An Introduction to Interest Rates
It is important that we have a good understanding of what interest rate is before we
proceed to discuss some of the key concepts and issues of money and banking. Interest
rates represent the prices of credit, which include both borrowing and lending. We are
familiar with different uses of credit.
(a) Borrowing: Home mortgages, car loans, credit cards, etc.
(b) Lending: Opening a savings account (i.e. “lending” to the bank), buying a
government bond (i.e. “lending” to the government), etc.
Interest rate plays a very important role in the economy because it determines the
spending and savings behaviors of individuals and companies.
(1) If interest rate rises, the “cost” of spending increases and hence individuals and
companies save more.
(2) If interest rate falls, the “cost” of spending decreases and hence individuals and
companies spend more.
When it comes to the definition of interest rate, the most basic definition is the price of
credit. However, that does not mean that there is only a single interest rate in an
economy.1 The term (or life) of the credit and the amount of the credit will have an
impact on the price of the credit (i.e. interest rate).2
In addition, the amount of interest paid/received depends on:
(1) The amount loaned or borrowed (i.e. the principal).
(2) The length of time the amount is loaned or borrowed (i.e. the term of the credit).
(3) The stated (or nominal) interest rate.
(4) The repayment schedule (i.e. how often the interest is paid and how the principal
should be paid back).
(5) The method used to calculate the interest payments.
Ways of calculating interest rates
There are many ways of determining the interest payments and some of them can be quite
1
This is not to be confused with the use of a single interest rate in most economics textbooks and lectures.
The reason for the single interest rate is to simplify any economic analysis involving the price of credit (i.e.
interest rate).
2
Since interest rate can be affected by many different factors, two federal laws have been passed to
minimize any confusion associated with interest rate. These two laws are the Truth in Lending Act (passed
in 1968) and the Truth in Savings Act (passed in 1991). The purpose of the Truth in Lending Act is to make
it easier for lenders to compare different interest rates when it comes to borrowing money; and the purpose
of the Truth in Savings Act is to make it easier for savers to compare different interest rates offered by
different depository institutions. The two acts achieved their objectives by requiring a standard
measurement of interest rate (in annual terms) regardless of the maturity, amount, or other terms of the
credit.
Chapter 3-1
complicated. The following are some of the more common ways of calculating interest
payments that you might encounter: (i) simple interest, (ii) bank discount, and (iii)
compound interest.
(i) Simple interest
The saver only receives interest payment on the principal for the time period it is in a
savings account. In other words, there will be no interest paid on the accumulated interest
remaining in the account.
Int. pmt .  principal Int. rate  no. of payment period
= p i  n
Total pmt .  principal+ int. pmt.
= p + p  i  n  p(1  i  n)
Example: Suppose you deposit $1,000 into a savings account that pay a simple interest
rate of 4% for 5 years, assuming that the bank pays its interest once a year. How much
interest will you earn at the end of the 5-year period? And how much money is in your
account (if you do not withdraw the interest)?
Interest  $1,000  0.04  5  $200
Final balance 1,000  200  $1,200
(ii) Bank discount
With the bank discount method, the full interest payment is first determined from the
amount of money borrowed, and then the borrower will receive the difference. In other
words, the borrower has to pay the interest payments up front when he/she first takes out
the loan. This is a very common method used by banks on short-term loans taken out by
businesses.3 It is important to note that the interest rate a borrower effectively paid is
different from the nominal interest rate quoted. This is best illustrated with the following
example.
Example: Suppose Orange, Inc. took out a $10,000 loan from St. Nick Federal for 1 year
at 7% interest and the interest payment is determined on a bank discount method. What is
the interest rate Orange, Inc. has effectively paid for that loan?
Interest payment  0.07  10,000  $700
Amount received  10,000  700  $9,300
700
Effectiveinterest rate 
 7.53%
9300
From the example above, we know that even though Orange, Inc. is quoted an annual rate
of 7%, it effectively is paying an interest of 7.53% because it is only “getting” $9,300
from the bank rather than $10,000 (and yet it is paying interest on the full $10,000).
“Interest payments” of Treasury bills (offered by the federal government) are based on the bank discount
method. In other words, an individual buys the T-bills at a discount and receives the principal in full when it
matures.
3
Chapter 3-2
(iii) Compound interest rate
Unlike the simple interest rate, if the saver decides to keep the interest payment received
in the savings account, interest will be paid on those interest payments. As a result:
Total pmt. = p(1  i ) n
The compound interest rate technique is a more common practice than the simple interest
rate technique in the real word.
Example: Suppose the bank decides to pay compound interest rather than simple interest,
how would that change your answers to the previous example?
Interest  1000 1.04 5  1000  $216 .65
Final balance  $1,216 .65
Similar to the bank discount method, the nominal interest rate quoted under a compound
interest method usually differs from the interest rate the borrower and lender effectively
paid and received, respectively, due to the compounding factor.
We need to look at the effective annual interest rate in order to determine the actual
interest rate received or paid for a year.
Effectiveannual rate  (1 
i m
) 1
m
where r = quoted annual rate
m = number of times interest is compounded
Example: In the previous example, the bank is compounding its interest once a year.
Suppose the bank starts paying its interest once every six months, how much interest will
you actually earn?
0.04 2
) 1
2
 0.0404
Effectiveannual rate  (1 
Interest rate (or return) of money market and fixed-income capital instruments
So far, we have looked at interest as the price of borrowing and lending between
individuals, and between an individual and an institution (e.g. a saver and a bank). There
are other forms borrowing and lending which involve the exchange of a security. For
example, an individual can “lend” the U.S. Federal government by buying a U.S. treasury
securities. We will divide such securities into two different categories: (1) money market
instruments and (2) fixed income debt instruments.
1. Money market instruments
Money market instruments are financial securities that have very short life spans, and
they are very homogenous financial instruments, i.e. they share very similar features:
a. They are short-term instruments with life spans of less than a year.
b. They are very high quality securities, i.e. very low default risk.
Chapter 3-3
c. They have very high denominations (or face value).
d. They are usually sold at discount, i.e. they are sold for less than their face values.
Quoting prices for the financial instruments
Before we proceed further in discussing the various financial instruments and the interest
rates (or returns) associated with them, we need to first understand how their prices are
quoted in the newspaper and by the brokers who buy and sell them. When you ask your
broker for the price of a particular security, you are usually quoted either the bid price or
the ask price (depending on whether you are selling or buying the security).
Bid price: the price that a broker (or dealer) is willing to buy
Ask price: the price that a broker (or dealer) is willing to sell
In this case, you will be paying the ask price when you buy a security and receiving the
bid price when you are selling a security.
There are 8 general types of money market instruments:
(i) Treasury bills (T-bills)
The T-bill is the shortest security offered by the federal government. It usually has a life
of 91 days (13 weeks or 3 months), 182 days (26 weeks or 6 months), and 52 weeks (1
year); and has a face value of $10,000.
New issues of the T-bills are auctioned off at the Federal reserve banks, and individual
and institutional investors can bid for them. 3 months and 6 months T-bills are auctioned
off every week (on Monday), and 1 year T-bills are auctioned off every month (on the
fourth Thursday). Existing issues of T-bills can be purchased directly from the Fed or
from brokers (that are government securities dealer).
Pricing of a T-bill
Since T-bills do not pay interest, you buy the T-bills on discount (i.e. paying a price less
than the face value), and you receive the face value on maturity. Since T-bills are bought
and sold at discount, their prices are not quoted directly. You are quoted the discount that
you will receive/pay for the T-bills. In addition, the discount is quoted on an annual basis
so you will need to adjust the discount based on the life of the T-bill. To actually
determine the bid or asked price of the T-bill, you need to use the following formula:

