Table based calculations of actuarial present values

Factual Questions
1

It seems that modern course books still show table based calculations of the APVs of mthly life contingencies being done using lx tables which only have lx tabulated for integer values of x, leading, if m>1, to the calculation being approximated by either:

Calculating the APV of the mthly life contingency using the table, interpolating in the cases of the non integers within the calculation.

Or first calculating the APV of the corresponding (same x, n (if temporary) and v) yearly life annuity due using the table, and then applying the Euler-Mclaurin based “Woolhouse approximation formula” to this, thus obtaining the approximate APV of the corresponding (same x, m, n (if temporary) and v) mthly life annuity due, and finally, if needed, applying to this, the standard identity formula (making the APV of the mthly life contingency the subject of this formula) which connects the APV of the mthly life contingency to the APV of the corresponding (same x, m, n (if temporary) and v) mthly life annuity due.

Accurate table based calculations of the APVs of mthly life contingencies would certainly, of course, be achieved by doing these calculations using accurate lx tables which have lx tabulated for x in steps of 1/m, and, in fact, because the only practical values for m are m=1, 2, 3, 4, 6 or 12 and 1/1=12/12, ½=6/12, 1/3=4/12, ¼=3/12, 1/6=2/12 and 1/12=1/12, accurate table based calculations of the APVs of mthly life contingencies would be achieved by doing these calculations using accurate lx tables which have lx tabulated for x in steps of 1/12. Is it not possible to produce such lx tables, and if not, then why not?

2

It sounds like you could fit something (parametric or nonparametric) to the integer age values to get something reasonable; these guys have a specific way of doing that, but their book costs money.

ETA: though I wonder at the value of this, given lifespans and from how far forward insurers are projecting.

3

Thank you. I winder, though, if, perhaps, it is because data is only collected yearly based rather than monthly based?

4

For mortality rates below age 65, the SSA uses death records by sex and age, so in theory they could create tables by day if they deemed it useful. But I doubt even having it by month (as you suggested) would be that useful (for their purposes), or they’d be doing it.

5

Thank you for your answers, much appreciated.

6

Probably in the same vein as my last question, why can’t ALL actuarial early retirement pension factors be calculated accurately instead of only discrete accurate factors being tabulated with ‘in-between’ ones being interpolated?

7

It aeems like an over-refinement to me, and interpolation may be perfectly fine, but I’m no expert. @puzzlegal has mentioned being a young pension actuary at one time & may have a more informed opinion.

8

You are seeking a level of accuracy which is beyond spurious. I can’t even conceive what ‘more accurate early retirement factors’ could mean.

Actuaries don’t do accuracy in the sense an accountant would recognise. We give opinions. Which all boil down to ‘this is as good a number as any to use, given your stated purpose’.

I’m getting better. If for any reason my wife asks me to confirm a value like 3,856 I hardly ever reply ‘five thousand’ any more. She finds that disturbing.

To answer your specific query ‘why do actuaries do this (and not that)?’, ‘because this is good enough to meet our client’s needs, and that (or anything else) would be no better’.

This is not actuarial advice. Should you require actuarial advice, please consalt an actuary qualified in your jurisdiction.

9

Thank you!

10

Thsnk you!

Actuarial early retirement pension factors
11

While the data exists to calculate monthly, or daily, life factors, as @Maserschmidt suggests, i don’t think anyone bothers. It used to be mathematically burdensome, and people got in the habit of interpolating. And now that it’s feasible, there’s no good reason to modify the software.

For a random project, i once played with various interpolations for expected number of people in a cohort to die between age ~20 and age ~60. The various interpolations and approximations made essentially no difference.

(Between ages 0 and 1 the mortality is high and varies significantly by month. You can find precise monthly data from the US census bureau for those increments.)

As for pension calculations, many plans actually specify the interpolations to use. There are also a variety of life tables, and most plans specify which life table they will use. I can assure you that the difference between using the unisex table and the by-sex table matters more than the interpolation. For that matter, the difference between the 2020 table and the 2010 table probably matters more, too. (i haven’t looked to see what years have published tables these days, but detailed tables weren’t published annually back when i practiced as a pension actuary. Maybe they are now. The covid effect on recent table is surely greater than any difference due to interpolation at typical retirement ages, though.)

Basically,

This is true. Actuaries deal with uncertainties and estimation as a daily practice, and tend not to get excited about tiny differences in calculations. That’s one of the most significant differences between actuaries and accountants.

12

This reminds me a bit of an epic thread with a fellow who was trying to write a cookbook using both metric and US customary units of measure. And this fellow Could. Not. Understand that using a 10-decimal place conversion factor wasn’t useful, and was in fact harmful.

He could not comprehend that when a recipe says “1 cup of diced carrots” that really means “About 1 cup plus/minus a few percent. The actual weight and volume of material will also vary depending on dice size, container shape, agitation or packing of the container, freshness of the produce, etc.”

In his mind that was 1.0000000000 cups of diced carrots and should be converted to 0.2365882365 L of diced carrots and not a 10th of a nanoliter more or less.

13

Thank you.

14

Exactly - I have to wonnder what meaning someone can get out of a statistic to apply to a single instance. Whether you have a 1 in 165 or 1 in 163 chance of dying this year is effectively meaningless, and does not take into account other factors (tendency to jaywalk living next to busy street? Like to frequent crowded spots during an epidemic - will or won’t get a vaccine?) that are specific to an individual. Calculating pensions when the market or interest rates can jump around significantly (or inflation makes a difference) is a waste of time beyond a certain point.

IIRC my pension plan was based on thousands of people, and what was considered an appropriate amount for the total fund varied more based on (varying by year) assumed future growth rates than whether someone would perhaps live an extra 6 months. In bad years, the company scrambled to add extra, in good years, they coasted on market growth.