Information about Time Value Of Money
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 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:
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).
To get the PV of a growing annuity due, multiply the above equation by (1+r).
A single payment C at future time i has the following future value at future time n:
Summing over all payments from time 1 to time n, then reversing the order of terms and substituting
:
, the multiplicative factor being 
, with 
terms. Applying the formula for geometric series, we get
The present value of the annuity (PVA) is obtained by simply dividing by
:
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:
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:
Initially, before any payments, the present value of the system is just the endowment principal (
). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (
). Plugging this back into the equation:
as the only term remaining.
The number of monthly payments is
and the monthly interest rate is
The annuity formula for (A/P) calculates the monthly payment:
Using the algrebraic identity that if:
then
The present value formula can be rearranged such that:
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 time period needed.
The present value formula restated in terms of the interest rate is:
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):
These steps can be combined into a single formula:
For example, stocks are commonly noted as trading at a certain price/earnings 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:
And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).
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:
If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:
See below for formulaic equivalents of the time value of money formulae with continuous compounding.
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.
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The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.
..... Click the link for more information.
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.
..... Click the link for more information.
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 spreadsheet 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
- n is the number of payment periods.
- :
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.
To get the PV of a growing annuity due, multiply the above equation by (1+r).
Present value of a perpetuity
The PV of a perpetuity (a perpetual annuity) formula is simple division.
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. True.
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 time period.A single payment C at future time i has the following future value at future time n:
Summing over all payments from time 1 to time n, then reversing the order of terms and substituting
:
, the multiplicative factor being , with terms. Applying the formula for geometric series, we get
The present value of the annuity (PVA) is obtained by simply dividing by
:
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:
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:
Initially, before any payments, the present value of the system is just the endowment principal (
). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (). Plugging this back into the equation:
Perpetuity derivation
Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:
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:
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
and the monthly interest rate is
The annuity formula for (A/P) calculates the monthly payment:
- :
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:
then
The present value formula can be rearranged such that:
(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 time 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:
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:
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):
These steps can be combined into a single formula:
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 price/earnings 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:
And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).
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:
If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:
Time value of money formulae with continuous compounding
Rates are sometimes converted into the continous compound interest rate equivalent because the continuous equivalent is more convenient (for example, more easily differentiated). Each of the formulae above may be restated in their continuous equivalents. For example, the present value of a future payment can be restated in the following way, where e is the base of the natural logarithm:
See below for formulaic equivalents of the time value of money formulae 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
- 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
External links
- Time Value of Money from studyfinance.com at the University of Arizona
An investor is any party that makes an investment.
The term has taken on a specific meaning in finance to describe the particular types of people and companies that regularly purchase equity or debt securities for financial gain in exchange for funding an expanding company.
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The term has taken on a specific meaning in finance to describe the particular types of people and companies that regularly purchase equity or debt securities for financial gain in exchange for funding an expanding company.
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Present value is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk.
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bank is a commercial or state institution that provides financial services , including issuing money in various forms, receiving deposits of money, lending money and processing transactions and the creating of credit.
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This article or section needs copy editing for grammar, style, cohesion, tone and/or spelling.
You can assist by [ editing it] now. A how-to guide is available, as is general .
This article has been tagged since February 2007.
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You can assist by [ editing it] now. A how-to guide is available, as is general .
This article has been tagged since February 2007.
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The arm's length principle (ALP) is the condition or the fact that the parties to a transaction are independent and on an equal footing.
The principle is often invoked to avoid undue government influence over other bodies, such as the legal system, the press, or the arts.
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The principle is often invoked to avoid undue government influence over other bodies, such as the legal system, the press, or the arts.
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Financial Capital vs. Real Capital
Financial capital refers to the funds provided by lenders (and investors) to businesses to purchase real capital like equipment for producing goods/services.
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Financial capital refers to the funds provided by lenders (and investors) to businesses to purchase real capital like equipment for producing goods/services.
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In essence financial risk is any risk associated with money.
A common concern with any investment is that you may lose the money you invest - your capital.
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Investment related
Depending on the nature of the investment, the type of 'investment' risk will vary.A common concern with any investment is that you may lose the money you invest - your capital.
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Inflation is measured as the growth of the money supply in an economy, without a commensurate increase in the supply of goods and services. This results in a rise in the general price level as measured against a standard level of purchasing power.
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For other uses, see Annuity.
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.
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Simple Description
A perpetuity is an annuity that has no definite end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence (although the British government has issued them in the past, and they are known and still trade as..... Click the link for more information.
spreadsheet is a rectangular table (or grid) of information, often financial information. The word came from "spread" in its sense of a newspaper or magazine item (text and/or graphics) that covers two facing pages, extending across the center fold and treating the two pages as one
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Microsoft Excel (full name Microsoft Office Excel) is a spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS. It features calculation, graphing tools, pivot tables and a macro programming language called VBA (Visual Basic for
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OpenOffice.org Calc is the spreadsheet component of the OpenOffice.org software package.
Calc is similar to Microsoft Excel, with a roughly equivalent range of features. Calc is capable of opening and saving spreadsheets in Microsoft Excel file format.
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Calc is similar to Microsoft Excel, with a roughly equivalent range of features. Calc is capable of opening and saving spreadsheets in Microsoft Excel file format.
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bond is a debt security, in which the authorized issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date, termed maturity.
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Maturity may refer to:
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- Sexual maturity
- Mature technology, a term indicating that a technology has been in use and development for long enough that most of its initial problems have been overcome
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Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (i.e. interest is compounded).
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spreadsheet is a rectangular table (or grid) of information, often financial information. The word came from "spread" in its sense of a newspaper or magazine item (text and/or graphics) that covers two facing pages, extending across the center fold and treating the two pages as one
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Present value is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk.
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Future value measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate; this value does not include corrections for inflation or other factors that affect the true value of money in the future.
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For other uses, see Annuity.
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.
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For other uses, see Annuity.
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time.
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geometric series is a series with a constant ratio between successive terms. For example, the series
is geometric, because each term is equal to half of the previous term.
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is geometric, because each term is equal to half of the previous term.
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In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time or halving time.
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In finance, rate of return (ROR) or return on investment (ROI), or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested.
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Gordon growth model is a variant of the discounted dividend model, a method for valuing a stock or business. Often used to provide difficult-to-resolve valuation issues for litigation, tax planning, and business transactions that are currently off market.
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Earnings per share (EPS) are the earnings returned on the initial investment amount.
In the US, the Financial Accounting Standards Board (FASB) requires companies' income statements to report EPS for each of the major categories of the income statement: continuing
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In the US, the Financial Accounting Standards Board (FASB) requires companies' income statements to report EPS for each of the major categories of the income statement: continuing
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Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (i.e. interest is compounded).
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e is the unique real number such that the value of the derivative (slope of the tangent line) of f(x) = ex at the point x = 0 is exactly 1.
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The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
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Net present value (NPV) is a standard method for the financial appraisal of long-term projects. Used for capital budgeting, and widely throughout economics, it measures the excess or shortfall of cash flows, in present value (PV) terms, once financing charges are met.
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