Net Present Value Npv

I will provide an NPV formula to calculate net present value of a perpetuity, annuity, project, future cash flows and loan illustrated with NPV Calculation. Net present value is the sum of discounted future cash flows brought down to reflect their worth as of present day. Any amount of money that is offered to be paid to you in future has lower worth than it`s face value. The decline in money`s face value may be attributed to interest rate and inflation.

NPV of Perpetuity

A perpetuity is defined as a never ending stream of payments or receipts. A classic example of perpetuity is that of a perpetual bond that never matures and makes periodic interest payments for an indefinite time period. You can opt to sell the bond at a future date at a price that is discounted at the market rate of return. To find the present value of the perpetuity, you have to divide the perpetuity amount by the discount rate. Yet if you were considering the net present value of perpetuity that considered the price you paid for the perpetuity then you would have to subtract the present value of perpetuity from the price you paid for it. Let me illustrate this with a example calculation for a perpetuity that you purchased for a $100 and it promised to pay 9% annual interest for indefinite period of time. See the following calculations for present value and net present value of this perpetuity.

PV of Perpetuity = Payment/Interest rate = $10/9% = 10/0.09 = $111 NPV of Perpetuity = -Cost + Payment/Interest rate = -$100 + $10/9% = -100 + 10/0.09 = -100 + 111.11 = $11.11

NPV of a lump sum

Even though the correct technical term is finding present value of a lump sum that is discounted at a discount rate i% for n number of periods, yet I have noticed a large number of queries this page attracts seem to be trying to find net present value of a lump sum. Disregarding the difference in terminology, I will explain to you how a single amount of money is discounted at an interest rate for given number of years to reflect it`s worth as of present. The formula for finding net present value of a lump sum is the product of the money amount and the present value interest factor as shown below:

PV = R x PVIF(i%, n)

here R is the money amount due at a future date and PVIF is the present value interest factor which brings down the value of future value to it`s present. There are two variations of the PVIF formula, the first is used when interest is compounded per period and the second one is applicable when interest is compounded continuously. I will show you both these formulas now The formula for PVIF when interest is compounded discretely is as follows

PVIF(i%, n) = (1+i)-n

And the formula for PVIF when interest is compounded continuously is as follows

PVIF(i%, n) = e-in

As an example I will explain finding net present value of $10,000 due 10 years from now given the discount rate of 10%. For the sake of complete argument I will make use of both variations of the PVIF formula to find net present value of $10,000.

Let us start with assuming that interest rate is compounded discretely thus we have the following npv calculation: NPV = $10,000 x PVIF(10%, 10) NPV = $10,000 x (1+10%)-10 NPV = $10,000 x (1.10)-10 NPV = $10,000 x 0.3855433 NPV = $3,855.43

Now let`s have look at net present value calculation when interest is compounded continuously NPV = $10,000 x PVIF(10%, 10) NPV = $10,000 x e-10%x10 NPV = $10,000 x e-1 NPV = $10,000 x 2.7182818284590452353602874713527-1 NPV = $10,000 x 0.36787944 NPV = $3,678.79

You may have noticed that net present value is lower when interest is compounded continuously as compared to when interest is compounded discretly.

NPV of an annuity

An annuity is a series of periodic payments or receipts for a fixed or definite period of time. Annuity payments or receipts may occur at either start of period or end of period. Payments in to a pension fund, provident fund, 401K plan, or a savings account require you to make start of period payments; this sort of an annuity is referred to as an annuity due. Mortgage payments and loan repayments require you to make end of period payments and this sort of an annuity is referred to as an ordinary annuity. To find net present value of an annuity, we discount each of payments at an interest rate for the time period t where t ranges from 1 to n for an ordinary annuity and from 0 to n-1 for an annuity due. If the annuity payments or receipts are in constant amounts we can use the shortened formula listed below to find the NPV of an annuity with constant or uniform series of cash flows

Annuity NPV Formula

The first of these formula is for finding NPV of an ordinary annuity where payments or receipts in constant amounts are expected at the end of the period. An example of this type of receipt would be a payment from a pension fund at the end of each month or a payment for a home mortgage at the end of each month or quarter.

NPV = R [1 - {1/(1+i%)^n}]/i%

Here we discount each of the payments or receipts from the annuity at the interest rate, for the time period starting at 1 and ending at n.

Annuity Due NPV Formula

The second of these formula is for an annuity due where payments or receipts in constant amounts are expected at the start of the period. An example of this would be monthly housing rent payment or lease payment for machinery.

NPV = R [1 - {1/(1+i%)^n}](1+i%)/i%

Here each payment or receipt is discounted at interest rate i for time period t-1