Four Types of Credit Market Instruments

In terms of the timing of their payments, there are four basic types of credit market
instruments.
1. A simple loan, which we have already discussed, in which the lender provides
the borrower with an amount of funds, which must be repaid to the lender at the
maturity date along with an additional payment for the interest. Many money market
instruments are of this type: for example, commercial loans to businesses.
2. A fixed-payment loan (which is also called a fully amortized loan) in which the
lender provides the borrower with an amount of funds, which must be repaid by making
the same payment every period (such as a month), consisting of part of the principal
and interest for a set number of years. For example, if you borrowed $1,000, a
fixed-payment loan might require you to pay $126 every year for 25 years. Installment
loans (such as auto loans) and mortgages are frequently of the fixed-payment type.
3. A coupon bond pays the owner of the bond a fixed interest payment (coupon
payment) every year until the maturity date, when a specified final amount (face
value or par value) is repaid. The coupon payment is so named because the bondholder
used to obtain payment by clipping a coupon off the bond and sending it to
the bond issuer, who then sent the payment to the holder. Nowadays, it is no longer
necessary to send in coupons to receive these payments. A coupon bond with $1,000
face value, for example, might pay you a coupon payment of $100 per year for ten
years, and at the maturity date repay you the face value amount of $1,000. (The face
value of a bond is usually in $1,000 increments.)
A coupon bond is identified by three pieces of information. First is the corporation
or government agency that issues the bond. Second is the maturity date of the bond. Third is the bond’s coupon rate, the dollar amount of the yearly coupon payment
expressed as a percentage of the face value of the bond. In our example, the
coupon bond has a yearly coupon payment of $100 and a face value of $1,000. The
coupon rate is then $100/$1,000  0.10, or 10%. Capital market instruments such
as U.S. Treasury bonds and notes and corporate bonds are examples of coupon bonds.
4. A discount bond (also called a zero-coupon bond) is bought at a price below
its face value (at a discount), and the face value is repaid at the maturity date. Unlike
a coupon bond, a discount bond does not make any interest payments; it just pays off
the face value. For example, a discount bond with a face value of $1,000 might be
bought for $900; in a year’s time the owner would be repaid the face value of $1,000.
U.S. Treasury bills, U.S. savings bonds, and long-term zero-coupon bonds are examples
of discount bonds.
These four types of instruments require payments at different times: Simple loans
and discount bonds make payment only at their maturity dates, whereas fixed-payment
loans and coupon bonds have payments periodically until maturity. How would you
decide which of these instruments provides you with more income? They all seem so
different because they make payments at different times. To solve this problem, we use
the concept of present value, explained earlier, to provide us with a procedure for
measuring interest rates on these different types of instruments.

Measuring Interest Rates

Different debt instruments have very different streams of payment with very different
timing. Thus we first need to understand how we can compare the value of one kind
of debt instrument with another before we see how interest rates are measured. To do
this, we make use of the concept of present value.
The concept of present value (or present discounted value) is based on the commonsense
notion that a dollar paid to you one year from now is less valuable to you than
a dollar paid to you today: This notion is true because you can deposit a dollar in asavings account that earns interest and have more than a dollar in one year.
Economists use a more formal definition, as explained in this section.
Let’s look at the simplest kind of debt instrument, which we will call a simple
loan. In this loan, the lender provides the borrower with an amount of funds (called
the principal) that must be repaid to the lender at the maturity date, along with an
additional payment for the interest. For example, if you made your friend, Jane, a simple
loan of $100 for one year, you would require her to repay the principal of $100
in one year’s time along with an additional payment for interest; say, $10. In the case
of a simple loan like this one, the interest payment divided by the amount of the loan
is a natural and sensible way to measure the interest rate. This measure of the socalled
simple interest rate, i, is:
If you make this $100 loan, at the end of the year you would have $110, which
can be rewritten as:
$100  (1  0.10)  $110
If you then lent out the $110, at the end of the second year you would have:
$110  (1  0.10)  $121
or, equivalently,
$100  (1  0.10)  (1  0.10)  $100  (1  0.10)2  $121
Continuing with the loan again, you would have at the end of the third year:
$121  (1  0.10)  $100  (1  0.10)3  $133
Generalizing, we can see that at the end of n years, your $100 would turn into:
$100  (1  i )n
The amounts you would have at the end of each year by making the $100 loan today
can be seen in the following timeline:
This timeline immediately tells you that you are just as happy having $100 today
as having $110 a year from now (of course, as long as you are sure that Jane will pay
you back). Or that you are just as happy having $100 today as having $121 two years
from now, or $133 three years from now or $100  (1  0.10)n, n years from now.
The timeline tells us that we can also work backward from future amounts to the present:
for example, $133  $100  (1  0.10)3 three years from now is worth $100
today, so that:
The process of calculating today’s value of dollars received in the future, as we have
done above, is called discounting the future. We can generalize this process by writing
$100 
$133
(1  0.10)3
$100  (1  0.10)n
Year
n
Today
0
$100 $110
Year
1
$121
Year
2
$133
Year
3
i 
$10
$100
 0.10  10%
62 PA RT I I Financial Markets
today’s (present) value of $100 as PV, the future value of $133 as FV, and replacing
0.10 (the 10% interest rate) by i. This leads to the following formula:

How Reliable Are the Money Data?

The difficulties of measuring money arise not only because it is hard to decide what
is the best definition of money, but also because the Fed frequently later revises earlier
estimates of the monetary aggregates by large amounts. There are two reasons why
the Fed revises its figures. First, because small depository institutions need to report
the amounts of their deposits only infrequently, the Fed has to estimate these amounts
until these institutions provide the actual figures at some future date. Second, the
adjustment of the data for seasonal variation is revised substantially as more data
become available. To see why this happens, let’s look at an example of the seasonal
variation of the money data around Christmas-time. The monetary aggregates always
rise around Christmas because of increased spending during the holiday season; the
rise is greater in some years than in others. This means that the factor that adjusts the
data for the seasonal variation due to Christmas must be estimated from several years
of data, and the estimates of this seasonal factor become more precise only as more
data become available. When the data on the monetary aggregates are revised, the seasonal
adjustments often change dramatically from the initial calculation.
Table 2 shows how severe a problem these data revisions can be. It provides the
rates of money growth from one-month periods calculated from initial estimates of
the M2 monetary aggregate, along with the rates of money growth calculated from a
major revision of the M2 numbers published in February 2003. As the table shows,
for one-month periods the initial versus the revised data can give a different picture
of what is happening to monetary policy. For January 2003, for example, the initial
data indicated that the growth rate of M2 at an annual rate was 2.2%, whereas the
revised data indicate a much higher growth rate of 5.4%.
A distinctive characteristic shown in Table 2 is that the differences between the
initial and revised M2 series tend to cancel out. You can see this by looking at the last
row of the table, which shows the average rate of M2 growth for the two series and
the average difference between them. The average M2 growth for the initial calculation
of M2 is 6.5%, and the revised number is 6.5%, a difference of 0.0%. The conclusion
we can draw is that the initial data on the monetary aggregates reported by
the Fed are not a reliable guide to what is happening to short-run movements in the
money supply, such as the one-month growth rates. However, the initial money data
are reasonably reliable for longer periods, such as a year. The moral is that we probably
should not pay much attention to short-run movements in the money supply
numbers, but should be concerned only with longer-run movements.

Measuring Money

The definition of money as anything that is generally accepted in payment for goods
and services tells us that money is defined by people’s behavior. What makes an asset
money is that people believe it will be accepted by others when making payment. As
we have seen, many different assets have performed this role over the centuries, ranging
from gold to paper currency to checking accounts. For that reason, this behavioral
definition does not tell us exactly what assets in our economy should be considered
money. To measure money, we need a precise definition that tells us exactly what
assets should be included.
The Federal Reserve System (the Fed), the central banking authority responsible for
monetary policy in the United States, has conducted many studies on how to measure
money. The problem of measuring money has recently become especially crucial
because extensive financial innovation has produced new types of assets that might
properly belong in a measure of money. Since 1980, the Fed has modified its measures
of money several times and has settled on the following measures of the money

