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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.

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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 $FVA_{due}$:

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

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