Deferred Annuities

calendar_today October 26, 2025 article 301 words access_time 2 min replay Modified October 26, 2025

Table of Contents

Concept

Asked to evaluate:
1. Present value before annuity’s starting period
2. Accumulated value after annuity’s ending period
3. Current value during an annuity
We will break each of these into specific subcases

Assume there are $(m+1)$ periods before the first payment
Also assume there are $n$ periods during the annuity (therefore $(m + n)$ periods total)
We are asked to find the present value at $t=0$
- This is equal to $\nu^m a_{\left< n \right>}$
- In other words, it’s the present value at the beginning of the annuity $(t=m)$ discounted by $m$ time periods
Also valid to think of it as a subtraction of annuities; by the addition identity for annuities, we have:

$$\text{PV} = a_{\left< m + n \right>} - a_{\left< m \right>} = \nu^m a_{\left< n \right>}$$

It’s the same concept; just accumulate the annuity, take the result, and accumulate it for $m$ periods

$$\text{FV} = s_{\left< m + n \right>} - s_{\left< m \right>} = (1+i)^m s_{\left< n \right>}$$

Suppose the annuity has a term of $n$ periods and we are varying $m$ within the term
Then, the present value of the annuity at the $m$‘th payment date (also known as the current value) is:

$$(1+i)^m a_{\left< n \right>} = \nu^{n-m} s_{\left< n \right>}$$

Basically: you have to accumulate the present value and discount the future value of the annuity to $t=m$
A relation with less entropy that is equivalent to the LHS and RHS of the above equation:
- This basically means you can find the current value by taking the sum of:
  1. The future value of the annuity at $t=m$
  2. The present value of the annuity at $t=n-m$ (the value you must subtract $n$ by to get $m$)

$$s_{\left< m \right>} + a_{\left< n - m \right>}$$

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