 n 
P  10,000 1  rBD 

 360 

where rBD = bid discount rate if selling or asked discount rate if buying
n = number of days to maturity
Example: The following table is taken from the Feb 19, 1997 Treasury Bills table in the
Wall Street Journal. As an investor, you are interested in buying one Jun 5, 1997 T-bill
and selling one Jun 19, 1997 T-bill from your portfolio. What price will you pay for the
Jun 5, 1997 T-bill? And what price will you receive for the Jun 19, 1997 T-bill?
Maturity
Days to mat.
Bid
Asked
Chapter 3-4
Chg.
Ask Yld.
Jun 05 ’97
Jun 12 ’97
Jun 19 ’97
Jun 26 ’97
106
113
120
127
4.96
4.95
4.98
4.98
4.94
4.93
4.96
4.96
+0.01
+0.01
+0.03
+0.03
5.08
5.08
5.11
5.12
(i) Price paid for Jun 5, 1997 T-bill (asked price)
 
106 
Asked price= 10 ,000 1   0.0494 
  $9,854 .54
360 
 
(ii) Price received for Jun 19, 1997 T-bill (bid price)
 
127 
Bid price= 10 ,000 1   0.0498 
  $9,824 .32
360 
 
The return (or yield) of a T-bill
As an investor, you are interested in the return earned by a particular T-bill you have
bought (hence the asked yield). However, you have to make certain adjustments before
you can compare the T-bill’s yield rate to another financial instrument’s yield rate for the
following two reasons:
(i) T-bill does not pay interest payment. The “interest payment” receives by an investor
comes in the form of the discount receives.
(ii) The price of a T-bill is computed using a 360-days calendar year rather than a 365days calendar year, which is common in computing the yield of most financial
instruments.
In order to compare the return of a T-bill to other financial instruments, we need to
compute the T-bill’s bond equivalent yield ( rBEY ) using the following formula:
rBEY 
10,000  P 365

P
n
The above formula is very tedious because we need to first solve for the T-bill’s asked
price. We can simplify it by substituting the formula for P we have discussed earlier. As
a result, the bond equivalent yield is computed as follows:
rBEY 
365  rBD
360  ( n  rBD )
Example: Using the same T-bill price sheet, verify that the asked yield (i.e. bond
equivalent yield) of a Jun 19, 1996 T-bill is 5.11%.
rBEY 
365  0.0496
 0.0511  5.11 %
360  (120  0.0496 )
(ii) Federal funds
All commercial banks and depository institutions are required by law to hold reserves for
the deposits at its branches with its district Federal Reserve Bank. Each institution is
Chapter 3-5
required to keep a certain percentage of its total deposits as reserves. This is like keeping
a non-interest-bearing checking account with a Federal Reserve Bank.
If a bank is temporarily short of its required reserve, it can borrow from other banks that
have excess reserves. In other words, the banks can trade reserves among themselves. The
amount of money traded for this purpose is known as federal funds. They are usually
traded (or borrowed) for a very short period of time: overnight to 3 days.
(iii) Commercial paper
One of the short-term financing sources for a firm to meet its current obligations is the
commercial paper. They are short-term (usually less than 270 days) unsecured (i.e. not
backed by any asset) promissory notes with a fixed maturity. Commercial papers usually
have a very large face value (with the smallest denomination at $100,000), and they are
traded at discount (i.e. selling at less than the face value).
(iv) Negotiable certificates of deposits (CD)
Most of you have some experience dealing with CDs at a local bank. That is when you
put aside a certain amount of money for a fixed period of time (and there is a penalty for
early withdrawal). Such CDs are non-negotiable, i.e. you cannot sell this CD to another
person.
The CDs which can be classified as money market instruments are those issued by banks
and S&Ls. They have a very large denomination (or face value) in the amount of $1
million per CD. Unlike most other money market instruments, such CDs do pay interest
to their holders on the maturity date. In addition, they are negotiable, i.e. they can be sold
to another person. These negotiable CDs have a maturity date of at least 14 days.
However, most of them are issued at 30 and 60 days.
(v) Eurodollar deposits
Eurodollar deposits are US dollar-denominated deposits kept at banks outside the United
States. These are large time deposits with a maturity of less than 6 months. Most of these
deposits are concentrated in London, which is one of the world’s and Europe’s largest
financial market.
(vi) Banker’s acceptance
Banker’s acceptance dates back to the 12th century when they are used to finance
international trades. It is simply a promissory note issued by a credit worthy bank
guaranteeing the exporter will make the payment once the shipment has been received.
Banker’s acceptances are usually traded at discount once it is issued. The last holder of
the banker’s acceptance will receive the face value from the exporter’s bank.
Chapter 3-6
(vii) Repurchasing agreements (Repos)
A repurchasing agreement is simply the sale of securities (usually treasury securities) with
the promise of buying them back at a higher price at a later date. They are usually issued
by corporations, state and local governments, and some other big non-bank institutions.
The concept of a repurchasing agreements market is very similar to a federal funds
market. In that case, why does a repurchasing agreements market exist? That is because
the federal funds market is only open to depository institutions.
(viii) Broker’s calls
As an investor, you do not have to pay the full amount for the securities you purchased
through a broker. You can set up a margin account. In other words, you can borrow part
of the money from your broker. In other to loan you the money, the broker can borrow the
money from a bank. The amount borrow is the broker’s call.
As we have discussed earlier, money market instruments are very homogenous financial
securities. As a result, the rates (or yields) for each type of money market instruments are
pretty standardized regardless of the issuers. You can find the rates for the different
money market instruments in the Money Rate table of WSJ’s Credit Markets section.
2. Fixed income capital market instruments
Unlike money market instruments, fixed income capital market instruments do pay
interest payment on a regular basis. That is why they are called fixed income instruments,
because holders of these instruments receive a fixed amount of interest payments on a
regular basis and a known face value on the maturity date. They are capital market
instruments because they have a maturity of more than a year (and some go up to 30
years). There are 5 general types of fixed income capital market instruments:
(i) T-notes and T-bonds
T-notes and T-bonds have longer life spans than T-bills: T-notes are usually 2 to 10 years,
and T-bonds are usually 10 to 30 years. Unlike T-bills, T-notes and T-bonds pay interest
periodically on a semi-annual basis.
You have to be very careful when reading the price quotes for T-notes and T-bonds. Their
“prices” are quoted as a percentage of the face value (which is usually $1000). In
addition, they are quoted in 1/32, e.g. 100: 16 is 100 16/32. Do not confuse the colon with
a period. Why is this the case? That is because 1/32 used to be the Spanish dollar units.
As a result, the actual bid or asked price is determined as follows:
P
r
1,000  10  r
100
where r = “price” quoted as percentage of face value
Chapter 3-7
Example: The following table is taken from the Nov 6, 1995 Govt. Bonds & Notes table
in the Wall Street Journal. As an investor, you are interested in buying one 7 7/8 % Aug
