Method 0 is the “traditional method”. Calculations of all functions (annuity, life insurance, commutation functions, etc.) are done at whole ages, and linear interpolation is used to handle fractional ages. This is the least precise method, but is reasonably good for most purposes.

For example, the present value of a life annuity at age 𝑥+𝑡 is based on the calculation at age 𝑥 and at age 𝑥 + 1, and then we do a straight line interpolation:

for integral 𝑥 and 0 ≤ 𝑡 ≤ 1

Formula 3

 

A proof of Formula 3 is provided below.

Note:

We also interpolate on whole numbers for other fractional values, such as: number of years certain, deferred commencement age, stop age, etc.

Fractional Ages

Interpolation on whole numbers in Formula 4 is based on the assumptions that  and its inverse  are linear functions on 𝑡 for integral 𝑥 and 0 ≤ 𝑡 ≤ 1, hence  is linear as well. This is based on both UDD and the Balducci hypotheses.

for integral 𝑥 and 0 ≤ 𝑡 ≤ 1

Formula 4

 

Then the present value of a life annuity of 1 payable annually, at the beginning of the year, to a life aged 𝑥+𝑡 is the sum of a discounted expected cash flow, which is expressed as:

Substituting for in Formula 5 gives:

.

Payments more frequently that annually

For payments more frequently that annually, we use the Woolhouse method to calculate .