Ideally refined actuarial tables

[Richard Purvey]

Ideally refined actuarial tables and calculating APVs based on them

Imagine if you could get an accurate decrement table, such as a life table, for any given date of birth and effective date, tabulated for every date from this effective date to the terminal date inclusive.

Before using such an ideally refined life table to calculate the APV of an mthly life contingency, you would have to find the dates which are r/m years after the effective date (r either equals 1 to mn or 0 to mn-1, depending on what the mthly life contingency is, and the only practical values for m are 1, 2, 3, 4, 6 or 12).

Now, r multiplied by 1/m years is equal to 12r/m multiplied by 1/12 years.  This means that these future dates could be calculated, using a spreadsheet, by adding 12r/m months to the effective date.

Such a spreadsheet could comprise: the ideally refined life table (tables if more than one life, say, in the case of an mthly type joint life status APV) followed by an inputted value for m followed by an inputted value for v followed by a column for the values for r followed by a column which calculates the dates which are 12r/m months, i.e. r/m years, after the effective date followed by column(s) which would look these dates up in the ideally refined life table(s).  The rest of the calculation, such as the multiplying by v^(r/m), would be done in the subsequent columns.

A final interesting point is that such ideally refined decrement tables would vastly improve the accuracy of an estimate of the APV of an accrued age retirement benefit like the example shown in David Dickson’s book, life contingent risks/section 9.6.2 (career average plans)/page 313/example 9.8, part b.  For example, the APVs of the mthly life annuities could be calculated in increments of one day.

A spreadsheet to do this part of the calculation could start with a column with the first cell left blank (see later, why) and the rest of the cells in this column being populated by the dates starting from the effective date down to and including the date which is one day before the normal retirement date.  The next two columns could be the ideally refined life table followed by an inputted value for m followed by an inputted value for v followed by a column for the values for r followed by a column which calculates the dates which are 12r/m months, i.e. r/m years, after the macro inputted date (see later) followed by a column which would look these dates up in the ideally refined life table followed by a column which would multiply these look ups by v^(r/m).  You would make the first available blank cell in this last column sum these values.  Finally, you would write a macro which would copy each date in the first column then paste it into the initial blank cell at the top of this column and then paste the summation result in the last column into the next available cell in a column to the right of these columns.  Hence, after running the macro, you would have a list of the APVs of the mthly life annuities calculated in increments of one day.

 

Richard Purvey November 2024

 

 

 

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