Module 2
Future Value, Present Value, and Interest Rates Assignment
a. Valuing Monetary Payments Now and in the Future
To compare the value of payments made on different dates, we need a set of
tools called future value and present value. We’ll use them to see how and why the
promise to make a payment on one date is more or less valuable than the promise to
make a payment on a different date. For example, we already know that if you want to
borrow $100 today, your repayment needs to be bigger if you promise to make it in a
year than if you promise to make it in a month. But how much more will you have to
pay? The answer depends on both the date of payment and the interest rate. For the
time being, we’re going to assume that we know for sure that you will repay the loan.
What is the future value of one dollar deposited in an interest-bearing account
today? To answer this question, let’s start with a definition: Future value is the value
on some future date of an investment made today. Say that today you invest $100 in a
savings account that guarantees 5 percent interest per year. After one year, you’ll have
$105 (the investment at its present value of $100 plus $5 in interest). So the future
value of $100 one year from now at an interest rate of 5 percent is $105. We could
also say that the $100 investment yields $5, which explains why an interest rate is
sometimes called a yield.
To generalize this concept so that we can handle different interest rates and
initial investments of any size, we can express it mathematically. First we need to
convert the percentage interest rate into a decimal, so that 5 percent becomes 0.05.
Note that in this expression, as in all mathematical manipulations, the interest rate is
expressed in decimal terms. Now we can express future value as an equation. If the
present value of your initial investment is $100 and the interest rate is 5 percent, then
the future value one year from now is $100 + $100(0.05) = $105 Present value of the
investment + Interest = Future value in one year It is essential to convert all interest
rates to decimals before doing any computation. This is consistent with the fact that
we quote interest rates as “parts per 100,” so 54percent means 5 parts per 100, or 0.05.
This expression shows us immediately that the higher the interest rate, the higher the
future value. If the interest rate were to rise to 6 percent, then the future value of $100
would be $100 + $100(0.06) = $106 In general, the future value, FV, of an investment
with a present value, PV, invested at an interest rate i is FV = PV + PV × i = PV × (1 +
i) (1) Future value in one year = Present value of the investment today × (One plus the
interest rate).
But this example is too simple. Most financial instruments don’t make single
payments in exactly one year, so we need to figure out what happens when the time to
repayment varies. Computing the future value of an investment to be repaid two years
from now is straightforward, so let’s do that first. But because we quote interest rates
on a yearly basis, we need to be careful. Using one-year interest rates to compute the
value of an investment that will be repaid more than one year from now requires
applying the concept of compound interest, which is interest on the interest. If you
leave an investment in an interest-bearing account for two years, during the second
year you will receive interest not only on your initial investment but also on the
interest you earned for the first year (because they both have an opportunity cost).
Getting back to our example, let’s say that you leave your $100 deposit in the
bank for two years at 5 percent interest per year. The future value of this investment
has four parts. The first three are straightforward. They are the initial investment of
$100, the interest on that investment in the first year, and the interest on it in the
second year. But because you left the interest from the first year in the bank during the
second year, it is as if you made a new deposit at the beginning of the second year,
and that earns interest too. So the fourth part of the future value is the interest you
receive during the second year on the interest you received in the first year. That’s
compounding. With an initial deposit of $100 and an interest rate of 5 percent, we can
add up these four parts to compute your investment’s future value in two years. $100
+ $100(0.05) + $100(0.05) + $5(0.05) = $110.25 Present value of the initial
investment + Interest on the initial investment in first year + Interest on the initial
investment in second year + Interest on the interest from first year in second year =
Future value in two years.
We can use a small amount of algebra2 to show that this equals $100(1.05)
(1.05) = $100(1.05)2 Extending it to three years, four years, or more just means
multiplying by (1.05) over and over again. The multiplication takes care of the
compounding. The final line shows that after 10 years, a deposit with a present value
of $100 becomes $162.89. That is, it earns $62.89 in interest. If we had ignored
compounding and just multiplied 5 percent by 10 years to get 50 percent, the answer
would have been $150. Compounding produced an additional $12.89 in interest over
10 years. To put it as clearly as possible, multiplying the number of years times the
annual interest rate gives the wrong answer! we can derive a general formula for
future value. FVn = PV × (1 + i)n (2) Future value in n years = Present value of the
investment × (One plus the interest rate) raised to n.
So to compute future value, all we need to do is calculate one plus the interest
rate (measured as a decimal) raised to the nth power and multiply it by the present
value. Before we go any further, we should stop to consider an important problem.
What if you want to put your $100 into a bank for six months, or 2½ years, or any
amount of time that is not a round number of years? The answer is that the formula
still works. You can compute the future value using equation (2) regardless of whether
n is a whole number. There is one pitfall, however. In computing future value, both
the interest rate and n must be measured in the same time units. We have been
measuring interest rates as the percentage per year, so we were careful to measure n in
years as well. So, if we want the future value in half of one year, n would be ½; if we
wanted it in one month, n would be 1/12; and if we wanted the future value in one
day, n would be 1/365.
As you can see, taking advantage of the future-value formula requires an
understanding of the transformations needed to convert time from years to months or
vice versa. Converting n from years to months is easy—everyone knows there are
124months in a year—but converting the interest rate is harder. If the annual interest
rate is 5 percent, what is the interest rate for one month? To figure out the answer,
we’ll start with the future-value formula, but in months. Remember that compounding
means you cannot just multiply the monthly interest rate by 12 to get the annual
interest rate. Instead, if i m is the one-month interest rate and n is the number of
months, then a deposit made for one year will have a future value of $100(1 + i m)12.
We know that this amount equals $100(1.05), so figuring out the answer means
equating the two amounts, (1 + im)12 = (1.05) and raising each side to the one-twelfth
power: (1 + im) = (1.05)1/12 = 1.0041.
