TABLE OF CONTENTS
LESSON 1: FINANCIAL MARKETS ARITHMETIC.................................................................................1
SECTION 1: SIMPLE AND COMPOUND INTEREST ........................................................................................... 1
SECTION 2: THE TIME VALUE OF MONEY ..................................................................................................... 5
LESSON 2: PRICING TREASURY BONDS ................................................................................................7
SECTION 1: BOND PRICING .......................................................................................................................... 7
SECTION 2: VALUING SEMI-ANNUAL CASH FLOWS ................................................................................... 16
SECTION 3: VALUING ZERO-COUPON AND PERPETUAL BONDS .................................................................. 18
LESSON 3: VALUING A BOND BETWEEN COUPON PAYMENTS ..................................................... 20
SECTION 1: UNDERSTANDING THE CONCEPT OF ACCRUED INTEREST ....................................................... 20
SECTION 2: COMPUTING THE FULL PRICE................................................................................................. 21
SECTION 3: COMPUTING THE ACCRUED INTEREST AND THE CLEAN PRICE ............................................... 22
LESSON 4: BOND PRICE AND YIELD RELATIONSHIP ...................................................................... 24
SECTION 1: SOURCES OF RETURN .............................................................................................................. 24
SECTION 2: YIELD MEASURES ................................................................................................................... 24
SECTION 3: BOND PRICE AND YIELD RELATIONSHIP .................................................................................. 26
LESSON 5: PRICING A TREASURY BILL .............................................................................................. 27
SECTION 1: UNDERSTANDING TREASURY BILLS ........................................................................................ 27
SECTION 2: PRICING TREASURY BILLS ...................................................................................................... 27
REFERENCES ............................................................................................................................................. 29
GLOSSARY OF TERMS ............................................................................................................................. 30
A. Learning Objective
This module focuses on valuation of government securities. It will introduce the concepts of
present and future values and how to calculate these using compounding and discounting
techniques. The module will also outline how various yields (such as the current yield and the
yield to maturity) are calculated, and explains the relationship between bond prices and yields
as well as how to identify whether a bond is trading at par, at a premium, or at a discount. Each
concept is reinforced with practical and applied exercises and case studies.
B. Learning Outcomes
By the end of this module, participants will have a solid understanding of how government
securities are priced. Participants will also understand the relationship between bond prices and
yields. With this knowledge, participants will be able to discern and use secondary market
information to inform judgement on expected cost of meeting government’s financing
requirements. The skills are also necessary for other post-issuance strategies such as liability
management operations.
i
Lesson 1: Financial Markets Arithmetic
Section 1: Simple and compound interest
The principles of financial market arithmetic have long been used to illustrate that $1 received
today is not the same as $1 received at a point in the future. Faced with a choice between
receiving $1 today or $1 in one year’s time, we would not be indifferent, given a rate of interest
of say, 10% and provided that this rate is equal to our required nominal rate. Our choice would
be between $1 today or $1 plus 10 cents – the interest on $1 for one year at 10% per annum.
The notion that money has a time value is a basic concept in the analysis of financial
instruments. Money has time value because of the opportunity to invest it at a rate of interest.
A. Simple interest
A security that has one interest payment on maturity is said to be accruing simple interest. On
short-term instruments, there is usually one interest payment on maturity, hence simple interest
is paid when the instrument expires. The terminal value of an instrument with simple interest
is given by:
𝑭𝑽 = 𝑷𝑽(𝟏 + 𝒓)……………Equation 1
Where:
FV - Terminal value or future value
PV - Initial bond proceeds or present value
r - Interest rate
If PV is $100, r is 5% and the bond is a one year bond, then future value is calculated as:
FV = 100(1 + 0.05) = $105.
B. Compound interest
Let us now consider a $100 par value 3-year bond, paying a coupon rate of 6% per annum. At
the end of the first year, we assume that the $6 coupon that government was supposed to pay
out to the investor is instead accrued on the principal sum of $100.
The applicable principal amount at the end of year 2 will be $106 (that is, $100 original
principal amount plus $6 coupon accrued in year1). In the second year, the bond accrues a
coupon of $6.36.
This illustrates how compounding works, which is the principle of paying interest on interest.
What will the terminal value of our $100 three-year bond be?
In compounding we are seeking to find a future value given a present value, a time period and
an interest rate.
If $100 is issued today (at time 𝑡0 ) at 6%, then one year later (𝑡1 ) the government will owe:
$100 𝑥 (1 + 0.06)𝑥 (1 + 0.06) = $100 𝑥 (1 + 0.06)2
= $100 𝑥 (1.06)2
= $112.36
1
The outcome of the process of compounding is the future value of the initial amount. We do
not have to calculate the terminal value long-hand as we can use the expression:
𝐅𝐕 = (𝟏 + 𝐫)𝐧 ……………….. Equation 2
Where
𝑟 is the periodic rate of interest (expressed as a decimal)
𝑛 is the number of periods (in years) for which the bond will remain outstanding.
