CHAPTER 6

Time Value of Money

Learning Objectives

After completing this chapter the student should be able to:

 Define and provide examples of the time value of money.
 Distinguish between simple and compound interest, and use compound interest in
engineering economic analysis.
 Determine equivalence of cash flows in terms of present and future worth.
 Solve problems using the single payment compound interest formulas.
 Solve problems modeled by the uniform series compound interest formulas.

 Use arithmetic and geometric gradients to solve appropriately modeled problems.

 Apply nominal and effective interest rates.
 Use discrete and continuous compounding in appropriate contexts

5.1 Concepts of Money

Money is anything that is generally accepted as payment for goods and services and
repayment of debts. The main uses of money are as a medium of exchange, a unit of
account, and a store of value. Some authors explicitly require money to be a standard of
deferred payment. The dominant form of money is currency. The term "price system" is
sometimes used to refer to methods using commodity valuation or money accounting
systems.
Economic characteristics: Money is generally considered to have the following
characteristics, which are summed up in a rhyme found in older economics textbooks
Medium of exchange: Money is used as an intermediary for trade, in order to avoid the
inefficiencies of a barter system, which are sometimes referred to as the 'double coincidence
of wants problem'. Such usage is termed a medium of exchange.
Unit of account: A unit of account is a standard numerical unit of measurement of the market
value of goods, services, and other transactions. It is also known as a "measure" or "standard"
of relative worth and deferred payment.
Store of value: To act as a store of value, a commodity, a form of money, or financial.
Capital must be able to be reliably saved, stored, and retrieved — and be predictably
useful when it is so retrieved.

5.2 Time Value of Money (Cost of Money)

The costs are paid and the benefits are received during different periods of the life of
the system. Money can have different values at different times. This is because money
can be used to earn more money between the different instances of time. Obviously,
$10,000 now is worth more than $10,000 a year from now even if there is no inflation.
This is because it can earn money during the interval. One could deposit the money in
the bank and earn interest on it. This is the earning power of money over time and is

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called time value of money, that is, $10,000 now has more value than $10,000 six
months from now. Because the interest rate is the more identifiable and accepted
measure of the earning power of money, it is usually accepted as the time value of
money and indication of its earning power. We have to be careful not to confuse the
earning power of money, which is related to interest rate, with the buying power of
money, which is related to inflation.
The time value of money seems like a sophisticated concept, yet it is one that you
grapple with every day. Should you buy something today or save your money and buy it
later? Here is a simple example of how your buying behaviour can have varying results:
Pretend you have $100 and you want to buy a $100 refrigerator for your dorm room. If
you buy it now, you end up broke. But if you invest your money at 6% annual interest,
then in a year you can still buy the refrigerator, and you will have $6 left over. However,
if the price of the refrigerator increases at an annual rate of 8% due to inflation, then
you will not have enough money (you will be $2 short) to buy the refrigerator a year
from now. In that case, you probably are better off buying the refrigerator now (Case 1
in Figure 5.1). If the inflation rate is running at only 4%, then you will have $2 left over if
you buy the refrigerator a year from now (Case 2 in Figure 4.1).

Figure 5.1 Gains achieved or losses incurred by delaying consumption

Clearly, the rate at which you earn interest should be higher than the inflation rate in
order to make any economic sense of the delayed purchase. In other words, in an
inflationary economy, your purchasing power will continue to decrease as you further
delay the purchase of the refrigerator. In order to make up this future loss in purchasing
power, the rate at which you earn interest should be sufficiently larger than the
anticipated inflation rate.
After all, time, like money, is a finite resource. There are only 24 hours in a day, so time
has to be budgeted, too. What this example illustrates is that we must connect earning
power and purchasing power to the concept of time. The way interest operates reflects
the fact that money has a time value. This is why amounts of interest depend on lengths

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of time; interest rates. For example, are typically given in terms of a percentage per
year. We may define the principle of the time value of money as follows: The economic
value of a sum depends on when the sum is received. Because money has both earning
power and purchasing power over time (i.e., it can be put to work, earning more
money for its owner). A dollar received today has a greater value than a dollar received
at some future time. When we deal with large amounts of money, long periods of time,
or high interest rates, and the change in the value of a sum of money over time becomes
extremely significant. For example, at a current annual interest rate of 10%, $1 million
will earn $100,000 in interest in a year; thus, to wait a year to receive $1 million clearly
involves a significant sacrifice. When deciding among alternative proposals, we must
take into account the operation of interest and the time value of money in order to make
valid comparisons of different amounts at various times.
When financial institutions quote lending or borrowing interest rates in the
marketplace, those interest rates reflect the desired earning rate, as well as any
protection from loss in the future purchasing power of money because of inflation.
Interest rates, adjusted for inflation, rise and fall to balance the amount saved with the amount
borrowed, which affects the allocation of scarce resources between present and future uses.

5.3 Interest Rate

When money is borrowed, it has to be paid back. In addition to the amount of the loan,
an extra amount of money is paid to the lender for the use of money during the period of
a loan, just as you pay a rent on a house or a car. The rate of interest, i, is the percentage
of the money you pay for its use over a time period. The interest rate is referred to by
different names such as rent, cost of money, and value of money. In investment
terminology, it is called the minimum acceptable rate of return or MARR.
Money can be loaned and repaid in many ways, and. equally, money can earn interest in
many different ways. Usually, however, at the end of each interest period, the interest
earned on the principal amount is calculated according to a specified interest rate. The
two computational schemes for calculating this earned interest yield either simple
interest or compound interest. Engineering economic analysis uses the compound-
interest scheme exclusively, as it is most frequently practiced in the real world.
The first scheme considers interest earned on only the principal amount during each
interest period. In other words, under simple interest, the interest earned during each
interest period does not earn additional interest in the remaining periods, even though
you do not withdraw it. In general, for a deposit of P dollars at a simple interest rate of i
for N periods, the total earned interest I would be

Simple interest is commonly used with add-on loans or bonds.

Under a compound-interest scheme, the interest earned in each period is calculated
based on the total amount at the end of the previous period. This total amount includes

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the original principal plus the accumulated interest that has been left in the account. In
this case, you are in effect increasing the deposit amount by the amount of interest
earned. In general, if you deposited (invested) P dollars at an interest rate i, you would
have P + iP = P( 1 + i ) dollars at the end of one interest period. If the entire amount
(principal and interest) were reinvested at the same rate i for another period, you
would have at the end of the second period

To compare the value of money at different points in time, we need to use an acceptable
interest rate. The interest rate will depend on the position in time that the money is
needed and the length of time it is required. If money is borrowed for a long period,
then the uncertainty of the economy will introduce a risk factor and influence the
interest rate. For short periods, it can be assumed that the economy is stable and the
risk is predictable. Availability of money in the financial market also has an effect on the
interest rate. If the banks have more money than people need to borrow, then the
interest rate is low and vice versa. Money, like any other commodity, obeys the laws of
supply and demand.

Elements of Transactions Involving Interest

Many types of transactions involve interest-e.g., borrowing money, investing money, or
purchasing machinery on credit-but certain elements are common to all of these types
of transactions:
1. Principal (P) the initial amount of money invested or borrowed in transactions.
2. The interest rate (i) measures the cost or price of money and is expressed as a
percentage per period of time.
3. Interest period (n) A period of time which determines how frequently interest is
calculated.
4. A specified length of time marks the duration of the transaction and thereby
establishes a certain number of interest periods (N).
5. A plan for receipts or disbursements (An) that yields a particular cash flow
pattern over a specified length of time. (For example, we might have a series of equal
monthly payments that repay a loan.)
6. A future amount of money (F) results from the cumulative effects of the interest
rate over a number of interest periods.

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5.4 Economic Equivalence
The central factor in deciding among alternative cash flows involves comparing their
economic worth. This would be a simple matter if. in the comparison, we did not need to
consider the time value of money: We could simply add the individual payments within
a cash flow, treating receipts as positive cash flows and payments (disbursements) as
negative cash flows. The fact that money has a time value makes our calculations more
complicated. Calculations for determining the economic effects of one or more cash
flows are based on the concept of economic equivalence.
Economic equivalence exists between cash flows that have the same economic effect
and could therefore be traded for one another in the financial marketplace, which we
assume to exist. Economic equivalence refers to the fact that a cash flow whether a
single payment or a series of payments-can be converted to an equivalent cash flow at
any point in time. The important fact to remember about the present value of future
cash flows is that the present sum is equivalent in value to the future cash flows. It is
equivalent because if you had the present value today, you could transform it into the
future cash flows simply by investing it at the interest rate, also referred to as the
discount rate. The strict conception of equivalence may be extended to include the
comparison of alternatives. For example, we could compare the value of two proposals
by finding the equivalent value of each at any common point in time. If financial
proposals that appear to be quite different turn out to have the same monetary value,
then we can be economically indifferent to choosing between them. That is, in terms of
economic effect, one would be an even exchange for the other, so no reason exists to
prefer one over the other in terms of their economic value.

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A way to see the concepts of equivalence and economic indifference at work in the real world
is to note the variety of payment plans offered by lending institutions for consumer loans.
Equivalence calculations can be viewed as an application of the compound-interest
relationships we developed before. Suppose, for example, that we invest $1,000 at 12%
annual interest for five years. The formula developed for calculating compound interest, F =
P(1+ i)N , expresses the equivalence between some present amount P and a future amount F
for a given interest rate i and a number of interest periods, N. Therefore, at the end of the
investment period, our sums grow to

Thus, we can say that at 12% interest, $1,000 received now is equivalent to $1,762.34
received in five years, and thus we could trade $1,000 now for the promise of receiving
$1,762.34 in five years. Example 2.2 further demonstrates the application of this basic
technique. When selecting a point in time at which to compare the value of alternative cash
flows, we commonly use either the present time, which yields what is called the present
worth of the cash flows, or some point in the future, which yields their future worth. The
choice of the point in time often depends on the circumstances surrounding a particular
decision, or it may be chosen for convenience.

5.5 Equivalence of Single Cash Flows

Compound-Amount Factor: Given a present sum P invested for N interest periods at
interest rate i, what sum will have accumulated at the end of the N periods? You
probably noticed right away that this description matches the case we first encountered
in describing compound interest. To solve for F (the future sum), we use

Because of its origin in the compound-interest calculation, the factor (1 + i)N is known as
the compound-amount factor. Like the concept of equivalence, this factor is one of the
foundations of engineering economic analysis. Given this factor, all other important
interest formulas can be derived. This process of finding F is often called the
compounding process. We may also express that factor in a functional notation as
(F/P, i, N), which is read as "Find F, given i, and N." This factor is known as the single-
payment compound- amount factor. When we incorporate the table factor into the
formula, the formula is expressed as follows:

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Present-Worth Factor: Finding the present worth of a future sum is simply the reverse
of compounding and is known as the discounting process. From compounding factor
equation, we can see that if we need to find a present sum P, given a future sum F. we
simply solve for P:

The factor 1/(1 + i)N is known as the single-payment present-worth factor and is
designated (P/F, i, N). Tables have been constructed for P/F factors and for various
values of i and N. The interest rate i and the P/F factor are also referred to as the
discount rate and the discounting factor, respectively.

5.6 Uniform Series Compound Interest Formulae

Many times we will find uniform series of receipts or disbursements. Automobile loans,
house payments, and many other loans are based on a uniform payment series. It will
often be convenient to use tables based on a uniform series of receipts or
disbursements. Suppose we are interested in the future amount F of a fund to which we
contribute A dollars each period and on which we earn interest at a rate of i per period.
The contributions are made at the end of each of the N periods. These transactions are
graphically illustrated in below. Looking at this diagram, we see that, if an amount A is
invested at the end of each period for N periods, the total amount F that can be
withdrawn at the end of N periods will be the sum of the compound amounts of the
individual deposits.

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As shown above, the A dollars we put into the fund at the end of the first period will be
worth A(l + i)N-1 at the end of N periods. The A dollars we put into the fund at the end of
the second period will be worth A(l + i)N-2 and so forth. Finally, the last A dollars that we
contribute at the end of the Nth period will be worth exactly A dollars at that time. This
means we have a series in the form

Multiplying the above Equation by (1+i) results in

Subtracting the F value to eliminate common terms gives us

Cash flow diagram of the relationship between A and F

The bracketed term in the above equation is called the equal-payment-series

compound amount factor, or the uniform-series compound-amount factor; its
factor notation is (F/A, i, N).
If we solve the Equation for A, we obtain

The term within the brackets is called the equal-payment-series sinking-fund factor,
or just sinking-fund factor, and is referred to with the notation (A/F, i, N). A sinking
fund is an interest-bearing account into which a fixed sum is deposited each interest
period; it is commonly established for the purpose of replacing fixed assets.