Converting from decimals to a percentage, the one-month interest rate is 0.41
percent. We can handle any mismatch between the time units of i and n in a similar
way (see Tools of the Trade: Computing Compound Annual Rates on page 82). These
fractions of percentage points, like 0.41 percent, are so important in discussing
interest rates that they have their own name, basis points. A basis point is one one
hundredth of a percentage point. That is, one basis point equals 0.01 percent. You’re
probably wondering how useful all this discussion of future value really is. To see,
consider the following question: If you put $1,000 per year into the bank at 44percent
interest, how much would you have saved after 40 years? The answer is $98,826—
more than twice the $40,000 you deposited. Figuring out the exact answer is
complicated because we need to add up the future values of forty $1,000 deposits,
each made in a different year, but doing so uses the concept of future value. The first
$1,000 is deposited for 40 years, so its future value is $1,000(1.04)40 = $4,801.02 The
second $1,000 is deposited for 39 years, so its future value is $1,000(1.04)39 =
$4,616.37 and so on. The practical implication of this calculation is that buying one
less soda or candy bar per day isn’t just good for your physical health; it’s good for
your financial health, too.
It’s easy to see why future value is important. We often want to know what
savings and investments will be worth in the future. But that isn’t the only thing we
need to know. There is another, somewhat different task that we face with some
regularity. We need to be able to figure out how much a payment promised in the
future is worth today. Say you agree to make a $225 loan, and the borrower offers to
repay you either $100 a year for three years or $125 a year for two years. Which offer
should you take? Answering this question means figuring out the current value of the
promised payments on the dates when they will be made. To do that, we’ll use the
concept of present value, sometimes referred to as present discounted value.
In our discussion of future value, we used the term present value to mean the
initial amount invested or deposited. The way we used the term suggests its technical
definition: Present value is the value today (in the present) of a payment that is
promised to be made in the future. Put another way, present value is the amount that
must be invested today in order to realize a specific amount on a given future date.
Financial instruments promise future cash payments, so we need to know how to
value those payments. Present value is an integral component of the computation of
the price of all financial instruments.
To understand the calculation of present value, go back to future value.
Remember that at a 5 percent interest rate, the future value one year from now of a
$100 investment today is $105. It follows that at this same 5 percent interest rate, the
present value of $105 one year from now is $100. All we did was invert the future
value calculation. Reversing the calculation in general terms is just as easy. Start with
the fact that the future value of a payment equals the current investment times one
plus the interest rate: FV = PV × (1 + i) [equation (1)]. Divide both sides of this
expression by (14+ i) to get an expression for how much we need to invest today to
realize the future value one year from today. The result is PV = ______ FV (1 + i) (3)
Present value = Future value of the payment divided by (One plus the interest rate) In
our example, we see that ______ FV (1 + i) = ______ $105 (1.05) = $100 so the
present value of $105 one year from now, at a 5 percent interest rate, is indeed $100.
While future value tells us what today’s investment will be worth in the future,
present value tells us what promised future payments are worth today. This means that
the properties of present value mirror those of future value. In the same way that
future value rises as the interest rate rises, present value falls as the interest rate rises.
To see this, first look at the calculation of the present value of $105 in one year at an
interest rate of 6 percent. The answer is _____ $105 1.06 = $99.06 which is less than
the $100 needed when the interest rate is only 5 percent. Present value falls as the
interest rate rises. What happens if the payment is going to be made in two years
instead of one? What is the present value of $105 in two years at an interest rate of 5
percent? Again, we can compute the answer using the future-value formula by asking
what present value has a future value of $105 in two years at an interest rate of 5
percent. This is the solution to $105 = PV(1.05)2 The answer is PV = _____ $105
1.052 = $95.24. We can generalize this process by looking at the future value in n
years of an investment today: FVn = PV(1 + i)n. Dividing both sides of this
expression by (1 + i)n, we get the general formula for present value: PV = FV
_______ n (1 + i)n (4) Present value = Future value of a payment made in n years
divided by (One plus the interest rate) raised to n.
We’re going to use equation (4) over and over again. It is the single most
important relationship in our study of financial instruments, so we have highlighted it
in a red rectangle. Once we can figure out the present value of any future payment,
then we understand the fundamentals of mortgages, credit cards, car loans, and even
stocks. We will spend the rest looking at how present value changes when we change
the various components of the formula, and how to use it more generally. But before
we do, it is important to note one final similarity between present value and future
value. Recall that to calculate future value, n need not be measured in years. We can
do the computation even when n is the number of months, so long as the interest rate
is measured in months as well. The same is true of present value. So long as we
measure n and i in the same time unit, and the interest rate is expressed as a decimal,
the formula works.
It is useful to go through each of the three properties of present value, looking
at the impact of changing each one: the size of the future payment (FVn), the time
until the payment is made (n), and the interest rate (i). Starting with FVn, we see that
doubling the future value of the payment, without changing the time of the payment
or the interest rate, doubles the present value. For example, at a 54percent interest rate,
a $100 payment made in two years has a present value of $90.70. Doubling the
payment to $200 doubles the present value to $181.40. In fact, increasing or
decreasing FVn by any percentage will change PV by the same percentage, in the
same direction. We have already seen that the sooner a payment is to be made, the
more it is worth. How much more? To see, let’s return to the example of a $100
payment at 54percent interest. How sensitive is the present value of this payment to
the time until it is made? Plugging some numbers into the general present-value
formula [equation (4)], and allowing the time to go from 0 to 30 years, which shows
that the present value of the payment is worth $100 if it is made immediately but
declines gradually to $23 for a payment made in 30 years.
The rate of decline in the present value is related to the same phenomenon that
gives us the rule of 72 (described in Your Financial World: How Long Does Your
Investment Take to Double? on page 77). Consider this question: At a 5 percent
interest rate, how long into the future must a payment of $100 be made for it to be
worth the same as $50 received today? The answer is 14.4 years. That is, at 5 percent
interest, the present value of $100 paid in 14.4 years is $50. Note that 14.4 equals 72
divided by 5, so it is also the number of years an investment takes to double in value
when the return is 5 percent per year. We can repeat the computation to see that the
investment takes 28.8 years to double twice, which tells us that the present value of
$100 paid 28.8 years from now is $25.
