The EAC PV Excel add-in employs various calculation methods to determine the present value of a life annuity. This provides the user with flexibility in matching the results produced by other systems. The two general methods are based on (1) the traditional “commutation function” approach, and (2) a “cash flow” approach.

·        The “commutation function” approach follows traditional actuarial methodology based on certain simplifying assumptions for speed and ease of calculation.

·        The “discounted cash flow” approach that is a projection of future cash flow discounted with interest and probability of payment. This approach facilitates the use of complications such as generational mortality, spot rates and forward rates while producing a theoretically more accurate calculation.

The methods differ in the manner in which they handle (1) fractional ages and (2) payments that are made more frequently than once per year.

For present value calculations, if age 𝑥 is a whole number and payments are annual, the present value calculation is relatively straightforward. But if that is not the case, complexities arise because mortality tables have 𝑙𝑥 values for only whole ages, necessitating the use of approximation methods. Several present value calculation methods are offered in the EAC Tool, allowing flexibility for user in matching results that may have been calculated by other systems.

Simplifying assumptions are shown in the table below, depending on which calculation method is used.

Calculation Method	Type of Calculation	Fractional Ages For integral 𝑥, 0 ≤ 𝑡 ≤ 1	Frequency of Payments For 𝑚 > 1	Comments
0	Discounted expected cash flow	·    𝑙𝑥+𝑡 is  linear (UDD) ·    𝑣𝑡 is linear ·    1/𝑙𝑥+𝑡 is linear (Balducci) ·    𝑝𝑥+t is linear	Woolhouse	This is the “traditional method” that does calculations using whole numbers and interpolation to handle factional values interpolation on whole ages, for example,  and  are linear functions of 𝑡 for integral 𝑥 and 0 ≤ 𝑡 ≤ 1: Also, interpolate on whole numbers for fractional number of years certain. This is the least precise method but is reasonably good for most purposes.
1		𝑙𝑥+𝑡 is linear (UDD)	Exact	Straightforward, theoretically most correct. Easily handles fractional ages with mid-year events. Calculation is -times slower than other methods, but still very fast in the EAC add-in.
2		𝑙𝑥+𝑡 is linear (UDD)	Woolhouse	A faster calculation than Method 1, more precise that Method 0.
8	Commutation functions	𝑙𝑥+𝑡 is linear (UDD)	Woolhouse	Interpolate on 𝑙𝑥+𝑡 to get 𝐷𝑥+𝑡 and 𝑁𝑥+𝑡. Do not interpolate on 𝑣𝑡.
9		𝑣𝑡 𝑙𝑥+𝑡 is linear	Woolhouse	Interpolate on 𝑣𝑥+𝘵 and 𝑙𝑥+𝑡 to get 𝐷𝑥+𝑡 and 𝑁𝑥 +𝑡.

 

In general, for a person age 𝑥, and payment frequency of 𝑚 payments per year:

·       If 𝑥 is a whole number, and 𝑚 = 1, then:

All methods produce the same result.

·       If 𝑥 is a whole number, and 𝑚 ≠ 1, then:

Method 1 produces the most accurate result.

All other methods produce the same result, which is slightly different than Method 1.

·       If 𝑥 is not a whole number, then:

Method 1 produces the most accurate result.

All other methods produce slightly different results.

Note:

Method 1 always produces the most accurate result. The other methods use simplifying assumptions, producing a slightly less accurate result, but good enough for practical purposes.