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The time value of money is based on 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 particular, if one received the payment today, one can then earn interest on the money until that specified future date.

All of the standard calculations are based on the most basic formula, the present value of a future sum, "discounted" to the present. 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.

- Present Value of a Annuity (PVA) is the present value of a stream of (equally-sized) future payments, such as a mortgage.

- Present Value of a Perpetuity is the value of a regular stream of payments that lasts "forever", or at least indefinitely.

- Future Value (FV) of an amount invested (such as in a deposit account) now at a given rate of interest.

- 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.

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 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).

- 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 (not necessarily an integer)

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

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

- $PV\; =\; sum\_\{t=0\}^\{n\}\; frac\{FV\_\{t\}\}\{(1+i)^t\}$

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

- PVA the value of the annuity at time=0
- A 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.
- :$PV(A)\; ,=,frac\{A\}\{i\}\; cdot\; left[\{1-frac\{1\}\{left(1+iright)^n\}\}\; right]$

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

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

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

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

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

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

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

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

- 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
- i, the interest rate that would be compounded for each period of time
- n, the number of payment periods
- :$FV(A)\; ,=,Acdotfrac\{left(1+iright)^n-1\}\{i\}$

- The future value of a growing annuity (FVA) formula has five variables, each of which can be solved for:
- FV(A), the value of the annuity at time = n
- A, the value of initial payment at time 0
- i, the interest rate that would be compounded for each period of time
- g, the growing rate that would be compounded for each period of time
- n, the number of payment periods

Where i <> g :

- :$FV(A)\; ,=,Acdotfrac\{left(1+iright)^n-left(1+gright)^n\}\{i-g\}$

Where i = g :

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

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 $k\; =\; n-m$:

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

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

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:

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

- $text\{Principal\}\; =\; C\; /\; i$

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:

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

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

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

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

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

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

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:

- $A\; =\; P\; times\; left(A\; /\; P\; right)\; =\; P\; times\; \{\; i\; (1+i)^n\; over\; (1+i)^n\; -\; 1\; \}$

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

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.

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

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

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

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:

- $FV\; ,=,Acdotfrac\{1-frac\{1\}\{left(1+iright)^n\}\}\{i\}\; cdot\; (1+i)^n\; ,=,Acdotfrac\{left(1+iright)^n-1\}\{i\}$

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:

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

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

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

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

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

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

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

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

- Net present value
- Option time value
- Discounting
- Discounted cash flow
- Exponential growth
- Hyperbolic discounting
- Internal rate of return
- Perpetuity
- Real versus nominal value
- Time preference
- Earnings growth

- Time Value of Money CalculatorFarsight Calculator
- Time Value of Money from studyfinance.com at the University of Arizona
- Time Value of Money Ebook

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Last updated on Sunday August 31, 2008 at 05:06:49 PDT (GMT -0700)

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Last updated on Sunday August 31, 2008 at 05:06:49 PDT (GMT -0700)

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