Editing Time value of money (section)

==Derivations==

===Annuity derivation===
The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where ''C'' is the payment amount and ''n'' the period.

A single payment ''C'' at future time ''m'' has the following future value at future time ''n'':

:<math>FV \ = C(1+i)^{n-m}</math>

Summing over all payments from time 1 to time ''n'', then reversing the order of terms and substituting ''k'' = ''n'' − ''m'':

:<math>FVA \ = \sum_{m=1}^n C(1+i)^{n-m} \ = \sum_{k=0}^{n-1} C(1+i)^k</math>

Note that this is a [[geometric series]], with the initial value being ''a'' = ''C'', the multiplicative factor being 1 + ''i'', with ''n'' terms. Applying the formula for geometric series, we get:

:<math>FVA \ = \frac{ C ( 1 - (1+i)^n )}{1 - (1+i)} \ = \frac{ C ( 1 - (1+i)^n )}{-i} </math>

The present value of the annuity (PVA) is obtained by simply dividing by <math>(1+i)^n</math>:

:<math>PVA \ = \frac{FVA}{(1+i)^n} = \frac{C}{i} \left( 1 - \frac{1}{(1+i)^n} \right)</math>

Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:

:<math>\text{Principal} \times i = C</math>
:<math>\text{Principal} = \frac{C}{i} + \text{goal} </math>

Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:
:<math>FV = PV(1+i)^n</math>

Initially, before any payments, the present value of the system is just the endowment principal, <math>PV = \frac{C}{i}</math>. At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (<math>FV = \frac{C}{i} + FVA</math>). Plugging this back into the equation:
:<math>\frac{C}{i} + FVA = \frac{C}{i} (1+i)^n</math>
:<math>FVA = \frac{C}{i} \left[ \left(1+i \right)^n - 1 \right]</math>

===Perpetuity derivation===
Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:
:<math> \left({1 - {1 \over { (1+i)^n } }}\right) </math>
can be seen to approach the value of 1 as ''n'' grows larger. At infinity, it is equal to 1, leaving <math> {C \over i} </math> as the only term remaining.