The time value of money is the value of money with a given amount of interest earned or inflation accrued over a given amount of time. The ultimate principle suggests that a certain amount of money today has different buying power than the same amount of money in the future. This notion exists both because there is an opportunity to earn interest on the money and because inflation will drive prices up, thus changing the “value” of the money.
The time value of money is the central concept in finance theory.For example, £100 of today’s money invested for one year and earning 5% interest will be worth £105 after one year. Therefore, £100 paid now or £105 paid exactly one year from now both have the same value to the recipient who assumes 5% interest; using time value of money terminology, £100 invested for one year at 5% interest has a future value of £105. This notion dates at least to Martín de Azpilcueta (1491–1586) of the School of Salamanca.
The method also allows the valuation of a likely stream of income in the future, in such a way that the annual incomes are discounted and then added together, thus providing a lump-sum “present value” of the entire income stream.
All of the standard calculations for time value of money derive from the most basic algebraic expression for the present value of a future sum, “discounted” to the present by an amount equal to the time value of money. For example, a sum of FV to be received in one year is discounted (at the rate of interest r) to give a sum of PV at present: PV = FV − r·PV = FV/(1+r).
Some standard calculations based on the time value of money are:
Present value The current worth of a future sum of money or stream of cash flows given a specified rate of return.
Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they be earnings or obligations.Present value of an annuity An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of each period for an ordinary annuity while they occur at the beginning of each period for an annuity due.
Present value of a perpetuity is an infinite and constant stream of identical cash flows.
Future value is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today.
Future value of an annuity (FVA) is the future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.
Calculations
There are several basic equations that represent the equalities listed above. The solutions may be found using (in most cases) the formulas, a financial calculator or a spreadsheet. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).
For any of the equations below, the formula may also be rearranged to determine one of the other unknowns. In the case of the standard annuity formula, however, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).
These equations are frequently combined for particular uses. For example, bonds can be readily priced using these equations. A typical coupon bond is composed of two types of payments: a stream of coupon payments similar to an annuity, and a lump-sum return of capital at the end of the bond’s maturity – that is, a future payment. The two formulas can be combined to determine the present value of the bond.
An important note is that the interest rate i is the interest rate for the relevant period. For an annuity that makes one payment per year, i will be the annual interest rate. For an income or payment stream with a different payment schedule, the interest rate must be converted into the relevant periodic interest rate. For example, a monthly rate for a mortgage with monthly payments requires that the interest rate be divided by 12 (see the example below). See compound interest for details on converting between different periodic interest rates.
The rate of return in the calculations can be either the variable solved for, or a predefined variable that measures a discount rate, interest, inflation, rate of return, cost of equity, cost of debt or any number of other analogous concepts. The choice of the appropriate rate is critical to the exercise, and the use of an incorrect discount rate will make the results meaningless.
For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due). If you are using a financial calculator or a spreadsheet, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the Present Value of an annuity due simply multiply the PV of an ordinary annuity by (1 + i).
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