The interest rate is the third important determinant of the present value of a
future payment. To see how important it is, let’s look at the present value of a $100
payment made 1, 5, 10, and 20 years from now at various interest rates. The general
formula [equation (4)] allows us to do this series of computations. Note what happens
as the interest rate increases—that is, as you read down a column in the table or move
to the right in the figure. You can see immediately that higher interest rates are
associated with lower present values, no matter what the size or timing of the
payment. Conversely, lower interest rates are associated with higher present values.
Note, too, that at any fixed interest rate, an increase in the time until a payment
is made reduces its present value. Read across any row of the table and you will see
that as the time increases from 1 to 5 to 10 to 20 years, the present value goes down.
The final lesson to take away from these calculations has to do with how present
value changes with both time and the interest rate. Table 4.2 shows what happens to
the present value of a payment as the interest rate increases. You can see that if the
payment is to be made in one year (column two), as the interest rate increases from 1
percent to 54percent, the present value falls from $99.01 to $95.24.
This concept illustrates a fundamental principle in finance known as the
relationship between interest rates and present value, highlighting how changes in
interest rates can impact the value of future cash flows. When discussing present
value (PV) and interest rates, it's essential to delve into how these variables interact
and influence financial decisions across various contexts.
Present value is a concept used to determine the current worth of a future sum
of money, discounted back at a specified rate of return (or interest rate). It reflects the
principle that money received in the future is worth less than money received today
due to the opportunity cost of waiting and the risk associated with receiving future
cash flows.
For instance, consider an investment or a financial instrument that promises to
pay a certain amount of money in the future. The present value of that future cash
flow is calculated by discounting it back to the present using an appropriate discount
rate, which is typically the prevailing interest rate for similar investments or
securities.
The relationship between interest rates and present value is inverse: as interest
rates increase, present value decreases, and vice versa. This is because higher interest
rates imply a higher discount rate, which reduces the present value of future cash
flows. Conversely, lower interest rates lead to a lower discount rate, increasing the
present value of future cash flows.
When evaluating investment opportunities, analysts use present value
calculations to assess the attractiveness of potential investments. Higher discount rates
(or interest rates) decrease the present value of expected future cash flows, making
investments less attractive. Conversely, lower discount rates increase present value
and can make investments appear more favorable. In fixed-income investments like
bonds, present value calculations determine the current market price of the bond.
Bond prices are inversely related to interest rates: when interest rates rise, bond prices
fall because the discounted present value of future coupon payments decreases.
Businesses use present value calculations to evaluate capital projects and
investment proposals. By discounting the expected future cash flows back to their
present value, companies can determine whether the project will generate a positive
net present value (NPV). Projects with a positive NPV are typically considered
worthwhile investments. Borrowers and lenders use present value calculations to
determine loan terms and mortgage payments. Higher interest rates increase the cost
of borrowing by reducing the present value of future loan payments. This relationship
influences decisions on refinancing, loan duration, and terms. Individuals use present
value calculations to make financial decisions, such as determining the value of future
pension payments, assessing the cost of financing a home or car purchase, or
evaluating the impact of inflation on retirement savings.
In summary, the relationship between interest rates and present value is
fundamental to understanding financial decision-making and economic analysis.
Changes in interest rates directly affect the present value of future cash flows,
influencing investment decisions, bond pricing, capital budgeting, loan terms, and
personal financial planning. By applying present value concepts, individuals and
businesses can make informed decisions that maximize returns, manage risks, and
achieve their financial goals effectively.
Now look at the present value of a payment that will be made in 10 years. As
the interest rate goes from 1 to 5 percent, the present value of a $100 payment 10
years from now falls from $90.53 to $61.39. This is a decline of $29.14, or more than
30 percent. Not only does the present value of a future payment fall with the interest
rate; the further in the future the promised payment is to be made, the more the
present value falls. As a result, a change in interest rates has a much greater impact on
the present value of a payment made far in the future than it has on one to be made
soon.
b. Applying Present Value
All of our examples thus far have focused on computing the present value of a
single payment on a given future date. Thinking of present value in this way gives us
enormous flexibility. It means that we can compute the present value not just of a
single payment but also of any group of payments made on any number of dates. As
we saw earlier, to use present value in practice, we need to look at sequences, or
streams of payments. And valuing a stream of payments means summing their present
values. That is, the value of the whole is the sum of the value of its parts.
Present value is a fundamental concept in finance that allows us to evaluate
the current worth of future cash flows. It's based on the principle that money received
or paid in the future is worth less than the same amount received or paid today due to
the opportunity cost of not having that money available for investment or
consumption immediately.
When we talk about present value being additive, we mean that we can sum up
the present values of individual cash flows to determine the total present value of a
stream of payments. This principle is crucial in various financial applications,
including internal rate of return (IRR) calculations and bond valuations.
IRR is a metric used to evaluate the profitability of an investment. It represents
the discount rate at which the net present value (NPV) of all cash flows (both positive
and negative) from a project or investment equals zero. In other words, it's the rate of
return at which the present value of expected future cash flows equals the initial
investment cost. By calculating the present value of each cash flow using the IRR, we
can determine whether an investment is likely to be profitable or not.
Bonds are debt instruments issued by governments, municipalities, or
corporations to raise capital. They typically promise periodic interest payments
(coupons) and repayment of the principal amount at maturity. The value of a bond
today is the present value of its future cash flows, which include both the periodic
coupon payments and the repayment of principal at maturity. Investors discount these
future cash flows back to the present using an appropriate discount rate (such as the
yield to maturity), reflecting the risk and time value of money. By summing up the
present values of all future cash flows, we arrive at the fair market value of the bond.
In both applications—IRR calculations and bond valuations—the additive
nature of present value allows us to break down complex cash flow streams into
manageable components and evaluate their worth in today's terms. This concept is
foundational in financial decision-making, enabling investors and analysts to compare
different investment opportunities and make informed choices based on their expected
returns and risks.
