Differences in calculating the present value and future value of a lump sum, annuity, perpetuity and a series of unequal (multiple) cash flows

We can highlight the diffrances and the similarity between all of them are as follow

In present value calculation we calculate at present value of future cash flow as against in calculation of the future value we calculate the future cash flow i.e what will be the amount at the future point of time

Lump Sum

The lump sum has the following characteristics:

Formula for of FV and PV for lump sum

(1) FV = PV * (1 + r)N
(2) PV = FV * { 1 }
(1 + r)N

Where: FV = future value of a single sum of money,
PV = present value of a single sum of money, R = annual interest rate,
and N = number of years

We can see that to calculate present value and Future value of lump sum amount is different

Now let us find out the formula for annuity

Future Value Annuity Factor = ((1 + r)n – 1)/r

Present Value Annuity Factor = (1 – (1 + r)-n /r

Where r = interest rate and N = number of payment

The FV annuity factor formula gives the future total dollar amount of a series of $1 payments, but in problems there will likely be a periodic cash flow amount given .PV of an annuity: the formula listed above shows today’s value of a series of $1 payments to be received in the future. To calculate the PV of an annuity, multiply the annuity amount A by the present value annuity factor.The FV and PV annuity factor formulas work with an ordinary annuity

In order to call a series of cash flows an annuity and use the PVA and FVA equations, the following must be true:

1. All payments must be equal and consecutive.
2. Payments last for a certain period.
3. The first payment starts one period from the beginning of the time line (time period for PV).

The last payment is at the end of time line (time period for FV).

FV and PV of Similar or Uneven Cash Flows
The PV and FV annuity formulas consider the level and sequential cash flows, but if a problem breaks this assumption, the annuity formulas no longer apply. To solve problems with uneven cash flows, each cash flow must be discounted back to the present (for PV problems) or compounded to a future date (for FV problems); then the sum of the present (or future) values of all cash flows is taken. In practice, particularly if there are many cash flows, this exercise is usually completed by using a spreadsheet.

For this type of TVOM, there are many cash flows over the time line. Cash flows can be all the same or all different. Cash flows are not necessarily consecutive.

To solve for the Future Value (FV) of multiple cash flows, simply treat each cash flow as a lump sum and then add them up:

FV = FV of C1 + FV of C2 + FV of C3
FV = C1 (1 + r)2 + C2 (1 + r)1 + C3 (1 + r)0

Similarly, to solve for the Present Value (PV) of multiple cash flows:
PV = PV of C1 + PV of C2 + PV of C3