C. Compounding more than once a year
Now let us consider a 1-year bond with par value of $100, again at our rate of 6% but with
quarterly interest payments. Such a bond would accrue interest of $6 in the normal way but
$1.50 would accrue every quarter, and this would then benefit from compounding. The total
obligation at the end of the year will be:
This gives us, $100 𝑥 1.06136, a terminal value of $106.136. This is some 13 cents more than
the terminal value using annual compounded interest.
In general, if compounding takes place 𝑚 times per year, then at the end of 𝑛 years, 𝑚𝑛 interest
payments will have been made and the future value of the principal is given by:
𝐫
𝐅𝐕 = 𝐏𝐕(𝟏 + 𝐦)𝐦𝐧 ………………. Equation 3
The formula for different compounding frequencies are shown below:
2
D. Effective interest rates
The interest rate quoted on a bond is usually the flat rate. However, we are often required to
compare two interest rates which apply for similar bond maturities but have different interest
payment frequencies. For example, a two-year bond with interest paid quarterly compared to a
two-year bond with semi-annual interest payments.
This is normally done by comparing equivalent annualised rates. The annualised rate is the
interest rate with annual compounding that results in the same rate at the end of the period as
the rate we are comparing. The concept of the effective interest rate allows us to state that:
………….Equation 4
where 𝑎𝑒𝑟 is the equivalent annual rate. Therefore, if 𝑟 is the interest rate quoted which pays
𝑛 interest payments per year, the 𝑎𝑒𝑟 is given by:
………….Equation 5
The equivalent annual interest rate 𝑎𝑒𝑟 is known as the effective interest rate. We have already
referred to the quoted interest rate as the “nominal” interest rate. We can rearrange equation 5
above to give us equation 6 which allows us to calculate nominal rates.
………….Equation 6
We can see then that the effective rate will be greater than the flat rate if compounding takes
place more than once a year. The effective rate is sometimes referred to as the annualised
percentage rate or APR.
Example
 The Government of Mozambique issued a 1-year bond with a coupon rate quoted at 5%,
payable in semi-annual instalments. What is the effective rate that the Government pays
at the end of the period?
 The Government of Mozambique issued another 1-year bond with a quoted nominal
coupon rate of 6.4% payable at maturity. What is the equivalent rate for the same 1-year
bond that pays coupon on a monthly basis?
3
4
Section 2: The time value of money
A. Present values with single payments
In section 1, we saw how a future value could be calculated given a known present value and
rate of interest. For example, if a government issues a $100 bond today for one year at a coupon
rate of 6%, it will pay 100 × (1 + 0.06) = $106 at the end of the year. The future value of $100
in this case is $106. We can also say that $100 is the present value of $106 in our example.
We established the following future value relationship:
𝐅𝐕 = (𝟏 + 𝐫)𝐧 .
By reversing this expression, we arrive at the present value form:
𝑷𝑽 =
𝑭𝑽
(𝟏+𝒓)𝒏
……………………..Equation 7
where terms are as before. Equation 7 applies in the case of annual interest payments and
enables us to calculate the present value of a known future sum.
To calculate the present value for a short-term instrument of less than one year, we will need
to adjust what would have been the interest paid for a whole year by the proportion of days of
the bond will run. Rearranging the equation, the present value of a known future value is:
………….Equation 8
Given a present value and a future value at maturity, what then is the interest rate paid? We
can rearrange the basic equation again to solve for the yield.
………….Equation 9
Using equation 9 will give us the interest rate for the actual period. We can then convert this to
an effective interest rate using equation 10.
………….Equation 10
When interest is compounded more than once a year, the formula for calculating present value
is modified, as shown by equation 11:
………….Equation 11
Where, as before 𝐹𝑉 is the cash flow at the end of year 𝑛, 𝑚 is the number of times a year
interest is compounded, and 𝑟 is the rate of interest or discount rate.
Illustrating this therefore, the present value of $100 that is received at the end of 5 years at a
rate of interest rate of 5%, with quarterly compounding is:
5
B. Present values with multiple discounting
Present values for short-term instruments of under one year maturity often involve a single
interest payment. If there is more than one interest payment, then any discounting needs to take
this into account. If discounting takes place m times per year, then we derive the present value
formula as follows:
………….Equation 12
For example, what is the present value of the sum of $1000 that is to be paid in 5 years where
the discount rate is 5% and there is semi-annual discounting? Using equation 12 above, we see
that:
The effect of more frequent discounting is to lower the present value.
