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The **time value of money** is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.

In economics, **time preference** is the current relative valuation placed on receiving a good at an earlier date compared with receiving it at a later date.

- History
- Calculations
- Formula
- Future value of a present sum
- 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 an annuity
- Future value of a growing annuity
- Formula table
- Derivations
- Annuity derivation
- Perpetuity derivation
- Continuous compounding
- Examples
- Differential equations
- See also
- Notes
- References
- External links

The time value of money explains why interest is paid or earned: interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the time value of money.

**Interest**, in finance and economics, is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum, at a particular rate. It is distinct from a fee which the borrower may pay the lender or some third party. It is also distinct from dividend which is paid by a company to its shareholders (owners) from its profit or reserve, but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs.

**Debt** is when something, usually money, is owed by one party, the borrower or debtor, to a second party, the lender or creditor. Debt is a deferred payment, or series of payments, that is owed in the future, which is what differentiates it from an immediate purchase. The debt may be owed by sovereign state or country, local government, company, or an individual. Commercial debt is generally subject to contractual terms regarding the amount and timing of repayments of principal and interest. Loans, bonds, notes, and mortgages are all types of debt. The term can also be used metaphorically to cover moral obligations and other interactions not based on economic value. For example, in Western cultures, a person who has been helped by a second person is sometimes said to owe a "debt of gratitude" to the second person.

It also underlies investment. Investors are willing to forgo spending their money now only if they expect a favorable return on their investment in the future, such that the increased value to be available later is sufficiently high to offset the preference to have money now.

To **invest** is to allocate money in the expectation of some benefit in the future.

An **investor** is a person that allocates capital with the expectation of a future financial return. Types of investments include: equity, debt securities, real estate, currency, commodity, token, derivatives such as put and call options, futures, forwards, etc. This definition makes no distinction between the investors in the primary and secondary markets. That is, someone who provides a business with capital and someone who buys a stock are both investors. An investor who owns a stock is a shareholder.

In finance, **return** is a profit on an investment. It comprises any change in value of the investment, and/or cash flows which the investor receives from the investment, such as interest payments or dividends. It may be measured either in absolute terms or as a percentage of the amount invested. The latter is also called the holding period return.

The Talmud (~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." ^{ [1] }

The **Talmud** 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 Talmud was the centerpiece of Jewish cultural life and was foundational to "all Jewish thought and aspirations", serving also as "the guide for the daily life" of Jews.

**Makot** is a tractate of the Mishnah and Talmud. It is the fifth volume of the order of Nezikin. Makkot deals primarily with laws of the jewish courts and the punishments which they may administer, and may be regarded as a continuation of tractate Sanhedrin, of which it originally formed part.

The notion was later described by Martín de Azpilcueta (1491–1586) of the School of Salamanca.

**Martín de Azpilcueta**, or **Doctor Navarrus**, was an important Spanish canonist and theologian in his time, and an early economist, the first to develop monetarist theory.

The **School of Salamanca** is the Renaissance of thought in diverse intellectual areas by Spanish and Portuguese theologians, rooted in the intellectual and pedagogical work of Francisco de Vitoria. From the beginning of the 16th century the traditional Catholic conception of man and of his relation to God and to the world had been assaulted by the rise of humanism, by the Protestant Reformation and by the new geographical discoveries and their consequences. These new problems were addressed by the School of Salamanca. The name refers to the University of Salamanca, where de Vitoria and other members of the school were based.

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.^{ [2] }

This principle allows for the valuation of a likely stream of income in the future, in such a way that annual incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream; all of the standard calculations for time value of money derive from the most basic algebraic expression for the present value of a future sum, "discounted" to the present by an amount equal to the time value of money. For example, the future value sum to be received in one year is discounted at the rate of interest to give the present value sum :

Some standard calculations based on the time value of money are:

*Present value*: The current worth of a future sum of money or stream of cash flows, given a specified rate of return. Future cash flows are "discounted" at the*discount rate;*the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to valuing future cash flows properly, whether they be earnings or obligations.^{ [3] }*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.^{ [4] }

*Present value of a perpetuity*is an infinite and constant stream of identical cash flows.^{ [5] }

*Future value*: The value of an asset or cash at a specified date in the future, based on the value of that asset in the present.^{ [6] }*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. The formulas are programmed into most financial calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).^{ [7] }

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 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, 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, 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*).

The following formula use these common variables:

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

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

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 by numerical methods:

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

Note that this series can be summed for a given value of *n*, or when *n* is ∞.^{ [8] } This is a very general formula, which leads to several important special cases given below.

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 by numerical methods:

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

In this case each cash flow grows by a factor of (1+*g*). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of *g* as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.

**Where i ≠ g :**

**Where i = g :**

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

A perpetuity 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.

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:

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 for stock valuation.

The future value (after *n* periods) of an annuity (FVA) formula has four variables, each of which can be solved for by numerical methods:

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

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 :**

**Where i = g :**

The following table summarizes the different formulas commonly used in calculating the time value of money.^{ [9] } These values are often displayed in tables where the interest rate and time are specified.

Find | Given | Formula |
---|---|---|

Future value (F) | Present value (P) | |

Present value (P) | Future value (F) | |

Repeating payment (A) | Future value (F) | |

Repeating payment (A) | Present value (P) | |

Future value (F) | Repeating payment (A) | |

Present value (P) | Repeating payment (A) | |

Future value (F) | Initial gradient payment (G) | |

Present value (P) | Initial gradient payment (G) | |

Fixed payment (A) | Initial gradient payment (G) | |

Future value (F) | Initial exponentially increasing payment (D) Increasing percentage (g) | (for i ≠ g) (for i = g) |

Present value (P) | Initial exponentially increasing payment (D) Increasing percentage (g) | (for i ≠ g) (for i = g) |

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.

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*:

Summing over all payments from time 1 to time n, then reversing t

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

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:

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

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.

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:

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 of the continuously compounded rate *r*(*t*):

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, as detailed below.

Using continuous compounding yields the following formulas for various instruments:

- Annuity
- Perpetuity
- Growing annuity
- Growing perpetuity
- Annuity with continuous payments

These formulas assume that payment A is made in the first payment period and annuity ends at time t.^{ [10] }

Ordinary and partial differential equations (ODEs and PDEs) – equations involving derivatives and one (respectively, multiple) variables are ubiquitous in more advanced treatments of financial mathematics. 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 ( Carr & Flesaker 2006 , pp. 6–7).

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 as:

This states that values decreases (−) over time (∂_{t}) at the discount rate (*r*(*t*)). Applied to a function it yields:

For an instrument whose payment stream is described by *f*(*t*), the value *V*(*t*) satisfies the inhomogeneous first-order ODE ("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 functions, 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

The Green's function for the value at time *t* of a £1 cash flow at time *u* is

where *H* is the Heaviside step function – the notation "" 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, ) of the future discount rates ( for future, *r*(*v*) for discount rates), while past cash flows are worth 0 (), 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, this simplifies to

where is "time remaining until cash flow".

Thus for a stream of cash flows *f*(*u*) ending by time *T* (which can be set to for no time horizon) the value at time *t,* is given by combining the values of these individual cash flows:

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.

- ↑ "Makkot 3a William Davidson Talmud online".Cite web requires
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(help) - ↑ Carther, Shauna (3 December 2003). "Understanding the Time Value of Money".Cite web requires
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(help) - ↑ Staff, Investopedia (25 November 2003). "Present Value - PV".Cite web requires
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(help) - ↑ "Present Value of an Annuity".Cite web requires
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(help) - ↑ Staff, Investopedia (24 November 2003). "Perpetuity".Cite web requires
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(help) - ↑ Staff, Investopedia (23 November 2003). "Future Value - FV".Cite web requires
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(help) - ↑ Hovey, M. (2005). Spreadsheet Modelling for Finance. Frenchs Forest, N.S.W.: Pearson Education Australia.
- ↑ http://mathworld.wolfram.com/GeometricSeries.html Geometric Series
- ↑ "NCEES FE exam".Cite web requires
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(help) - ↑ "Annuities and perpetuities with continuous compounding".Cite web requires
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(help)

In finance, **discounted cash flow** (**DCF**) analysis is a method of valuing a project, company, or asset using the concepts of the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate development, corporate financial management and patent valuation. It was used in industry as early as the 1700s or 1800s, widely discussed in financial economics in the 1960s, and became widely used in U.S. Courts in the 1980s and 1990s.

**Discounting** is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The **discount**, or **charge**, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

In finance, 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 rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.

The **internal rate of return** (**IRR**) is a measure of an investment’s rate of return. The term *internal* refers to the fact that the calculation excludes external factors, such as the risk-free rate, inflation, the cost of capital, or various financial risks.

In economics and finance, **present value** (**PV**), also known as **present discounted value**, is the value of an expected income stream determined as of the date of valuation. The present value is always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of the borrowed funds is less than the total amount of money paid to the lender.

In finance, an **interest rate swap** (**IRS**) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a linear IRD and one of the most liquid, benchmark products. It has associations with forward rate agreements (FRAs), and with zero coupon swaps (ZCSs).

A **perpetuity** is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as consols and were all finally redeemed in 2015. Real estate and preferred stock are among some types of investments that affect the results of a perpetuity, and prices can be established using techniques for valuing a perpetuity. Perpetuities are but one of the time value of money methods for valuing financial assets. Perpetuities are a form of ordinary annuities.

**Future value** is the value of an asset at a specific date. It 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, or more generally, rate of return; it is the present value multiplied by the accumulation function. The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money calculations.

**Rational pricing** is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

**Bond valuation** is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected cash flows to the present using an appropriate discount rate.

In finance, the **duration** of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.

In finance, **bond convexity** is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance.

**Actuarial notation** is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.

**Stock duration** of an equity stock is the average of the times until its dividends are received, weighted by their present values.

The **actuarial present value** (**APV**) is the expected value of the present value of a contingent cash flow stream. Actuarial present values are typically calculated for the benefit-payment or series of payments associated with life insurance and life annuities. The probability of a future payment is based on assumptions about the person's future mortality which is typically estimated using a life table.

The **modified internal rate of return** (**MIRR**) is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of equal size. As the name implies, MIRR is a modification of the internal rate of return (IRR) and as such aims to resolve some problems with the IRR.

The **marginal efficiency of capital** (MEC) is that rate of discount which would equate the price of a fixed capital asset with its present discounted value of expected income.

Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) **continuous repayment mortgage** is a mortgage loan paid by means of a continuous annuity.

**Public Market Equivalent** designs a set of analyses used in the Private Equity Industry to evaluate the performance of a Private Equity Fund against a public benchmark or index. The analysis is also referred by the acronym ICM for Index Comparison Method. More than a specific calculation method, PME encompasses a number of different analyses sharing the same objective. The first PME analysis was proposed by Long and Nickels in 1996.

An **annuity** is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time.

- Carr, Peter; Flesaker, Bjorn (2006),
*Robust Replication of Default Contingent Claims (presentation slides)*(PDF), Bloomberg LP, archived from the original (PDF) on 2009-02-27. See also Audio Presentation and paper.Cite uses deprecated parameter`|deadurl=`

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

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