![Economics of climate change chapter3 discounting curves](https://upload.wikimedia.org/wikipedia/commons/9/99/Economics_of_climate_change_chapter3_discounting_curves.png)
The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of
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 primary functions which distinguish money are as ...
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 or some cash at an earlier date compared with receiving it at a later ...
.
The
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
value of money is among the factors considered when weighing the
opportunity cost
In microeconomic theory, the opportunity cost of a particular activity is the value or benefit given up by engaging in that activity, relative to engaging in an alternative activity. More effective it means if you chose one activity (for example ...
s of spending rather than
saving
Saving is income not spent, or deferred consumption. Methods of saving include putting money aside in, for example, a deposit account, a pension account, an investment fund, or as cash. Saving also involves reducing expenditures, such as recur ...
or
investing
Investment is the dedication of money to purchase 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 i ...
money. As such, it is among the reasons why
interest
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct ...
is paid or earned: interest, whether it is on a
bank deposit
A deposit account is a bank account maintained by a financial institution in which a customer can deposit and withdraw money. Deposit accounts can be savings accounts, current accounts or any of several other types of accounts explained below.
...
or
debt
Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another party, the creditor. Debt is a deferred payment, or series of payments, which differentiates it from an immediate purchase. The ...
, 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
Return may refer to:
In business, economics, and finance
* Return on investment (ROI), the financial gain after an expense.
* Rate of return, the financial term for the profit or loss derived from an investment
* Tax return, a blank document or t ...
on their investment in the future, such that the increased
value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
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
The discounted cash flow (DCF) analysis is a method in finance of valuing a security, 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 devel ...
.
History
The
Talmud
The Talmud (; he, , Talmūḏ) 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 cente ...
(~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
Martín de Azpilcueta
Martín de Azpilcueta (Azpilkueta in Basque) (13 December 1491 – 1 June 1586), or Doctor Navarrus, was an important Spanish canonist and theologian in his time, and an early economist who independently formulated the quantity theory of mone ...
(1491–1586) of the
School of Salamanca
The School of Salamanca ( es, Escuela de Salamanca) is the Renaissance of thought in diverse intellectual areas by Spanish theologians, rooted in the intellectual and pedagogical work of Francisco de Vitoria. From the beginning of the 16th cen ...
.
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 are
discounted
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.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficient ...
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 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 inte ...
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
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 inte ...
'': 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:
*a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...
, given a specified
rate of return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
. 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
In investment, 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, mo ...
'': 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 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 conso ...
'' is an infinite and constant stream of identical cash flows.
* ''
Future value
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 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 computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
. 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
Return of capital (ROC) refers to principal payments back to "capital owners" (shareholders, partners, unitholders) that exceed the growth (net income/taxable income) of a business or investment. It should not be confused with Rate of Return (ROR ...
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
A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
, 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
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
at which the amount compounds each period
* ''g'' is the growing rate of payments over each time period
Future value of a present sum
The
future value
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 ...
(''FV'') formula is similar and uses the same variables.
:
Present value of a future sum
The present value formula is the core formula for the time value of money; each of the other formulae is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.
The
present value
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 inte ...
(''PV'') formula has four variables, each of which can be solved for by
numerical methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
:
:
The cumulative present value of future cash flows can be calculated by summing the contributions of ''FV
t'', the value of cash flow at time ''t'':
:
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.
Present 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 an
annuity
In investment, 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, mo ...
(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'').
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 :
:
Where i = g :
:
To get the PV of a growing
annuity due, multiply the above equation by (1 + ''i'').
Present value of a perpetuity
A
perpetuity
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 conso ...
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.
:
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:
:
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 In finance and investing, the dividend discount model (DDM) is a method of valuing the price of a company's stock based on the fact that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. In ...
used for
stock valuation
In financial markets, stock valuation is the method of calculating theoretical values of companies and their stocks. The main use of these methods is to predict future market prices, or more generally, potential market prices, and thus to profit fr ...
.
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:
:
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 :
:
Where i = g :
:
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'':
:
Summing over all payments from time 1 to time n, then reversing t
:
Note that this is a
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
, 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:
:
: