Citibank: basic of corporate finance
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UNIT 9: FIXED INCOME SECURITIES
INTRODUCTION
In Unit Four, we introduced the pricing of fixed income securities (bonds) and explained
how to compute the present value of coupon payments and principal repayment. The
important point to remember is that a bond in the open market trades for a price equal to
its present value.
The focus of this unit is on the calculations that are used to price bonds and determine their
yields. Of all debt instruments available worldwide, U.S. Treasury securities (debt
instruments) represent the largest market, are the most liquid, and are considered to be
of the highest credit quality (i.e., free from default risk). Furthermore, many companies
throughout the world use U.S. Treasuries as part of their investment portfolios.
Throughout this unit, U.S. Treasury securities are used in pricing and yield examples
designed to illustrate the various principles and calculations. An understanding of U.S.
Treasuries will provide a foundation for your continuing study of more complex and risky
debt securities. It is important to remember that given matching characteristics, the
calculations used to price U.S. Treasury instruments are applicable to any other type of debt
security.
UNIT OBJECTIVES
When you have successfully completed this unit, you will be able to:
• Select the correct formula for computing the price of a bond
• Recognize the relevant data needed to calculate the price, discount rate, and
yield of Treasury securities and other debt instruments
• Convert yields of various debt instruments to bond equivalent yields (BEY)
• Calculate the duration of a bond and a portfolio of bonds
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INTRODUCTION TO BOND PRICING AND YIELD MATHEMATICS
Nonmarketable
and marketable
securities
Through the Treasury Department, the United States Government
issues many types of debt securities for purchase by investors.
Instruments that do not have a secondary market (e.g. U.S. savings
bonds and retirement bonds) are considered to be nonmarketable.
They are purchased directly from the government and held until
maturity.
In this unit, we will discuss the pricing of debt instruments that are
classified as marketable — that is, they are bought and sold in a
secondary market. Specifically, we will discuss:
• Treasury bills
• Treasury notes
• Treasury bonds
• Eurobonds
U.S. Treasury Bills
Zero-coupon
instruments
with maturity
under a year
The shortest term debt instruments are U.S. Treasury Bills.
These bills are issued in maturities of 13, 26, and 52 weeks. The
government holds an auction every week for the purchase of 13- and
26-week bills, which are zero-coupon instruments. Coupon payments
are not made during the life of these bills, and they are bought and
sold on a discount basis — often referred to as the discount rate or
yield of the bill.
The 52-week bills are auctioned every four weeks. Semi-annual
interest payments are made on these bills.
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Pricing U.S. Treasury Bills
Non-interest
bearing
security
To calculate the price of zero-coupon Treasury bills or any non-
interest bearing security, we use the following formula:
P = F [1 - ((R x D
M
) / 360)]
Where:
P = Current price of the security
F = Face value of the security
D
M
= Number of days until the security matures
(U.S. Treasury bills assume a 360-day year)
R = Discount yield (yield to maturity)
Example
To illustrate the use of this formula, consider the following question.
At what price would we expect a $1,000 Treasury bill to be trading in
the market with 64 days to maturity and a 6.5% discount yield to
maturity? Apply the formula and perform the computations. In this
example, D
M
= 64 and R = 0.065.
P = F [1 - ((R x D
M
) / 360)]
P = $1,000 [1 - ((0.065 x 64) / 360)]
P = $1,000 [1-0.01166]
P = $1,000 [0.98844]
P = $988.44
We would expect the bill to sell for $988.44 in the bond market.
This price would appear in the financial section of The Wall Street
Journal as 98.84. The quoted price is a percentage of the selling
price.
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Calculating the Discount Rate
There may be times when the price of the security is known but the
discount rate used in pricing the instrument is unknown. To calculate
the rate, we use the pricing formula and solve for R. The variables
have the same value as defined for the pricing formula. The formula
for calculating the discount rate is:
R = [(F - P) / F] x (360 / D
M
)
Example
For example, suppose that a 45-day, $1,000 Treasury bill is priced at
$988. Find the discount rate that was used to price the security. To
solve, let F = $1,000, P = $988, and D
M
= 45 days.
R = [(F - P) / F] x (360 / D
M
)
R = [($1,000 - $988) / $1,000] x (360 / 45)
R = [0.0120] x (8.00)
R = 0.0960 or 9.60%
Yield to
maturity
A discount rate of 9.60% was used to calculate the price of the
Treasury bill ($988). This rate is often referred to as the yield to
maturity of the security. Some investors might say, "The security is
priced to yield 9.60%."
