In the case of the calculation of the future value of an annuity-due interest it means that each term is interest bearing one period longer than in the case of annuity-immediate interest.

This matters is we want to calculate the future value of #n# yearly terms of #P# annuity-due interest payments with a growth rate per year of #i#.

The balance is then calculated as follows:

\[FV=P \cdot (1+i)^{n} + P \cdot (1+i)^{n-1} + P \cdot (1+i)^{n-2} + \cdots + P \cdot (1+i)^2 + P \cdot (1+i)\]

We can move #(1+i)# outside the brackets:

\[FV=(1+i)\cdot \left(P \cdot (1+i)^{n-1} + P \cdot (1+i)^{n-2} + P \cdot (1+i)^{n-3} + \cdots + P \cdot (1+i) + P\right)\]

This now says that the future value with annuity-due interest is equal to #(1+i)# multiplied by the future value with annuity-immediate interest.

Hence, \[FV=P \cdot (1+i) \cdot \frac{(1+i)^n-1}{i}\]

For the calculation of the present value on the other hand holds that each term is one period less interest bearing. This can be seen in the following calculation.

If we want to calculate the present value of #n# terms of interest payments #P# with growth rate #i# we find:

\[PV=P \cdot \frac{1}{(1+i)^{n-1}} + P \cdot \frac{1}{(1+i)^{n-2}} + \ldots + P \cdot \frac{1}{1+i} + P\]

This is equal to:

\[PV=(1+i) \cdot \left(P \cdot \frac{1}{(1+i)^{n}} + P \cdot \frac{1}{(1+i)^{n-1}} + \ldots +P \cdot \frac{1}{(1+i)^2} + P \cdot \frac{1}{1+i}\right)\]

We recognize this as #(1+i)# multiplied by the present value of the situation with annuity-immediate interest.

Hence, we find:

\[PV = P \cdot (1+i) \cdot \frac{1-(1+i)^{-n}}{i}\]