Time Value of Money
The time value of money is the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal.
In other words, the present value of a certain amount a of money is greater than the present value of the right to receive the same amount of money at time t in the future. This is because the amount a could be deposited in an interest-bearing bank account (or otherwise invested) from now to time t and yield interest. (Consequently, lenders acting at arm's length demand interest payments for use of their financial capital. Additional motivations for demanding interest are to compensate for the risk of borrower default and the risk of inflation, as well as other, more technical considerations.)
All of the standard calculations are based on the most basic formula, the present value of a future sum, "discounted" to a present value. For example, a sum of FV to be received in one year is discounted (at the appropriate rate of r) to give a sum of PV at present.
Some standard calculations based on the time value of money are:
Present Value (PV) of an amount that will be received in the future.
Future Value (FV) of an amount invested (such as in a deposit account) now at a given rate of interest.
Present Value of an Annuity (PVA) is the present value of a stream of (equally-sized) future payments, such as a mortgage.
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.
Present Value of a Perpetuity is the value of a regular stream of payments that lasts "forever", or at least indefinitely.
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 program such as Microsoft Office Excel or OpenOffice.org Calc. The formulas are programmed into most financial calculators and several spreadsheat functions (such as PV, FV, RATE, NPER, and PMT).
For any of the equations below, the formulae 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 r is the interest rate for the relevant period. For an annuity that makes one payment per year, r 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 time period (known as an ordinary annuity), or at the beginning of each time 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+r).
Formulas
Present Value of a Future Sum
The present value formula is the core formula for the time value of money; each of the other formulae is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.
The present value (PV) formula has four variables, each of which can be solved for:
PV is the value at time=0
FV is the value at time=n
i is the rate at which the amount will be compounded each period
n is the number of periods
Future Value of a Present Sum
The future value (FV) formula is similar and uses the same variables.
Present Value of an Annuity
The present value of an annuity (PVA) formula has four variables, each of which can be solved for:
PVA the value of the annuity at time=0
A the value of the individual payments in each compounding period
r equals the interest rate that would be compounded for each period of time
n is the number of payment periods.
Future Value of an Annuity
The future value of an annuity (FVA) formula has four variables, each of which can be solved for:
FV(A) the value of the annuity at time=n
A the value of the individual payments in each compounding period
r equals the interest rate that would be compounded for each period of time
Present Value of a Growing Annuity
Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of G as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.
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