2001 T-bond and selling one 7½% Nov 2001 T-bond from your portfolio. What price will
you pay for the Aug 2001 T-bond? And what price will you receive for the Nov 2001 Tbond?
Rate
7 7/8
8
13 3/8
7 1/2
Maturity
Mth/Yr.
Aug 01n
Aug 96-01
Aug 01
Nov 01n
Bid
Asked
Chg.
Ask
Yld.
110:09
101:24
136:31
108:23
110:11
101:28
137:03
108:25
-4
+1
-5
-2
5.74
5.49
5.74
5.75
(i) Price paid for 7 7/8 % Aug 2001 T-bond (asked price)
Asked price= 10  110
11
 $1,103 .44
32
(ii) Price received for 7½% Nov 2001 T-bond (bid price)
Bid price= 10 108
23
 $1,087 .19
32
(ii) Agency debts
There are a number of federally sponsored agencies (which are privately owned entities)
that make loans to a certain class of borrowers. The reason such agencies exist is because
the Congress believes that the supply of credit is too limited, too variable or too
expensive for certain class of borrowers. These agencies are set up to provide dependable
sources of credit at the lowest possible cost. These agencies issue their own debt
instruments to raise money to make various types of loans.
The following are examples of some federally sponsored agencies:
1. Agencies issuing mortgage related debt
a. Federal Home Loan Bank (FHLB)
b. Federal National Mortgage Association (FNMA) or Fannie Mae
c. Government National Mortgage Association (GNMA) or Ginnie Mae
d. Federal Home Loan Mortgage Corporation (FHLMC) or Freddie Mac
2. Agencies issuing farm related debt
a. Farm Credit Financial Assistance Corporation
b. Federal Land Banks
c. Federal Intermediate Credit Banks
Pricing of agency debts
The pricing of agency debts is very similar to the pricing of T-notes and T-bonds: it uses
the “:” to represent 1/32.
Chapter 3-8
(iii) Municipal bonds
In addition to the federal government, there are other governmental units in the United
States which can be group as states, counties, municipalities, townships, school districts
and special districts. Many of these governmental units raise money by issuing municipal
bonds (or Munis).
There are three different types of municipal bonds:
1. Revenue bonds: These are bonds issued to financed a particular project. Revenues
generated by the project will be used to pay the interest payments and repay the
principal.
2. General Obligation bonds (GOs): These are bonds that are backed simply by the full
faith and credit of the governmental units that they will make the interest payments and
repay the principal.
3. Industrial Development bonds (IDBs): These are bonds used to finance the purchase or
construction of industrial facilities that will be leased to firms at favorable rates.
(iv) Corporate bonds
As we have discussed earlier, a firm can raise money by issuing debt instruments. They
will issue commercial papers for short-term needs, and corporate bonds for long-term
needs. Issuing corporate bonds is more complicated than issuing commercial papers
because the firm needs to prepare a lengthy legal document and seek approval from the
Securities and Exchange Commission (SEC).
Pricing of corporate bonds
The pricing of corporate bonds is different from the pricing of the debt instruments
because it does not use 1/32 as an explicit unit. In addition, only one price (quoted as
percentage of face value) is published in the press rather than both the bid and asked
prices.
Example: The following table is taken from the Nov 6, 1995 Corporation Bonds table in
the Wall Street Journal. Suppose you are interested in buying a 43/8% AT&T bond
maturing in 2000. What price will you be paying for that bond?
Bonds
ATT 43/800
ATT 43/801
ATT 6s01
Cur
Yld
Vol
Close
Net
Chg
4.4
4.6
6.0
34
71
41
987/16
95¼
99½
- 1/32
+¼
+¼
Price paid for the AT&T bond:
Price = 98
7
 10  $984 .38
16
Chapter 3-9
The interest rate (or return) of fixed-income capital instruments
The fixed income capital instruments that we have discussed so far are also known as
coupon-bearing bonds, i.e. they pay interest (or coupon) payments to their holders. As a
result, we have to be very careful when we are trying to determine interest rate (or return)
associated with such coupon-bearing instruments. There are several “interest-like”
interest and we need to know which is the true one, i.e. the one that represents the return
of a fixed-income capital instrument.
(i) Coupon rate
For a bond that pays interest payments on a periodic basis is known as a coupon bond.
Each coupon bond specifies a coupon rate that is expressed as a percentage of the face
value of the bond.
Couponrate 
annual coupon
facevalue
For example, a coupon bond with a 10% coupon rate will pay the holder a $100 a year if
the face value is $1,000. The coupon rate is predetermined and it is not affected by any
economic conditions once the bond is issued.
(ii) Current yield
Unlike the coupon rate, the current yield of a bond reflects the current economic
condition rather than focuses on the face value. The current yield is defined as:
Current yield =
annual coupon
market price of bond
Example: In the previous example, we know the price of a 43/8% AT&T bond maturing
in 2000 is $984.38. As a result, we can verify the current yield of the bond to be 4.4% as
follows:
Current yield =
43.75
 0.044  4.4%
984 .38
You can look up the current market price of a coupon bond (both government and
corporate) in the Wall Street Journal.
(iii) Yield to maturity
The problem with current yield is that ignores the components of the interest payments
received from the coupon bonds. These coupon payments can be reinvested into the
market. As a result, the yield of a bond changes when you factor in the reinvestment of
the coupon payments. The calculation for the yield to maturity is very complicated
without the aid of a financial calculator. You need to solve for YTM in the following
equation given the current market price of the bond, P0 :
Chapter 3-10
T
P0  
t 1
Ct
Par