Rate of Return
True (actual)
yield
Because an investor pays less than face value for a zero-coupon
(discounted) security, the investor earns a rate of return on the
investment. This rate of return, which is based on the difference
between the purchase price and sale price, is always greater than the
discount rate, which is based on the face value of the security.
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One convention is to state this rate of return on a 365-day year basis.
The rate of return is the simple interest earned on the investment,
which is often referred to as the true (actual) yield of the bill. The
formula is:
i = [(P
s
- P
b
) / P
b
] x [365 / D
H
]
Where:
i = Simple annual interest (rate of return) earned
on a 365-day year basis
P
s
= Price at which security was sold
P
b
= Price at which security was bought
D
H
= Number of days security was held
Example
Suppose an investor purchases a $10,000, 90-day Treasury bill for
$9,600 and then sells it after 30 days for $9,700. What is the annual
rate of return (simple interest) earned by the investor? Apply the
formula with these values: P
s
= $9,700, P
b
= $9,600, and D
H
= 30.
i = [(P
s
- P
b
) / P
b
] x [365 / D
H
]
i = [($9,700 - $9,600 ) / $9,600 ] x [365 / 30]
i = [0.0104] x [12.1667]
i = 0.1267 or 12.67%
The investor has earned an annual rate of return of 12.67% for a
30-day investment in the Treasury bill. If the investor holds the bill to
maturity, then the face value of the security is used as the selling
price when calculating the simple interest earned.
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Bond
equivalent
yield
The true or actual yield is sometimes called the bond equivalent
yield (BEY) of the Treasury bill. This yield allows for direct
comparison to a Treasury note with bond yields that are priced
on a 365-day year basis and make semi-annual coupon payments.
Comparisons should be made between instruments with the same
amount of time to maturity. For example, the yield on a Treasury bill
with 30 days to maturity can be compared to the yield on a Treasury
note or bond also with 30 days until maturity.
The calculation of a bond equivalent yield for Treasury bills with
maturities between six months and one year is complicated by the
fact that Treasury notes and bonds make coupon payments during the
period. That formula will not be given here, but be aware that an
adjustment is necessary for the comparison to be valid.
Money market
yield
The term money market yield (MMY) refers to the rate of return
earned during a 360-day year, rather than a 365-day year. The
calculation is identical to the true yield calculation, except that we
substitute 360 for 365 in the formula. Certificates of deposit (CD)
and other short-term instruments are often quoted on a money market
yield basis.
Summary
U.S. Treasury Bills are short-term (up to one year) debt securities
issued through the U.S. Treasury Department. They are marketable
securities because they are bought and sold in a secondary market.
Zero-coupon securities are those for which no coupon payments are
made during the life of the securities. They are bought and sold on a
discounted basis.
The price of a non-interest bearing security is calculated with this
formula:
P = F [1 - ((R x D
M
) / 360)]
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Yield to maturity is the discount rate of a security. The formula for
calculating the discount rate is:
R = [(F - P) / F] x (360 / D
M
)
The rate of return on a zero-coupon security is greater than the
discount rate. The true (actual) yield is based on a 365-day year and
is calculated as:
i = [(P
S
- P
b
) / P
b
] x (365 / D
H
)
Money market yield (MMY) refers to a rate of return based on a
360-day year.
Before we continue to the next section regarding U.S. Treasury notes and bonds, please
complete the Practice Exercise which follows.
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PRACTICE EXERCISE 9.1
Directions: Use the list of formulas to solve Exercises 1- 4. Perform the calculations
to find the correct solutions. Check your solutions with those of the
Answer Key.
List of Formulas
(1) P = F [1 - ((R x D
M
) / 360)]
(2) i = [(P
s
- P
b
) / P
b
] x [365 / D
H
]
(3) R = [(F - P) / F] x (360/D
M
)
Data Solutions
1) Compute the price of a 26-week Treasury bill that has:
n A face value of $10,000
n A yield to maturity of 4.75%
n 60 days until maturity
1a) Now, consider that this bill is held until maturity.
Compute its actual yield.
2) Compute the price on a 13-week Treasury bill that has:
n 24 days until maturity
n A 3.25% discount rate
n A $1,000 face value
3) Compute the discount rate used to price a 26-week
Treasury bill that has:
n A $10,000 face value
n A price at time of issue of $9,400
4) Compute the yield to maturity of a 13-week Treasury
bill that has:
n A $1,000 face value
n 45 days until maturity
n A current price of $987.50
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