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Capital-Recovery Factor (Annuity Factor): We can determine the amount of a
periodic payment, A, if we know P, i, and N. To relate P to A, recall the relationship
between P and F in equation: F = P(l + i)N. By replacing F by P(1 + i)N, we get

Cash flow diagram of the relationship between P and A

Now we have an equation for determining the value of the series of end-of-period
payments, A, when the present sum P is known. The portion within the brackets is
called the equal-payment-series capital-recovery factor, or simply capital-recovery
factor, which is designated (A/P, i, N). In finance, this A/P factor is referred to as the
annuity factor. The annuity factor indicates a series of payments of a fixed, or constant,
amount for a specified number of periods.
What would you have to invest now in order to repay A dollars at the end of each of the
next N periods? We now face just the opposite of the equal-payment capital-recovery
factor situation: A is known, but P has to be determined. With the help of capital-
recovery factor equation, solving for P gives us

The bracketed term is referred to as the equal-payment-series present-worth factor

and is designated (P/A, i, N).

5.7 Present Value of Perpetuities

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Perpetuity is a stream of cash flows that continues forever. A good example is a share of
preferred stock that pays a fixed cash dividend each period (usually a quarter-year) and
never matures. An interesting feature of any perpetual annuity is that you cannot
compute the future value of its cash flows, because it is infinite. However, it has a well-
defined present value. It appears counterintuitive that a series of cash flows that lasts
forever can have a finite value today.
Another difficulty in present worth analysis arises when we encounter an infinite
analysis period (N=0). In governmental analyses, a service or condition sometimes must
be maintained for an infinite period. The need for roads, dams, pipelines, and other
components of national, state, or local infrastructure is sometimes considered to be
permanent. In these situations a present worth of cost analysis would have an infinite
analysis period. We call this particular analysis capitalized cost. Capitalized cost is the
present sum of money that would need to be set aside now, at some interest rate, to
yield the funds required to provide the service (or whatever)indefinitely. To accomplish
this, the money set aside for future expenditures must not decline. Thus, for any initial
present sum P, there can be an end-of-period withdrawal of A equal to iP each period,
and these withdrawals may continue forever without diminishing the initial sum P. This
gives us the basic relationship:
For N = ∞, A = Pi
This relationship is the key to capitalized cost calculations. Earlier we defined
capitalized cost as the present sum of money that would need to be set aside at some
interest rate to yield the funds to provide the desired task or service forever. Capitalized
cost is therefore the P in the equation A = i P. It follows that:

If we can resolve the desired task or service into an equivalent A, the capitalized cost
may be computed.

5.8 Gradient Series Cash Flows

1. Arithmetic (Linear) Gradient Series
It frequently happens that the cash flow series is not of constant amount A. Instead,
there is a uniformly increasing series as shown: Sometimes cash flows will vary linearly,
that is, they increase or decrease by a set amount, G, the gradient amount. This type of
series is known as a strict gradient series, as shown below. Note that each payment is
An = (n - 1)G. Note also that the series begins with a zero cash flow at the end of period
zero. If G > 0, the series is referred to as an increasing gradient series. If G < 0, it is
referred to as a decreasing gradient series. A typical problem involving a linear gradient
series includes an initial payment during period one that increases by G during some
number of interest periods.

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In order to use the strict gradient series to solve typical problems, we must view cash
flows as shown above as a composite series, or a set of two cash flows, each
corresponding to a form that we can recognize and easily solve. The above figure
illustrates that the form in which we find a typical cash flow can be separated into two
components: a uniform series of N payments of amount A, and a gradient series of
increments of a constant amount G. The need to view cash flows that involve linear
gradient series as composites of two series is very important for the solution of
problems, as we shall now see. How much would you have to deposit now in order to

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withdraw the gradient amounts specified in figure below? To find an expression for the
present amount P, we apply the single-payment present-worth factor to each term of
the series.

The resulting factor in brackets is called the gradient-series present-worth factor and
is designated by the notation (P/G, i, N).

2. Geometric Gradient Series

Another kind of gradient series is formed when the series in a cash flow is determined
not by some fixed amount like $500, but by some fixed rate expressed as a percentage.
Many engineering economic problems, particularly those relating to construction costs
or maintenance costs, involve cash flows that increase or decrease over time by a
constant percentage (geometric), a process that is called compound growth. Price
changes caused by inflation are a good example of such a geometric series. If we use g to
designate the percentage change in a payment from one period to the next, the
magnitude of the nth payment, An is related to the first payment Al as follows:

The g can take either a positive or a negative sign, depending on the type of cash flow.

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If g > 0, the series will increase: if g < 0, the series will decrease. The figure below
illustrates the cash flow diagram for this situation. Notice that the present worth Pn of
any cash flow An at an interest rate i is

To find an expression for the present amount for the entire series, P, we apply the
single-payment present-worth factor to each term of the series:

The expression in above equation has the following closed expression

The factor within brackets is called the geometric-gradient-series present-worth

factor and is designated (P / A1, g, i, N).

A geometrically increasing or decreasing gradient series

5.9 Nominal and Effective Interest

Take a closer look at the billing statement from any credit card, or, if you financed a new
car recently, examine the loan contract. You should be able to find the interest that the
bank charges on your unpaid balance. Even if a financial institution uses a unit of time
other than a year-for example, a month or a quarter, when calculating interest payments
and in other matters, the institution usually quotes the interest rate on an annual basis.
Many banks, for example, state the interest arrangement for credit cards in the
following manner: "18% compounded monthly."

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This statement means simply that each month the bank will charge 1.5% interest (12
months per year x 1.5% per month = 18% per year) on the unpaid balance. As shown in
figure below, we say that 18% is the nominal interest rate or annual percentage rate
(APR) and that the compounding frequency is monthly (12 times per year). Although
the APR is commonly used by financial institutions and is familiar to many customers, it
does not explain precisely the amount of interest that will accumulate in a year. To
explain the true effect of more frequent compounding on annual interest amounts, we
will introduce the term effective interest rate, commonly known as annual effective yield,
or annual percentage yield (APY).

Relationship between APR and interest period

The annual effective yield (or effective annual interest rate) is the one rate that
truly represents the interest earned in a year. On a yearly basis, you are looking for a
cumulative rate-1.5% each month for 12 times. This cumulative rate predicts the actual
interest payment on your outstanding credit card balance.
We could calculate the total annual interest payment for a credit card debt of $1,000 by
using the formula given before. If P = $1,000, i = 1.5%, and N = 12, we obtain

Clearly, the bank is earning more than 18% on your original credit card debt. In fact, you
are paying $195.62.The implication is that, for each dollar owed, you are paying an
equivalent annual interest of 19.56 cents. In terms of an effective annual interest rate
(ia), the interest payment can be rewritten as a percentage of the principal amount:

In other words, paying 1.5% interest per month for 12 months is equivalent to paying
19.56% interest just one time each year. Certainly more frequent compounding
increases the amount of interest paid over a year at the same nominal interest rate.
Assuming that the nominal interest rate is r and that M compounding periods during the
year, the annual effective yield ia can be calculated as follows:

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When M = 1, we have special case of annual compounding. Substituting M = 1, reduces
the above equation into ia = r. That is when compounding takes place once annually,
effective interest equal to nominal interest.
The table in below shows effective interest rates at various compounding intervals for
4%-12% APRs. As you can see, depending on the frequency of compounding, the
effective interest earned or paid by the borrower can differ significantly from the APR.
Therefore, truth-in-lending laws require that financial institutions quote both nominal
and effective interest rates when you deposit or borrow money.

Discrete Compounding: If cash flow transactions occur quarterly, but interest is

compounded monthly, we may wish to calculate the effective interest rate on a quarterly
basis. To consider this situation, we may redefine the above equation as

Where
M = the number of interest periods per year,
C = the number of interest periods per payment period, and
K = the number of payment periods per year.
Note that M = CK
Example: Suppose that you make quarterly deposits into a savings account that earns 8%
interest compounded monthly. Compute the effective interest rate per quarter.

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The annual effective interest rate i, is (1 + 0.02013)4 = 8.24%. For the special case of
annual payments with annual compounding, we obtain
i =ia with C = M and K = 1.

Continuous Compounding: To be competitive on the financial market, or to entice

potential depositors, some financial institutions offer more frequent compounding. As
the number of compounding periods (M) becomes very large, the interest rate per
compounding period (r/M) becomes very small. As M approaches infinity and r/M
approaches zero. We approximate the situation of continuous compounding. By
taking limits on the right side of the equation, we obtain the effective interest rate per
payment period as

In sum, the effective interest rate per payment period is

To calculate the effective annual interest rate for continuous compounding, we set K
equal to 1, resulting in

As an example, the effective annual interest rate For a nominal interest rate of 12%
compounded continuously is ia = e0.12 - 1 = 12.7497%.

Find the effective interest rate per quarter at a nominal rate of 8% compounded
(a) weekly. (b) daily, and (c) continuously.

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