t
(1  YTM )
(1  YTM ) T
where T represents the life of the bond.
However, we can approximate the YTM of the bond with the following equation:
Annual coupon payment+
YTM 
Face value  Market price
Years until maturity
1
Face value+ (2  Market price)
3
Since the yield to maturity is the most accurate measure of the interest rate, the two terms
are interchangeable in most contexts. In the future, when we refer to the term “interest
rate”, we are actually referring to the yield to maturity.
Example: We can determine the yield to maturity of the 43/8% AT&T bond maturing in
2000 (hence a remaining life of 3 years) as follows:
1000  984 .38
3
YTM 
 0.0495  4.95%
1
1000  (2  984 .38 )
3
43 .75 
The distinction between yield to maturity, rate of return and coupon rate
Since the yield to maturity of a bond is used to represent the “market” interest rate, most
people use that to represent the rate of return earned from a bond. This is only true if the
bond is held on to maturity. However, most of the time bonds are sold prior to maturity.
As a result, the rate of return of a bond differs from the yield to maturity. In order to
determine the rate of return of a bond, we need to use the following formula:
HPRT  RT 
PT  P0   Ct
P0
In the above situation, the rate of return of the bond ( RT ) is also known as the holding
period return of the bond ( HPR T ). This represents the return earned while holding on to
the bond.
We can simplify this scenario by looking at a case where the bond is held for only one
period (i.e. from time t to t+1):
Rt 1 

Pt 1  Pt  C t
Pt
Pt 1  Pt C t

 Capital gain + Current yield
Pt
Pt
Chapter 3-11
The only time the rate of return of a bond is equivalent to its yield to maturity is when the
bond is bought and held one period prior to its maturity date:
Lets assume that the bond matures at time t+1, and the bond is bought at time t. As a
result, the price of the bond at time t+1 has to be equal to the face value of the bond. In
this case:
Pt 
Ct
Pt 1

1  YTM 1  YTM
1  YTM 
YTM 
C t Pt 1

Pt
Pt
Ct Pt 1
C P  Pt

 1  t  t 1
Pt
Pt
Pt
Pt
In addition, we have to be very careful with the terms we are using. It is very easy to mix
up the terms yield to maturity (i.e. interest rate), rate of return, and coupon rate if we do
not understand them very well. It is important that you know the distinction of those three
terms.
The relationship between interest rate and bond price
From the above discussion, we know that there is a relationship between interest rate and
bond prices. The following are some of the most common and important relationships:
(1) There is an inverse relationship between the market interest rate and the price of a
bond, i.e. if the market interest rate goes up (down), then the price of the bond will go
down (up).
(2) The price of the bond is affected by the relationship between its coupon rate and the
market interest rate.
Coupon rate > market rate  market price > par  sold at premium
Coupon rate = market rate  market price = par  sold at par
Coupon rate < market rate  market price < par  sold at discount
(2) The lower the coupon rate of a bond (with the same term to maturity), the more
sensitive it is to interest changes.
(4) Longer-term bonds are more sensitive to interest rate changes than shorter-term bonds
(with the same coupon rate). As a result, longer-term bonds are considered to be
riskier than shorter-term bonds.
From the above four relationships, we know that certain “features” of a bond greatly
influence the impact of interest rate changes on a bond’s price. Such impact is known as
interest rate risk.
The distinction between real and nominal interest rate
As we have discussed earlier, the interest rate is usually represented by the yield to
Chapter 3-12
maturity of a bond. However, the interest rate (or yield to maturity) mentioned is usually
the nominal interest rate. In other words, it is measured in current dollars. This is different
from the real interest rate, which is adjusted for inflation.
The relationship between the nominal and real interest rates can be expressed as follows:
1  i  (1  ir )  (1   e )
where i  the nominal interest rate
ir  the real interest rate
 e  the expected inflation rate
The above equation can be rewritten as:
i  ir   e  (ir  e )
Since (ir   e ) is so small, the above equation can be rewritten as:
i  ir   e
which is more commonly known as the Fisher Equation of interest rate. It simply states
that the real interest rate is simply the nominal interest rate corrected for inflation:
ir  i   e
It is important to remember that the real interest rate is measured in real terms rather than
in current dollars. In other words, it tells you how much goods and services you can
actually get. As a result, it is more important to look at the real interest rate rather than the
nominal interest rate, because the real interest rate reflects the true cost of borrowing and
lending.
Example: Suppose you are interested in investing in a bond that will provide you an
interest rate of 6% in the upcoming year. Is this a good investment if you expect the
inflation rate to be 4% in that same year?
i r  6%  4%  2%
Suppose the inflation rate is forecasted to be around 8% in the upcoming year. In that
situation, is the bond an attractive investment?
i r  6%  8%  2%
Chapter 3-13