Why Present and Future Values of an Annuity Due Use the Same Growth Factor

Summary

The document addresses the relationship between present and future values for ordinary annuities and annuities due. It corrects the premise that the two annuity-due formulas require unrelated adjustments: once the present value is known, its future value is obtained by moving that amount forward by the accumulation factor for the number of periods. The same conversion applies to either payment timing convention.

The answer derives the ordinary-annuity future-value formula by multiplying its present-value expression by the period growth factor. It then states that the annuity-due future value follows the same present-to-future relationship. The timing difference between ordinary and due annuities is reflected in when payments occur and hence in the present-value and future-value formulas themselves. The explanation gives the general algebraic principle, but does not expand the annuity-due formulas or discuss special cases such as zero interest rates, irregular payment intervals, or differing compounding conventions.

Key ideas

• Future value is obtained from present value by applying the accumulation factor over the relevant periods.
• The present-to-future conversion relationship holds for both ordinary annuities and annuities due.
• Payment timing changes the annuity valuation formulas, while the conversion between their present and future values remains consistent.
• Compounding and timing conventions must be specified when applying annuity formulas.

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# Present Value versus. Future Value of an Annuity Due


# Present Value versus. Future Value of an Annuity Due












To determine the present value of an annuity due, 1 is added to the discount factor of the ordinary annuity. However, to determine the future value of an annuity due, 1 is removed from the discount factor of the ordinary annuity.

What is the mathematical principle and/or intuition for this difference? I understand that broadly, PV and FV are inverses/opposites of one another but I need to know what exactly is going.

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/41669

I do not follow your analysis. In the case of either type of annuity the FV is equal to the PV times $(1+r)^n$. This factor is simply the factor which translates any amount in period 0 into an equivalent amount in period n.

For an ordinary annuity: $$PVA=PMT \frac{1}{r}[1-\frac{1}{(1+r)^n}]$$

When this value is "transferred" to period $n$ by multiplying by the $n$ period growth factor $(1+r)^n$ we get the FV formula:

$$FVA=PVA (1+r)^n=PMT\frac{1}{r}[(1+r)^n-1]$$

The exact same relationship exists between $PVA_{due}$ and $FVA_{due}$:

$$FVA_{due}=PVA_{due} (1+r)^n$$

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.