6
Lesson 2: Pricing Treasury bonds
In Lesson 1, we reviewed the concept of time value of money. It is important to understand the
principles of present and future value, compound interest and discounting, because they are all
connected with bond pricing.
Section 1: Bond Pricing
The fundamental principle of pricing is that the value of a bond is equal to the present value
of expected future cash flows.
Refresh: Discounting converts value of future cash flows to present value
Rate used to discount future cash flows is Discount Rate
Discount rate represents alternative cost of borrowing in the market or simply, opportunity cost
or required rate of return. The Yield to Maturity (YTM) is normally used to discount cash flows
The discount rate used determines the present value of future cash-flows, hence, determining
an appropriate Discount Rate is key to fair valuation or pricing of Bonds
Bond valuation involves the following steps:
7
Step 1 and 2: Determining Cash flows and their Timing
A conventional bond has two cash flow sources:


coupon payment over the life of the instrument
principal (face or par value) repayment usually at maturity
The par value of a bond is the amount that the issuer has agreed to repay the bondholder at
maturity date.
The coupon rate, also called the nominal rate, is the interest rate that the issuer agrees to pay
each year. The annual interest amount paid to bondholders during the life of the bond is called
coupon.
Coupon = 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑥 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒
If the coupon is paid semi-annually, this means exactly half the coupon is paid as interest every
six months.
To illustrate the determination of expected cash flows, consider a simple bond that matures in
four years, has a coupon rate of 10%, and has a maturity value of $100. For simplicity, let’s
assume the bond pays interest annually. The cash flow for this bond is:
Year
1
2
3
4
Cash Flow
10
10
10
110
The time between coupon payments for any bond is counted as 1 period, so there are 4 periods
between the first and last cash flows for the bond in our example. The maturity payment is
received 4 periods from issue date.
Step 3: Determining the Appropriate Discount Rate (s)
Once the cash flows for a fixed income security are estimated, the next step is to determine the
appropriate interest rate to be used to discount the cash flows. The interest rate that is used to
discount a bond’s cash flows (therefore called the discount rate) is the rate required by the
bondholder. It is therefore known as the bond’s yield.
8
Step 4 and 5: Discounting the Expected Cash Flows
Given expected (estimated) cash flows and the appropriate interest rate to discount the cash
flows, the final step in the valuation process is to value the cash flows.
What is the value of a single cash flow to be received in the future? It is the amount of money
that must be invested today to generate that future value. The resulting value is called the
present value of a cash flow. It is also called the discounted value.
Present value is the discounted value of a stream of future cash flows. Discounted value means
the current worth of future cash flows. Future cash flows are adjusted to account for the time
value of money. Money has time value – its value decreases with time!
Figure 1: Discounting Cash Flows
Discounting converts future value to present value, while compounding converts present
value to future value.
The discount rate used as part of the present value (price) calculation is key to everything, as it
reflects where the bond is trading in the market and how the market perceives it.
The present value of a cash flow will depend on (i) when a cash flow will be received (i.e., the
timing of a cash flow) and (ii) the interest rate used to discount the cash flows.
First, we calculate the present value for each expected cash flow. Then, to determine the value
of the bond, we sum the present values (i.e., for all of the bond’s expected cash flows).
Therefore, when pricing a bond, we need to calculate the present value of all the coupon interest
payments and the present value of the redemption payment, and sum these.
9
The price of a conventional bond that pays annual coupons can therefore be given by:
𝑷=
𝑪
𝑪
𝑪
𝑪
𝑴
+
+
+ ⋯+
+
𝟐
𝟑
𝑵
(𝟏 + 𝒓) (𝟏 + 𝒓)
(𝟏 + 𝒓)
(𝟏 + 𝒓)
(𝟏 + 𝒓)𝑵
𝑵
= ∑
𝒏=𝟏
𝑪
𝑪
+
(𝟏 + 𝒓)𝑵 (𝟏 + 𝒓)𝑵
…Equation 13
For long-hand calculation purposes, the first half of equation 13 is usually simplified and
expressed in two ways as shown in equation 14:
………….Equation 14
The price of a bond that pays semi-annual coupons is given by the expression in equation 15,
which is our earlier expression modified to allow for the twice-yearly discounting:
………….Equation 15
Note how we set 2𝑁 as the power to which to raise the discount factor, as there are two interest
payments every year for a bond that pays semi-annually. Therefore a more convenient function
to use might be the number of interest periods in the life of the bond, as opposed to the number
of years to maturity, which we could set as 𝑛, allowing us to alter the equation for a semiannually paying bond as:
………….Equation 16
The formula in equation 16 calculates the fair price on a coupon payment date, so that there is
no accrued interest incorporated into the price. It also assumes that there is an even number of
coupon payments dates remaining before maturity. The date used as the point for calculation
is the settlement date for the bond, the date on which a bond will change hands after it is
traded.
10
For a new issue of bonds, the settlement date is the day when the stock is delivered to investors
and payment is received by the bond issuer. The settlement date for a bond traded in the
secondary market is the day that the buyer transfers payment to the seller of the bond and when
the seller transfers the bond to the buyer.
Different markets will have different settlement conventions, but most settle one business day
after the trade date (the notation used in bond markets is T+1).
Example 1
To illustrate the present value formula, consider a simple bond that matures in 4 years, has a
coupon rate of 10%, and has a maturity value of $100. For simplicity, let’s assume the bond
pays interest annually. The discount rate is 8%. The cash flow for this bond is:
Year
1
2
3
4
Cash Flow
10
10
10
110
The present value of each cash flow is:
The value of this security is then the sum of the present values of the four cash flows. That is,
the present value is $106.6243 = $9.2593 + $8.5734 + $7.9383 + $80.8533.
This example is demonstrated in the accompanying excel sheet, Ex.1. Practice using this
sheet. Is this bond trading at discount, par or premium?
11
A. Present Value Properties
An important property about the present value can be seen from the above illustration. For the
first three years, the cash flow is the same ($10) and the discount rate is the same (8%). The
present value decreases as we go further into the future.
This is an important property of the present value: for a given discount rate, the further into
the future a cash flow is received, the lower its present value. As 𝑡 increases, 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒𝑡
decreases.
Suppose that instead of a discount rate of 8%, a 12% discount rate is used for each cash flow.
Then, the present value of each cash flow is:
The value of this security is then $93.9253 = $8.9286 + $7.9719 + $7.1178 + $69.9070.
The security’s value is lower if a 12% discount rate is used compared to an 8% discount rate
($93.9253 versus $106.6243). This is another general property of present value: the higher the
discount rate, the lower the present value. Since the value of a security is the present value of
the expected cash flows, this property carries over to the value of a security: the higher the
discount rate, the lower a security’s value. The reverse is also true: the lower the discount rate,
the higher a security’s value.
B. Relationship between Coupon rate, Discount Rate, and Price Relative to Par Value
Now that we know how to price a bond, we can demonstrate the relationship between the
coupon rate, discount rate, and price relative to par value. The coupon rate on our hypothetical
bond is 10%. When an 8% discount rate is used, the bond’s value is $106.6243. That is, the
price is greater than par value (premium). This is because the coupon rate (10%) is greater
than the required yield (the 8% discount rate).
12
We also showed that when the discount rate is 12% (i.e., greater than the coupon rate of 10%),
the price of the bond is $93.9253. That is, the bond’s value is less than par value when the
coupon rate is less than the required yield (discount). When the discount rate is the same as
the coupon rate, 10%, the bond’s value is equal to par value as shown below:
Year
1
2
3
4
Cash-Flow
10
10
10
110
Total
PV at 10%
9.0909
8.2645
7.5131
75.1315
100.0000
Using the features of bonds presented in the Excel Sheet Ex.3, calculated the price
for each bond. Which bond is trading at discount, par and premium?
C. Change in a Bond’s Value Toward Maturity
As a bond moves closer to its maturity date, its value changes. More specifically, assuming that
the discount rate does not change, a bond’s value:



decreases over time if the bond is selling at a premium
increases over time if the bond is selling at a discount
is unchanged if the bond is selling at par value
At the maturity date, the bond’s value is equal to its par value. So, over time, as the bond moves
toward its maturity date, its price will move to its par value, a characteristic sometimes referred
to as a “pull to par value”.
To illustrate what happens to a bond selling at a premium, consider once again the 4-year 10%
coupon bond. When the discount rate is 8%, the bond’s price is 106.6243. Suppose that one
year later, the discount rate is still 8%. There are only three cash flows remaining since the
bond is now a 3-year security.
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The cash flow and the present value of the cash flows are:
Year
1
2
3
Cash-Flow
10
10
110
Total
PV at 8%
9.2593
8.5734
87.3215
105.1542
The price has declined from $106.6243 to $105.1542.
Now suppose that the bond’s price is initially below par value. For example, as stated earlier,
if the discount rate is 12%, the 4-year 10% coupon bond’s value is $93.9253. Assuming the
discount rate remains at 12%, one year later the cash flow and the present value of the cash
flow would be:
Year
1
2
3
Cash-Flow
10
10
110
Total
PV at 12%
8.9286
7.9719
78.2958
95.1963
The bond’s price increases from $93.9253 to $95.1963.
Using the features of bonds presented in the Excel Sheet Ex.4, calculate the price
for each bond and see if it changes with time.
