# Time value of money Wikipedia the free encyclopedia

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From Wikipedia, the free encyclopedia

A **time value of money** calculation is one which solves for one of several variables in a financial problem. In a typical case, the variables might be: a balance (the real or nominal value of a debt or a financial asset in terms of monetary units); a periodic rate of interest; the number of periods; and a series of cash flows (in the case of a debt, these are payments against principal and interest; in the case of a financial asset, these are contributions to or withdrawals from the balance). More generally, the cash flows may not be periodic but may be specified individually. Any of the variables may be the independent variable (the sought-for answer) in a given problem. For example, one may know that: the interest is 0.5% per period (per month, say); the number of periods is 60 (months); the initial balance (of the debt, in this case) is 25,000 units; and the final balance is 0 units. The unknown variable may be the monthly payment that the borrower will need to pay.

For example, £100 invested for one year, earning 5% interest, will be worth £105 after one year; therefore, £100 paid now *and* £105 paid exactly one year later *both* have the same value to a recipient who expects 5% interest. That is, £100 invested for one year at 5% interest has a *future value* of £105. [ 1 ] This notion dates back at least to Martín de Azpilcueta (1491–1586) of the School of Salamanca .

This principle allows for the valuation of a likely stream of income in the future, in such a way that 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, the future value sum to be received in one year is discounted at the rate of interest to give the present value sum :

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;*the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to valuing future cash flows properly, whether they be earnings or obligations. [ 2 ]

**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. [ 3 ]

**Present value of a perpetuity** is an infinite and constant stream of identical cash flows. [ 4 ]

**Future value**. The value of an asset or cash at a specified date in the future, based on the value of that asset in the present. [ 5 ]

**Future value of an annuity (FVA)**. The future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.

## Contents

## Calculations [ edit ]

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). [ 6 ]

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* ).

## Formula [ edit ]

The following formula use these common variables:

*PV*is the value at time=0 (present value)*FV*is the value at time=n (future value)*A*is the value of the individual payments in each compounding period*n*is the number of periods (not necessarily an integer)*i*is the discount rate. or the interest rate at which the amount will be compounded each period*g*is the growing rate of payments over each time period

### Future value of a present sum [ edit ]

The future value (FV) formula is similar and uses the same variables.

### Present value of a future sum [ edit ]

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:

The cumulative present value of future cash flows can be calculated by summing the contributions of *FV _{t}* . the value of cash flow at time

*t*

Note that this series can be summed for a given value of *n*. or when *n* is ∞. [ 7 ] This is a very general formula, which leads to several important special cases given below.

### Present value of an annuity for n payment periods [ edit ]

In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for:

To get the PV of an annuity due. multiply the above equation by (1 + *i* ).

### Present value of a growing annuity [ edit ]

In this case each cash flow grows by a factor of (1+g). 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.

**Where i ≠ g :**

**Where i = g :**

To get the PV of a growing annuity due. multiply the above equation by (1 + *i* ).

### Present value of a perpetuity [ edit ]

A perpetuity is payments of a set amount of money that occur on a routine basis and continues forever. When *n* → ∞, the *PV* of a perpetuity (a perpetual annuity) formula becomes simple division.

Present Value of Int Factor Annuity

### Present value of a growing perpetuity [ edit ]

When the perpetual annuity payment grows at a fixed rate (g) the value is theoretically determined according to the following formula. In practice, there are few securities with precise characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.

### Future value of an annuity [ edit ]

The future value of an annuity (FVA) formula has four variables, each of which can be solved for:

### Future value of a growing annuity [ edit ]

The future value of a growing annuity (FVA) formula has five variables, each of which can be solved for: