Time value of money: Definition from Answers.com

Are you indifferent between receiving $1,000 today and receiving $1,000 one year from today? If your intuition prefers receiving the funds today, rather than one year from today, then your intuition recognizes the time value of money. Owners of cash can permit borrowers to rent the use of their cash. Interest is payment for the use of cash.

Expenditures for an investment most often precede the receipts produced by that investment. Cash received later has less value than cash received sooner. The difference in timing affects whether making an investment will earn a profit. Amounts of cash received at different times have different values. We use interest calculations to make valid comparisons among amounts of cash paid or received at different times.

Concepts

Businesses typically state interest cost as a percentage of the amount borrowed per unit of time. Examples are 12 percent per year and 1 percent per month. When the statement of interest cost includes no time period, then the rate applies to a year; thus "interest at the rate of 12 percent" means 12 percent per year.

The amount borrowed or loaned is the principal. Compound interest means that the amount of interest earned during a period increases the principal, which is then larger for the next interest period.

If you deposit $1,000 in a savings account that pays compound interest at the rate of 6 percent per year, you will earn $60 by the end of one year. If you do not withdraw the $60, then $1,060 will earn interest during the second year. During the second year your principal of $1,060 will earn $63.60 interest; $60 on the initial deposit of $1,000 and $3.60 on the $60 earned the first year. By the end of the second year, you will have $1,123.60. Compounded annually at 8 percent, cash doubles itself in nine years. If a twenty-five-year old invests $2,000 each year which earns 8 percent a year, the retirement fund will grow to more than $425,000 by the time that person reaches age sixty-five.

When only the original principal earns interest during the entire life of the loan, the interest due at the time the borrower repays the loan is simple interest. Simple interest calculations ignore interest on previously earned interest. Nearly all economic calculations, however, involve compound interest.

Problems involving the time value of money generally fall into two groups:

We want to know the future value of cash invested or loaned today.
We want to know the present value, or today's value, of cash to be received or paid at later dates.

Future Value

If you invest $1 today at 12 percent compounded annually, it will grow to $1.12000 at the end of one year, $1.25440 at the end of two years, $1.40493 at the end of three years, and so on, according to the formula:

Fn; = P (1 + r)_n

where

F_n = accumulation or future value

P = one-time investment today

r = interest rate per period

n = number of periods from today

The amount F_n is the future value of the present payment, P, compounded at r percent per period for n periods.

Example. How much will $2,000 deposited today at 8 percent compounded annually be worth 40 years from now? $2,000 will grow to $2,000 × (1.08)⁴⁰ = $2000 × 21.72452 = $434,490 * *(While you can compute 1.08 can be raised to the 40th power manually, future value tables, calculation, and computers can remove the tedium of such computations.)

Present Value

Now, consider how much principal, P, you must invest today in order to have a specified amount, F_n, at the end of n periods. You know the future amount, F_n, the interest rate, r, and the number of periods, n; you want to find P. In order to have $1 one year from today when deposits earn 8 percent, you must invest P of $.92593 today. That is, F₁ = P (1.08)¹ "or" $1 = $.92593 × 1.08.

The number (1 + r) - _n [= 1/(1 + r)ⁿ] equals the present value of $1 to be received after n periods when interest accrues at r percent per period. The discounted present value of $1 to be received n periods in the future is (1 + r) - n when the discount rate is r percent per period for n periods.

Example What is the present value of $1 due 10 years from now if the interest rate (equivalently, the discount rate) r is 8 percent per year? (1 + .08)-¹⁰ × $1 = $.46319 * *(Present value tables and computers simplify such a calculation)

Changing the Compounding Period: Nominal and Effective Rates

"Twelve percent, compounded annually" states the price for a loan; this means that interest increases principal once a year at the rate of 12 percent. Often, however, the price for a loan states that compounding is to take place more than once a year. A savings bank may advertise that it pays 6 percent, compounded quarterly. This means that at the end of each quarter the bank credits savings accounts with interest calculated at the rate 1.5 percent (= 6%/4).

$10,000 invested today at 12 percent, compounded annually, grows to a future value one year later of $11,200. If the rate of interest is 12 percent compounded semiannually, the bank adds 6 percent interest to the principal every six months. At the end of the first six months, $10,000 will have grown to $10,600; that amount will grow to $10,600 × 1.06 = $11,236 by the end of the year. Notice that 12 percent compounded semiannually is equivalent to 12.36 percent compounded annually. At 12 percent compounded monthly, $1 will grow to $1 × (1.01)12 = $1.12683 and $10,000 will grow to $11,268. Thus, 12 percent compounded monthly provides the same ending amount as 12.68 percent compounded annually. Common terminology would say that 12percent compounded monthly has an effective rate of 12.68 percent compounded annually or is equivalent to 12.68 percent compounded annually. If a nominal rate, r, compounds m times per year, the effective rate equals (1 + r/m)^m - 1.

Bibliography

Stickney, Clyde P., and Weil, Roman L. (2000). Financial Accounting: An Introduction to Concepts, Methods and Uses, 9th ed. Ft. Worth, TX: Dryden.

[Article by: ROMAN L. WEIL]

This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (March 2008)

The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time.

For example, 100 dollars of today's money invested for one year and earning 5 percent interest will be worth 105 dollars after one year. Therefore, 100 dollars paid now or 105 dollars paid exactly one year from now both have the same value to the recipient assuming 5 percent interest; using time value of money terminology, 100 dollars invested for one year at 5 percent interest has a future value of 105 dollars.^[1] 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^[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 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^[5].

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.

1 Calculations
2 Formula
3 Derivations
- 3.1 Annuity derivation
- 3.2 Perpetuity derivation
4 Examples
5 Time value of money formulas with continuous compounding
6 See also
7 References
8 External links

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)^[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

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:

$PV \ = \ \frac{FV}{(1+i)^n}$

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 (not necessarily an integer)

The cumulative present value of future cash flows can be calculated by summing the contributions of $F V t$ , the value of cash flow at time=t

$PV \ = \ \sum_{t=0}^{n} \frac{FV_{t}}{(1+i)^t}$

Note that this series can be summed for a given value of n, or when n is $\infty$ .^[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

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:

$PV(A) \,=\,\frac{A}{i} \cdot \left[ {1-\frac{1}{\left(1+i\right)^n}} \right]$

PV(A) is the value of the annuity at time=0
A is the value of the individual payments in each compounding period
i equals the interest rate that would be compounded for each period of time
n is the number of payment periods.

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

Present value of a growing annuity

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 :

$PV\,=\,{A \over (i-g)}\left[ 1- \left({1+g \over 1+i}\right)^n \right]$

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

Where i = g :

$PV\,=\,{A*n \over 1+i}$

Present value of a perpetuity

When $n \rightarrow\infty$ , the PV of a perpetuity (a perpetual annuity) formula becomes simple division.

$PV(P) \ = \ { A \over i }$

When this is an increasing perpetuity, this i becomes i’ 1+i’=(1+i)/(1+g) i’=(i-g)/(1+g)

so A/i’ = A x (1+g)/(i-g) not (A/(i-g))

Present value of a growing perpetuity

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 precisely these 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.

$PVGP \ = \ { A \over (i-g) }$

This is the well known Gordon Growth model used for stock valuation.

Future value of a present sum

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

$FV \ = \ PV \cdot (1+i)^n$

Future value of an annuity

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

$FV(A) \,=\,A\cdot\frac{\left(1+i\right)^n-1}{i}$

FV(A) is the value of the annuity at time = n
A is the value of the individual payments in each compounding period
i is the interest rate that would be compounded for each period of time
n is the number of payment periods

Future value of a growing annuity

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

Where i ≠ g :

$FV(A) \,=\,A\cdot\frac{\left(1+i\right)^n-\left(1+g\right)^n}{i-g}$

Where i = g :

$FV(A) \,=\,A\cdot n(1+i)^{n-1}$

FV(A) is the value of the annuity at time = n
A is the value of initial payment at time 0
i is the interest rate that would be compounded for each period of time
g is the growing rate that would be compounded for each period of time
n is the number of payment periods

Derivations

Annuity derivation

The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.