Evolution of the Payments System

We can obtain a better picture of the functions of money and the forms it has taken
over time by looking at the evolution of the payments system, the method of conducting
transactions in the economy. The payments system has been evolving over
centuries, and with it the form of money. At one point, precious metals such as gold
were used as the principal means of payment and were the main form of money. Later,
paper assets such as checks and currency began to be used in the payments system
and viewed as money. Where the payments system is heading has an important bearing
on how money will be defined in the future.
To obtain perspective on where the payments system is heading, it is worth exploring
how it has evolved. For any object to function as money, it must be universally acceptable;
everyone must be willing to take it in payment for goods and services. An object
that clearly has value to everyone is a likely candidate to serve as money, and a natural
choice is a precious metal such as gold or silver. Money made up of precious metals
or another valuable commodity is called commodity money, and from ancient
times until several hundred years ago, commodity money functioned as the medium
of exchange in all but the most primitive societies. The problem with a payments system
based exclusively on precious metals is that such a form of money is very heavy
and is hard to transport from one place to another. Imagine the holes you’d wear in
your pockets if you had to buy things only with coins! Indeed, for large purchases
such as a house, you’d have to rent a truck to transport the money payment.
The next development in the payments system was paper currency (pieces of paper
that function as a medium of exchange). Initially, paper currency carried a guarantee
that it was convertible into coins or into a quantity of precious metal. However, currency
has evolved into fiat money, paper currency decreed by governments as legal
tender (meaning that legally it must be accepted as payment for debts) but not convertible
into coins or precious metal. Paper currency has the advantage of being much
lighter than coins or precious metal, but it can be accepted as a medium of exchange
only if there is some trust in the authorities who issue it and if printing has reached a
sufficiently advanced stage that counterfeiting is extremely difficult. Because paper
currency has evolved into a legal arrangement, countries can change the currency that
they use at will. Indeed, this is currently a hot topic of debate in Europe, which has
adopted a unified currency (see Box 1).
Major drawbacks of paper currency and coins are that they are easily stolen and
can be expensive to transport in large amounts because of their bulk. To combat this
problem, another step in the evolution of the payments system occurred with the
development of modern banking: the invention of checks.
A check is an instruction from you to your bank to transfer money from your account
to someone else’s account when she deposits the check. Checks allow transactions to

Facts about limits of sequences

In this section we will go over some basic results about the limits of sequences. We start with
looking at how sequences interact with inequalities.
2.2.1 Limits and inequalities
A basic lemma about limits is the so called squeeze lemma. It allows us to show convergence of
sequences in difficult cases if we can find two other simpler convergent sequences that “squeeze”
the original sequence.
Lemma 2.2.1 (Squeeze lemma). Let fang, fbng, and fxng be sequences such that
an  xn  bn for all n 2 N:
Suppose that fang and fbng converge and
lim
n!¥
an = lim
n!¥
bn:
Then fxng converges and
lim
n!¥
xn = lim
n!¥
an = lim
n!¥
bn:
The intuitive idea of the proof is best illustrated on a picture, see Figure 2.1. If x is the limit of
an and bn, then if they are both within e=3 of x, then the distance between an and bn is at most 2e=3.
As xn is between an and bn it is at most 2e=3 from an. Since an is at most e=3 away from x, then xn
must be at most e away from x. Let us follow through on this intuition rigorously.
an x xn bn
Figure 2.1: Squeeze lemma in picture.
Proof. Let x := lim an = lim bn. Let e > 0 be given.
Find an M1 such that for all n  M1 we have that jan􀀀xj < e=3, and an M2 such that for all
n  M2 we have jbn􀀀xj < e=3. Set M := maxfM1;M2g. Suppose that n  M. We compute
jxn􀀀anj = xn􀀀an  bn􀀀an
= jbn􀀀x+x􀀀anj
 jbn􀀀xj+jx􀀀anj
<
e
3
+
e
3
=
2e
3
:
48 CHAPTER 2. SEQUENCES AND SERIES
Armed with this information we estimate
jxn􀀀xj = jxn􀀀x+an􀀀anj
 jxn􀀀anj+jan􀀀xj
<
2e
3
+
e
3
= e:
And we are done.
Example 2.2.2: A simple example of how to use the squeeze lemma is to compute limits of
sequences using limits that are already known. For example, suppose that we have the sequence
f 1
n
p
ng. Since
p
n  1 for all n 2 N we have
0 
1
n
p
n