Imagine that you run a sports equipment factory. As part of your strategic
planning, you are considering buying a new machine that makes tennis rackets. The
machine costs $1 million and can produce 3,000 rackets a year. If you can sell the
rackets for $50 apiece (wholesale), the machine will generate $150,000 in revenue
each year. To simplify the analysis, we will assume that the machine is the only
necessary input in the production of tennis rackets; that we know the exact amount of
revenue it will produce (in reality, that has to be estimated); and that the machine will
last for exactly 104years, during which time it will work perfectly, without requiring
any maintenance.
At the end of the 10 years, the machine will abruptly cease to operate and will
have no resale value. Should you buy the machine? The answer is: It depends. If you
borrow the $1 million to pay for the machine, will the revenue from the machine,
$150,000 per year, be enough to cover the payments on the loan? If so and you have
something left over, then buying the machine may be a good idea. But if you can’t
make the payments, then buying the machine is a losing proposition. So you need to
figure out whether the machine’s revenue will be high enough to cover the payments
on the loan you would need to buy it.
The internal rate of return is the interest rate that equates the present value of
an investment with its cost. For the tennis racket machine, it is the interest rate at
which the present value of the revenue from the tennis rackets, $150,000 per year for
104years, equals the $1 million cost of the machine. To find the internal rate of return,
we take the sum of the present value of each of the yearly revenues (we can’t take the
present value of the total revenue) and equate it with the machine’s cost.
You can solve this equation using a financial calculator or spreadsheet. The
answer, 8.14 percent, is the internal rate of return on your investment. That is, the
annual rate of return for investing $1 million in the machine is 8.14 percent. But is
that rate of return high enough to justify your investment? That depends on the cost of
the $14million you need to buy the machine. There are two ways you can come up
with the $1 million. You can use your company’s retained earnings—the funds you’ve
saved from your past profits. Or you can borrow. In the first case, you need to figure
out if the machine is more profitable than other ways you might use the funds, just as
you might compare interest-bearing investments. The other main use for the retained
earnings is to lend them to someone at the same rate at which you could borrow.
Understanding the opportunity cost of an investment is crucial when
considering borrowing for a major purchase like a machine. It involves weighing the
benefits you could gain from alternative investments compared to what you would
gain from investing in the machine.
When borrowing money to buy the machine, you must carefully assess
whether the anticipated profits from using the machine will be sufficient to cover the
costs of the loan and still leave you with a net profit. This assessment typically
involves calculating the expected return on investment (ROI) based on factors such as
projected revenue increase, cost savings, and other financial benefits the machine
would bring to your business operations.
Moreover, opportunity cost extends beyond just financial considerations. It
also involves evaluating intangible factors such as time, effort, and risk. For instance,
investing in the machine may free up time for your employees, allowing them to focus
on more productive tasks or expanding your business operations. On the other hand,
choosing not to invest in the machine might mean missing out on potential growth
opportunities or losing competitive advantage in the market.
In making such decisions, it's essential to conduct a thorough cost-benefit
analysis, taking into account both the direct and indirect costs and benefits associated
with borrowing for the machine. Additionally, considering alternative financing
options, interest rates, repayment terms, and potential tax implications can provide a
more comprehensive picture of the opportunity cost involved in your investment
decision. By carefully evaluating these factors, you can make an informed choice that
aligns with your business goals and financial capabilities.
When considering borrowing, you need to calculate the expected return on
investment (ROI) of acquiring the machine. This includes projecting the additional
revenue or cost savings the machine will generate over its useful life. By comparing
these potential gains with the cost of financing (interest payments, fees), you can
determine if the investment is financially viable. It's crucial to consider not just the
immediate costs and benefits but also the long-term implications on your cash flow
and profitability.
Borrowing introduces financial risk, such as the obligation to make regular
loan payments regardless of business performance. Assessing your business's ability
to generate sufficient cash flow to cover these payments is essential. Additionally,
consider the risk of technological obsolescence or changes in market demand that
could affect the machine's utility and resale value over time.
Investing in the machine should align with your business strategy and long-
term goals. Evaluate how the machine enhances your operational efficiency, expands
production capacity, improves product quality, or enables you to offer new
products/services. These strategic benefits can contribute to sustainable growth and
competitive advantage in the market.
Beyond financial metrics, opportunity cost encompasses the potential benefits
foregone by choosing to allocate resources (money) to one investment (the machine)
over alternatives. This includes alternative investments that could yield higher returns
or strategic initiatives that could drive greater business value. For example, if you
invest in the machine, you may not have funds available for other growth
opportunities like marketing campaigns, research and development, or talent
acquisition.
Explore how borrowing and purchasing the machine affect your tax position.
Depending on your location and tax laws, you may be eligible for deductions related
to interest payments or depreciation of the machine. Factor in these potential tax
benefits when calculating the overall cost of borrowing and the net impact on your
business's financial health. Consider the impact of borrowing on your business's
overall financial stability and ability to secure future financing. Maintaining a healthy
balance between debt and equity is crucial for long-term sustainability and resilience
against economic fluctuations.
By conducting a thorough analysis of these factors, you can make an informed
decision about whether borrowing to purchase the machine aligns with your business
objectives and financial capabilities. Remember, seeking advice from financial
professionals and considering multiple scenarios can provide additional clarity and
confidence in your investment strategy.
As we would expect, when the interest rate rises, the payments rise too. At
what interest rate can you afford a loan to buy the tennis racket machine? Recall that
you have $150,000 a year in revenue, and your internal rate of return is 8.14 percent.
So as long as the interest rate is 8 percent or less, you know you can cover the
payments. But we can answer this question with more precision. To see why, notice
that the internal rate of return equation (5) is virtually identical to the loan
equation4(6). In fact, the internal rate of return is the interest rate at which $150,000 a
year for 10 years will exactly cover the loan. So we really needed to do this
computation only once to answer the question.
Investing in a tennis racket machine becomes economically viable when the
internal rate of return (IRR) surpasses the interest rate associated with financing it.
