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Relating the Present and Future Values of an Ordinary Annuity

Article Quant Q&A · Author: pabhp
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Summary

The document derives the relationship between the present value and future value of an ordinary annuity with equal payments. It writes present value as the sum of each payment discounted from its receipt date, and future value as the same payments accumulated to the final date. Because both expressions contain the same payment terms at different time positions, multiplying present value by the accumulation factor for the full term gives the future value.

For annual payments of amount K, rate r, and n periods, the stated result is FV = PV × (1 + r)^n. The derivation makes the timing assumption explicit: payments arrive at each period end, beginning in year one and ending in year n. It applies to a constant rate and matching annual period convention; different payment timing, compounding frequency, or varying rates would require adjusting the accumulation factors.

Key ideas

  • An ordinary annuity pays equal amounts at the end of each period.
  • Present value discounts every payment back to the valuation date.
  • Future value accumulates each payment to the final period.
  • With a constant rate and matching period convention, future value equals present value multiplied by the term accumulation factor.

Tags

Fixed incomeStatistics
Full text
# Find a relationship between the present value and future value of an annuity


# Find a relationship between the present value and future value of an annuity












The following is a previous examination question in Financial Mathematics:

> If $A, r, n, PV$ and $FV$ represents the ordinary annuity (annuity immediate) amount, rate of interest, number of years, the present value and the future value of the annuity respectively, find a relationship between $PV$ and $FV$.

I need some help in illustrating this "relationship" as this question was allocated 6 marks. I know that the present value of annuity immediate can be given by: $PV = K\left[ {\frac{{1 - {{\left( {1 + r} \right)}^{ - n}}}}{r}} \right]$ where $K$ is the amount per period; but how can this be used in this case?

I need some help in illustrating this "relationship" as this question was allocated 6 marks.

I know that the present value of annuity immediate can be given by: $PV = K\left[ {\frac{{1 - {{\left( {1 + r} \right)}^{ - n}}}}{r}} \right]$ where $K$ is the amount per period; but how can this be used in this case?

I know that the present value of annuity immediate can be given by: $PV = K\left[ {\frac{{1 - {{\left( {1 + r} \right)}^{ - n}}}}{r}} \right]$ where $K$ is the amount per period; but how can this be used in this case?

## Answer by Magic is in the chain (score 1, accepted)

https://quant.stackexchange.com/a/42388

Assuming the annuity pays K every year from year 1 to n, you can write it’s PV as follows:

$PV=K \left( \frac{1}{1+r} +\frac{1}{(1+r)^2 }+ \dots + \frac{1}{(1+r)^n} \right)$

And FV, by noting that the first K is invested for n-1 periods, and the last one is received at n:

$FV=K \left( (1+r)^{n-1} + (1+r)^{n-2}+ \dots+ 1 \right) $

Now one just needs to compare the expressions on the right hand side of both equations. Multiplying PV by $(1+r)^n$ will then make the rhs of the first equal to that of the second equation, and hence:

$FV=PV \,(1+r)^n$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.