fall. Nothing about the people changed; only the reward for waiting did. Every life insurance promise pays in the future, so all else equal a lower discount rate makes it dearer to fund.

There is no separate theory for each product. An annuity is a row of pure endowments, one per year of survival. A whole-of-life assurance is a term assurance whose term never ends. Every EPV a life actuary builds is amount × probability × discount factor, summed. Prove the scaling to yourself:

Build the pricing sheet

The term assurance table above is what a pricing actuary would put in a grid, and it is worth building once yourself: one row per year of the term, three columns for the three numbers, and two reconciliations at the bottom.