To understand how the price of a bond changes as it moves towards maturity, consider 3 bonds
with the following features:





the 3 bonds mature in 20 years and have a yield required by the market of 8%
coupon payments are annual
Bond 1 is a premium bond, paying a 10% coupon rate and priced at $119.6363
Bond 2 is a discount bond, paying 6% coupon rate and priced at $80.3637
Bond 3 is a par bond, paying coupon 8% coupon rate and priced $100.0000.
The premium bond with an initial price of 119.6363 decreases in price until it reaches par value
at the maturity date. The discount bond with an initial price of 80.3637 increases in price until
it reaches par value at the maturity date.
14
In practice, over time, the discount rate will change due to the change in discount rate and
change in cash flow as the bond moves toward maturity.
Table 1: Price Movement Bond Approaches Maturity
Figure 2: The Effect of Time on a Bond’s Price
Using the features of bonds presented in the Excel Sheet Ex.5, calculate the price
for each bond until maturity and plot the three bonds as shown in Figure 3 above.
Note: you can use the Excel syntax function to do the calculations. The syntax to use is PV
(rate, nper, [pmt], [fv], [type]). See Excel Sheet Ex. 5 for more guidance.
In practice, over time the discount rate will change. So, the bond’s value will change due to
both the change in the discount rate and the change in the cash flow as the bond moves toward
maturity.
15
For example, again suppose that the discount rate for the 4-year 10% coupon bond is 8% so
that the bond is selling for $106.6243. One year later, suppose that the discount rate appropriate
for a 3-year 10% coupon bond increases from 8% to 9%.
Then the cash flow and present value of the cash flows are shown below:
Year
1
2
3
Cash-Flow
10
10
110
Total
PV at 9%
9.1743
8.4168
84.9402
102.5313
The bond’s price will decline from $106.6243 to $102.5313. As shown earlier, if the discount
rate did not increase, the price would have declined to only $105.1542. The price decline of
$4.0930 ($106.6243 − $102.5313) can be decomposed as follows:
Section 2: Valuing Semi-annual Cash Flows
In our illustrations, we assumed coupon payments are paid once per year. For most bonds, the
coupon payments are semi-annual. This does not introduce any complexities into the
calculation. The procedure is to simply adjust the coupon payments by dividing the annual
coupon payment by 2 and adjust the discount rate by dividing the annual discount rate by 2.
The time t in the present value formula is treated in terms of 6-month periods rather than years.
For example, consider once again the 4-year 10% coupon bond with a maturity value of $100.
The cash flow for the first 3.5 years is equal to $5 ($10/2). The last cash flow is equal to the
final coupon payment ($5) plus the maturity value ($100). So the last cash flow is $105.
Now the tricky part. If an annual discount rate of 8% is used, how do we obtain the semi-annual
discount rate? We will simply use one-half the annual rate, 4% (or 8%/2). Given the cash flows
and the semi-annual discount rate of 4%, the present value of each cash flow is shown below:
16
The bond’s value is equal to the sum of the present value of the eight cash flows, $106.7327.
Notice that this price is greater than the price when coupon payments are annual ($106.6243).
This is because one-half the annual coupon payment is received six months sooner than when
payments are annual. This produces a higher present value for the semi-annual coupon
payments relative to the annual coupon payments.
The value of a non-amortizing bond can be divided into two components: (i) the present value
of the coupon payments and (ii) the present value of the maturity value. For a fixed-rate coupon
bond, the coupon payments represent an annuity.
A short-cut formula can be used to compute the value of a bond when using a single discount
rate: compute the present value of the annuity and then add the present value of the maturity
value. The present value of an annuity is:
For a bond with annual interest payments, 𝑖 is the annual discount rate and the “no. of periods”
is equal to the number of years. Applying this formula to a semi-annual paying bond, the
annuity payment is one half the annual coupon payment and the number of periods is double
the number of years to maturity. So, the present value of the coupon payments can be expressed
as:
where 𝑖 is the semi-annual discount rate (annual rate/2). Notice that in the formula, we use the
number of years multiplied by 2 since a period in our illustration is 6 months.
17
The present value of the maturity value is equal to:
To illustrate this computation, consider once again the 4-year 10% coupon bond with an annual
discount rate of 8% and a semi-annual discount rate of one half this rate (4%) for the reason
cited earlier. Then:
To determine the price, the present value of the maturity value must be added to the present
value of the coupon payments. The present value of the maturity value is:
The price is then $106.7327 ($33.6637 + $73.0690). This agrees with our previous calculation
for the price of this bond.
Using the features of bonds presented in the Excel Sheet Ex.6 and Ex.7, calculate
the price for each of the bonds that pay semi-annual coupons.
Section 3: Valuing Zero-Coupon and Perpetual bonds
A. Valuing Zero-Coupon bond
The previous section dealt with pricing for conventional coupon-bearing bonds. Bonds that do
not pay a coupon during their life are known as zero-coupon bonds. For a zero-coupon bond,
there is only one cash flow, the maturity value. The value of a zero-coupon bond that matures
N years from now is:
where i is the semi-annual discount rate.
It may seem surprising that the number of periods is double the number of years to maturity.
The important factor is to allow for the same number of interest periods as coupon bonds of
the same currency. That is, even though there are no actual coupons, we calculate prices and
yields on the basis of a quasi-coupon period. In computing the value of a zero-coupon bond,
the number of 6-month periods (i.e., “no. of years ×2”) is used in the denominator of the
18
formula. The rationale is that the pricing of a zero-coupon bond should be consistent with the
pricing of a semi-annual coupon bond. We have to note carefully the quasi-coupon periods in
order to maintain consistency with conventional bond pricing. Therefore, the use of 6-month
periods is required in order to have uniformity between the present value calculations.
To illustrate the application of the formula, the value of a 5-year zero-coupon bond with a
maturity value of $100 discounted at an 8% interest rate is $67.5564, as shown below:
B. Valuing Perpetual bond
There also exist perpetual or irredeemable bonds which have no redemption date, so that
interest on them is paid indefinitely. They are also known as undated bonds or consols. An
example of an undated bond is the 3½% War Loan, a gilt originally issued in 1916. Most
undated bonds date from a long time in the past and it is unusual to see them issued today. In
structure the cash flow from an undated bond can be viewed as a continuous annuity. The fair
price of such a bond is calculated by setting T = ∞, such that:
19
Lesson 3: Valuing a Bond between Coupon Payments
Section 1: Understanding the concept of Accrued Interest
Our discussion of bond pricing up to now has ignored coupon interest. All bonds (except zerocoupon bonds) accrue interest on a daily basis, and this is then paid out on the coupon date.
In most major bond markets, the convention is to quote price as a clean price (i.e. without
accrued) but settle on a dirty basis (i.e. with accrued). The clean price is the price of the bond
as given by the net present value of its cash flows, but excluding coupon interest that has
accrued on the bond since the last coupon payment. As all bonds accrue interest on a daily
basis, even if a bond is held for only one day, interest will have been earned by the bondholder.
When a bond is bought or sold midway through a coupon period, a certain amount of coupon
interest will have accrued. The coupon payment is always received by the person holding the
bond at the time of the coupon payment (as the bond will then be registered in his name).
Because he may not have held the bond throughout the coupon period, he will need to pay the
previous holder some ‘compensation’ for the amount of interest which accrued during his
ownership.
The interest earned by the seller is the interest that has accrued between the last coupon
payment date and the settlement 1 date. This interest is called accrued interest. Accrued means
that the interest is earned but not distributed to the bondholder. At the time of purchase, the
buyer must compensate the seller for the accrued interest. The buyer recovers the accrued
interest when the next coupon payment is received.
1
The settlement date for a bond traded in the secondary market is the day that the buyer transfers payment to the seller of the bond and when
the seller transfers the bond to the buyer. The term value date is sometimes used in place of settlement date, however the two te rms are not
strictly synonymous. A settlement date can only fall on a business date, so that a bond traded on a Friday will settle on a Monday. However a
value date can sometimes fall on a non-business day, for example when accrued interest is being calculated.
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Let us illustrate this in Figure 3:
Figure 3: Accrued interest
If a bond is bought at t = 0 and held until maturity, t = N, the buyer receives all the coupon
payments between t = 0 and t = N. However, bonds may be traded at any time before maturity,
and should the transaction date fall between two coupon payments, the new buyer will receive
the full interest payment at the next coupon date.
The upper graph shows a coupon period split into two fractions by the settlement date, w, and
1−w, with 0 < w ≤ 1. 𝑤 is equal to the ratio of the number of days between the settlement date
and the next coupon date to the total number of days in the coupon period. The new owner
receives the totality of the interest payment at time t=Next coupon, but is only entitled to the
interest compounded over w-th of the coupon period. The lower graph shows that interest
accrues daily until the coupon payment date, e.g., at times t = 1 and t=Next coupon, when it is
then paid in full.
When the price of a bond is computed using the present value calculations described earlier, it
is computed with accrued interest embodied in the price. This price is referred to as the full
price or dirty price. It is the full price that the buyer pays the seller. From the full price, the
accrued interest must be deducted to determine the price of the bond, sometimes referred to as
the clean price.
Below, we show how the present value formula is modified to compute the full price when a
bond is purchased between coupon periods.
Section 2: Computing the Full Price
To compute the full price, it is first necessary to determine the fractional periods between the
settlement date and the next coupon payment date. This is determined as follows:
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Then the present value of the expected cash flow to be received 𝑡 periods from now using a
discount rate 𝑖 assuming the first coupon payment is 𝑤 periods from now is:
To illustrate the calculation, suppose that there are five semi-annual coupon payments
remaining for a 10% coupon bond. Also assume the following:
i). 78 days between the settlement date and the next coupon payment date
ii). 182 days in the coupon period
Then 𝑤 is 0.4286 periods (= 78/182). The present value of each cash flow assuming that each
is discounted at 8% annual discount rate is:
The full price is the sum of the present value of the cash flows, which is $106.8192. Remember
that the full price includes the accrued interest that the buyer is paying the seller.
Section 3: Computing the Accrued Interest and the Clean Price
To find the price without accrued interest, called the clean price or simply price, the accrued
interest must be computed. To determine the accrued interest, it is first necessary to determine
the number of days in the accrued interest period. The number of days in the accrued interest
period is determined as follows:
The percentage of the next semi-annual coupon payment that the seller has earned as accrued
interest is found as follows:
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So, for example, returning to our illustration where the full price was computed, since there are
182 days in the coupon period and there are 78 days from the settlement date to the next coupon
payment, the days in the accrued interest period is 182 minus 78, or 104 days. Therefore, the
percentage of the coupon payment that is accrued interest is:
This is the same percentage found by simply subtracting 𝑤 from 1. In our illustration, 𝑤 was
0.4286, and then 1 − 0.4286 = 0.5714.
Given the value of w, the amount of accrued interest (AI) is equal to:
So, for the 10% coupon bond whose full price we computed, since the semi-annual coupon
payment per $100 of par value is $5 and 𝑤 is 0.4286, the accrued interest is:
The clean price is then:
In our illustration, the clean price 2 is:
Exercise
Suppose that a bond is purchased between coupon periods. The days between the settlement
date and the next coupon period is 115. There are 183 days in the coupon period. Suppose
that the bond purchased has a coupon rate of 7.4% and there are 10 semiannual coupon
payments remaining.
a. What is the dirty price for this bond if a 5.6% discount rate is used?
b. What is the accrued interest for this bond?
c. What is the clean price?
2
Notice that in computing the full price the present value of the next coupon payment is computed. However, the buyer pays
the seller the accrued interest now despite the fact that it will be recovered at the next coupon payment date.
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Lesson 4: Bond price and yield relationship
Section 1: Sources of return
A. Sources of Return
When an investor purchases a bond, s/he can expect to receive a dollar return from one or more
of the following sources:
a) the coupon interest payments made by the issuer
b) any capital gain (or capital loss) when the security matures, or is sold
c) income from reinvestment of interim cash flows (interest and/or principal payments
prior to stated maturity)
a) Coupon Interest Payments
The most obvious source of return on a bond is the periodic coupon interest payments. For
zero-coupon instruments, the return from this source is zero. By purchasing a security below
its par value and receiving the full par value at maturity, the investor in a zero-coupon
instrument is effectively receiving interest in a lump sum.
b) Capital Gain or Loss
An investor receives cash when a bond matures, or is sold before maturity. For a bond held to
maturity, there will be a capital gain if the bond is purchased below its par value. For example,
a bond with a par value of $100 but purchased for $94.17, will generate a capital gain of $5.83
($100 - $94.17) if held to maturity. Similarly, a capital loss is generated when the proceeds
received are less than the purchase price. For a bond held to maturity, there will be a capital
loss if the bond is purchased for more than its par value (i.e., purchased at a premium).
c) Reinvestment Income
With the exception of zero-coupon bonds, most bonds make periodic interest payments that
can be reinvested. The interest earned from reinvesting the interim cash flows (interest
payments) prior to final maturity is called reinvestment income.
Section 2: Yield Measures
A. Nominal Yield
Nominal yield, or the coupon rate, is the stated interest rate of the bond. This yield percentage
is the percentage of par value. It is calculated as:
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑌𝑖𝑒𝑙𝑑 =
𝐴𝑛𝑛𝑢𝑎𝑙 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃𝑎𝑦𝑚𝑒𝑛𝑡
𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒
B. Current yield
Because bonds trade in the secondary market, they may sell for less or more than par value,
which will yield an interest rate that is different from the nominal yield, called the current
yield, or current return.