This principle applies broadly across investments: if the IRR, which represents the
rate at which the net present value of future cash flows equals zero, exceeds the cost
of borrowing funds, the investment is typically considered profitable.
The decision to purchase such equipment involves assessing various financial
factors beyond IRR and borrowing costs. For instance, considering the machine's
expected lifespan, maintenance costs, and potential revenue generation through
increased training efficiency can provide a comprehensive financial analysis.
Additionally, market conditions, such as demand for tennis facilities or coaching
services, can influence the profitability and utilization rate of the machine, further
impacting its return on investment (ROI).
Moreover, financial institutions and investors often use IRR as a critical metric
for evaluating the attractiveness of investments. It helps determine whether the
expected returns compensate adequately for the associated risks and financing costs.
Understanding these financial metrics and their implications can guide sound
investment decisions, ensuring that investments not only meet but exceed financial
objectives and contribute positively to overall business or personal financial
strategies.
Before we go on, we can use the concept of internal rate of return to answer
the question at the beginning of the present-value section on page 78: If you agree to
make a $225 loan, and the borrower offers to repay either $100 a year for three years
or $125 a year for two years, which should you take? The first step in figuring out
what to do is to compute the internal rate of return of the two payment streams.
One of the most common uses of the concept of present value is in the
valuation of bonds. A bond is a promise to make a series of payments on specific
future dates. It is issued as part of an arrangement to borrow. In essence, the borrower,
or seller, gives an IOU to the lender, or buyer, in return for some amount of money.
Both governments and corporations need to borrow, so both issue bonds. Because
bonds create obligations, they are best thought of as legal contracts that (1) require the
borrower to make payments to the lender and (2) specify what happens if the
borrower fails to do so.
Because there are many different kinds of bonds, to focus our discussion, we’ll
look at the most common type, a coupon bond. Say a borrower who needs $100
“issues” or sells a $100 coupon bond to a lender. The bond issuer is required to make
annual payments, called coupon payments. The annual amount of those payments
(expressed as a percentage of the amount borrowed) is called the coupon rate. If the
coupon rate is 54percent, then the borrower/issuer pays the lender/bondholder $5 per
year per $100 borrowed. The yearly coupon payment equals the coupon rate times the
amount borrowed. The bond also specifies when the issuer is going to repay the initial
$100 and the payments will stop, called the maturity date or term to maturity.
The final payment, a repayment of the initial $100 loan, is often referred to as
the principal, face value, or par value of the bond. Before the advent of computers, an
investor buying a bond would receive a certificate with a number of dated coupons
attached. To claim the coupon payments, the investor would cut off the coupons and
mail them to the bond issuer. At maturity, the investor would redeem the certificate for
the final payment. The Reading Railroad Company bond pictured on page 88 still has
some coupons attached.3 You can see that the borrower who issues a bond is
promising to make a series of regular interest payments over the life of the bond, plus
a final payment on the maturity date.
Valuing a bond involves calculating its present value, which is derived from
the future cash flows it promises, namely the repayment of principal and periodic
coupon payments. This process allows investors to determine how much they should
be willing to pay for such an investment.
To begin with, the repayment of the principal is a crucial component of bond
valuation. The principal, also known as the face value or par value of the bond,
represents the amount that will be repaid to the bondholder at maturity. This
repayment is typically considered a future cash outflow discounted back to its present
value using an appropriate discount rate. The discount rate is often determined by
prevailing interest rates or the risk associated with the bond issuer.
In addition to the principal repayment, bonds also generate periodic coupon
payments. These payments represent the interest income that bondholders receive
throughout the bond's life. The value of these coupon payments is determined by
discounting each payment back to its present value using the same discount rate
applied to the principal repayment.
The present value of a bond is the sum of the present values of its principal
repayment and all future coupon payments. This calculation reflects the current worth
of the bond in today's dollars, considering the time value of money—where money
available now is worth more than the same amount in the future due to its potential
earning capacity.
Investors and financial analysts use bond valuation techniques to assess
whether a bond is fairly priced in the market or if it presents a potential opportunity
for either overvaluation or undervaluation. Factors influencing bond valuation include
prevailing interest rates, credit quality of the issuer, term to maturity, and market
demand for similar bonds.
Understanding bond valuation is essential for making informed investment
decisions, whether purchasing bonds directly or through bond funds. It provides a
framework for evaluating risk and return characteristics, helping investors allocate
their capital effectively within their broader investment strategies.
This example highlights two important properties of periodic fixed payments.
First, the longer the payments go on—the more of them there are—the higher their
total value. Even though the additional payments fall farther into the future, the
overall present value still grows. Because a long-term bond (one that lasts for 30
years, for instance) has more payments than a short-term maturity bond (one whose
final payment is made, say, in 5 years), the coupon payments on the long-term bond
will be worth more than the coupon payments on the short-term bond. Second, as is
always the case in present-value calculations, the higher the interest rate, the lower the
present value. Raising the interest rate from 6 to 7 percent, for example, lowers the
total value of the five future payments on our short-term bond from $42.12 to $41.00.
This formula looks complicated because it is. But we can learn two simple
facts just by looking at its parts. The value of the coupon bond, PCB, rises when (1)
the yearly coupon payments, C, rise and (2) the interest rate, i, falls. The first of these
conclusions follows from the fact that a higher coupon rate means larger payments,
and the present value of a larger payment is larger. The second follows directly from
the present-value relationship: The lower the interest rate, the higher the present value
of any and all future payments. The fact that lower interest rates mean higher bond
prices—and higher interest rates mean lower bond prices—is extremely important.
Because bonds promise fixed payments on future dates, the higher the interest rate,
the lower their present value.
The relationship between bond prices and interest rates is a fundamental
concept in fixed-income investing. The value of a bond is inversely related to the
interest rate used to calculate the present value of its promised payments. When
interest rates rise, the present value of a bond's future cash flows declines, leading to a
decrease in the bond's price. Conversely, when interest rates fall, the present value of
a bond's future cash flows increases, resulting in a higher bond price.
To understand this relationship, consider the mechanism of discounting future
cash flows. The present value of a bond is calculated by discounting its future
payments (both the principal repayment and the periodic coupon payments) back to
the present using a discount rate that reflects current interest rates. When interest rates
increase, the discount rate used in this calculation also rises. As a result, the present
value of each future payment is lower, leading to a decrease in the overall value of the
bond.
For example, if an investor holds a bond with a fixed coupon rate of 5% and
market interest rates increase to 6%, new bonds issued in the market will offer higher
returns to match the new interest rate environment. Consequently, the existing bond
with a lower coupon rate becomes less attractive to investors, causing its price to drop
so that its yield aligns with the new market rate. This adjustment ensures that the
bond's price reflects the new market conditions and provides a competitive yield to
potential buyers.
Conversely, when market interest rates decline, the discount rate used to
calculate the present value of the bond's future payments decreases. This means that
the present value of each future payment is higher, leading to an increase in the bond's
price. Investors will find the fixed coupon payments of the existing bond more
attractive compared to the lower yields available in the market, driving up its price.
This inverse relationship between bond prices and interest rates is a critical
consideration for bond investors. It highlights the interest rate risk inherent in bond
investments—the risk that changes in market interest rates will affect the value of
existing bonds. Investors need to be aware of this risk and consider the potential
impact of interest rate fluctuations on their bond portfolios.
Additionally, the sensitivity of a bond's price to interest rate changes is
influenced by its duration, a measure of the bond's weighted average time to receive
the bond's cash flows. Bonds with longer durations are generally more sensitive to
interest rate changes, exhibiting greater price volatility in response to fluctuations in
interest rates. Conversely, bonds with shorter durations are less sensitive to interest
rate changes and tend to be more stable in price.
Understanding the dynamics of bond pricing and interest rates allows investors
to make informed decisions about their bond investments, manage interest rate risk
effectively, and optimize their fixed-income portfolios in varying economic
conditions.
c. Real and Nominal Interest Rates
In calculating present value, our goal has been to assess the number of dollars
you would pay today for fixed dollar payments in the future. To do this, we used the
nominal interest rate, which is the interest rate expressed in current-dollar terms. We
did not worry about the possibility that inflation might change the purchasing power
of the dollars. Because borrowers and lenders care about the purchasing power of the
money they pay out and receive, they care about inflation. So we need to adjust the
return on a loan, looking not just at the nominal interest rate but at the inflation-
adjusted interest rate, called the real interest rate.
Think about a $100 loan made at a 5 percent interest rate for one year. The
borrower receives $100 at the beginning of the year and repays $105 at the end of the
year. If prices go up 5 percent during the year—that is, if the inflation rate is 54percent
—then the $105 returned to the lender at the end of the year will buy exactly what
$100 did at the beginning of the year. The lender’s inflation-adjusted return is zero.
No lender would be content with earning a zero return on their investment,
which underscores why it is unlikely for a lender to agree to a loan with a 5 percent
nominal interest rate if the expected inflation rate is also 5 percent. In such a scenario,
the real interest rate—calculated by subtracting the expected inflation rate from the
nominal interest rate—would effectively be zero. This means the lender would not
gain any real return on the loan after accounting for the eroding effects of inflation on
the purchasing power of the repaid funds.
Lenders typically seek compensation for the opportunity cost of lending their
money, the risk of default, and the loss of purchasing power due to inflation. A
nominal interest rate equal to the expected inflation rate fails to provide any real
return, making the loan unattractive. In essence, the lender would be no better off in
real terms at the end of the loan period than at the beginning, which contradicts the
fundamental principle of earning a return on investment.
Moreover, the possibility of the real interest rate being negative adds another
layer of concern for lenders. When the inflation rate exceeds the nominal interest rate,
the real interest rate becomes negative. For example, if the nominal interest rate is 5
percent and the actual inflation rate turns out to be 6 percent, the real interest rate
would be -1 percent. This situation implies that the lender's purchasing power
decreases over the loan period, leading to a loss in real terms.
Negative real interest rates are particularly problematic because they
essentially mean that the lender is paying the borrower to use their money. This
scenario is unsustainable for lenders, as it erodes their capital and reduces their
incentive to lend. Consequently, lenders strive to set nominal interest rates that not
only cover expected inflation but also provide a positive real return to justify the risks
and opportunity costs associated with lending.
To ensure a positive real return, lenders incorporate an inflation premium into
the nominal interest rate. This premium reflects the expected rate of inflation over the
loan period, ensuring that the nominal rate sufficiently compensates for the
anticipated decrease in the value of money. Additionally, lenders factor in the real
interest rate, which represents the true economic return they require for parting with
their funds.
For instance, if a lender requires a real return of 2 percent and expects
inflation to be 5 percent, they would set the nominal interest rate at 7 percent. This
rate ensures that after accounting for the 5 percent inflation, the lender still achieves a
2 percent real return. This approach protects the lender's investment from the eroding
effects of inflation and provides a fair compensation for the use of their funds.
Understanding this dynamic is crucial for borrowers as well. Borrowers need
to be aware that nominal interest rates reflect not just the cost of borrowing but also
the inflation expectations and the lender's required real return. When inflation
expectations rise, borrowers can expect nominal interest rates to increase
correspondingly, making borrowing more expensive. Conversely, in a low-inflation
environment, nominal rates might be lower, reducing the cost of borrowing.
The interaction between nominal interest rates, real interest rates, and inflation
expectations plays a significant role in the broader economy. Central banks, for
example, monitor these factors closely when setting monetary policy. By adjusting
nominal interest rates, central banks aim to influence real economic activity, control
inflation, and stabilize the economy.
In periods of high inflation, central banks may raise nominal interest rates to
curb inflationary pressures and ensure positive real returns for lenders, thereby
encouraging savings and reducing excessive borrowing. On the other hand, in low-
inflation or deflationary periods, central banks might lower nominal interest rates to
stimulate economic activity by making borrowing cheaper and encouraging
investment and consumption.
Financial markets also react to changes in nominal and real interest rates.
Bond prices, for instance, are inversely related to interest rates. When nominal interest
rates rise, bond prices typically fall, and vice versa. Investors and financial analysts
use these relationships to make informed decisions about portfolio allocation, risk
management, and investment strategies.
In summary, the relationship between nominal interest rates, expected
inflation, and real interest rates is a fundamental aspect of financial decision-making
for both lenders and borrowers. No lender would be satisfied with a zero or negative
real return, which is why nominal interest rates must exceed expected inflation rates
to ensure a positive real return. This understanding helps maintain the balance
between the cost of borrowing and the return on lending, supporting stable and
sustainable economic growth.
The point of this example is that borrowers look at the inflation-adjusted cost
of borrowing, while lenders focus on the inflation-adjusted return. No one cares only
about the number of dollars. People also care about what those dollars can buy. In
other words, everyone cares about real interest rates. This is why economists think of
the nominal interest rate as having two parts, the real interest rate and expected
inflation. Say that you want to borrow $100 for one year. You find a lender who is
willing to give you a loan, but the two of you need to agree on the interest rate. Both
of you care about the inflation rate over the coming year, which will affect the
purchasing power of the dollars you will use to repay the loan.
When entering into a loan agreement, both parties must acknowledge the
uncertainty surrounding future interest rates. This uncertainty necessitates forecasting
to establish a mutually agreeable nominal interest rate for the loan. The nominal
interest rate consists of two primary components: the expected inflation rate over the
term of the loan and the real interest rate.
The nominal interest rate is the total rate that borrowers will pay and lenders
will receive, without adjusting for inflation. To ensure a fair and beneficial agreement,
it must account for the anticipated rate of inflation during the loan period. Inflation
erodes the purchasing power of money over time, so both parties must consider how
inflation is likely to impact the value of future payments.
Expected inflation is a forecasted measure of how much prices in the economy
are expected to rise annually during the loan's term. This forecast is typically based on
various economic indicators, historical data, central bank policies, and market
expectations. Accurate inflation forecasting is crucial because it ensures that the
interest rate agreed upon reflects the anticipated economic environment, protecting
the real value of the payments for both lenders and borrowers.
In addition to expected inflation, the nominal interest rate includes the real
interest rate. The real interest rate represents the rate of return lenders require, above
and beyond the compensation for inflation, to justify the risks associated with lending
money. It reflects the real cost of borrowing to the borrower and the real yield to the
lender, providing a clearer picture of the loan's economic impact without the distortion
of inflation.
The real interest rate is influenced by various factors, including the supply and
demand for credit, the creditworthiness of the borrower, economic growth
expectations, and monetary policy. A higher real interest rate generally indicates a
higher cost of borrowing, reflecting increased risk or greater demand for credit.
Conversely, a lower real interest rate suggests a lower borrowing cost, often
associated with lower risk or increased supply of available funds.
To conclude a loan agreement, both parties must collaboratively estimate the
expected inflation rate and negotiate the real interest rate based on their respective
assessments of the economic landscape and risk factors. This negotiation process
ensures that the nominal interest rate agreed upon fairly compensates the lender for
the time value of money and the risk taken while ensuring that the borrower
understands the cost of the loan.
Moreover, this approach underscores the importance of economic forecasts
and risk assessments in financial agreements. Accurate forecasting helps manage
expectations and reduces the likelihood of disputes arising from unexpected economic
changes. It also allows both lenders and borrowers to make informed decisions that
align with their financial goals and risk tolerance.
In practice, financial professionals often use various tools and models to
forecast inflation and real interest rates, incorporating a wide range of economic data
and market insights. These forecasts are continually updated to reflect changing
economic conditions, helping both parties stay informed and adjust their strategies as
necessary.
Understanding and effectively managing the components of the nominal
interest rate—expected inflation and the real interest rate—are critical for ensuring
that loan agreements are fair, transparent, and economically sound. This knowledge
enables both borrowers and lenders to navigate the complexities of financial markets
with greater confidence and foresight.
This is called the Fisher equation after the early 20th-century economist Irving
Fisher. It shows that in general, the nominal interest rate is positively related to
expected inflation. The higher expected inflation, the higher the nominal interest rate.
While the relationship is not a tight one, higher nominal interest rates are usually
associated with higher inflation rates. In 1980 and 1981, for example, U.S. interest
rates were sky-high; the U.S. Treasury had to pay more than 154percent for its short-
term borrowing.
By 1986, interest rates had dropped to more reasonable levels, close to 5
percent. That’s a 10-percentage-point move in just five years! The figure shows that
as inflation fell, nominal interest rates also fell. In fact, the declines were almost
identical. Real interest rates didn’t change much during this period. The term real
interest rate can cause confusion. For the most part, the financial markets quote
nominal interest rates.5 When people use the term interest rate without qualification,
they are referring to the nominal interest rate, the one they see every day.
In our discussions and analyses, we will adhere to the standard convention of
using the term "interest rate" to refer specifically to the nominal interest rate. This
nominal interest rate represents the total percentage return expected by lenders or the
total cost of borrowing for borrowers, unadjusted for inflation. It includes the
compensation for both the real value of the loan and the expected erosion of
purchasing power due to inflation over time.
To clarify, the nominal interest rate is the rate quoted on financial instruments,
such as loans, bonds, and savings accounts, which incorporates the overall expected
return or cost. This rate is crucial for calculating actual payments and receipts in
financial transactions, providing a comprehensive view of the financial obligations or
benefits associated with borrowing or investing.
On the other hand, we will use the term "real interest rate" to refer specifically
to the nominal interest rate adjusted for expected inflation. The real interest rate is
calculated by subtracting the expected inflation rate from the nominal interest rate.
This adjustment provides a clearer measure of the true cost of borrowing or the actual
yield on an investment in terms of purchasing power. The real interest rate reflects the
real economic return, net of the inflationary effects that reduce the value of money
over time.
