Time value of money
Encyclopedia
The time value of money is the value of money figuring in a given amount of interest
earned over a given amount of time. The time value of money is the central concept in finance theory.
For example, $100 of today's money invested for one year and earning 5% interest will be worth $105 after one year. Therefore, $100 paid now or $105 paid exactly one year from now both have the same value to the recipient who assumes 5% interest; using time value of money terminology, $100 invested for one year at 5% interest has a future value of $105. This notion dates at least to Martín de Azpilcueta
(1491–1586) of the School of Salamanca
.
The method also allows the valuation of a likely stream of income in the future, in such a way that the 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, a sum of FV to be received in one year is discounted (at the rate of interest r) to give a sum of PV at present: PV = FV − r·PV = FV/(1+r).
Some standard calculations based on the time value of money are:
. 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, however, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).
These equations are frequently combined for particular uses. For example, bonds
can be readily priced using these equations. A typical coupon bond is composed of two types of payments: a stream of coupon payments similar to an annuity, and a lump-sum return of capital
at the end of the bond's maturity
- that is, a future payment. The two formulas can be combined to determine the present value of the bond.
An important note is that the interest rate 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, you must decide 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). If you are using a financial calculator or a spreadsheet
, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the Present Value of an annuity due simply multiply the PV of an ordinary annuity by (1 + i).
The present value
(PV) formula has four variables, each of which can be solved for:
The cumulative present value
of future cash flows can be calculated by summing the contributions of
, 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.
(PVA) formula has four variables, each of which can be solved for:
To get the PV of an annuity due, multiply the above equation by (1 + i).
Where i ≠ g :
To get the PV of a growing annuity due, multiply the above equation by (1 + i).
Where i = g :
, the PV of a perpetuity (a perpetual annuity) formula becomes simple division.
This is the well known Gordon Growth model
used for stock valuation
.
(FV) formula is similar and uses the same variables.
(FVA) formula has four variables, each of which can be solved for:
Where i ≠ g :
Where i = g :
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 the order of terms and substituting
:
Note that this is a geometric series, with the initial value being
, the multiplicative factor being 
, with 
terms. Applying the formula for geometric series, we get
The present value of the annuity (PVA) is obtained by simply dividing by
:
----
Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:
+ goal
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:
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.
So the present value of €100 one year from now at 5% is €95.24.
The number of monthly payments is
and the monthly interest rate is
The annuity formula for (A/P) calculates the monthly payment:
This is considering an interest rate compounding monthly. If the interest were only to compound yearly at 6%, the monthly payment would be significantly different.
Using the algrebraic identity that if:
then
The present value formula can be rearranged such that:
(years)
This same method can be used to determine the length of time needed to increase a deposit to any particular sum, as long as the interest rate is known. For the period of time needed to double an investment, the Rule of 72
is a useful shortcut that gives a reasonable approximation of the period needed.
is needed to accumulate a given amount from an investment. For example, $100 is invested today and $200 return is expected in five years; what rate of return (interest rate) does this represent?
The present value formula restated in terms of the interest rate is:
This does not sound like very much, but remember - this is future money discounted back to its value today; it is understandably lower. To calculate the future value (at the end of the twenty-year period):
These steps can be combined into a single formula:
.
For example, stocks are commonly noted as trading at a certain P/E ratio
. The P/E ratio is easily recognized as a variation on the perpetuity or growing perpetuity formulae - save that the P/E ratio is usually cited as the inverse of the "rate" in the perpetuity formula.
If we substitute for the time being: the price of the stock for the present value; the earnings per share
of the stock for the cash annuity; and, the discount rate of the stock for the interest rate, we can see that:
And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).
Of course, stocks may have increasing earnings. The formulation above does not allow for growth in earnings, but to incorporate growth, the formula can be restated as follows:
If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:
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
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
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.
Annuity:
Perpetuity:
Growing annuity:
Growing perpetuity:
Annuity with continuous payments:
and partial
differential equation
s (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 .