A single payment C at future time m has the following future value at future time n:

$FV \ = C(1+i)^{n-m}$

Summing over all payments from time 1 to time n, then reversing the order of terms and substituting $k = n - m$ :

$FVA \ = \sum_{m=1}^n C(1+i)^{n-m} \ = \sum_{k=0}^{n-1} C(1+i)^k$

Note that this is a geometric series, with the initial value being $a = C$ , the multiplicative factor being $1 + i$ , with $n$ terms. Applying the formula for geometric series, we get

$FVA \ = \frac{ C ( 1 - (1+i)^n )}{1 - (1+i)} \ = \frac{ C ( (1+i)^n - 1 )}{i}$

The present value of the annuity (PVA) is obtained by simply dividing by $(1 + i) n$ :

$PVA \ = \frac{FVA}{(1+i)^n} = \frac{C}{i} \left( 1 - \frac{1}{(1+i)^n} \right)$

Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:

$\text{Principal} \times i = C$

Principal = C / i

+ goal

Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:

F V = P V (1 + i) n

Initially, before any payments, the present value of the system is just the endowment principal ( $P V = C / i$ ). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments ( $F V = C / i + F V A$ ). Plugging this back into the equation:

$\frac{C}{i} + FVA = \frac{C}{i} (1+i)^n$

$FVA = \frac{C}{i} \left[ \left(1+i \right)^n - 1 \right]$

Perpetuity derivation

Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:

$\left({1 - {1 \over { (1+i)^n } }}\right)$

can be seen to approach the value of 1 as n grows larger. At infinity, it is equal to 1, leaving ${C \over i}$ as the only term remaining.

Examples

Example 1: Present value

One hundred euros to be paid 1 year from now, where the expected rate of return is 5% per year, is worth in today's money:

$P \ = \ F \times (P/F) \ = F \times \ { 1 \over (1+i)^n } \ = \ \frac{\ 100}{1.05} \ = \ 95.23$

So the present value of €100 one year from now at 5% is €95.23.

Example 2: Present value of an annuity — solving for the payment amount

Consider a 10 year mortgage where the principal amount P is $200,000 and the annual interest rate is 6%.

The number of monthly payments is

$n = 10 {\rm \ years} \times 12 {\rm \ months \ per \ year} = 120 {\rm \ months}$

and the monthly interest rate is

$i = { 6 {\rm \% \ per \ year} \over 12 {\rm \ months \ per \ year} } = 0.5 {\rm \% \ per \ month}$

The annuity formula for (A/P) calculates the monthly payment:

$A \ = \ P \times \left( A / P \right) \ = \ P \times { i (1+i)^n \over (1+i)^n - 1 } \ = \ \$200,000 \times { 0.005(1.005)^{120} \over (1.005)^{120} - 1 }$

$= \ \$200,000 \times 0.01110205 \ = \ \$2,220.41 {\rm \ per \ month}$

Example 3: Solving for the period needed to double money

Consider a deposit of $100 placed at 10% (annual). How many years are needed for the value of the deposit to double to $200?

Using the algrebraic identity that if:

$x \ = \ b^y$

then

$y \ = \ {\ln (x) \over \ln(b)}$

The present value formula can be rearranged such that:

$y \ = \ {\ln ({FV \over PV}) \over \ln(1+i)} \ = \ {\ln ({200 \over 100}) \over \ln(1.10)} \ =\ {0.693 \over 0.0953} \ =\ 7.27$ (years)

This same method can be used to determine the length of time needed to increase a deposit to any particular sum, as long as the interest rate is known. For the period of time needed to double an investment, the Rule of 72 is a useful shortcut that gives a reasonable approximation of the period needed.

Example 4: What return is needed to double money?

Similarly, the present value formula can be rearranged to determine what rate of return is needed to accumulate a given amount from an investment. For example, $100 is invested today and $200 return is expected in five years; what rate of return (interest rate) does this represent?

The present value formula restated in terms of the interest rate is:

$i \ = \ \left({FV \over PV}\right)^{1 \over n} - 1 \ = \ \left({200 \over 100}\right)^{1 \over 5} - 1 \ = \ 2^{0.20} - 1 \ = \ 0.15 \ = \ 15%$

Example 5: Calculate the value of a regular savings deposit in the future.

