Common mistake 1 with Life annuity

Using a population mortality table for annuity pricing - annuitant mortality is lighter, which raises the value.

Common mistake 2 with Life annuity

Forgetting to match the timing convention: quoting an annuity-due value for a product that actually pays in arrears overstates the liability.

Common mistake 3 with Life annuity

Ignoring future mortality improvement in a long-dated annuity; it makes the annuity more expensive than a static table implies.

Common mistake 4 with Life annuity

Valuing a joint-life annuity as two single lives. The joint benefit is worth less than the sum of the parts and more than either alone, and the difference depends on the correlation between the two lives, which a simple sum ignores entirely.

EconStatsCraft › Life annuity

Life annuity

The actuarial present value of a life annuity - whole life, temporary or deferred, annuity-due or immediate, payable annually or m times a year - built from a mortality law and an interest rate, with tables showing how the value falls with age and rises as interest falls.

The intuition

An annuity pays while you live, so its value is the sum of each future payment discounted for interest AND weighted by the probability you are alive to collect it. Two discounts, applied together: money later is worth less, and a payment you might not live to receive is worth less still. Everything else is bookkeeping about when in the year the payment falls.

When to use it

Pricing a pension or a purchased life annuity, valuing the premium stream of an insurance policy (premiums are a life annuity), or converting a lump sum into an income for life.

Where the estimator comes from

The expected present value of a whole-life annuity-due paying 1 at the start of each year is \ddot{a}_x=\sum_{k\ge0}v^k\,{}_kp_x with v=1/(1+i). The immediate version pays at the end, so a_x=\ddot{a}_x-1. Writing D_x=v^x\ell_x and N_x=\sum_{k\ge x}D_k gives the commutation form \ddot{a}_x=N_x/D_x, which is how these were computed before machines. The fundamental identity linking annuities and insurances is A_x=1-d\,\ddot{a}_x where d=i/(1+i): an insurance and an annuity are two views of the same cash flow, and any pair of values that violates it contains an arithmetic error.

The model

\ddot{a}_x = \sum_{k=0}^{\infty} v^{k}\,{}_{k}p_x,\qquad \ddot{a}_{x:\overline{n}|} = \sum_{k=0}^{n-1} v^{k}\,{}_{k}p_x,\qquad \ddot{a}_x^{(m)} \approx \ddot{a}_x - \tfrac{m-1}{2m}

What it assumes

A deterministic, level interest rate for all future years.
The mortality basis is the right one for the annuitant population - annuitants are self-selected to be healthier than average, so a population table understates the cost.
The m-thly value uses the two-term Woolhouse approximation, which is accurate for smooth mortality but drifts at very old ages.
The annuitant cannot lapse or surrender. An annuity in payment is irreversible by design, which is what makes the mortality assumption so consequential - there is no exit to correct a mispricing.

Statistical properties

The value falls as interest rises and rises as mortality improves. A pension scheme is therefore short interest rates and long longevity, which is why both are hedged.
The identity A = 1 - d times a-due is exact and holds for any mortality basis whatsoever. It is the fastest check on any set of computed values.
An annuity-due always exceeds an annuity-immediate by exactly 1, because the first payment is not discounted or survived for.
Duration is long, typically 12 to 18 years for a pension in payment, so a 1% fall in the discount rate raises the liability by roughly 15%.
Guarantee periods and joint-life features shorten the mortality discount and therefore raise the value; a joint-life-last-survivor annuity is the most expensive form.
Mortality improvement is the dominant source of error. Pricing on today's rates and ignoring projection has historically understated liabilities by 5 to 15%.

How to read the output

a-double-dot_x minus a_x is exactly 1: the difference between paying at the start and the end of each year.
The whole-life annuity value falls with age (fewer payments expected) and falls as the interest rate rises (future payments discounted harder) - the two charts show each effect on its own.
A deferred annuity n|a-double-dot_x plus the temporary a-double-dot_{x:n} equals the whole-life a-double-dot_x: the payments are just split at time n.
Multiply the annuity value by the yearly payment to get the single premium; divide a lump sum by it to get the income it buys.

A worked reading

Makeham mortality, age 45, interest 4%. The whole-life annuity-due is 18.1821 and the annuity-immediate 17.1821 - differing by exactly 1, because the only difference between them is the payment made today, which needs no discounting and no survival. Two more identities are visible in the same table and worth checking by hand, because they are what tells you the table is internally consistent: the 20-year temporary annuity (13.5517) plus the 20-year deferred annuity (4.6304) comes to 18.1821, the whole-life value, since together they cover every year exactly once. And the annuity payable monthly comes to the same 18.1821 under the Woolhouse two-term approximation at this age. The age ladder is the commercial point: 18.1821 at 45 falls to 11.8741 at 65 and 8.3241 at 75, so the same yearly pension costs a 45-year-old buyer more than half as much again as a 65-year-old, entirely because of the years of payment in between.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Annuity value above 1/d	Impossible: that is the perpetuity limit	Check the survival probabilities are below one
A + d times a-due not equal to 1	Inconsistent basis or an arithmetic slip	The identity is exact; something is wrong
Value barely changing with age	Mortality too light, or the horizon truncated	Check the table closes properly
Enormous sensitivity to the interest rate	Correct. Duration is long	Report the sensitivity alongside the value
Immediate and due values equal	The timing adjustment was not applied	They must differ by exactly 1
Deferred annuity cheaper than expected	Correct: survival to vesting is not certain	The deferral discount is mortality as well as interest

Common mistakes

Using a population mortality table for annuity pricing - annuitant mortality is lighter, which raises the value.
Forgetting to match the timing convention: quoting an annuity-due value for a product that actually pays in arrears overstates the liability.
Ignoring future mortality improvement in a long-dated annuity; it makes the annuity more expensive than a static table implies.
Valuing a joint-life annuity as two single lives. The joint benefit is worth less than the sum of the parts and more than either alone, and the difference depends on the correlation between the two lives, which a simple sum ignores entirely.

Where to read this properly

Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks (2020), ch. 5.
Bowers et al., Actuarial Mathematics (1997), ch. 5.
Continuous Mortality Investigation, CMI Mortality Projections Model (CMI_20xx series, reissued annually - cite the version you used), on improvement scales.

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Life annuity

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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