Understanding the distinction between nominal and real interest rates is crucial
for making informed financial decisions. The nominal interest rate gives a broad
overview of financial returns or costs, while the real interest rate provides deeper
insights into the actual economic benefits or expenses when considering the impact of
inflation.
For instance, when analyzing the potential returns on an investment,
considering the real interest rate is essential. While a high nominal interest rate might
initially appear attractive, if the expected inflation rate is equally high, the real interest
rate—and thus the actual purchasing power gain—could be quite low. Conversely, a
lower nominal interest rate in a low-inflation environment might yield a higher real
return, providing better real economic benefits.
Similarly, for borrowers, understanding the real interest rate is critical for
assessing the true cost of borrowing. A nominal interest rate might seem manageable,
but if expected inflation is high, the real cost of the loan could be significant, affecting
long-term financial planning and affordability.
In financial markets, the interplay between nominal and real interest rates is
influenced by various factors, including central bank policies, economic growth
expectations, and global economic conditions. Central banks, for example, often set
nominal interest rates to influence economic activity, aiming to control inflation and
stimulate or cool down the economy as needed. These policies, in turn, affect the real
interest rates, influencing borrowing, spending, and investment decisions across the
economy.
Investors and financial analysts pay close attention to real interest rates when
evaluating the attractiveness of different investment opportunities. Real interest rates
affect asset prices, including stocks, bonds, and real estate, as they determine the real
rate of return that investors can expect. For instance, if real interest rates are low,
investors might seek higher returns in riskier assets, driving up prices in equity
markets.
Moreover, real interest rates are a key factor in international finance,
influencing exchange rates and capital flows between countries. Higher real interest
rates in a country can attract foreign investment, appreciating the local currency and
impacting trade balances.
In summary, by following this convention and using the term "interest rate" to
mean the nominal rate and the term "real interest rate" to refer to the nominal rate less
expected inflation, we provide clarity and precision in our financial discussions. This
approach helps ensure that all parties involved have a clear understanding of the
financial concepts, enabling better decision-making and effective financial planning.
Because we know the nominal interest rate, i, measuring the real interest rate
means subtracting forecasted inflation. There are a number of sources for these
forecasts. Twice a year the Federal Reserve Bank of Philadelphia publishes
professional forecasts. Once a month, the Survey Research Center of the University of
Michigan computes consumer inflation expectations.
But because forecasts are often wrong, our estimate will usually differ from
the real interest rate that occurs. Someone who is making an economically important
decision will do so based on the expected real interest rate. Some time later, that
person will look back and compute the real interest rate actually paid or received. The
first of these is known as the ex ante real interest rate, meaning “before the fact.” The
second, or realized rate, is the ex post real interest rate, meaning “after the fact.”
In financial analysis, the distinction between ex post and ex ante real interest
rates is significant and warrants thorough understanding. The ex post real interest rate
can always be computed because it is based on historical data. Specifically, we know
the nominal interest rate that was applied and the actual inflation rate that occurred
over the period in question. By subtracting the actual inflation rate from the nominal
interest rate, we derive the ex post real interest rate, which reflects the realized real
return on an investment or the realized cost of borrowing after accounting for
inflation.
For example, if a loan had a nominal interest rate of 5% and the actual
inflation rate during the loan period was 2%, the ex post real interest rate would be
3%. This calculation provides a clear retrospective view of the real economic outcome
of the financial transaction. Investors and borrowers can use ex post real interest rates
to assess the effectiveness of their financial decisions and to understand the true
economic impact of past investments or loans.
However, it is the ex ante real interest rate that is of primary interest for
forward-looking financial planning and decision-making. The ex ante real interest rate
is the anticipated real return on an investment or the expected real cost of borrowing,
based on forecasted inflation rates. It represents the difference between the nominal
interest rate and the expected inflation rate over the future period of the investment or
loan.
Forecasting the ex ante real interest rate is crucial because it informs investors
and borrowers about the likely real economic outcomes of their financial decisions.
For instance, an investor considering purchasing a bond will want to know the ex ante
real interest rate to assess the bond's potential real return after accounting for expected
inflation. Similarly, a borrower looking to take out a loan will need to understand the
ex ante real interest rate to evaluate the real cost of the loan in terms of purchasing
power.
Accurately estimating the ex ante real interest rate involves predicting future
inflation, which is inherently uncertain and influenced by numerous factors, including
economic policies, market conditions, global economic trends, and unexpected events.
Economists and financial analysts use various models and indicators to forecast
inflation, but these predictions are always subject to potential errors and revisions.
The ex ante real interest rate is particularly important for central banks and
policymakers, who use it as a tool to guide monetary policy. By setting nominal
interest rates with an eye on expected inflation, central banks aim to influence real
economic activity. For instance, if inflation is expected to rise, central banks might
increase nominal interest rates to maintain a stable ex ante real interest rate, thereby
controlling inflationary pressures. Conversely, in a low-inflation environment, central
banks might lower nominal interest rates to stimulate economic growth.
Investors also rely on ex ante real interest rates to make informed decisions
about asset allocation. A higher ex ante real interest rate may lead investors to favor
fixed-income securities, while a lower ex ante real interest rate might drive them
towards equities or other assets that offer potentially higher real returns.
Furthermore, understanding the ex ante real interest rate is crucial for
international investments and currency valuation. Investors comparing bonds or other
securities from different countries must consider the ex ante real interest rates in each
market, which can influence capital flows and exchange rates. Higher ex ante real
interest rates in one country can attract foreign investment, appreciating the local
currency and impacting trade balances.
The relationship between ex post and ex ante real interest rates also provides
insights into market expectations and economic conditions. Discrepancies between the
two can indicate unexpected changes in inflation or economic shocks, prompting
adjustments in future financial strategies and policies.
In summary, while the ex post real interest rate offers valuable insights into the
actual outcomes of past financial transactions, the ex ante real interest rate is essential
for anticipating the real economic impacts of future investments and loans. Accurate
forecasting of the ex ante real interest rate supports better financial decision-making,
effective monetary policy, and informed investment strategies, ultimately contributing
to more stable and predictable economic outcomes.
