The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of

_{t}'', the value of cash flow at time ''t'':
:$PV\; \backslash \; =\; \backslash \; \backslash sum\_^\; \backslash frac$
Note that this series can be summed for a given value of ''n'', or when ''n'' is ∞. This is a very general formula, which leads to several important special cases given below.

_{''t''}) at the discount rate (''r''(''t'')). Applied to a function it yields:
:$\backslash mathcal\; f\; =\; -\backslash partial\_t\; f(t)\; +\; r(t)\; f(t).$
For an instrument whose payment stream is described by ''f''(''t''), the value ''V''(''t'') satisfies the

Time Value of Money hosted by the University of Arizona

Time Value of Money

ebook {{DEFAULTSORT:Time Value Of Money Engineering economics Actuarial science Interest Intertemporal economics Money

money
Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic context. The main functions of money are distinguished as: a ...

now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference
In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good
In most contexts, the concept of good denotes the conduct that ...

.
The time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

value of money is among the factors considered when weighing the opportunity cost
In microeconomic theory
Microeconomics (from Greek prefix ''mikro-'' meaning "small" + ''economics'') is a branch of economics
Economics () is the social science that studies how people interact with value; in particular, the Produ ...

s of spending rather than saving
Saving is income
In microeconomics
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that are generally accepted b ...

or investing
Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort.
In finance, the purpose of investing is to generate a Return ...

money. As such, it is among the reasons why interest
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availa ...

is paid or earned: interest, whether it is on a bank deposit
A deposit account is a bank account
A bank account is a financial account maintained by a bank or other financial institution in which the financial transaction
A financial transaction is an agreement Agreement may refer to:
Agreements b ...

or debt
Debt is an obligation that requires one party, the debtor
A debtor or debitor is a legal entity (legal person) that owes a debt
Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to ...

, compensates the depositor or lender for the loss of their use of their money. Investors are willing to forgo spending their money now only if they expect a favorable net return on their investment in the future, such that the increased value
Value or values may refer to:
* Value (ethics)
In ethics
Ethics or moral philosophy is a branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, E ...

to be available later is sufficiently high to offset both the preference to spending money now and inflation (if present); see required rate of return
In finance, discounted cash flow (DCF) analysis is a method of valuing a security (finance), security, project, company, or financial asset, asset using the concepts of the time value of money.
Discounted cash flow analysis is widely used in inves ...

.
History

TheTalmud
The Talmud (; he, תַּלְמוּד ''Tálmūḏ'') is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the ...

(~500 CE) recognizes the time value of money. In Tractate Makkos page 3a the Talmud discusses a case where witnesses falsely claimed that the term of a loan was 30 days when it was actually 10 years. The false witnesses must pay the difference of the value of the loan "in a situation where he would be required to give the money back (within) thirty days..., and that same sum in a situation where he would be required to give the money back (within) 10 years...The difference is the sum that the testimony of the (false) witnesses sought to have the borrower lose; therefore, it is the sum that they must pay."
The notion was later described by (1491–1586) of the School of Salamanca
The School of Salamanca ( es, Escuela de Salamanca) is the Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punc ...

.
Calculations

Time value of money problems involve the net value of cash flows at different points in time. 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, cash flows 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 these 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 must 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 assuming that inflation would be zero percent. That is, £100 invested for one year at 5% interest has a ''future value'' of £105 under the assumption that inflation would be zero percent. This principle allows for the valuation of a likely stream of income in the future, in such a way that annual incomes arediscounted
Discounting is a financial mechanism in which a debtor
A debtor or debitor is a legal entity (legal person) that owes a debt
Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another p ...

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
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behavi ...

of a future sum, "discounted" to the present by an amount equal to the time value of money. For example, the future value sum $FV$ to be received in one year is discounted at the rate of interest $r$ to give the present value sum $PV$:
: $PV\; =\; \backslash frac$
Some standard calculations based on the time value of money are:
* ''Present value
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behavi ...

'': The current worth of a future sum of money or stream of cash flows
A cash flow is a real or virtual movement of money
Image:National-Debt-Gillray.jpeg, In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in ...

, given a specified rate of return
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ...

. 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.
* ''Present value of an annuity
An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account
A savings acco ...

'': 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.
:''Present value of a perpetuity
A perpetuity is an annuity
An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings ...

'' is an infinite and constant stream of identical cash flows.
* ''Future value
Future value is the value
Value or values may refer to:
* Value (ethics)
In ethics
Ethics or moral philosophy is a branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metap ...

'': The value of an asset or cash at a specified date in the future, based on the value of that asset in the present.
* ''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.
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
A spreadsheet is a computer application for organization, analysis, and storage of data in table (information), tabular form. Spreadsheets were developed as computerized analogs of paper accounting Worksheet#Accounting, worksheets. The program op ...

. 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 formula may also be rearranged to determine one of the other unknowns. In the case of the standard annuity formula, 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 {{Unreferenced, date=February 2007
Return of capital (ROC) refers to principal
Principal may refer to:
Title or rank
* Principal (academia)
The principal is the chief executive and the chief academic officer of a university
A university ( l ...

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
Compound interest is the addition of interest
Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

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, it must be decided 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). When using a financial calculator or a spreadsheet
A spreadsheet is a computer application for organization, analysis, and storage of data in table (information), tabular form. Spreadsheets were developed as computerized analogs of paper accounting Worksheet#Accounting, worksheets. The program op ...

, it can usually be set for either calculation. The following formulas are for an ordinary annuity. For the answer for the present value of an annuity due, the PV of an ordinary annuity
can be multiplied by (1 + ''i'').
Formula

The following formula use these common variables: * ''PV'' is the value at time zero (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 theinterest rate
An interest rate is the amount of interest
In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ...

at which the amount compounds each period
* ''g'' is the growing rate of payments over each time period
Future value of a present sum

Thefuture value
Future value is the value
Value or values may refer to:
* Value (ethics)
In ethics
Ethics or moral philosophy is a branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metap ...

(''FV'') formula is similar and uses the same variables.
:$FV\; \backslash \; =\; \backslash \; PV\; \backslash cdot\; (1+i)^n$
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. Thepresent value
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behavi ...

(''PV'') formula has four variables, each of which can be solved for by numerical methods
Numerical analysis is the study of algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as ...

:
:$PV\; \backslash \; =\; \backslash \; \backslash frac$
The cumulative present value of future cash flows can be calculated by summing the contributions of ''FVPresent 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 anannuity
An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account
A savings acco ...

(PVA) formula has four variables, each of which can be solved for by numerical methods:
:$PV(A)\; \backslash ,=\backslash ,\backslash frac\; \backslash cdot\; \backslash left;\; href="/html/ALL/s/\_\backslash right.html"\; ;"title="\; \backslash right">\; \backslash right$
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 : :$PV(A)\backslash ,=\backslash ,\backslash left;\; href="/html/ALL/s/1-\_\backslash left(\backslash right)^n\_\backslash right.html"\; ;"title="1-\; \backslash left(\backslash right)^n\; \backslash right">1-\; \backslash left(\backslash right)^n\; \backslash right$ Where i = g : :$PV(A)\backslash ,=\backslash ,$ To get the PV of a growing annuity due, multiply the above equation by (1 + ''i'').Present value of a perpetuity

Aperpetuity
A perpetuity is an annuity
An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings ...

is payments of a set amount of money that occur on a routine basis and continue forever. When ''n'' → ∞, the ''PV'' of a perpetuity (a perpetual annuity) formula becomes a simple division.
:$PV(P)\; \backslash \; =\; \backslash $
Present value of a growing perpetuity

When the perpetual annuity payment grows at a fixed rate (''g'', with ''g'' < ''i'') the value is determined according to the following formula, obtained by setting ''n'' to infinity in the earlier formula for a growing perpetuity: :$PV(A)\backslash ,=\backslash ,$ 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. This is the well known Gordon growth model used forstock valuation In financial markets, stock valuation is the method of calculating theoretical values of companies and their share capital, stocks. The main use of these methods is to predict future market prices, or more generally, potential market prices, and thu ...

.
Future value of an annuity

The future value (after ''n'' periods) of an annuity (FVA) formula has four variables, each of which can be solved for by numerical methods: :$FV(A)\; \backslash ,=\backslash ,A\backslash cdot\backslash frac$ To get the FV of an annuity due, multiply the above equation by (1 + i).Future value of a growing annuity

The future value (after ''n'' periods) of a growing annuity (FVA) formula has five variables, each of which can be solved for by numerical methods: Where i ≠ g : :$FV(A)\; \backslash ,=\backslash ,A\backslash cdot\backslash frac$ Where i = g : :$FV(A)\; \backslash ,=\backslash ,A\backslash cdot\; n(1+i)^$Formula table

The following table summarizes the different formulas commonly used in calculating the time value of money. These values are often displayed in tables where the interest rate and time are specified. Notes: *''A'' is a fixed payment amount, every period *''G'' is the initial payment amount of an increasing payment amount, that starts at ''G'' and increases by ''G'' for each subsequent period. *''D'' is the initial payment amount of an exponentially (geometrically) increasing payment amount, that starts at ''D'' and increases by a factor of (1+''g'') each subsequent period.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'': :$FV\; \backslash \; =\; C(1+i)^$ Summing over all payments from time 1 to time n, then reversing t :$FVA\; \backslash \; =\; \backslash sum\_^n\; C(1+i)^\; \backslash \; =\; \backslash sum\_^\; C(1+i)^k$ Note that this is ageometric series
In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series
:\frac \,+\, \frac \,+\, ...

, with the initial value being ''a'' = ''C'', the multiplicative factor being 1 + ''i'', with ''n'' terms. Applying the formula for geometric series, we get
:$FVA\; \backslash \; =\; \backslash frac\; \backslash \; =\; \backslash frac$
The present value of the annuity (PVA) is obtained by simply dividing by $(1+i)^n$:
:$PVA\; \backslash \; =\; \backslash frac\; =\; \backslash frac\; \backslash left(\; 1\; -\; \backslash frac\; \backslash right)$
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:
:$\backslash text\; \backslash times\; i\; =\; C$
:$\backslash text\; =\; \backslash frac\; +\; \backslash text$
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, $PV\; =\; \backslash frac$. At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments ($FV\; =\; \backslash frac\; +\; FVA$). Plugging this back into the equation:
:$\backslash frac\; +\; FVA\; =\; \backslash frac\; (1+i)^n$
:$FVA\; =\; \backslash frac\; \backslash left;\; href="/html/ALL/s/\backslash left(1+i\_\backslash right)^n\_-\_1\_\backslash right.html"\; ;"title="\backslash left(1+i\; \backslash right)^n\; -\; 1\; \backslash right">\backslash left(1+i\; \backslash right)^n\; -\; 1\; \backslash right$Perpetuity derivation

Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term: :$\backslash left(\backslash right)$ can be seen to approach the value of 1 as ''n'' grows larger. At infinity, it is equal to 1, leaving $$ as the only term remaining.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 thenatural logarithm
The natural logarithm of a number is its logarithm
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

and r is the continuously compounded rate:
:$\backslash text\; =\; \backslash text\backslash cdot\; e^$
This can be generalized to discount rates that vary over time: instead of a constant discount rate ''r,'' one uses a function of time ''r''(''t''). In that case the discount factor, and thus the present value, of a cash flow at time ''T'' is given by the integral
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of the continuously compounded rate ''r''(''t''):
:$\backslash text\; =\; \backslash text\backslash cdot\; \backslash exp\backslash left(-\backslash int\_0^T\; r(t)\backslash ,dt\backslash right)$
Indeed, a key reason for using continuous compounding is to simplify the analysis of varying discount rates and to allow one to use the tools of calculus. Further, for interest accrued and capitalized overnight (hence compounded daily), continuous compounding is a close approximation for the actual daily compounding. More sophisticated analysis includes the use of differential equations
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...

, as detailed below.
Examples

Using continuous compounding yields the following formulas for various instruments: ;Annuity: :$\backslash \; PV\; \backslash \; =\; \backslash $ ;Perpetuity: :$\backslash \; PV\; \backslash \; =\; \backslash $ ;Growing annuity: :$\backslash \; PV\; \backslash \; =\; \backslash $ ;Growing perpetuity: :$\backslash \; PV\; \backslash \; =\; \backslash $ ;Annuity with continuous payments: :$\backslash \; PV\; \backslash \; =\; \backslash $ These formulas assume that payment A is made in the first payment period and annuity ends at time t.Differential equations

Ordinary
Ordinary or The Ordinary often refer to:
Music
* Ordinary (EP), ''Ordinary'' (EP) (2015), by South Korean group Beast
* Ordinary (Every Little Thing album), ''Ordinary'' (Every Little Thing album) (2011)
* Ordinary (Two Door Cinema Club song), "O ...

and partial
Partial may refer to:
Mathematics
*Partial derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in ...

differential equation
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

s (ODEs and PDEs) – equations involving derivatives and one (respectively, multiple) variables are ubiquitous in more advanced treatments of financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. See Quantitative analyst.
In general, there exist two separate branch ...

