Standard Ultimate Life Table calculator, at 5%

Enter an age from 20 to 100. You get the life table values Exam FAM uses at that age, as SOA prints them, with the working.

Annuity-due ä at age 65

13.5498

A whole life annuity-due of 1 a year, at 5% interest.

l₆₅: 94,579.7
q₆₅: 0.005915
A₆₅: 0.35477
²A₆₅: 0.15420
₁₀E₆₅: 0.55305

The working

Makeham's law: $\mu_x$ $= A + Bc^x$ with $A$ $= 0.00022$ , $B$ $= 0.0000027$ and $c$ $= 1.124$ , starting from $l_{20}$ $= 100{,}000$ .
$q_{65}$ $= 1$ ${}- \dfrac{l_{66}}{l_{65}}$ $= 1$ ${}- \dfrac{94{,}020.3}{94{,}579.7}$ $= 0.005915$
$\ddot{a}_{65}$ $= \sum_{k \ge 0} v^k \, {}_kp_{65}$ $= 13.5498$ at $i$ $= 5\%$
$A_{65}$ $= 1 - d\,\ddot{a}_{65}$ $= 1$ ${}- \dfrac{0.05}{1.05}(13.5498)$ $= 0.35477$
${}^{2}A_{65}$ sums the same death probabilities at $v^2$ : ${}^{2}A_{65}$ $= 0.15420$
${}_{10}E_{65}$ $= v^{10} \dfrac{l_{75}}{l_{65}}$ $= 1.05^{-10} \times \dfrac{85{,}203.5}{94{,}579.7}$ $= 0.55305$

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What each value means

lx	Lives still alive at age x, out of 100,000 at age 20
qx	The chance a life aged x dies within the year
ä	Whole life annuity-due of 1 a year, paid while the life is alive
A	Whole life insurance of 1, paid at the end of the year of death
2A	The same insurance at twice the force of interest, for the variance
10E	1 paid in 10 years if the life is then alive

On the exam

SOA shows the Standard Ultimate Life Table at i = 5% on screen under the Exhibit button, with its distribution inventories and a normal distribution calculator.

Knowing how the values combine saves time: a term insurance is a whole life value minus a deferred one, and a temporary annuity is a whole life annuity minus a pure endowment times a later one.

Try it on an exam question

Question 1

Mortality follows the Standard Ultimate Life Table, and $i$ $= 0.05$ . You are given:

A_{55}

| |---| | 0.23524 |

A whole life annuity-due on (55) pays $24,000 at the start of each year.

Calculate the expected present value of the annuity.

Question 2

Mortality follows the Standard Ultimate Life Table, and $i$ $= 0.05$ . You are given:

$A_{55}$	${}_{10}E_{55}$	$A_{65}$
0.23524	0.59342	0.35477

A 10-year deferred whole life insurance on (55) pays $100,000 at the end of the year of death, provided death occurs after age 65.

Calculate the expected present value of this insurance.

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Questions people ask

What is the Standard Ultimate Life Table?

A life table built on Makeham's law of mortality, with A = 0.00022, B = 0.0000027 and c = 1.124, starting from 100,000 lives at age 20. SOA shows it on screen during Exam FAM, at 5% interest, under the Exhibit button.

Can I bring the life table to Exam FAM?

No. SOA provides its tables on screen during the exam, and candidates may not bring their own. Practicing with the values here gets you used to reading them.

How are A and the annuity-due related?

For a whole life insurance of 1 paid at the end of the year of death and a whole life annuity-due of 1 on the same life, A = 1 − d·ä, where d = i ÷ (1 + i). At 5%, d = 0.047619.

What does 2A mean?

The second moment of the whole life insurance's present value: the same sum as A, at twice the force of interest. Its variance is 2A minus the square of A.

What is 10E?

The 10-year pure endowment: the present value of 1 paid in 10 years if the life is still alive then. It equals v to the 10th times the chance of surviving 10 years.