To calculate the future value of a stream of savings deposit in the future requires two steps, or, alternatively, combining the two steps into one large formula. First, calculate the present value of a stream of deposits of $1,000 every year for 20 years earning 7% interest:

$PVA \,=\,A\cdot\frac{1-\frac{1}{\left(1+i\right)^n}}{i} \ = \ 1000\cdot\frac{1-\frac{1}{\left(1+.07\right)^{20}}}{.07} \ = \ 1000\cdot {1- 0.258 \over .07} \ = \ 1000 * 10.594 \ = \ $10,594$

This does not sound like very much, but remember - this is future money discounted back to its value today; it is understandably lower. To calculate the future value (at the end of the twenty-year period):

$FV \ = \ PV (1+i)^n \ = \ $10,594 * (1+.07)^{20} \ = \ $10,594 * 3.87 \ = \ $40,995$

These steps can be combined into a single formula:

$FV \,=\,A\cdot\frac{1-\frac{1}{\left(1+i\right)^n}}{i} \cdot (1+i)^n \,=\,A\cdot\frac{\left(1+i\right)^n-1}{i}$

Example 6: Price/earnings (P/E) ratio

It is often mentioned that perpetuities, or securities with an indefinitely long maturity, are rare or unrealistic, and particularly those with a growing payment. In fact, many types of assets have characteristics that are similar to perpetuities. Examples might include income-oriented real estate, preferred shares, and even most forms of publicly-traded stocks. Frequently, the terminology may be slightly different, but are based on the fundamentals of time value of money calculations. The application of this methodology is subject to various qualifications or modifications, such as the Gordon growth model.

For example, stocks are commonly noted as trading at a certain P/E ratio. The P/E ratio is easily recognized as a variation on the perpetuity or growing perpetuity formulae - save that the P/E ratio is usually cited as the inverse of the "rate" in the perpetuity formula.

If we substitute for the time being: the price of the stock for the present value; the earnings per share of the stock for the cash annuity; and, the discount rate of the stock for the interest rate, we can see that:

${P \over E} \ = \ {1 \over i} \ = \ {PV \over A }$

And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).

${ 1 \over P/E } \ = \ i$

Of course, stocks may have increasing earnings. The formulation above does not allow for growth in earnings, but to incorporate growth, the formula can be restated as follows:

${ P \over E } \ = \ {1 \over (i-g)}$

If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:

$g \ = \ i - {E \over P}$

Time value of money formulas with continuous compounding

Rates are sometimes converted into the continuous compound interest rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulæ above may be restated in their continuous equivalents. For example, the present value at time 0 of a future payment at time t can be restated in the following way, where e is the base of the natural logarithm and r is the continuously compounded rate:

$\ PV \ = \ FVe^{-rt}$

See below for formulaic equivalents of the time value of money formulæ with continuous compounding.

Present value of an annuity

$\ PV \ = \ {A(1-e^{-rt}) \over e^r -1}$

Present value of a perpetuity

$\ PV \ = \ {A \over e^r - 1}$

Present value of a growing annuity

$\ PV \ = \ {A(1-e^{-(r-g)t}) \over e^{(r-g)} - 1}$

Present value of a growing perpetuity

$\ PV \ = \ {A \over e^{(r-g)} - 1}$

Present value of an annuity with continuous payments

$\ PV \ = \ { 1 - e^{(-rt)} \over r }$

References

Crosson, S.V., and Needles, B.E.(2008). Managerial Accounting (8th Ed). Boston: Houghton Mifflin Company.

^ http://www.investopedia.com/articles/03/082703.asp
^ http://www.investopedia.com/terms/p/presentvalue.asp
^ http://www.getobjects.com/Components/Finance/TVM/pva.html
^ http://www.investopedia.com/terms/p/perpetuity.asp
^ http://www.investopedia.com/terms/f/futurevalue.asp
^ Hovey, M. (2005). Spreadsheet Modelling for Finance. Frenchs Forest, N.S.W.: Pearson Education Australia.
^ http://mathworld.wolfram.com/GeometricSeries.html Geometric Series

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Time value of money

Contents

Calculations

Formula

Present value of a future sum

Present value of an annuity for n payment periods

Present value of a growing annuity

Present value of a perpetuity

Present value of a growing perpetuity

Future value of a present sum

Future value of an annuity

Future value of a growing annuity

Derivations

Annuity derivation

Perpetuity derivation

Examples

Example 1: Present value

Example 2: Present value of an annuity — solving for the payment amount

Example 3: Solving for the period needed to double money

Example 4: What return is needed to double money?

Example 5: Calculate the value of a regular savings deposit in the future.

Example 6: Price/earnings (P/E) ratio

Time value of money formulas with continuous compounding

Present value of an annuity

Present value of a perpetuity

Present value of a growing annuity

Present value of a growing perpetuity

Present value of an annuity with continuous payments

See also

References

External links