. While time value of money can be understood without using the framework of differential equations, the added sophistication sheds additional light on time value, and provides a simple introduction before considering more complicated and less familiar situations. This exposition follows .
The fundamental change that the differential equation perspective brings is that, rather than computing a ''number'' (the present value ''now''), one computes a ''function'' (the present value now or at any point in ''future''). This function may then be analyzed—how does its value change over time—or compared with other functions.
Formally, the statement that "value decreases over time" is given by defining the linear differential operator $\backslash mathcal$ as:
:$\backslash mathcal\; :=\; -\backslash partial\_t\; +\; r(t).$
This states that values decreases (−) over time (∂inhomogeneous
Homogeneity and heterogeneity are concepts often used in the sciences
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
...

first-order ODE $\backslash mathcalV\; =\; f$ ("inhomogeneous" is because one has ''f'' rather than 0, and "first-order" is because one has first derivatives but no higher derivatives) – this encodes the fact that when any cash flow occurs, the value of the instrument changes by the value of the cash flow (if you receive a £10 coupon, the remaining value decreases by exactly £10).
The standard technique tool in the analysis of ODEs is Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous ordinary differential equation, inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means th ...

s, from which other solutions can be built. In terms of time value of money, the Green's function (for the time value ODE) is the value of a bond paying £1 at a single point in time ''u'' – the value of any other stream of cash flows can then be obtained by taking combinations of this basic cash flow. In mathematical terms, this instantaneous cash flow is modeled as a Dirac delta function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

$\backslash delta\_u(t)\; :=\; \backslash delta(t-u).$
The Green's function for the value at time ''t'' of a £1 cash flow at time ''u'' is
:$b(t;u)\; :=\; H(u-t)\backslash cdot\; \backslash exp\backslash left(-\backslash int\_t^u\; r(v)\backslash ,dv\backslash right)$
where ''H'' is the Heaviside step function
325px, The Heaviside step function, using the half-maximum convention
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the valu ...

– the notation "$;u$" is to emphasize that ''u'' is a ''parameter'' (fixed in any instance—the time when the cash flow will occur), while ''t'' is a ''variable'' (time). In other words, future cash flows are exponentially discounted (exp) by the sum (integral, $\backslash textstyle$) of the future discount rates ($\backslash textstyle$ for future, ''r''(''v'') for discount rates), while past cash flows are worth 0 ($H(u-t)\; =\; 1\; \backslash text\; t\; <\; u,\; 0\; \backslash text\; t\; >\; u$), because they have already occurred. Note that the value ''at'' the moment of a cash flow is not well-defined – there is a discontinuity at that point, and one can use a convention (assume cash flows have already occurred, or not already occurred), or simply not define the value at that point.
In case the discount rate is constant, $r(v)\; \backslash equiv\; r,$ this simplifies to
:$b(t;u)\; =\; H(u-t)\backslash cdot\; e^\; =\; \backslash begin\; e^\; \&\; t\; <\; u\backslash \backslash \; 0\; \&\; t\; >\; u,\backslash end$
where $(u-t)$ is "time remaining until cash flow".
Thus for a stream of cash flows ''f''(''u'') ending by time ''T'' (which can be set to $T\; =\; +\backslash infty$ for no time horizon) the value at time ''t,'' $V(t;T)$ is given by combining the values of these individual cash flows:
:$V(t;T)\; =\; \backslash int\_t^T\; f(u)\; b(t;u)\backslash ,du.$
This formalizes time value of money to future values of cash flows with varying discount rates, and is the basis of many formulas in financial mathematics, such as the Black–Scholes formula with varying interest rates.
See also

*Actuarial science
Actuarial science is the discipline that applies mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which the ...

* Discounted cash flow
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available ...

* Earnings growth Earnings growth is the annual compound annual growth rate (CAGR) of earnings from investments.
For more general discussion see: Sustainable growth rate#From a financial perspective; Stock valuation#Growth rate; Valuation using discounted cash flows ...

* Exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change ...

* Financial management
Financial management may be defined as the area or function in an organization which is concerned with profitability, expenses, cash and credit, so that the "organization may have the means to carry out its objective as satisfactorily as possibl ...

* Hyperbolic discounting
In economics, hyperbolic discounting is a time-''inconsistent'' model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis is actively being studied by neuroeconomics researchers.
According to the disco ...

* Internal rate of return
Internal rate of return (IRR) is a method of calculating an investment
Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, ...

* Net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ra ...

* Option time valueIn finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available w ...

* Real versus nominal value (economics)
In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behav ...

* Snowball effect
Metaphor
A metaphor is a figure of speech
A figure of speech or rhetorical figure is a word or phrase that entails an intentional deviation from ordinary language use in order to produce a rhetoric
Rhetoric () is the Art (sk ...

Notes

References

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

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Time Value of Money

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