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 (
) at the discount rate (
). Applied to a function it yields:
For an instrument whose payment stream is described by
the value 
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 the use of Green's function
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 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, 
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
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.
Interest
Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....
earned over a given amount of time. The time value of money is the central concept in finance theory.
For example, $100 of today's money invested for one year and earning 5% interest will be worth $105 after one year. Therefore, $100 paid now or $105 paid exactly one year from now both have the same value to the recipient who assumes 5% interest; using time value of money terminology, $100 invested for one year at 5% interest has a future value of $105. This notion dates at least to Martín de Azpilcueta
Martín de Azpilcueta
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.-Life:...
(1491–1586) of the School of Salamanca
School of Salamanca
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...
.
The method also allows the valuation of a likely stream of income in the future, in such a way that the 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
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
of a future sum, "discounted" to the present by an amount equal to the time value of money. For example, a sum of FV to be received in one year is discounted (at the rate of interest r) to give a sum of PV at present: PV = FV − r·PV = FV/(1+r).
Some standard calculations based on the time value of money are:
- Present valuePresent valuePresent value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
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, and the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they be earnings or obligations. - Present value of an annuityAnnuity (finance theory)The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...
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 perpetuityPerpetuityA perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence...
is an infinite and constant stream of identical cash flows.
- Future valueFuture valueFuture 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...
is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. - Future value of an annuity (FVA) is the future value of a stream of payments (annuity), assuming the payments are invested at a given rate of interest.
Calculations
There are several basic equations that represent the equalities listed above. The solutions may be found using (in most cases) the formulas, a financial calculator or a spreadsheetSpreadsheet
A spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...
. 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, however, there is no closed-form algebraic solution for the interest rate (although financial calculators and spreadsheet programs can readily determine solutions through rapid trial and error algorithms).
These equations are frequently combined for particular uses. For example, bonds
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
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
Return of capital refers to payments back to "capital owners" that exceed the growth of a business. It should not be confused with return on capital which measures a 'rate of return'....
at the end of the bond's maturity
Maturity (finance)
In finance, maturity or maturity date refers to the final payment date of a loan or other financial instrument, at which point the principal is due to be paid....
- 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
Compound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding...
for details on converting between different periodic interest rates.
The rate of return in the calculations can be either the variable solved for, or a predefined variable that measures a discount rate, interest, inflation, rate of return, cost of equity, cost of debt or any number of other analogous concepts. The choice of the appropriate rate is critical to the exercise, and the use of an incorrect discount rate will make the results meaningless.
For calculations involving annuities, you must decide whether the payments are made at the end of each period (known as an ordinary annuity), or at the beginning of each period (known as an annuity due). If you are using a financial calculator or a spreadsheet
Spreadsheet
A spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...
, you can usually set it for either calculation. The following formulas are for an ordinary annuity. If you want the answer for the Present Value of an annuity due simply multiply the PV of an ordinary annuity by (1 + i).
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
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
(PV) formula has four variables, each of which can be solved for:
- PV is the value at time=0
- FV is the value at time=n
- i is the discount rateDiscount rateThe discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
, or the interest rate at which the amount will be compounded each period - n is the number of periods (not necessarily an integer)
The cumulative present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
of future cash flows can be calculated by summing the contributions of
, the value of cash flow at time=tNote 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 annuityAnnuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...
(PVA) formula has four variables, each of which can be solved for:
- PV(A) is the value of the annuity at time=0
- A is the value of the individual payments in each compounding period
- i equals the interest rate that would be compounded for each period of time
- n is the number of payment periods.
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 :
To get the PV of a growing annuity due, multiply the above equation by (1 + i).
Where i = g :
Present value of a perpetuity
When, the PV of a perpetuity (a perpetual annuity) formula becomes simple division.Present value of a growing perpetuity
When the perpetual annuity payment grows at a fixed rate (g) the value is theoretically determined according to the following formula. In practice, there are few securities with 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
Gordon model
The Gordon growth model is a variant of the discounted cash flow model, a method for valuing a stock or business. Often used to provide difficult-to-resolve valuation issues for litigation, tax planning, and business transactions that don't have an explicit market value. It is named after Myron J....
used for stock valuation
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 from price movement – stocks that are judged...
.
Future value of a present sum
The future valueFuture 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 present value multiplied by the accumulation...
(FV) formula is similar and uses the same variables.
Future value of an annuity
The future value of an annuityAnnuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...
(FVA) formula has four variables, each of which can be solved for:
- FV(A) is the value of the annuity at time = n
- A is the value of the individual payments in each compounding period
- i is the interest rate that would be compounded for each period of time
- n is the number of payment periods
Future value of a growing annuity
The future value of a growing annuity (FVA) formula has five variables, each of which can be solved for:Where i ≠ g :
Where i = g :
- FV(A) is the value of the annuity at time = n
- A is the value of initial payment paid at time 1
- i is the interest rate that would be compounded for each period of time
- g is the growing rate that would be compounded for each period of time
- n is the number of payment periods
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 the order of terms and substituting
:Note that this is a geometric series, with the initial value being
, the multiplicative factor being , with terms. Applying the formula for geometric series, we getThe 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:
+ goalNote that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:
Initially, before any payments, the present value of the system is just the endowment principal (
). At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (). Plugging this back into the equation:Perpetuity derivation
Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term: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.Example 1: Present value
One hundred euros to be paid 1 year from now, where the expected rate of return is 5% per year, is worth in today's money:So the present value of €100 one year from now at 5% is €95.24.
Example 2: Present value of an annuity — solving for the payment amount
Consider a 10 year mortgage where the principal amount P is $200,000 and the annual interest rate is 6%.The number of monthly payments is
and the monthly interest rate is
The annuity formula for (A/P) calculates the monthly payment:
This is considering an interest rate compounding monthly. If the interest were only to compound yearly at 6%, the monthly payment would be significantly different.
Example 3: Solving for the period needed to double money
Consider a deposit of $100 placed at 10% (annual). How many years are needed for the value of the deposit to double to $200?Using the algrebraic identity that if:
then
The present value formula can be rearranged such that:
(years)This same method can be used to determine the length of time needed to increase a deposit to any particular sum, as long as the interest rate is known. For the period of time needed to double an investment, the Rule of 72
Rule of 72
In finance, the rule of 72, the rule of 70 and the rule of 69 are methods for estimating an investment's doubling time. The rule number is divided by the interest percentage per period to obtain the approximate number of periods required for doubling...
is a useful shortcut that gives a reasonable approximation of the period needed.
Example 4: What return is needed to double money?
Similarly, the present value formula can be rearranged to determine what rate of returnRate of return
In finance, rate of return , also known as return on investment , rate of profit or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or...
is needed to accumulate a given amount from an investment. For example, $100 is invested today and $200 return is expected in five years; what rate of return (interest rate) does this represent?
The present value formula restated in terms of the interest rate is:
- see also Rule of 72Rule of 72In finance, the rule of 72, the rule of 70 and the rule of 69 are methods for estimating an investment's doubling time. The rule number is divided by the interest percentage per period to obtain the approximate number of periods required for doubling...
Example 5: Calculate the value of a regular savings deposit in the future.
To calculate the future value of a stream of savings deposit in the future requires two steps, or, alternatively, combining the two steps into one large formula. First, calculate the present value of a stream of deposits of $1,000 every year for 20 years earning 7% interest:This does not sound like very much, but remember - this is future money discounted back to its value today; it is understandably lower. To calculate the future value (at the end of the twenty-year period):
These steps can be combined into a single formula:
Example 6: Price/earnings (P/E) ratio
It is often mentioned that perpetuities, or securities with an indefinitely long maturity, are rare or unrealistic, and particularly those with a growing payment. In fact, many types of assets have characteristics that are similar to perpetuities. Examples might include income-oriented real estate, preferred shares, and even most forms of publicly-traded stocks. Frequently, the terminology may be slightly different, but are based on the fundamentals of time value of money calculations. The application of this methodology is subject to various qualifications or modifications, such as the Gordon growth modelGordon model
The Gordon growth model is a variant of the discounted cash flow model, a method for valuing a stock or business. Often used to provide difficult-to-resolve valuation issues for litigation, tax planning, and business transactions that don't have an explicit market value. It is named after Myron J....
.
For example, stocks are commonly noted as trading at a certain P/E ratio
P/E ratio
The P/E ratio of a stock is a measure of the price paid for a share relative to the annual net income or profit earned by the firm per share...
. The P/E ratio is easily recognized as a variation on the perpetuity or growing perpetuity formulae - save that the P/E ratio is usually cited as the inverse of the "rate" in the perpetuity formula.
If we substitute for the time being: the price of the stock for the present value; the earnings per share
Earnings per share
Earnings per share is the amount of earnings per each outstanding share of a company's stock.In the United States, the Financial Accounting Standards Board requires companies' income statements to report EPS for each of the major categories of the income statement: continuing operations,...
of the stock for the cash annuity; and, the discount rate of the stock for the interest rate, we can see that:
And in fact, the P/E ratio is analogous to the inverse of the interest rate (or discount rate).
Of course, stocks may have increasing earnings. The formulation above does not allow for growth in earnings, but to incorporate growth, the formula can be restated as follows:
If we wish to determine the implied rate of growth (if we are given the discount rate), we may solve for g:
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 eE (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
is the base of the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
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
In that case the discount factor, and thus the present value, of a cash flow at time T is given by the integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the continuously compounded rate
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.
Examples
Using continuous compounding yields the following formulas for various instruments:Annuity:
Perpetuity:
Growing annuity:
Growing perpetuity:
Annuity with continuous payments:
Differential equations
OrdinaryOrdinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
and partial
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s (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 .
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 (
) at the discount rate (). Applied to a function it yields:For an instrument whose payment stream is described by
the value 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 the use of Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
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 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
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
– 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, 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 towhere
is "time remaining until cash flow".Thus for a stream of cash flows
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.
See also
- Actuarial scienceActuarial scienceActuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries. Actuaries are professionals who are qualified in this field through education and experience...
- Net present valueNet present valueIn finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
- Option time valueOption time valueIn finance, the time value of an option is the premium a rational investor would pay over its current exercise value , based on its potential to increase in value before expiring. This probability is always greater than zero, thus an option is always worth more than its current exercise value...
- Discounting
- Discounted cash flowDiscounted cash flowIn finance, discounted cash flow analysis is a method of valuing a project, company, or asset using the concepts of the time value of money...
- Exponential growthExponential growthExponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
- Hyperbolic discountingHyperbolic discountingIn behavioral economics, hyperbolic discounting is a time-inconsistent model of discounting.Given two similar rewards, humans show a preference for one that arrives sooner rather than later. Humans are said to discount the value of the later reward, by a factor that increases with the length of the...
- Internal rate of returnInternal rate of returnThe internal rate of return is a rate of return used in capital budgeting to measure and compare the profitability of investments. It is also called the discounted cash flow rate of return or the rate of return . In the context of savings and loans the IRR is also called the effective interest rate...
- PerpetuityPerpetuityA perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence...
- Real versus nominal value (economics)
- Time preferenceTime preferenceIn economics, time preference pertains to how large a premium a consumer places on enjoyment nearer in time over more remote enjoyment....
- Earnings growthEarnings growthEarnings growth is the annual rate of growth of earnings from investments.-Overview:Generally, the greater the earnings growth, the better.When the dividend payout ratio is the same, the dividend growth rate is equal to the earnings growth rate....