Fast Easy Accounting
Marginal Revenue Vs.
Marginal Cost Reference
Guide
Present Value, Internal Rate of Return,
Discounted Cash Flows and Opportunity
Costs
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Contents
Articles
Present value
1
Internal rate of return
8
Discounted cash flow
13
Opportunity cost
18
Time value of money
21
Rate of return
31
Capital budgeting
41
References
Article Sources and Contributors
45
Image Sources, Licenses and Contributors
46
Article Licenses
License
47
Present value
Present value
Present value, also known as present discounted value, is a future amount of money that has been discounted to
reflect its current value, as if it existed today. 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.[] 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 their authority over the money, and
is compensated for it in the form of interest. The initial amount of the borrowed funds (the present value) is less than
the total amount of money paid to the lender.
Present value calculations, and similarly future value calculations, are used to evaluate loans, mortgages, annuities,
sinking funds, perpetuities, and more. These calculations are used to make comparisons between cash flows that
don’t occur at simultaneous times.[] The idea is much like algebra, where variable units must be consistent in order to
compare or carry out addition and subtraction; time dates must be consistent in order to make comparisons between
values or carry out simple calculations. When deciding between projects in which to invest, the choice can be made
by comparing respective present values discounted at the same interest rate, or rate of return. The project with the
least present value, i.e. that costs the least today, should be chosen.
Background
If offered a choice between 100 today or 100 in one year and there is a positive real interest rate throughout the year
ceteris paribus, a rational person will choose 100 today. This is described by economists as time preference. Time
preference can be measured by auctioning off a risk free security—like a US Treasury bill. If a 100 note, payable in
one year, sells for 80 now, then 80 is the present value of the note that will be worth 100 a year from now. This is
because money can be put in a bank account or any other (safe) investment that will return interest in the future.
An investor who has some money has two options: to spend it right now or to save it. But the financial compensation
for saving it (and not spending it) is that the money value will accrue through the compound interest that he will
receive from a borrower (the bank account on which he has the money deposited).
Therefore, to evaluate the real value of an amount of money today after a given period of time, economic agents
compound the amount of money at a given (interest) rate. Most actuarial calculations use the risk-free interest rate
which corresponds to the minimum guaranteed rate provided by a bank's saving account for example. To compare
the change in purchasing power, the real interest rate (nominal interest rate minus inflation rate) should be used.
The operation of evaluating a present value into the future value is called a capitalization (how much will 100 today
be worth in 5 years?). The reverse operation—evaluating the present value of a future amount of money—is called a
discounting (how much will 100 received in 5 years—at a lottery for example—be worth today?).
It follows that if one has to choose between receiving 100 today and 100 in one year, the rational decision is to
choose the 100 today. If the money is to be received in one year and assuming the savings account interest rate is
5%, the person has to be offered at least 105 in one year so that two options are equivalent (either receiving 100
today or receiving 105 in one year). This is because if 100 is deposited in a savings account, the value will be 105
after one year.
1
Present value
2
Interest Rates
Interest is the additional amount of money gained between the beginning and the end of a time period. Interest
represents the time value of money, and can be thought of as rent that is required of a borrower in order to use
money from a lender.[][] For example, when an individual takes out a bank loan, they are charged interest.
Alternatively, when an individual deposits money into a bank, their money earns interest. In this case, the bank is the
borrower of the funds and is responsible for crediting interest to the account holder. Similarly, when an individual
invests in a company (through corporate bonds, or through stock), the company is borrowing funds, and must pay
interest to the individual (in the form of coupon payments, dividends, or stock price appreciation).[] The interest rate
is the change, expressed as a percentage, in the amount of money during one compounding period. A compounding
period is the length of time that must transpire before interest is credited, or added to the total.[] For example, interest
that is compounded annually is credited once a year, and the compounding period is one year. Interest that is
compounded quarterly is credited 4 times a year, and the compounding period is three months. A compounding
period can be any length of time, but some common periods are annually, semiannually, quarterly, monthly, daily,
and even continuously.
There are several types and terms associated with interest rates.
Compound Interest
Compound interest is multiplicative. Interest is earned on the interest that has already accrued (credited) in addition
to the principal (initial amount).[]
The term
is the factor often used to make various calculations concerning compound interest[]
Simple Interest
Simple interest is additive. Simple interest is earned only on the principal amount, and there is no interest earned on
interest already accrued.[]
The term
is the factor used to make various calculations when simple interest is applied. Simple interest is
often used when calculating interest earned during a fraction of the year. Simple interest earns a higher return during
the first year, and compound interest earns a higher return after the first year. At the end of the first year, simple
interest and compound interest earn the same return.[]
Effective Annual Rate of Interest
Effective annual rate of interest is the percentage increase in an amount of money after a year of accumulating
interest. This does not depend on what happens during the year. It can be calculated by
Where
represents the amount of money at time
.[]
Present value
3
Nominal Annual Rate of Interest
Nominal annual rate of interest is NOT the same as effective annual rate of interest. Instead, nominal annual interest
is compounded more than once a year, say,
times. For instance, if the nominal annual interest rate is 8% and
interest is compounded quarterly (4 times a year, or every three months), then the interest rate for ¼ of the year is
(8/4)%=2%. Because it is compounded more often, the effective annual interest rate is actually more than 8%[]
The formula to convert between effective annual interest rate and nominal annual interest rate is
Where
is the effective annual interest rate and
is the nominal annual interest rate, and
is the number of
times interest is compounded in a year[]
Rate of Discount
The rate of discount,
, refers to interest that is payable in advance, before funds have been transferred.
There also exists a nominal annual discount rate,
, which is analogous to nominal interest rate
The formula to convert between effective annual interest rate,
Where
is the effective annual interest rate, and
and effective annual discount rate
is
is the effective annual discount rate[]
Continuous Compounding
Continuous compounding is nominal interest that is compounded infinitely throughout the year. It is the
instantaneous growth rate in the principal amount, i.e. the interest rate can change as a function of time. A force of
interest, which describes the accumulated amount of money as a function of time, characterizes continuous
compounding:
Where
is the force of interest, and
is the amount of money as a function of time,
The formula to convert between effective annual interest rate,
, and its derivative.
, and the constant force of interest,
, is
[]
Real Rate of Interest
Real rate of interest refers to the interest rate that has been adjusted to account for inflation; it is the percent change
in buying power due to inflation.[] The real return refers to the surplus associated with an interest rate compared to an
investment rate that matches the inflation rate.
Where
is the inflation rate.[]
Present value
4
Calculation
The operation of evaluating a present sum of money some time in the future called a capitalization (how much will
100 today be worth in 5 years?). The reverse operation—evaluating the present value of a future amount of
money—is called discounting (how much will 100 received in 5 years be worth today?).[]
Spreadsheets commonly offer functions to compute present value. In Microsoft Excel, there are present value
functions for single payments (=NPV) and series of equal, periodic payments (=PV). Programs will calculate present
value flexibly for any cash flow and interest rate, or for a schedule of different interest rates at different times.
Present Value of a Lump Sum
The most commonly applied model of present valuation uses compound interest. The standard formula is:
Where
is the future amount of money that must be discounted,
the present date and the date where the sum is worth
,
is the number of compounding periods between
is the interest rate for one compounding period (the end
of a compounding period is when interest is applied, for example, annually, semiannually, quarterly, monthly, daily).
The interest rate, , is given as a percentage, but expressed as a decimal in this formula.
Often,
is referred to as the Present Value Factor []
This is also found from the formula for the future value with negative time.
For example if you are to receive $1000 in 5 years, and the effective annual interest rate during this period is 10% (or
0.10), then the present value of this amount is
The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving
$1000 in 5 years, or $620.92 today.[]
The purchasing power in today's money of an amount
the same formula, where in this case
of money,
years into the future, can be computed with
is an assumed future inflation rate.
Net Present Value of a stream of Cash Flows
A cash flow is an amount of money that is either paid out or received, differentiated by a negative or positive sign, at
the end of a period. Conventionally, cash flows that are received are denoted with a positive sign (total cash has
increased) and cash flows that are paid out are denoted with a negative sign (total cash has decreased). The cash flow
for a period represents the net change in money of that period.[] Calculating the net present value,
, of a
stream of cash flows consists of discounting each cash flow to the present, using the present value factor and the
appropriate number of compounding periods, and combining these values.[]
For example, if a stream of cash flows consists of +$100 at the end of period one, -$50 at the end of period two, and
+$35 at the end of period three, and the interest rate per compounding period is 5% (0.05) then the present value of
these three Cash Flows are
respectively
Thus the net present value would be
Present value
5
There are a few considerations to be made.
• The periods might not be consecutive. If this is case, the exponents will change to reflect the appropriate number
of periods
• The interest rates per period might not be the same. The cash flow must be discounted using the interest rate for
the appropriate period: if the interest rate changes, the sum must be discounted to the period where the change
occurs using the second interest rate, then discounted back to the present using the first interest rate.[] For
example, if the cash flow for period one is $100, and $200 for period two, and the interest rate for the first period
is 5%, and 10% for the second, then the net present value would be:
• The interest rate must necessarily coincide with the payment period. If not, either the payment period or the
interest rate must be modified. For example, if the interest rate given is the effective annual interest rate, but cash
flows are received (and/or paid) quarterly, the interest rate per quarter must be computed. This can be done by
converting effective annual interest rate, , to nominal annual interest rate compounded quarterly:
[]
Here,
is the nominal annual interest rate, compounded quarterly, and the interest rate per quarter is
Present Value of an Annuity
Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including
annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules;
payments of the same amount at regular time intervals. The term "annuity" is often used to refer to any such
arrangement when discussing calculation of present value. The expressions for the present value of such payments
are summations of geometric series.
There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate, payments are
received (or paid) at the end of each period, at times 1 through , while for an annuity due, payments are
received (or paid) at the beginning of each period, at times 0 through
.[] This subtle difference must be
accounted for when calculating the present value.
An annuity due is an annuity immediate with one more interest-earning period. Thus, the two present values differ
by a factor of
:
[]
The present value of an annuity immediate is the value at time 0 of the stream of cash flows:
where:
= number of periods,
= amount of cash flows,
= effective periodic interest rate or rate of return.
Present value
6
Present Value of a Perpetuity
A perpetuity refers to periodic payments, receivable indefinitely, although few such instruments exist. The present
value of a perpetuity can be calculated by taking the limit of the above formula as n approaches infinity.
Formula (2) can also be found by subtracting from (1) the present value of a perpetuity delayed n periods, or directly
by summing the present value of the payments
which form a geometric series.
Again there is a distinction between a perpetuity immediate – when payments received at the end of the period – and
a perpetuity due – payment received at the beginning of a period. And similarly to annuity calculations, a perpetuity
due and a perpetuity immediate differ by a factor of
:
[]
PV of a Bond
A corporation issues a bond, an interest earning debt security, to an investor to raise funds.[] The bond has a face
value, , coupon rate, , and maturity date which in turn yields the number of periods until the debt matures and
must be repaid. A bondholder will receive coupon payments semiannually (unless otherwise specified) in the amount
of
, until the bond matures, at which point the bondholder will receive the final coupon payment and the face
value of a bond,
. The present value of a bond is the purchase price.[] The purchase price is equal to the
bond's face value if the coupon rate is equal to the current interest rate of the market, and in this case, the bond is
said to be sold 'at par'. If the coupon rate is less than the market interest rate, the purchase price will be less than the
bond's face value, and the bond is said to have been sold 'at a discount', or below par. Finally, if the coupon rate is
greater than the market interest rate, the purchase price will be greater than the bond's face value, and the bond is
said to have been sold 'at a premium', or above par.[] The purchase price can be computed as:
The present value of a bond can also be computed after coupon payments have already begun. This happens if a
bondholder wishes to sell the bond after having received a number of coupons, but the bond still hasn't matured. In
this case, let denote the number of periods that have passed (or the number of coupon payments that have been
received).[] Then the PV of the bond will be
Technical details
Present value is additive. The present value of a bundle of cash flows is the sum of each one's present value.
In fact, the present value of a cashflow at a constant interest rate is mathematically one point in the Laplace
transform of that cashflow, evaluated with the transform variable (usually denoted "s") equal to the interest rate. The
full Laplace transform is the curve of all present values, plotted as a function of interest rate. For discrete time, where
payments are separated by large time periods, the transform reduces to a sum, but when payments are ongoing on an
almost continual basis, the mathematics of continuous functions can be used as an approximation.
These calculations must be applied carefully, as there are underlying assumptions:
• That it is not necessary to account for price inflation, or alternatively, that the cost of inflation is incorporated into
the interest rate.
Present value
• That the likelihood of receiving the payments is high — or, alternatively, that the default risk is incorporated into
the interest rate.
See time value of money for further discussion.
Variants/Approaches
There are mainly two flavors of PV. Whenever there will be uncertainties in both timing and amount of the cash
flows, the expected present value approach will often be the appropriate technique.
• Traditional Present Value Approach - in this approach a single set of estimated cash flows and a single interest
rate (commensurate with the risk, typically a weighted average of cost components) will be used to estimate the
fair value.
• Expected Present Value Approach - in this approach multiple cash flows scenarios with different/expected
probabilities and a credit-adjusted risk free rate are used to estimate the fair value.
Choice of interest rate
The interest rate used is the risk-free interest rate if there are no risks involved in the project. The rate of return from
the project must equal or exceed this rate of return or it would be better to invest the capital in these risk free assets.
If there are risks involved in an investment this can be reflected through the use of a risk premium. The risk premium
required can be found by comparing the project with the rate of return required from other projects with similar risks.
Thus it is possible for investors to take account of any uncertainty involved in various investments.
Present Value Method of Valuation
An investor, the lender of money, must decide the financial project in which to invest their money, and present value
offers one method of deciding.[] A financial project requires an initial outlay of money, such as the price of stock or
the price of a corporate bond. The project claims to return the initial outlay, as well as some surplus (for example,
interest, or future cash flows). An investor can decide which project to invest in by calculating each projects’ present
value (using the same interest rate for each calculation) and then comparing them. The project with the smallest
present value – the least initial outlay – will be chosen because it offers the same return as the other projects for the
least amount of money.[]
References
7
Internal rate of return
Internal rate of return
The internal rate of return (IRR) or economic rate of return (ERR) 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
(DCFROR) or the rate of return (ROR).[1] In the context of savings and loans the IRR is also called the effective
interest rate. The term internal refers to the fact that its calculation does not incorporate environmental factors (e.g.,
the interest rate or inflation).
Definition
The internal rate of return on an investment or project is the "annualized effective compounded return rate" or "rate
of return" that makes the net present value (NPV as NET*1/(1+IRR)^year) of all cash flows (both positive and
negative) from a particular investment equal to zero. It can also be defined as the discount rate at which the present
value of all future cash flow is equal to the initial investment or in other words the rate at which an investment
breaks even.
In more specific terms, the IRR of an investment is the discount rate at which the net present value of costs (negative
cash flows) of the investment equals the net present value of the benefits (positive cash flows) of the investment.
IRR calculations are commonly used to evaluate the desirability of investments or projects. The higher a project's
IRR, the more desirable it is to undertake the project. Assuming all projects require the same amount of up-front
investment, the project with the highest IRR would be considered the best and undertaken first.
A firm (or individual) should, in theory, undertake all projects or investments available with IRRs that exceed the
cost of capital. Investment may be limited by availability of funds to the firm and/or by the firm's capacity or ability
to manage numerous projects.
Uses
Because the internal rate of return is a rate quantity, it is an indicator of the efficiency, quality, or yield of an
investment. This is in contrast with the net present value, which is an indicator of the value or magnitude of an
investment.
An investment is considered acceptable if its internal rate of return is greater than an established minimum
acceptable rate of return or cost of capital. In a scenario where an investment is considered by a firm that has equity
holders, this minimum rate is the cost of capital of the investment (which may be determined by the risk-adjusted
cost of capital of alternative investments). This ensures that the investment is supported by equity holders since, in
general, an investment whose IRR exceeds its cost of capital adds value for the company (i.e., it is economically
profitable).
As per Hansen, 2004.The rate of return that equates the present value of a project’s cash inflows with the present
value of its cash outflows i.e. it sets out the net present value equal to zero. Internal rate of return is basically used to
measure the efficiency of capital investment. Internal rate of return is generally required low cost of capital to accept
the project.
8
Internal rate of return
9
Calculation
Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows from the net
present value as a function of the rate of return. A rate of return for which this function is zero is an internal rate of
return.
Given the (period, cash flow) pairs (
the net present value
,
) where
is a positive integer, the total number of periods
, the internal rate of return is given by
, and
in:
The period is usually given in years, but the calculation may be made simpler if is calculated using the period in
which the majority of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals)
and converted to a yearly period thereafter.
Any fixed time can be used in place of the present (e.g., the end of one interval of an annuity); the value obtained is
zero if and only if the NPV is zero.
In the case that the cash flows are random variables, such as in the case of a life annuity, the expected values are put
into the above formula.
Often, the value of
used.
cannot be found analytically. In this case, numerical methods or graphical methods must be
Example
If an investment may be given by the sequence of cash flows
Year (
then the IRR
)
Cash flow (
0
-1000000
1
300000
2
500000
3
500000
)
is given by
In this case, the answer is 13.19% (in the calculation, that is, r = .1319).
Numerical solution
Since the above is a manifestation of the general problem of finding the roots of the equation
many numerical methods that can be used to estimate
where
This
is considered the
th
. For example, using the secant method,
, there are
is given by
approximation of the IRR.
can be found to an arbitrary degree of accuracy. An accuracy of 0.00001% is provided by Microsoft Excel.
The convergence behaviour of the sequence is governed by the following:
• If the function
• If the function
has a single real root
has real roots
, then the sequence will converge reproducibly towards .
, then the sequence will converge to one of the roots and
changing the values of the initial pairs may change the root to which it converges.
Internal rate of return
• If function
Having
10
has no real roots, then the sequence will tend towards +∞.
when
or
when
may speed up convergence of
to
.
Numerical solution for single outflow and multiple inflows
Of particular interest is the case where the stream of payments consists of a single outflow, followed by multiple
inflows occurring at equal periods. In the above notation, this corresponds to:
In this case the NPV of the payment stream is a convex, strictly decreasing function of interest rate. There is always
a single unique solution for IRR.
Given two estimates
and
for IRR, the secant method equation (see above) with
will always produce
an improved estimate
. This is sometimes referred to as the Hit and Trial (or Trial and Error) method. More
accurate interpolation formulas can also be obtained: for instance the secant formula with correction
,
(which is most accurate when
) has been shown to be almost 10 times more accurate
than the secant formula for a wide range of interest rates and initial guesses. For example, using the stream of
payments {−4000, 1200, 1410, 1875, 1050} and initial guesses
and
the secant formula with
correction gives an IRR estimate of 14.2% (0.7% error) as compared to IRR = 13.2% (7% error) from the secant
method. Other improved formulas may be found in [2]
If applied iteratively, either the secant method or the improved formula will always converge to the correct solution.
Both the secant method and the improved formula rely on initial guesses for IRR. The following initial guesses may
be used:
where
Internal rate of return
Decision criterion
If the IRR is greater than the cost of capital, accept the project.
If the IRR is less than the cost of capital, reject the project.
Problems with using internal rate of return
As an investment decision tool, the calculated IRR should not be used to rate mutually exclusive projects, but only to
decide whether a single project is worth investing in.
In cases where one project has a higher
initial investment than a second
mutually exclusive project, the first
project may have a lower IRR
(expected return), but a higher NPV
(increase in shareholders' wealth) and
should thus be accepted over the
second project (assuming no capital
constraints).
IRR assumes reinvestment of interim
cash flows in projects with equal rates
of return (the reinvestment can be the
same project or a different project).
Therefore, IRR overstates the annual
equivalent rate of return for a project
NPV vs discount rate comparison for two mutually exclusive projects. Project 'A' has a
whose interim cash flows are
higher NPV (for certain discount rates), even though its IRR (= x-axis intercept) is lower
reinvested at a rate lower than the
than for project 'B' (click to enlarge)
calculated IRR. This presents a
problem, especially for high IRR
projects, since there is frequently not another project available in the interim that can earn the same rate of return as
the first project.
When the calculated IRR is higher than the true reinvestment rate for interim cash flows, the measure
will overestimate — sometimes very significantly — the annual equivalent return from the project. The
formula assumes that the company has additional projects, with equally attractive prospects, in which to
invest the interim cash flows.[3]
This makes IRR a suitable (and popular) choice for analyzing venture capital and other private equity investments, as
these strategies usually require several cash investments throughout the project, but only see one cash outflow at the
end of the project (e.g., via IPO or M&A).
Since IRR does not consider cost of capital, it should not be used to compare projects of different duration. Modified
Internal Rate of Return (MIRR) does consider cost of capital and provides a better indication of a project's efficiency
in contributing to the firm's discounted cash flow.
In the case of positive cash flows followed by negative ones and then by positive ones (for example, + + − − − +) the
IRR may have multiple values. In this case a discount rate may be used for the borrowing cash flow and the IRR
calculated for the investment cash flow. This applies for example when a customer makes a deposit before a specific
machine is built.
In a series of cash flows like (−10, 21, −11), one initially invests money, so a high rate of return is best, but then
receives more than one possesses, so then one owes money, so now a low rate of return is best. In this case it is not
even clear whether a high or a low IRR is better. There may even be multiple IRRs for a single project, like in the
11
Internal rate of return
example 0% as well as 10%. Examples of this type of project are strip mines and nuclear power plants, where there
is usually a large cash outflow at the end of the project.
In general, the IRR can be calculated by solving a polynomial equation. Sturm's theorem can be used to determine if
that equation has a unique real solution. In general the IRR equation cannot be solved analytically but only
iteratively.
When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the benefits
reinvested.[3] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost
of capital.
It has been shown[4] that with multiple internal rates of return, the IRR approach can still be interpreted in a way that
is consistent with the present value approach provided that the underlying investment stream is correctly identified as
net investment or net borrowing.
See also [5] for a way of identifying the relevant value of the IRR from a set of multiple IRR solutions.
Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV.[6] Apparently,
managers find it easier to compare investments of different sizes in terms of percentage rates of return than by
dollars of NPV. However, NPV remains the "more accurate" reflection of value to the business. IRR, as a measure of
investment efficiency may give better insights in capital constrained situations. However, when comparing mutually
exclusive projects, NPV is the appropriate measure.
Mathematics
Mathematically, the value of the investment is assumed to undergo exponential growth or decay according to some
rate of return (any value greater than −100%), with discontinuities for cash flows, and the IRR of a series of cash
flows is defined as any rate of return that results in a net present value of zero (or equivalently, a rate of return that
results in the correct value of zero after the last cash flow).
Thus, internal rate(s) of return follow from the net present value as a function of the rate of return. This function is
continuous. Towards a rate of return of −100% the net present value approaches infinity with the sign of the last cash
flow, and towards a rate of return of positive infinity the net present value approaches the first cash flow (the one at
the present). Therefore, if the first and last cash flow have a different sign there exists an internal rate of return.
Examples of time series without an IRR:
• Only negative cash flows — the NPV is negative for every rate of return.
• (−1, 1, −1), rather small positive cash flow between two negative cash flows; the NPV is a quadratic function of
1/(1 + r), where r is the rate of return, or put differently, a quadratic function of the discount rate r/(1 + r); the
highest NPV is −0.75, for r = 100%.
In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, consider
the total value of the cash flows converted to a time between the negative and the positive ones. The resulting
function of the rate of return is continuous and monotonically decreasing from positive infinity to negative infinity,
so there is a unique rate of return for which it is zero. Hence, the IRR is also unique (and equal). Although the
NPV-function itself is not necessarily monotonically decreasing on its whole domain, it is at the IRR.
Similarly, in the case of a series of exclusively positive cash flows followed by a series of exclusively negative ones
the IRR is also unique.
Finally, by Descartes' rule of signs, the number of internal rates of return can never be more than the number of
changes in sign of cash flow.
12
Internal rate of return
13
References
[1] Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269
[2] Moten, J. and Thron, C., Improvements on Secant Method for Estimating Internal Rate of Return, International Journal of Applied
Mathematics and Statistics, to appear.
[3] Internal Rate of Return: A Cautionary Tale (http:/ / www. cfo. com/ article. cfm/ 3304945)
[4] Hazen, G. B., "A new perspective on multiple internal rates of return," The Engineering Economist 48(2), 2003, 31–51.
[5] Hartman, J. C., and Schafrick, I. C., "The relevant internal rate of return," The Engineering Economist 49(2), 2004, 139–158.
[6] Pogue, M.(2004). Investment Appraisal: A New Approach. Managerial Auditing Journal.Vol. 19 No. 4, 2004. pp. 565–570
Further reading
1. Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0-471-26849-6
External links
• Economics Interactive Lecture from University of South Carolina (http://hspm.sph.sc.edu/courses/Econ/irr/
irr.html)
Discounted cash flow
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. All future cash flows are estimated and discounted to
give their present values (PVs)—the sum of all future cash flows, both
incoming and outgoing, is the net present value (NPV), which is taken
as the value or price of the cash flows in question.
Using DCF analysis to compute the NPV takes as input cash flows and
a discount rate and gives as output a price; the opposite
process—taking cash flows and a price and inferring a discount rate, is
called the yield.
Discounted cash flow analysis is widely used in investment finance,
real estate development, and corporate financial management.
Spreadsheet uses Free cash flows to estimate
stock's Fair Value and measure the sensibility of
WACC and Perpetual growth
Discount rate
The most widely used method of discounting is exponential discounting, which values future cash flows as "how
much money would have to be invested currently, at a given rate of return, to yield the cash flow in future." Other
methods of discounting, such as hyperbolic discounting, are studied in academia and said to reflect intuitive
decision-making, but are not generally used in industry.
The discount rate used is generally the appropriate Weighted average cost of capital (WACC), that reflects the risk of
the cashflows. The discount rate reflects two things:
1. Time value of money (risk-free rate) – according to the theory of time preference, investors would rather have
cash immediately than having to wait and must therefore be compensated by paying for the delay
2. Risk premium – reflects the extra return investors demand because they want to be compensated for the risk that
the cash flow might not materialize after all
Discounted cash flow
History
Discounted cash flow calculations have been used in some form since money was first lent at interest in ancient
times. As a method of asset valuation it has often been opposed to accounting book value, which is based on the
amount paid for the asset. Following the stock market crash of 1929, discounted cash flow analysis gained popularity
as a valuation method for stocks. Irving Fisher in his 1930 book The Theory of Interest and John Burr Williams's
1938 text The Theory of Investment Value first formally expressed the DCF method in modern economic terms.
Mathematics
Discounted cash flows
The discounted cash flow formula is derived from the future value formula for calculating the time value of money
and compounding returns.
Thus the discounted present value (for one cash flow in one future period) is expressed as:
where
• DPV is the discounted present value of the future cash flow (FV), or FV adjusted for the delay in receipt;
• FV is the nominal value of a cash flow amount in a future period;
• i is the interest rate, which reflects the cost of tying up capital and may also allow for the risk that the payment
may not be received in full;
• d is the discount rate, which is i/(1+i), i.e., the interest rate expressed as a deduction at the beginning of the year
instead of an addition at the end of the year;
• n is the time in years before the future cash flow occurs.
Where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows:
for each future cash flow (FV) at any time period (t) in years from the present time, summed over all time periods.
The sum can then be used as a net present value figure. If the amount to be paid at time 0 (now) for all the future
cash flows is known, then that amount can be substituted for DPV and the equation can be solved for i, that is the
internal rate of return.
All the above assumes that the interest rate remains constant throughout the whole period.
14
Discounted cash flow
Continuous cash flows
For continuous cash flows, the summation in the above formula is replaced by an integration:
where FV(t) is now the rate of cash flow, and λ = log(1+i).
DF (r/1+r)-1
Example DCF
To show how discounted cash flow analysis is performed, consider the following simplified example.
• John Doe buys a house for $100,000. Three years later, he expects to be able to sell this house for $150,000.
Simple subtraction suggests that the value of his profit on such a transaction would be $150,000 − $100,000 =
$50,000, or 50%. If that $50,000 is amortized over the three years, his implied annual return (known as the internal
rate of return) would be about 14.5%. Looking at those figures, he might be justified in thinking that the purchase
looked like a good idea.
1.1453 x 100000 = 150000 approximately.
However, since three years have passed between the purchase and the sale, any cash flow from the sale must be
discounted accordingly. At the time John Doe buys the house, the 3-year US Treasury Note rate is 5% per annum.
Treasury Notes are generally considered to be inherently less risky than real estate, since the value of the Note is
guaranteed by the US Government and there is a liquid market for the purchase and sale of T-Notes. If he hadn't put
his money into buying the house, he could have invested it in the relatively safe T-Notes instead. This 5% per annum
can therefore be regarded as the risk-free interest rate for the relevant period (3 years).
Using the DPV formula above (FV=$150,000, i=0.05, n=3), that means that the value of $150,000 received in three
years actually has a present value of $129,576 (rounded off). In other words we would need to invest $129,576 in a
T-Bond now to get $150,000 in 3 years almost risk free. This is a quantitative way of showing that money in the
future is not as valuable as money in the present ($150,000 in 3 years isn't worth the same as $150,000 now; it is
worth $129,576 now).
Subtracting the purchase price of the house ($100,000) from the present value results in the net present value of the
whole transaction, which would be $29,576 or a little more than 29% of the purchase price.
Another way of looking at the deal as the excess return achieved (over the risk-free rate) is (114.5 - 105)/(100 + 5) or
approximately 9.0% (still very respectable).
But what about risk?
We assume that the $150,000 is John's best estimate of the sale price that he will be able to achieve in 3 years time
(after deducting all expenses, of course). There is of course a lot of uncertainty about house prices, and the outcome
may end up higher or lower than this estimate.
(The house John is buying is in a "good neighborhood," but market values have been rising quite a lot lately and the
real estate market analysts in the media are talking about a slow-down and higher interest rates. There is a
probability that John might not be able to get the full $150,000 he is expecting in three years due to a slowing of
price appreciation, or that loss of liquidity in the real estate market might make it very hard for him to sell at all.)
Under normal circumstances, people entering into such transactions are risk-averse, that is to say that they are
prepared to accept a lower expected return for the sake of avoiding risk. See Capital asset pricing model for a further
discussion of this. For the sake of the example (and this is a gross simplification), let's assume that he values this
particular risk at 5% per annum (we could perform a more precise probabilistic analysis of the risk, but that is
beyond the scope of this article). Therefore, allowing for this risk, his expected return is now 9.0% per annum (the
arithmetic is the same as above).
15
Discounted cash flow
And the excess return over the risk-free rate is now (109 - 105)/(100 + 5) which comes to approximately 3.8% per
annum.
That return rate may seem low, but it is still positive after all of our discounting, suggesting that the investment
decision is probably a good one: it produces enough profit to compensate for tying up capital and incurring risk with
a little extra left over. When investors and managers perform DCF analysis, the important thing is that the net present
value of the decision after discounting all future cash flows at least be positive (more than zero). If it is negative, that
means that the investment decision would actually lose money even if it appears to generate a nominal profit. For
instance, if the expected sale price of John Doe's house in the example above was not $150,000 in three years, but
$130,000 in three years or $150,000 in five years, then on the above assumptions buying the house would actually
cause John to lose money in present-value terms (about $3,000 in the first case, and about $8,000 in the second).
Similarly, if the house was located in an undesirable neighborhood and the Federal Reserve Bank was about to raise
interest rates by five percentage points, then the risk factor would be a lot higher than 5%: it might not be possible
for him to predict a profit in discounted terms even if he thinks he could sell the house for $200,000 in three years.
In this example, only one future cash flow was considered. For a decision which generates multiple cash flows in
multiple time periods, all the cash flows must be discounted and then summed into a single net present value.
Methods of appraisal of a company or project
This is necessarily a simple treatment of a complex subject: more detail is beyond the scope of this article.
For these valuation purposes, a number of different DCF methods are distinguished today, some of which are
outlined below. The details are likely to vary depending on the capital structure of the company. However the
assumptions used in the appraisal (especially the equity discount rate and the projection of the cash flows to be
achieved) are likely to be at least as important as the precise model used.
Both the income stream selected and the associated cost of capital model determine the valuation result obtained
with each method. This is one reason these valuation methods are formally referred to as the Discounted Future
Economic Income methods.
• Equity-Approach
• Flows to equity approach (FTE)
Discount the cash flows available to the holders of equity capital, after allowing for cost of servicing debt capital
Advantages: Makes explicit allowance for the cost of debt capital
Disadvantages: Requires judgement on choice of discount rate
• Entity-Approach:
• Adjusted present value approach (APV)
Discount the cash flows before allowing for the debt capital (but allowing for the tax relief obtained on the debt
capital)
Advantages: Simpler to apply if a specific project is being valued which does not have earmarked debt capital
finance
Disadvantages: Requires judgement on choice of discount rate; no explicit allowance for cost of debt capital, which
may be much higher than a "risk-free" rate
• Weighted average cost of capital approach (WACC)
Derive a weighted cost of the capital obtained from the various sources and use that discount rate to discount the
cash flows from the project
Advantages: Overcomes the requirement for debt capital finance to be earmarked to particular projects
16
Discounted cash flow
Disadvantages: Care must be exercised in the selection of the appropriate income stream. The net cash flow to total
invested capital is the generally accepted choice.
• Total cash flow approach (TCF)Wikipedia:Please clarify
This distinction illustrates that the Discounted Cash Flow method can be used to determine the value of various
business ownership interests. These can include equity or debt holders.
Alternatively, the method can be used to value the company based on the value of total invested capital. In each case,
the differences lie in the choice of the income stream and discount rate. For example, the net cash flow to total
invested capital and WACC are appropriate when valuing a company based on the market value of all invested
capital.[1]
Shortcomings
Commercial banks have widely used discounted cash flow as a method of valuing commercial real estate
construction projects. This practice has two substantial shortcomings. 1) The discount rate assumption relies on the
market for competing investments at the time of the analysis, which would likely change, perhaps dramatically, over
time, and 2) straight line assumptions about income increasing over ten years are generally based upon historic
increases in market rent but never factors in the cyclical nature of many real estate markets. Most loans are made
during boom real estate markets and these markets usually last fewer than ten years. Using DCF to analyze
commercial real estate during any but the early years of a boom market will lead to overvaluation of the asset.
Discounted cash flow models are powerful, but they do have shortcomings. DCF is merely a mechanical valuation
tool, which makes it subject to the principle "garbage in, garbage out". Small changes in inputs can result in large
changes in the value of a company. Instead of trying to project the cash flows to infinity, terminal value techniques
are often used. A simple annuity is used to estimate the terminal value past 10 years, for example. This is done
because it is harder to come to a realistic estimate of the cash flows as time goes oninvolves calculating the period of
time likely to recoup the initial outlay.[2]
References
External links
• Continuous compounding/cash flows (http://ocw.mit.edu/courses/nuclear-engineering/
22-812j-managing-nuclear-technology-spring-2004/lecture-notes/lec03slides.pdf)
• The Theory of Interest (http://www.econlib.org/library/YPDBooks/Fisher/fshToI.html) at the Library of
Economics and Liberty.
• Monography about DCF (including some lectures on DCF) (http://www.wacc.biz).
• Foolish Use of DCF (http://www.fool.com/news/commentary/2005/commentary05032803.htm). Motley
Fool.
• Getting Started With Discounted Cash Flows (http://www.thestreet.com/university/personalfinance/
10385275.html). The Street.
• International Good Practice: Guidance on Project Appraisal Using Discounted Cash Flow (http://www.ifac.
org/Members/DownLoads/Project_Appraisal_Using_DCF_formatted.pdf), International Federation of
Accountants, June 2008, ISBN 978-1-934779-39-2
• Equivalence between Discounted Cash Flow (DCF) and Residual Income (RI) (http://papers.ssrn.com/sol3/
papers.cfm?abstract_id=381880) Working paper; Duke University - Center for Health Policy, Law and
Management
17
Discounted cash flow
18
Further reading
• International Association of CPAs, Attorneys, and Management (IACAM) (http://www.iacam.org/) (Free DCF
Valuation E-Book Guidebook)
• International Federation of Accountants (2007). Project Appraisal Using Discounted Cash Flow.
• Copeland, Thomas E.; Tim Koller, Jack Murrin (2000). Valuation: Measuring and Managing the Value of
Companies. New York: John Wiley & Sons. ISBN 0-471-36190-9.
• Damodaran, Aswath (1996). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset.
New York: John Wiley & Sons. ISBN 0-471-13393-0.
• Rosenbaum, Joshua; Joshua Pearl (2009). Investment Banking: Valuation, Leveraged Buyouts, and Mergers &
Acquisitions. Hoboken, NJ: John Wiley & Sons. ISBN 0-470-44220-4.
• James R. Hitchnera (2006). Financial Valuation: Applications and Models. USA: Wiley Finance.
ISBN 0-471-76117-6.
• Chander Sawhney (2012). Discounted Cash Flow –The Prominent Income Approach to Valuation. INDIA: (http:/
/corporatevaluations.in/static-1047-22-oth -Articles and Research Hub). ISBN NA Check |isbn= value
(help).
Opportunity cost
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In microeconomic theory, the opportunity cost of a choice is the value of the best alternative forgone, in a situation
in which a choice needs to be made between several mutually exclusive alternatives given limited resources.
Assuming the best choice is made, it is the "cost" incurred by not enjoying the benefit that would be had by taking
the second best choice available.[1] The New Oxford American Dictionary defines it as "the loss of potential gain
from other alternatives when one alternative is chosen". Opportunity cost is a key concept in economics, and has
been described as expressing "the basic relationship between scarcity and choice".[2] The notion of opportunity cost
plays a crucial part in ensuring that scarce resources are used efficiently.[3] Thus, opportunity costs are not restricted
to monetary or financial costs: the real cost of output forgone, lost time, pleasure or any other benefit that provides
utility should also be considered opportunity costs.
Opportunity cost
History
The term was coined in 1914 by Austrian economist Friedrich von Wieser in his book Theorie der gesellschaftlichen
Wirtschaft.[4] It was first described in 1848 by French classical economist Frédéric Bastiat in his essay "What Is Seen
and What Is Not Seen [5]".
Opportunity costs in consumption
Opportunity cost may be expressed in terms of anything which is of value. For example, an individual might decide
to use a period of vacation time for travel rather than to do household repairs. The opportunity cost of the trip could
be said to be the forgone home renovation.[citation needed]
Opportunity costs in production
Opportunity costs may be assessed in the decision-making process of production. If the workers on a farm can
produce either one million pounds of wheat or two million pounds of barley, then the opportunity cost of producing
one pound of wheat is the two pounds of barley forgone (assuming the production possibilities frontier is linear).
Firms would make rational decisions by weighing the sacrifices involved.
Explicit costs
Explicit costs are opportunity costs that involve direct monetary payment by producers. The opportunity cost of the
factors of production not already owned by a producer is the price that the producer has to pay for them. For
instance, a firm spends $100 on electrical power consumed, their opportunity cost is $100. The firm has sacrificed
$100, which could have been spent on other factors of production.
Implicit costs
Implicit costs are the opportunity costs in factors of production that a producer already owns. They are equivalent to
what the factors could earn for the firm in alternative uses, either operated within the firm or rent out to other firms.
For example, a firm pays $300 a month all year for rent on a warehouse that only holds product for six months each
year. The firm could rent the warehouse out for the unused six months, at any price (assuming a year-long lease
requirement), and that would be the cost that could be spent on other factors of production.
Non-monetary opportunity costs
Opportunity costs are not always monetary units or being able to produce one good over another. The opportunity
cost can also be unknown, or spawn a series of infinite sub opportunity costs. For instance, an individual could
choose not to ask a girl out on a date, in an attempt to make her more interested ("playing hard to get"), but the
opportunity cost could be that they get ignored - which could result in other opportunities being lost.
Evaluation
Note that opportunity cost is not the sum of the available alternatives when those alternatives are, in turn, mutually
exclusive to each other – it is the value of the next best use. The opportunity cost of a city's decision to build the
hospital on its vacant land is the loss of the land for a sporting center, or the inability to use the land for a parking lot,
or the money which could have been made from selling the land. Use for any one of those purposes would preclude
the possibility to implement any of the other.
20
Opportunity cost
References
[4]
•
[5] http:/ / www. econlib. org/ library/ Bastiat/ basEss1. html
External links
• The Opportunity Cost of Economics Education (http://www.nytimes.com/2005/09/01/business/01scene.
html) by Robert H. Frank
• Opportunity Cost Example & Analysis (http://www.youtube.com/watch?v=ezOdQUzLVAo)
Time value of money
The time value of money is the value of money with a given amount of interest earned or inflation accrued over a
given amount of time. The ultimate principle suggests that a certain amount of money today has different buying
power than the same amount of money in the future. This notion exists both because there is an opportunity to earn
interest on the money and because inflation will drive prices up, thus changing the "value" of the money. 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.[1] 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:
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, 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.[2]
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.[3]
Present value of a perpetuity is an infinite and constant stream of identical cash flows.[4]
Future value 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.[5]
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.
21
Time value of money
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 spreadsheet. The formulas are programmed into most financial
calculators and several spreadsheet functions (such as PV, FV, RATE, NPER, and PMT).[6]
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).
Formula
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 (PV) formula has four variables, each of which can be solved for:
1.
2.
3.
4.
PV is the value at time=0
FV is the value at time=n
i is the discount rate, 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 of future cash flows can be calculated by summing the contributions of FVt, the value
of cash flow at time t
22
Time value of money
Note that this series can be summed for a given value of n, or when n is ∞.[7] 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 (PVA)
formula has four variables, each of which can be solved for:
1.
2.
3.
4.
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 n → ∞, the PV of a perpetuity (a perpetual annuity) formula becomes simple division.
Present Value of Int Factor Annuity
Investment = 1000 Int 6.90% Compounded Qtrly (4 Times in Year) Tenure Yrs 5
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 used for stock valuation.
23
Time value of money
24
Future value of a present sum
The future value (FV) formula is similar and uses the same variables.
Future value of an annuity
The future value of an annuity (FVA) formula has four variables, each of which can be solved for:
1.
2.
3.
4.
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
To get the FV of an annuity due, multiply the above equation by (1 + i).
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 :
1.
2.
3.
4.
5.
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
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, 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
:
Time value of money
25
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:
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.
Examples
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.
Time value of money
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:
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
short-cut 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 return 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 72
26
Time value of money
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 model.
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:
27
Time value of money
Continuous compounding
Rates are sometimes converted into the continuous compound interest rate equivalent because the continuous
equivalent is more convenient (for example, more easily differentiated). Each of the formulæ above may be restated
in their continuous equivalents. For example, the present value at time 0 of a future payment at time t can be restated
in the following way, where e is the base of 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.
Examples
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.[8]
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
28
Time value of money
29
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 the use of 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
at time t,
for no time horizon) the value
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.
Time value of money
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
http:/ / www. investopedia. com/ articles/ 03/ 082703. asp
http:/ / www. investopedia. com/ terms/ p/ presentvalue. asp
http:/ / www. getobjects. com/ Components/ Finance/ TVM/ pva. html
http:/ / www. investopedia. com/ terms/ p/ perpetuity. asp
http:/ / www. investopedia. com/ terms/ f/ futurevalue. asp
Hovey, M. (2005). Spreadsheet Modelling for Finance. Frenchs Forest, N.S.W.: Pearson Education Australia.
http:/ / mathworld. wolfram. com/ GeometricSeries. html Geometric Series
http:/ / baselineeducation. blogspot. co. uk/ 2012/ 10/ annuities-and-perpetuities-with. html
• Carr, Peter; Flesaker, Bjorn (2006), Robust Replication of Default Contingent Claims (presentation slides) (http:/
/www.usc.edu/schools/business/FBE/seminars/papers/FMath_10-17-06_CARR_slides.pdf), Bloomberg LP.
See also Audio Presentation (http://www.fields.utoronto.ca/audio/06-07/finance_seminar/flesaker/) and
paper (http://www.usc.edu/schools/business/FBE/seminars/papers/FMath_10-17-06_CARR_RRDCC3.
pdf).
• Crosson, S.V., and Needles, B.E.(2008). Managerial Accounting (8th Ed). Boston: Houghton Mifflin Company.
External links
• Present Value Annuity Calculator (http://www.money-zine.com/Calculators/Retirement-Calculators/
Present-Value-Annuity-Calculator/)
• Future Value of an Annuity (http://www.investopedia.com/calculator/AnnuityFV.aspx)
• Time Value of Money Calculator by Farsight Calculator (http://www.farsightsoft.com/financial-calculator/
time-value-of-money.html)
• Time Value of Money hosted by the University of Arizona (http://www.studyfinance.com/lessons/timevalue/
index.mv)
• Time Value of Money Ebook (http://www.swlearning.com/finance/brigham/ifm8e/web_chapters/
webchapter28.pdf)
• Inflation calculator (http://data.bls.gov/cgi-bin/cpicalc.pl)
30
Rate of return
31
Rate of return
In finance, rate of return (ROR), also known as return on investment (ROI), rate of profit or sometimes just
return, is the ratio of money gained or lost (whether realized or unrealized) 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 net
income/loss. The money invested may be referred to as the asset, capital, principal, or the cost basis of the
investment. ROI is usually expressed as a percentage.
Calculation
The initial value of an investment,
, does not always have a clearly defined monetary value, but for purposes of
measuring ROI, the expected value must be clearly stated along with the rationale for this initial value. Similarly, the
final value of an investment,
, also does not always have a clearly defined monetary value, but for purposes of
measuring ROI, the final value must be clearly stated along with the rationale for this final value.[citation needed]
The rate of return can be calculated over a single period, or expressed as an average over multiple periods of time.
Single-period
Arithmetic return
The arithmetic return is:
is sometimes referred to as the yield. See also: effective interest rate, effective annual rate (EAR) or annual
percentage yield (APY).
Logarithmic or continuously compounded return
The logarithmic return or continuously compounded return, also known as force of interest, is defined as:
or
where: R = Returns P = Principal amount r = rate t = time period
Multiperiod average returns
Arithmetic average rate of return
The arithmetic average rate of return over n periods is defined as:
Geometric average rate of return
The geometric average rate of return, also known as the annualized return, over n periods is defined as:
Rate of return
32
Importantly, the annualized return is less than the average annual return (or equal if all annual returns are equal), as a
consequence of the AM–GM inequality. In fact, the difference between the annualized return and average annual
return is proportional to variance (square root of volatility) – the more volatile the performance, the greater the
difference, in proportion to the variance.[1] As a basic example, a return of +10%, followed by −10%, has an average
return of 0%, but the overall result is
for an overall return of −1%. For a return of +20%,
followed by −20%, this again has an average return of 0%, but an overall return of −4%. In the extreme, a return of
+100%, followed by −100%, has an average return of 0%, but an overall return of −100%, as the value ends at 0. In
cases of leveraged investments, even more extreme results are possible: a return of +200%, followed by −200%, has
an average return of 0%, but an overall return of −300%. In financial mathematics, the infinitesimal version of this
discrepancy between the average return and the annualized return, meaning when the year-long periods are replaced
by shorter periods (in the limit infinitely short periods), is Itō's lemma for geometric Brownian motion.
In the presence of external flows, such as cash or securities moving into or out of the portfolio, the overall return
must be calculated gross of these movements, which is done by the True Time-Weighted Rate of Return (TWRR).
Time-weighted rates of return are important because they eliminate the impact of cash flows. This is helpful when
assessing the job that a money manager did for his/her clients, where typically the clients control these cash flows.[2]
Internal rate of return
The internal rate of return (IRR), also known as the dollar-weighted rate of return or the money-weighted rate
of return (MWRR), is defined as the value(s) of that satisfies the following equation:
where:
• NPV = net present value of the investment
•
= cashflow at time
When the cost of capital
is smaller than the IRR rate
, the investment is profitable, i.e.,
.
Otherwise, the investment is not profitable.
MWRR are helpful in that they take cash flows into consideration. This is especially helpful when evaluating cases
where the money manager controls cash flows (for private equity investments, for example, as well as sub-portfolio
rates of return) as well as to provide the investor with their return. Contrast with TWRR.
Comparisons between various rates of return
Arithmetic and logarithmic return
The value of an investment is doubled over a year if the annual ROR
$100) = ln(2) = 69.3%. The value falls to zero when
= -100%, that is, if
= +100%, that is, if
= ln($200 /
= -∞.
Arithmetic and logarithmic returns are not equal, but are approximately equal for small returns. The difference
between them is large only when percent changes are high. For example, an arithmetic return of +50% is equivalent
to a logarithmic return of 40.55%, while an arithmetic return of -50% is equivalent to a logarithmic return of
-69.31%.
Logarithmic returns are often used by academics in their research. The main advantage is that the continuously
compounded return is symmetric, while the arithmetic return is not: positive and negative percent arithmetic returns
are not equal. This means that an investment of $100 that yields an arithmetic return of 50% followed by an
arithmetic return of -50% will result in $75, while an investment of $100 that yields a logarithmic return of 50%
followed by a logarithmic return of -50% it will remain $100.
Rate of return
33
Comparison of arithmetic and logarithmic returns for initial investment of $100
Initial investment,
$100
$100
$100
$100
$100
Final investment,
$0
$50
$100
$150
$200
−$100
−$50
$0
$50
$100
−100%
−50%
0%
50%
100%
−∞
−69.31%
0%
Profit/loss,
Arithmetic return,
Logarithmic return,
40.55% 69.31%
Arithmetic average and geometric average rates of return
Both arithmetic and geometric average rates of returns are averages of periodic percentage returns. Neither will
accurately translate to the actual dollar amounts gained or lost if percent gains are averaged with percent losses.[3] A
10% loss on a $100 investment is a $10 loss, and a 10% gain on a $100 investment is a $10 gain. When percentage
returns on investments are calculated, they are calculated for a period of time – not based on original investment
dollars, but based on the dollars in the investment at the beginning and end of the period. So if an investment of $100
loses 10% in the first period, the investment amount is then $90. If the investment then gains 10% in the next period,
the investment amount is $99.
A 10% gain followed by a 10% loss is a 1% loss. The order in which the loss and gain occurs does not affect the
result. A 50% gain and a 50% loss is a 25% loss. An 80% gain plus an 80% loss is a 64% loss. To recover from a
50% loss, a 100% gain is required. The mathematics of this are beyond the scope of this article, but since investment
returns are often published as "average returns", it is important to note that average returns do not always translate
into dollar returns.
Example #1 Level Rates of Return
Year 1
Year 2
Year 3
Year 4
Rate of Return
5%
5%
5%
5%
Geometric Average at End of Year
5%
5%
5%
5%
Capital at End of Year
$105.00 $110.25 $115.76 $121.55
Dollar Profit/(Loss)
$5.00
Compound Yield
5%
$10.25
$15.76
$21.55
5.4%
Example #2 Volatile Rates of Return, including losses
Year 1
Year 2
Rate of Return
50%
-20%
30%
-40%
Geometric Average at End of Year
50%
9.5%
16%
-1.6%
Capital at End of Year
Year 3 Year 4
$150.00 $120.00 $156.00 $93.60
Dollar Profit/(Loss)
($6.40)
Compound Yield
-1.6%
Rate of return
34
Example #3 Highly Volatile Rates of Return, including losses
Year 1 Year 2 Year 3
Rate of Return
-95%
Geometric Average at End of Year
-95%
Capital at End of Year
$5.00
0%
0%
-77.6% -63.2%
$5.00
$5.00
Year 4
115%
-42.7%
$10.75
Dollar Profit/(Loss)
($89.25)
Compound Yield
-22.3%
Annual returns and annualized returns
Care must be taken not to confuse annual and annualized returns. An annual rate of return is a single-period return,
while an annualized rate of return is a multi-period, arithmetic average return.
An annual rate of return is the return on an investment over a one-year period, such as January 1 through December
31, or June 3, 2006 through June 2, 2007. Each ROI in the cash flow example above is an annual rate of return.
An annualized rate of return is the return on an investment over a period other than one year (such as a month, or two
years) multiplied or divided to give a comparable one-year return. For instance, a one-month ROI of 1% could be
stated as an annualized rate of return of 12.7% = ((1+0.01)12 - 1). Or a two-year ROI of 10% could be stated as an
annualized rate of return of 4.88% = ((1+0.1)(12/24) - 1).
In the cash flow example below, the dollar returns for the four years add up to $265. The annualized rate of return for
the four years is: $265 ÷ ($1,000 x 4 years) = 6.625%.
Uses
• ROI is a measure of cash[citation needed] generated by or lost due to the investment. It measures the cash flow or
income stream from the investment to the investor, relative to the amount invested. Cash flow to the investor can
be in the form of profit, interest, dividends, or capital gain/loss. Capital gain/loss occurs when the market value or
resale value of the investment increases or decreases. Cash flow here does not include the return of invested
capital.
Cash Flow Example on $1,000 Investment
Year 1 Year 2 Year 3 Year 4
Dollar Return
$100
$55
$60
$50
ROI
10%
5.5%
6%
5%
• ROI values typically used for personal financial decisions include Annual Rate of Return and Annualized Rate
of Return. For nominal risk investments such as savings accounts or Certificates of Deposit, the personal investor
considers the effects of reinvesting/compounding on increasing savings balances over time. For investments in
which capital is at risk, such as stock shares, mutual fund shares and home purchases, the personal investor
considers the effects of price volatility and capital gain/loss on returns.
• Profitability ratios typically used by financial analysts to compare a company’s profitability over time or
compare profitability between companies include Gross Profit Margin, Operating Profit Margin, ROI ratio,
Dividend yield, Net profit margin, Return on equity, and Return on assets.[4]
• During capital budgeting, companies compare the rates of return of different projects to select which projects to
pursue in order to generate maximum return or wealth for the company's stockholders. Companies do so by
considering the average rate of return, payback period, net present value, profitability index, and internal rate of
Rate of return
35
return for various projects.[5]
• A return may be adjusted for taxes to give the after-tax rate of return. This is done in geographical areas or
historical times in which taxes consumed or consume a significant portion of profits or income. The after-tax rate
of return is calculated by multiplying the rate of return by the tax rate, then subtracting that percentage from the
rate of return.
• A return of 5% taxed at 15% gives an after-tax return of 4.25%
0.05 x 0.15 = 0.0075
0.05 - 0.0075 = 0.0425 = 4.25%
• A return of 10% taxed at 25% gives an after-tax return of 7.5%
0.10 x 0.25 = 0.025
0.10 - 0.025 = 0.075 = 7.5%
Investors usually seek a higher rate of return on taxable investment returns than on non-taxable investment returns.
• A return may be adjusted for inflation to better indicate its true value in purchasing power. Any investment with a
nominal rate of return less than the annual inflation rate represents a loss of value, even though the nominal rate
of return might well be greater than 0%. When ROI is adjusted for inflation, the resulting return is considered an
increase or decrease in purchasing power. If an ROI value is adjusted for inflation, it is stated explicitly, such as
“The return, adjusted for inflation, was 2%.”
• Many online poker tools include ROI in a player's tracked statistics, assisting users in evaluating an opponent's
profitability.
Cash or potential cash returns
Time value of money
Investments generate cash flow to the investor to compensate the investor for the time value of money.
Except for rare periods of significant deflation where the opposite may be true, a dollar in cash is worth less today
than it was yesterday, and worth more today than it will be worth tomorrow. The main factors that are used by
investors to determine the rate of return at which they are willing to invest money include:
• estimates of future inflation rates
• estimates regarding the risk of the investment (e.g. how likely it is that investors will receive regular
interest/dividend payments and the return of their full capital)
• whether or not the investors want the money available (“liquid”) for other uses.
The time value of money is reflected in the interest rates that banks offer for deposits, and also in the interest rates
that banks charge for loans such as home mortgages. The “risk-free” rate is the rate on U.S. Treasury bills, because
this is the highest rate available without risking capital.
The rate of return which an investor expects from an investment is called the Discount Rate. Each investment has a
different discount rate, based on the cash flow expected in future from the investment. The higher the risk, the higher
the discount rate (rate of return) the investor will demand from the investment.
Rate of return
36
Compounding or reinvesting
Compound interest or other reinvestment of cash returns (such as interest and dividends) does not affect the discount
rate of an investment, but it does affect the Annual Percentage Yield, because compounding/reinvestment increases
the capital invested.
For example, if an investor put $1,000 in a 1-year Certificate of Deposit (CD) that paid an annual interest rate of 4%,
compounded quarterly, the CD would earn 1% interest per quarter on the account balance. The account balance
includes interest previously credited to the account.
Compound Interest Example
1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
Capital at the beginning of the period
$1,000
$1,010
$1,020.10
$1,030.30
Dollar return for the period
$10
$10.10
$10.20
$10.30
Account Balance at end of the period
$1,010.00
$1,020.10
$1,030.30
$1,040.60
Quarterly ROI
1%
1%
1%
1%
The concept of 'income stream' may express this more clearly. At the beginning of the year, the investor took $1,000
out of his pocket (or checking account) to invest in a CD at the bank. The money was still his, but it was no longer
available for buying groceries. The investment provided a cash flow of $10.00, $10.10, $10.20 and $10.30. At the
end of the year, the investor got $1,040.60 back from the bank. $1,000 was return of capital.
Once interest is earned by an investor it becomes capital. Compound interest involves reinvestment of capital; the
interest earned during each quarter is reinvested. At the end of the first quarter the investor had capital of $1,010.00,
which then earned $10.10 during the second quarter. The extra dime was interest on his additional $10 investment.
The Annual Percentage Yield or Future value for compound interest is higher than for simple interest because the
interest is reinvested as capital and earns interest. The yield on the above investment was 4.06%.
Bank accounts offer contractually guaranteed returns, so investors cannot lose their capital. Investors/Depositors lend
money to the bank, and the bank is obligated to give investors back their capital plus all earned interest. Because
investors are not risking losing their capital on a bad investment, they earn a quite low rate of return. But their capital
steadily increases.
Returns when capital is at risk
Capital gains and losses
Many investments carry significant risk that the investor will lose some or all of the invested capital. For example,
investments in company stock shares put capital at risk. The value of a stock share depends on what someone is
willing to pay for it at a certain point in time. Unlike capital invested in a savings account, the capital value (price) of
a stock share constantly changes. If the price is relatively stable, the stock is said to have “low volatility.” If the price
often changes a great deal, the stock has “high volatility.” All stock shares have some volatility, and the change in
price directly affects ROI for stock investments.
Stock returns are usually calculated for holding periods such as a month, a quarter or a year.
Rate of return
37
Reinvestment when capital is at risk: rate of return and yield
Example: Stock with low volatility and a regular quarterly dividend, reinvested
End of:
1st Quarter 2nd Quarter 3rd Quarter 4th Quarter
Dividend
$1
$1.01
$1.02
$1.03
Stock Price
$98
$101
$102
$99
Shares Purchased
0.010204
0.01
0.01
0.010404
Total Shares Held
1.010204
1.020204
1.030204
1.040608
Investment Value
$99
$103.04
$105.08
$103.02
Quarterly ROI
-1%
4.08%
1.98%
-1.96%
Yield is the compound rate of return that includes the effect of reinvesting interest or dividends.
To the right is an example of a stock investment of one share purchased at the beginning of the year for $100.
• The quarterly dividend is reinvested at the quarter-end stock price.
• The number of shares purchased each quarter = ($ Dividend)/($ Stock Price).
• The final investment value of $103.02 is a 3.02% Yield on the initial investment of $100. This is the compound
yield, and this return can be considered to be the return on the investment of $100.
To calculate the rate of return, the investor includes the reinvested dividends in the total investment. The investor
received a total of $4.06 in dividends over the year, all of which were reinvested, so the investment amount increased
by $4.06.
• Total Investment = Cost Basis = $100 + $4.06 = $104.06.
• Capital gain/loss = $103.02 - $104.06 = -$1.04 (a capital loss)
• ($4.06 dividends - $1.04 capital loss ) / $104.06 total investment = 2.9% ROI
The disadvantage of this ROI calculation is that it does not take into account the fact that not all the money was
invested during the entire year (the dividend reinvestments occurred throughout the year). The advantages are: (1) it
uses the cost basis of the investment, (2) it clearly shows which gains are due to dividends and which gains/losses are
due to capital gains/losses, and (3) the actual dollar return of $3.02 is compared to the actual dollar investment of
$104.06.
For U.S. income tax purposes, if the shares were sold at the end of the year, dividends would be $4.06, cost basis of
the investment would be $104.06, sale price would be $103.02, and the capital loss would be $1.04.
Since all returns were reinvested, the ROI might also be calculated as a continuously compounded return or
logarithmic return. The effective continuously compounded rate of return is the natural log of the final investment
value divided by the initial investment value:
•
is the initial investment ($100)
•
is the final value ($103.02)
.
Rate of return
Mutual fund and investment company returns
Mutual funds, exchange-traded funds (ETFs), and other equitized investments (such as unit investment trusts or
UITs, insurance separate accounts and related variable products such as variable universal life insurance policies and
variable annuity contracts, and bank-sponsored commingled funds, collective benefit funds or common trust funds)
are essentially portfolios of various investment securities such as stocks, bonds and money market instruments which
are equitized by selling shares or units to investors. Investors and other parties are interested to know how the
investment has performed over various periods of time.
Performance is usually quantified by a fund's total return. In the 1990s, many different fund companies were
advertising various total returns—some cumulative, some averaged, some with or without deduction of sales loads or
commissions, etc. To level the playing field and help investors compare performance returns of one fund to another,
the U.S. Securities and Exchange Commission (SEC) began requiring funds to compute and report total returns
based upon a standardized formula—so called "SEC Standardized total return" which is the average annual total
return assuming reinvestment of dividends and distributions and deduction of sales loads or charges. Funds may
compute and advertise returns on other bases (so-called "non-standardized" returns), so long as they also publish no
less prominently the "standardized" return data.
Subsequent to this, apparently investors who'd sold their fund shares after a large increase in the share price in the
late 1990s and early 2000s were ignorant of how significant the impact of income/capital gain taxes was on their
fund "gross" returns. That is, they had little idea how significant the difference could be between "gross" returns
(returns before federal taxes) and "net" returns (after-tax returns). In reaction to this apparent investor ignorance, and
perhaps for other reasons, the SEC made further rule-making to require mutual funds to publish in their annual
prospectus, among other things, total returns before and after the impact of U.S federal individual income taxes. And
further, the after-tax returns would include 1) returns on a hypothetical taxable account after deducting taxes on
dividends and capital gain distributions received during the illustrated periods and 2) the impacts of the items in #1)
as well as assuming the entire investment shares were sold at the end of the period (realizing capital gain/loss on
liquidation of the shares). These after-tax returns would apply of course only to taxable accounts and not to
tax-deferred or retirement accounts such as IRAs.
Lastly, in more recent years, "personalized" brokerage account statements have been demanded by investors. In other
words, the investors are saying more or less that the fund returns may not be what their actual account returns are,
based upon the actual investment account transaction history. This is because investments may have been made on
various dates and additional purchases and withdrawals may have occurred which vary in amount and date and thus
are unique to the particular account. More and more funds and brokerage firms are now providing personalized
account returns on investor's account statements in response to this need.
With that out of the way, here's how basic earnings and gains/losses work on a mutual fund. The fund records
income for dividends and interest earned which typically increases the value of the mutual fund shares, while
expenses set aside have an offsetting impact to share value. When the fund's investments increase in market value, so
too does the value of the fund shares (or units) owned by the investors. When investments increase (decrease) in
market value, so too the fund shares value increases (or decreases). When the fund sells investments at a profit, it
turns or reclassifies that paper profit or unrealized gain into an actual or realized gain. The sale has no effect on the
value of fund shares but it has reclassified a component of its value from one bucket to another on the fund
books—which will have future impact to investors. At least annually, a fund usually pays dividends from its net
income (income less expenses) and net capital gains realized out to shareholders as an IRS requirement. This way,
the fund pays no taxes but rather all the investors in taxable accounts do. Mutual fund share prices are typically
valued each day the stock or bond markets are open and typically the value of a share is the net asset value of the
fund shares investors own.
38
Rate of return
Total returns
This section addresses only total returns without the impact of U.S. federal individual income and capital gains taxes.
Mutual funds report total returns assuming reinvestment of dividend and capital gain distributions. That is, the
dollar amounts distributed are used to purchase additional shares of the funds as of the reinvestment/ex-dividend
date. Reinvestment rates or factors are based on total distributions (dividends plus capital gains) during each period.
•
•
•
•
•
•
Average annual total return (geometric)
US mutual funds are to compute average annual total return as prescribed by the U.S. Securities and Exchange
Commission (SEC) in instructions to form N-1A (the fund prospectus) as the average annual compounded rates of
return for 1-year, 5-year and 10-year periods (or inception of the fund if shorter) as the "average annual total return"
for each fund. The following formula is used:[6]
Where:
P = a hypothetical initial payment of $1,000.
T = average annual total return.
n = number of years.
ERV = ending redeemable value of a hypothetical $1,000 payment made at the beginning of the 1-, 5-, or 10-year
periods at the end of the 1-, 5-, or 10-year periods (or fractional portion).
Solving for T gives
Example
39
Rate of return
40
Example: Balanced mutual fund during boom times with regular annual dividends,
reinvested at time of distribution, initial investment $1,000 at end of year 0, share price
$14.21
Year 1
Year 2
Year 3
Year 4
Year 5
Dividend per share
$0.26
$0.29
$0.30
$0.50
$0.53
Capital gain distribution per share
$0.06
$0.39
$0.47
$1.86
$1.12
Total Distribution Per Share
$0.32
$0.68
$0.77
$2.36
$1.65
Share Price At End Of Year
$17.50
$19.49
$20.06
$20.62
$19.90
Reinvestment factor
1.01829 1.03553 1.03975 1.11900 1.09278
Shares owned before distribution
70.373
71.676
74.125
76.859
84.752
Total distribution
$22.52
$48.73
$57.10
$181.73 $141.60
Share price at distribution
$17.28
$19.90
$20.88
$22.98
$21.31
Shares purchased
1.303
2.449
2.734
7.893
6.562
Shares owned after distribution
71.676
74.125
76.859
84.752
91.314
• Total return = (($19.90 × 1.09278) / $14.21) - 1 = 53.04%
• Average annual total return (geometric) = ((($19.90 × 91.314) / $1,000) ^ (1 / 5)) - 1 = 12.69%
Using a Holding Period Return calculation, after five years, an investor who reinvested owned 91.314 shares valued
at $19.90 per share. ((($19.90 × 91.314) / $1,000) - 1) / 5 = 16.34% return. An investor who did not reinvest received
total cash payments of $5.78 per share. ((($19.90 + $5.78) / $14.21) - 1) / 5 = 16.14% return.
Mutual funds include capital gains as well as dividends in their return calculations. Since the market price of a
mutual fund share is based on net asset value, a capital gain distribution is offset by an equal decrease in mutual fund
share value/price. From the shareholder's perspective, a capital gain distribution is not a net gain in assets, but it is a
realized capital gain.
Summary: overall rate of return
Rate of Return and Return on Investment indicate cash flow from an investment to the investor over a specified
period of time, usually a year.
ROI is a measure of investment profitability, not a measure of investment size. While compound interest and
dividend reinvestment can increase the size of the investment (thus potentially yielding a higher dollar return to the
investor), Return on Investment is a percentage return based on capital invested.
In general, the higher the investment risk, the greater the potential investment return, and the greater the potential
investment loss.
Rate of return
41
Notes
[1] This statement and the example calculations all follow from the difference of squares formula,
For
the terms have average 100% but
product less than 100%.
[3] Damato, Karen. Doing the Math: Tech Investors' Road to Recovery is Long. Wall Street Journal, pp.C1-C19, May 18, 2001
References
Further reading
• A. A. Groppelli and Ehsan Nikbakht. Barron’s Finance, 4th Edition. New York: Barron’s Educational Series, Inc.,
2000. ISBN 0-7641-1275-9
• Zvi Bodie, Alex Kane and Alan J. Marcus. Essentials of Investments, 5th Edition. New York: McGraw-Hill/Irwin,
2004. ISBN 0073226386
• Richard A. Brealey, Stewart C. Myers and Franklin Allen. Principles of Corporate Finance, 8th Edition.
McGraw-Hill/Irwin, 2006
• Walter B. Meigs and Robert F. Meigs. Financial Accounting, 4th Edition. New York: McGraw-Hill Book
Company, 1970. ISBN 0-07-041534-X
• Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0-471-26849-6
Capital budgeting
Corporate finance
Working capital
•
Cash conversion cycle
•
Return on capital
•
Economic Value Added
•
Just-in-time
•
Economic order quantity
•
Discounts and allowances
•
Factoring
Capital budgeting
•
•
•
Capital investment decisions
The investment decision
The financing decision
Sections
•
Managerial finance
•
Financial accounting
•
Management accounting
•
Mergers and acquisitions
•
Balance sheet analysis
•
Business plan
•
Corporate action
Capital budgeting
42
Societal components
•
Financial market
•
Financial market participants
•
Corporate finance
•
Personal finance
•
Public finance
•
Banks and banking
•
Financial regulation
•
Clawback
Capital budgeting (or investment appraisal) is the planning process used to determine whether an organization's
long term investments such as new machinery, replacement machinery, new plants, new products, and research
development projects are worth pursuing. It is budget for major capital, or investment, expenditures.[1]
Many formal methods are used in capital budgeting, including the techniques such as
•
•
•
•
•
•
•
•
Accounting rate of return
Payback period
Net present value
Profitability index
Internal rate of return
Modified internal rate of return
Equivalent annuity
Real options valuation
These methods use the incremental cash flows from each potential investment, or project. Techniques based on
accounting earnings and accounting rules are sometimes used - though economists consider this to be improper such as the accounting rate of return, and "return on investment." Simplified and hybrid methods are used as well,
such as payback period and discounted payback period.
Capital Budgeting Definition
Capital budgeting is a long-term economics decision making. Each potential project's value should be estimated
using a discounted cash flow (DCF) valuation, to find its net present value (NPV). (First applied to Corporate
Finance by Joel Dean in 1951; see also Fisher separation theorem, John Burr Williams: Theory.) This valuation
requires estimating the size and timing of all the incremental cash flows from the project. (These future cash highest
NPV(GE).) The NPV is greatly affected by the discount rate, so selecting the proper rate—sometimes called the
hurdle rate—is critical to making the right decision. The hurdle rate is the Minimum acceptable rate of return on an
investment. This should reflect the riskiness of the investment, typically measured by the volatility of cash flows,
and must take into account the financing mix. Managers may use models such as the CAPM or the APT to estimate a
discount rate appropriate for each particular project, and use the weighted average cost of capital (WACC) to reflect
the financing mix selected. A common practice in choosing a discount rate for a project is to apply a WACC that
applies to the entire firm, but a higher discount rate may be more appropriate when a project's risk is higher than the
risk of the firm as a whole.
Capital budgeting
Internal rate of return
The internal rate of return (IRR) is defined as the discount rate that gives a net present value (NPV) of zero. It is a
commonly used measure of investment efficiency.
The IRR method will result in the same decision as the NPV method for (non-mutually exclusive) projects in an
unconstrained environment, in the usual cases where a negative cash flow occurs at the start of the project, followed
by all positive cash flows. In most realistic cases, all independent projects that have an IRR higher than the hurdle
rate should be accepted. Nevertheless, for mutually exclusive projects, the decision rule of taking the project with the
highest IRR - which is often used - may select a project with a lower NPV.
In some cases, several zero NPV discount rates may exist, so there is no unique IRR. The IRR exists and is unique if
one or more years of net investment (negative cash flow) are followed by years of net revenues. But if the signs of
the cash flows change more than once, there may be several IRRs. The IRR equation generally cannot be solved
analytically but only via iterations.
One shortcoming of the IRR method is that it is commonly misunderstood to convey the actual annual profitability
of an investment. However, this is not the case because intermediate cash flows are almost never reinvested at the
project's IRR; and, therefore, the actual rate of return is almost certainly going to be lower. Accordingly, a measure
called Modified Internal Rate of Return (MIRR) is often used.
Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV[citation needed],
although they should be used in concert. In a budget-constrained environment, efficiency measures should be used to
maximize the overall NPV of the firm. Some managers find it intuitively more appealing to evaluate investments in
terms of percentage rates of return than dollars of NPV.
Equivalent annuity method
The equivalent annuity method expresses the NPV as an annualized cash flow by dividing it by the present value of
the annuity factor. It is often used when assessing only the costs of specific projects that have the same cash inflows.
In this form it is known as the equivalent annual cost (EAC) method and is the cost per year of owning and operating
an asset over its entire lifespan.
It is often used when comparing investment projects of unequal lifespans. For example if project A has an expected
lifetime of 7 years, and project B has an expected lifetime of 11 years it would be improper to simply compare the
net present values (NPVs) of the two projects, unless the projects could not be repeated.
The use of the EAC method implies that the project will be replaced by an identical project.
Alternatively the chain method can be used with the NPV method under the assumption that the projects will be
replaced with the same cash flows each time. To compare projects of unequal length, say 3 years and 4 years, the
projects are chained together, i.e. four repetitions of the 3 year project are compare to three repetitions of the 4 year
project. The chain method and the EAC method give mathematically equivalent answers.
The assumption of the same cash flows for each link in the chain is essentially an assumption of zero inflation, so a
real interest rate rather than a nominal interest rate is commonly used in the calculations.
Real options
Real options analysis has become important since the 1970s as option pricing models have gotten more
sophisticated. The discounted cash flow methods essentially value projects as if they were risky bonds, with the
promised cash flows known. But managers will have many choices of how to increase future cash inflows, or to
decrease future cash outflows. In other words, managers get to manage the projects - not simply accept or reject
them. Real options analysis try to value the choices - the option value - that the managers will have in the future and
adds these values to the NPV.
43
Capital budgeting
Ranked Projects
The real value of capital budgeting is to rank projects. Most organizations have many projects that could potentially
be financially rewarding. Once it has been determined that a particular project has exceeded its hurdle, then it should
be ranked against peer projects (e.g. - highest Profitability index to lowest Profitability index). The highest ranking
projects should be implemented until the budgeted capital has been expended.
Funding Sources
When a corporation determines its capital budget, it must acquire said funds. Three methods are ge stock have no
financial risk but dividends, including all in arrears, must be paid to the preferred stockholders before any cash
disbursements can be made to common stockholders; they generally have interest rates higher than those of
corporate bonds. Finally, common stocks entail no financial risk but are the most expensive way to finance capital
projects.The Internal Rate of Return is very important.
Need For Capital Budgeting
1. As large sum of money is involved which influences the profitability of the firm making capital budgeting an
important task.
2. Long term investment once made can not be reversed without significance loss of invested capital. The
investment becomes sunk and mistakes, rather than being readily rectified,must often be borne until the firm can
be withdrawn through depreciation charges or liquidation. It influences the whole conduct of the business for the
years to come.
3. Investment decision are the base on which the profit will be earned and probably measured through the return on
the capital. A proper mix of capital investment is quite important to ensure adequate rate of return on investment,
calling for the need of capital budgeting.
4. The implication of long term investment decisions are more extensive than those of short run decisions because
of time factor involved, capital budgeting decisions are subject to the higher degree of risk and uncertainty than
short run decision.[2]
External links and references
• Capital Budgeting (http://forex-management-online.blogspot.com/2009/04/what-is-capital-budgeting.html)
• International Good Practice: Guidance on Project Appraisal Using Discounted Cash Flow (http://www.ifac.
org/Members/DownLoads/Project_Appraisal_Using_DCF_formatted.pdf), International Federation of
Accountants, June 2008, ISBN 978-1-934779-39-2
• Prospective Analysis: Guidelines for Forecasting Financial Statements (http://papers.ssrn.com/sol3/papers.
cfm?abstract_id=1026210), Ignacio Velez-Pareja, Joseph Tham, 2008
• To Plug or Not to Plug, that is the Question: No Plugs, No Circularity: A Better Way to Forecast Financial
Statements (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1031735&rec=1&srcabs=1026210),
Ignacio Velez-Pareja, 2008
• A Step by Step Guide to Construct a Financial Model Without Plugs and Without Circularity for Valuation
Purposes (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1138428&rec=1&srcabs=1031735), Ignacio
Velez-Pareja, 2008
• Long-Term Financial Statements Forecasting: Reinvesting Retained Earnings (http://papers.ssrn.com/sol3/
papers.cfm?abstract_id=1286542), Sergei Cheremushkin, 2008
44
Article Sources and Contributors
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Tobias Bergemann, Too Old, Unsolicited, Versus22, Wcspaulding, WereSpielChequers, Wikiinstructor, William hauser, Wordsmith, Wtshymanski, Yamara, ZeroOne, 387 anonymous edits
Rate of return Source: http://en.wikipedia.org/w/index.php?oldid=553268515 Contributors: 7ema7, Ahoerstemeier, AlbertaSunwapta, Alkarex, Altruistguy, Amirab, Amizzi, Andrés
Djordjalian, AnnuitSophia, Aylad, BTLizard, Bluerasberry, CRoetzer, Chamal N, Chrisgil25, Clear memory, Closedmouth, Costkiller, Doronp, Drunken Pirate, Echion2, Emrek4z, Encyclops,
Ewlyahoocom, Feibels, Finnancier, Flowanda, Gabi S., Gaius Cornelius, Gary King, Giler, Glane23, GraemeL, Gregalton, GregorB, Ground Zero, Heirpixel, Hoegholm, Hubbardaie, Hydrogen
Iodide, Ian Pitchford, Ilya Voyager, IraChesterfield, IvanLanin, JMSwtlk, James086, Junming999, Kalbasa, Kameyama, Kenmayer, Kuru, Levineps, Linuxerist, Lotje, Lunchscale, Magator,
Magister Mathematicae, MarceloB, Mark ok7, Marra, Mathdeveloper, Matt1111111, Mboverload, Michaelas10, Mild Bill Hiccup, Mindmatrix, MinorContributor, Miquonranger03, Mnmngb,
Mrholybrain, Music Sorter, Musiphil, Namazu-tron, Nbarth, NoticeBored, Nschuma, O18, Octopus-Hands, Ohnoitsjamie, Patrick, Paul.Paquette, Plasticup, Pol098, Protonk, Psb777,
QuiteUnusual, Radagast83, Reedy, Rickhev1, Rjwilmsi, Robykiwi, Roy Fultun, Saurael, Seaphoto, Secular mind, Shpleeurnck, Siktath, Simonj23, SirIsaacBrock, Smallbones, Smee, SnakeType,
South Bay, SueHay, Svetovid, TECH-NOIR, The Thing That Should Not Be, The Utahraptor, Thinktwins, Uncle Dick, UncleDouggie, Versus22, Wgroth, White Trillium, Wikipelli, Wkbeh,
YahyaAlfa, Zfeinst, 335 anonymous edits
Capital budgeting Source: http://en.wikipedia.org/w/index.php?oldid=552257283 Contributors: AdjustShift, Agbr, Alansohn, Barek, Bhoola Pakistani, Capricorn42, Chris the speller,
ChrisGualtieri, Cometstyles, Drewwiki, Elonka, Esun, Expurgator, Farhanmukadam, Faweekee, Fintor, Future Perfect at Sunrise, Fæ, Garemoko, Gareth Griffith-Jones, Guy M, Hans Plantinga,
Jaya-ss, Jonkerz, Katieh5584, Kenckar, Kkumaresan26, Kortaggio, Kotra, Kuru, Lamro, Laptop.graham, Les boys, LittleWink, Malik Shabazz, MarceloB, MikeLynch, Mikespedia, Mitsuhirato,
Mtmelendez, PatentSearch, Patstuart, PaulHanson, Peteinterpol, Pine, Pocopocopocopoco, Pondster123, Reinoutr, Romistrub, SVCherry, Sandymok, Seaphoto, Skapur, Skater, Smallbones,
Smallman12q, Spancar, Srajan01, Stathisgould, StaticGull, SueHay, TastyPoutine, Tommy2010, Uncle G, Uncle Milty, Utcursch, Whitejay251, Winhunter, Woohookitty, 140 anonymous edits
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Image Sources, Licenses and Contributors
Image Sources, Licenses and Contributors
Image:exclusive investments.png Source: http://en.wikipedia.org/w/index.php?title=File:Exclusive_investments.png License: Public Domain Contributors: Grieger
File:DCFM Calculator.JPG Source: http://en.wikipedia.org/w/index.php?title=File:DCFM_Calculator.JPG License: GNU General Public License Contributors: Design and developed by
Octávio Viana to ATM Investor Association
file:GDP PPP Per Capita IMF 2008.svg Source: http://en.wikipedia.org/w/index.php?title=File:GDP_PPP_Per_Capita_IMF_2008.svg License: Creative Commons Attribution 3.0
Contributors: Powerkeys
File:Emblem-money.svg Source: http://en.wikipedia.org/w/index.php?title=File:Emblem-money.svg License: GNU General Public License Contributors: perfectska04
File:Panorama clip3.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Panorama_clip3.jpg License: GNU Free Documentation License Contributors: Avron, Ichabod, Jatkins,
Körnerbrötchen, Nikopoley, Rob T Firefly, ゆ い し あ す, 5 anonymous edits
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License
License
Creative Commons Attribution-Share Alike 3.0 Unported
//creativecommons.org/licenses/by-sa/3.0/
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