Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up.
Sign up to join this community
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.
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 $FVA_{due}$:
$$FVA_{due}=PVA_{due} (1+r)^n$$