1
n
:
for all n 2 N. We already know that lim1=n = 0. Hence, using the constant sequence f0g and the
sequence f1=ng in the squeeze lemma, we conclude that
lim
n!¥
1
n
p
n
= 0:
Limits also preserve inequalities.
Lemma 2.2.3. Let fxng and fyng be convergent sequences and
xn  yn;
for all n 2 N. Then
lim
n!¥
xn  lim
n!¥
yn:
Proof. Let x := lim xn and y := lim yn. Let e > 0 be given. Find an M1 such that for all n  M1 we
have jxn􀀀xj < e=2. Find an M2 such that for all n  M2 we have jyn􀀀yj < e=2. In particular, for
n  maxfM1;M2g we have x􀀀xn < e=2 and yn􀀀y < e=2. We add these inequalities to obtain
yn􀀀xn+x􀀀y < e; or yn􀀀xn < y􀀀x+e:
Since xn  yn we have 0  yn􀀀xn and hence
0 < y􀀀x+e; or 􀀀e < y􀀀x:
In other words, x􀀀y<e for all e >0. That means that x􀀀y0, as we have seen that a nonnegative
number less than any positive e is zero. Therefore x  y.
2.2. FACTS ABOUT LIMITS OF SEQUENCES 49
We give an easy corollary that can be proved using constant sequences and an application of
Lemma 2.2.3. The proof is left as an exercise.
Corollary 2.2.4.
i) Let fxng be a convergent sequence such that xn  0, then
lim
n!¥
xn  0:
ii) Let a;b 2 R and let fxng be a convergent sequence such that
a  xn  b;
for all n 2 N. Then
a  lim
n!¥
xn  b:
Note in Lemma 2.2.3 you cannot simply replace all the non-strict inequalities with strict
inequalities. For example, let xn := 􀀀1=n and yn := 1=n. Then xn < yn, xn < 0, and yn > 0 for all n.
However, these inequalities are not preserved by the limit operation as we have lim xn = lim yn = 0.
The moral of this example is that strict inequalities may become non-strict inequalities when limits
are applied. That is, if we know that xn < yn for all n, we can only conclude that
lim
n!¥
xn  lim
n!¥
yn:
This issue is a common source of errors.

Monotone sequences

The simplest type of a sequence is a monotone sequence. Checking that a monotone sequence
converges is as easy as checking that it is bounded. It is also easy to find the limit for a convergent
monotone sequence, provided we can find the supremum or infimum of a countable set of numbers.
Definition 2.1.9. A sequence fxng is monotone increasing if xn  xn+1 for all n 2 N. A sequence
fxng is monotone decreasing if xn  xn+1 for all n 2 N. If a sequence is either monotone increasing
or monotone decreasing, we simply say the sequence is monotone. Some authors also use the word
monotonic.
Theorem 2.1.10. A monotone sequence fxng is bounded if and only if it is convergent.
Furthermore, if fxng is monotone increasing and bounded, then
lim
n!¥
xn = supfxn j n 2 Ng:
If fxng is monotone decreasing and bounded, then
lim
n!¥
xn = inffxn j n 2 Ng:
Proof. Let us suppose that the sequence is monotone increasing. Suppose that the sequence is
bounded. That means that there exists a B such that xn  B for all n, that is the set fxn j n 2 Ng is
bounded. Let
x := supfxn j n 2 Ng:
Let e > 0 be arbitrary. As x is the supremum, then there must be at least one n0 2 N such that
xn0 > x􀀀e (because x is the supremum). As fxng is monotone increasing, then it is easy to see (by
induction) that xn  xn0 for all n  n0. Hence
jxn􀀀xj = x􀀀xn  x􀀀xn0 < e:
2.1. SEQUENCES AND LIMITS 43
Hence the sequence converges to x. We already know that a convergent sequence is bounded, which
completes the other direction of the implication.
The proof for monotone decreasing sequences is left as an exercise.
Example 2.1.11: Take the sequence fp1
ng.
First we note that p1
n > 0 and hence the sequence is bounded from below. Let us show that it
is monotone decreasing. We start with
p
n+1 
p
n (why is that true?). From this inequality we
obtain.
1
p
n+1

1
p
n
:
So the sequence is monotone decreasing, bounded from below (and hence bounded). We can apply
the theorem to note that the sequence is convergent and that in fact
lim
n!¥
1
p
n
= inf

1
p
n

:
We already know that the infimum is greater than or equal to 0, as 0 is a lower bound. Suppose we
take a number b  0 such that b  p1
n for all n. We can square both sides to obtain
b2 
1
n
;
for all n 2 N. We have seen before that this implies that b2  0 (a consequence of the Archimedean
property). As we also have b2  0, then b2 = 0 and hence b = 0. Hence b = 0 is the greatest lower
bound and hence the limit.
Example 2.1.12: Be careful however. You have to show that a monotone sequence is bounded
in order to use Theorem 2.1.10. For example, take the sequence f1+1=2+


+1=ng. This is a
monotone increasing sequence that grows very slowly. We will see, once we get to series, that this
sequence has no upper bound and so does not converge. It is not at all obvious that this sequence
has no bound.
A common example of where monotone sequences arise is the following proposition. The proof
is left as an exercise.
Proposition 2.1.13. Let S  R be a nonempty bounded set. Then there exist monotone sequences
fxng and fyng such that xn;yn 2 S and
sup S = lim
n!¥
xn and inf S = lim
n!¥
yn: