Time value of money

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The time value of money refers to the fact that there is normally a greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.

Contents

• 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 refers to the observation that it is better to receive money sooner than later. Money you have today can be invested to earn a positive rate of return, producing more money tomorrow. Therefore, a dollar today is worth more than a dollar in the future. ^[1]

The time value of money is among the factors considered when weighing the opportunity costs of spending rather than saving or investing money. As such, it is among the reasons why interest is paid or earned: interest, whether it is on a bank deposit or debt, 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 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.

History

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." ^[2]

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

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. ^[3]

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 ${\displaystyle FV}$ to be received in one year is discounted at the rate of interest ${\displaystyle r}$ to give the present value sum ${\displaystyle PV}$:

${\displaystyle PV={\frac {FV}{(1+r)}}}$

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

• Present value : The current worth of a future sum of money or stream of cash flows, given a specified rate of return. Future cash flows are "discounted" at the discount rate; 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. ^[4]
• 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.

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).

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 the interest rate at which the amount compounds each period
• g is the growing rate of payments over each time period

Formula table

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)	${\displaystyle F=P\cdot (1+i)^{n}}$
Present value (P)	Future value (F)	${\displaystyle P=F\cdot (1+i)^{-n}}$
Repeating payment (A)	Future value (F)	${\displaystyle A=F\cdot {\frac {i}{(1+i)^{n}-1}}}$
Repeating payment (A)	Present value (P)	${\displaystyle A=P\cdot {\frac {i(1+i)^{n}}{(1+i)^{n}-1}}}$
Future value (F)	Repeating payment (A)	${\displaystyle F=A\cdot {\frac {(1+i)^{n}-1}{i}}}$
Present value (P)	Repeating payment (A)	${\displaystyle P=A\cdot {\frac {(1+i)^{n}-1}{i(1+i)^{n}}}}$
Future value (F)	Initial gradient payment (G)	${\displaystyle F=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}}}}$
Present value (P)	Initial gradient payment (G)	${\displaystyle P=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}(1+i)^{n}}}}$
Fixed payment (A)	Initial gradient payment (G)	${\displaystyle A=G\cdot \left[{\frac {1}{i}}-{\frac {n}{(1+i)^{n}-1}}\right]}$
Future value (F)	Initial exponentially increasing payment (D) Increasing percentage (g)	${\displaystyle F=D\cdot {\frac {(1+g)^{n}-(1+i)^{n}}{g-i}}}$  (for i ≠ g) ${\displaystyle F=D\cdot {\frac {n(1+i)^{n}}{1+g}}}$  (for i = g)
Present value (P)	Initial exponentially increasing payment (D) Increasing percentage (g)	${\displaystyle P=D\cdot {\frac {\left({1+g \over 1+i}\right)^{n}-1}{g-i}}}$  (for i ≠ g) ${\displaystyle P=D\cdot {\frac {n}{1+g}}}$  (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.

Derivations

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 formulas 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:

${\displaystyle {\text{PV}}={\text{FV}}\cdot e^{-rt}}$

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):

${\displaystyle {\text{PV}}={\text{FV}}\cdot \exp \left(-\int _{0}^{T}r(t)\,dt\right)}$

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, as detailed below.

Differential equations

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 ${\displaystyle {\mathcal {L}}}$ as:

${\displaystyle {\mathcal {L}}:=-\partial _{t}+r(t).}$

This states that value decreases (−) over time (∂_t) at the discount rate (r(t)). Applied to a function it yields:

${\displaystyle {\mathcal {L}}f=-\partial _{t}f(t)+r(t)f(t).}$

For an instrument whose payment stream is described by f(t), the value V(t) satisfies the inhomogeneous first-order ODE ${\displaystyle {\mathcal {L}}V=f}$ ("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 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 ${\displaystyle \delta _{u}(t):=\delta (t-u).}$

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

${\displaystyle b(t;u):=H(u-t)\cdot \exp \left(-\int _{t}^{u}r(v)\,dv\right)}$

where H is the Heaviside step function – the notation "${\displaystyle ;u}$" 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, ${\displaystyle \textstyle {\int }}$) of the future discount rates (${\displaystyle \textstyle {\int _{t}^{u}}}$ for future, r(v) for discount rates), while past cash flows are worth 0 (${\displaystyle H(u-t)=1{\text{ if }}t<u,0{\text{ if }}t>u}$), 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, ${\displaystyle r(v)\equiv r,}$ this simplifies to

${\displaystyle b(t;u)=H(u-t)\cdot e^{-(u-t)r}={\begin{cases}e^{-(u-t)r}&t<u\\0&t>u,\end{cases}}}$

where ${\displaystyle (u-t)}$ is "time remaining until cash flow".

Thus for a stream of cash flows f(u) ending by time T (which can be set to ${\displaystyle T=+\infty }$ for no time horizon) the value at time t,${\displaystyle V(t;T)}$ is given by combining the values of these individual cash flows:

${\displaystyle V(t;T)=\int _{t}^{T}f(u)b(t;u)\,du.}$

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

• Discounted cash flow

• Earnings growth

• Exponential growth

• Financial management

• Hyperbolic discounting

• Internal rate of return

• Net present value

• Option time value

• Real versus nominal value (economics)

• Return on time invested

• Snowball effect

Notes

Related Research Articles

The discounted cash flow (DCF) analysis, in financial analysis, is a method used to value a security, project, company, or asset, that incorporates the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate development, corporate financial management, and patent valuation. Used in industry as early as the 1700s or 1800s, it was widely discussed in financial economics in the 1960s, and U.S. courts began employing the concept in the 1980s and 1990s.

In finance, discounting is a 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. This transaction is based on the fact that most people prefer current interest to delayed interest because of mortality effects, impatience effects, and salience effects. 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.

The net present value (NPV) or net present worth (NPW) is a way of measuring the value of an asset that has cashflow by adding up the present value of all the future cash flows that asset will generate. The present value of a cash flow depends on the interval of time between now and the cash flow because of 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.

Internal rate of return (IRR) is a method of calculating 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 financial risk.

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 usually less than 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 equal or 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 than tomorrow. 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 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).

In finance, 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.

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 – and hence asset pricing models – 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 process by which an investor arrives at an estimate of the theoretical fair value, or intrinsic worth, 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, such as 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.

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

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 that investment over a specified time period, such as interest payments, coupons, cash dividends and stock 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 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.

In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan according to an amortization schedule, typically through equal payments.

In finance, the weighted-average life (WAL) of an amortizing loan or amortizing bond, also called average life, is the weighted average of the times of the principal repayments: it's the average time until a dollar of principal is repaid.

In financial economics, the dividend discount model (DDM) is a method of valuing the price of a company's capital stock or business value based on the assertion that intrinsic value is determined by the sum of future cash flows from dividend payments to shareholders, discounted back to their present value. The constant-growth form of the DDM is sometimes referred to as the Gordon growth model (GGM), after Myron J. Gordon of the Massachusetts Institute of Technology, the University of Rochester, and the University of Toronto, who published it along with Eli Shapiro in 1956 and made reference to it in 1959. Their work borrowed heavily from the theoretical and mathematical ideas found in John Burr Williams 1938 book "The Theory of Investment Value," which put forth the dividend discount model 18 years before Gordon and Shapiro.

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.

The public market equivalent (PME) is a collection of performance measures developed to assess private equity funds and to overcome the limitations of the internal rate of return and multiple on invested capital measurements. While the calculations differ, they all attempt to measure the return from deploying a private equity fund's cash flows into a stock market index.

In investment, 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. Annuities may be calculated by mathematical functions known as "annuity functions".

References

External links

• Time Value of Money ebook