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The future value determines what a sum of money will be worth at a period in the future, given various rates of interest, compounding periods, and time. The FV is calculated by multiplying the present value by the accumulation function. The value does not include corrections for inflation or other factors that affect the true value of money in the future. The process of finding the FV is often called capitalization.
On the other hand, the present value (PV) is the value on a given date of a payment or series of payments made at other times. The process of finding the PV from the FV is called discounting.
PV and FV are related, as reflected in the following compounding interest formula.
PV and FV vary directly; when one increases, the other increases, assuming that the interest rate and number of periods remain constant.
The interest rate (or discount rate) and the number of periods are the two other variables that affect the FV and PV. The higher the interest rate, the lower the PV and the higher the FV. The same relationships apply for the number of periods. The more time that passes, or the more interest accrued per period, the higher the FV will be if the PV is constant, and vice versa.
The formula implicitly assumes that there is only a single payment. If there are multiple payments, the PV is the sum of the present values of each payment and the FV is the sum of the future values of each payment.
A perpetuity is a unique type of annuity. Whereas an annuity has a specified end, a perpetuity is a stream of cash flow payments that has no end – it continues forever. Essentially, they are ordinary annuities, but have no end date. There aren’t many actual perpetuities, but the United Kingdom has issued them in the past.
Since there is no end date, the annuity formulas we have explored don’t apply here. There is no end date, so there is no future value formula. To find the FV of a perpetuity would require setting a number of periods which would mean that the perpetuity up to that point can be treated as an ordinary annuity.
There is, however, a PV formula for perpetuities. The PV is simply the payment size (A) divided by the interest rate (r). Notice that there is no n, or number of periods. More accurately, it is what results when you take the limit of the ordinary annuity PV formula as n → ∞.
It is also possible that an annuity has payments that grow at a certain rate per period. The rate at which the payments change is fittingly called the growth rate (g). The PV of a growing perpetuity is represented with the following formula:
It is essentially the same except that the growth rate is subtracted from the interest rate. Another way to think about it is that for a normal perpetuity, the growth rate is just 0, so the formula boils down to the payment size divided by r.
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