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October 4, 2024 at 12:00 pm #25415
Recently I asked the following question on a number of question and answer websites;
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 EulerMclaurin 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?
The many helpful answers tell me that actually it IS possible to produce such lx tables. One of these helpful answers said ” …the data exists to calculate monthly, or daily,…”
This answer, telling me that it is even possible to produce accurate lx tables which have lx tabulated for x in days, got me thinking: early retirement pension factors must be, in some way, related to lx tables, meaning that as long as they are related to such a table, they too can be accurately tabulated for ages, or terms early, in days. And so, naturally, I asked the same people about this. And, again, I got helpful answers telling me that this, too, is possible. They also gave me reasons as to why such lx tables and early retirement pension factor tables are not in widespread use. The two main reasons, they say, are; Reason number one, the improved accuracy which would result would be insignificant (personally, though, I would still want any pension or life insurance calculations, relating to me, done as accurately as possible, but that is maybe just me, I have a background in chemistry and other true sciences). Reason number two, inertia, one answer telling me “I don’t think anyone bothers. It used to be mathematically burdensome, and people got in the habit of interpolating”. At least lack of technology can’t be used as an excuse, since spreadsheets, able to store thousands upon thousands of rows, have been around for at least a couple of decades now!
Ah, well, maybe (hopefully) one day (soon) such lx tables and early retirement pension factor tables will be in widespread use.
Richard Purvey
October 2024

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