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The simplest measure of yield on a bond is current yield, also known as flat, interest or
running yield. The current yield relates the annual dollar coupon interest to a bond’s market
price. The formula for the current yield is:
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑 =
𝐴𝑛𝑛𝑢𝑎𝑙 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑃𝑎𝑦𝑚𝑒𝑛𝑡
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑀𝑎𝑟𝑘𝑒𝑡 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐵𝑜𝑛𝑑
The current yield for an 8-year bond paying a coupon of 7% and whose price is $94.17 is:
𝐴𝑛𝑛𝑢𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 0.07 𝑥 $100 = $7
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑 =
$7
= 0.0743 𝑜𝑟 7.43%
94.17
Bonds selling for less than par value are said to be selling at a discount. If the market interest
rate of a new bond issue is lower than what you are getting, then you will be able to sell your
bond for more than par value, you will be selling your bond at a premium. For a bond selling
at par, the current yield will be equal to the coupon rate.
The drawback of current yield is that it considers only the coupon interest and no other source
for an investor’s return. No consideration is given to the capital gain an investor will realize
when a bond purchased at a discount is held to maturity; nor is there any recognition of the
capital loss an investor will realize if a bond purchased at a premium is held to maturity. No
consideration is given to reinvestment income.
C. Yield to Maturity
The yield to maturity (YTM) or gross redemption yield is the total return earned if a bond is
held until maturity. It takes into account all coupon payments plus any capital gain or loss
realized during the life of the bond. It is equivalent to the discount rate or internal rate of return
on the bond: the rate that equates discounted cash flows on the bond to current price. The
formula for YTM is essentially that for calculating the price of a bond. For a bond paying
annual coupons, the YTM is calculated by solving the following equation:
Note that the number of interest periods in an annual-coupon bond are equal to the number of
years to maturity, and so for these bonds n is equal to the number of years to maturity. We can
simplify the equation using the summation sign, Σ:
25
Note that this expression has two variable parameters, the price, 𝑃𝑑 and yield, 𝑟𝑚. It cannot be
rearranged to solve for yield 𝑟𝑚 explicitly and must be solved using numerical iteration. The
process involves estimating a value for 𝑟𝑚 and calculating the price associated with the
estimated yield. If the calculated price is higher (lower) than the price of the bond at the time,
the yield estimate is lower (higher) than the actual yield, and so it must be adjusted until it
converges to the level that corresponds with the bond price.
To calculate YTM for a semi-annual coupon bond, adjustment to the formula is made to allow
for the semi-annual payments as:
where 𝑛 is the number of interest periods in the life of the bond and therefore equal to the
number of years to maturity multiplied by 2.
When a bond is bought at a discount, YTM will always be greater than current yield because
there will be a gain when the par value is received. When a bond is bought at a premium, YTM
will always be less than current yield because there will be a loss when par value is received.
Section 3: Bond price and yield relationship
A fundamental principle of bond pricing is that market interest rates and bond prices move in
opposite directions. When market interest rates rise, prices of fixed-rate bonds fall, and this is
reflected in the equation below.
Let us see the intuition behind this relationship.
Suppose you bought an 8% coupon bond at issue for $1,000. A year later, you want to sell the
bond but interest rates have risen to 10%. You will have to sell the bond for less than the price
you bought is for, that is, for less than $1,000. Why? Because no rational investor would pay
$1,000 for a bond that pays 8%, when they can buy a similar bond with an equal credit rating
and earn 10%. The bond will have to be priced such that the $80 the new buyer will receive
per year is 10% of buying price, and in this case, $800 ($200 less than what you paid).
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Lesson 5: Pricing a Treasury bill
Section 1: Understanding Treasury bills
Treasury bills, also called T-bills, are discounted short-term debt securities with maturities of
up to one year, and is one of the most common money market instruments. For a government,
the issue of Treasury bills is a way to cover short-term state budget deficits. Moreover, bills
represent an important instrument of governmental fiscal policy and the central bank's
monetary policy.
Treasury bills are issued at a discount to par value, have no coupon rate, mature at par value,
and have a maturity date of less than 12 months. As discount securities, Treasury bills do not
pay coupon interest; the return to the investor is the difference between the maturity value and
the purchase price. The most common bills are three-month (91-day) maturity instruments,
although in theory any maturity between one-month and twelve-month may be selected.
In debt capital markets the yield on a domestic government T-bill is usually considered to
represent the risk-free interest rate, since it is a short-term instrument guaranteed by the
government. This makes the T-bill rate, in theory at least, the most secure and liquid investment
in the market.
Treasury bills are quoted differently in the market as they are money market instruments. They
are discount instruments and so the price quote is on a discount basis. Different markets use
different day count conventions in calculating Treasury-bill yields. The US money markets
assume a 360-day count basis, similar to the Euro money markets but different to UK money
markets, which use a 365-day basis. In most SSA markets (Namibia, Tanzania, Zambia,
Zimbabwe, Uganda and Kenya), the 365 convention is used.
Section 2: Pricing Treasury bills
The convention in the Treasury bill market is to calculate a bill’s yield on a discount basis. This
is the amount of discount expressed as an annualised percentage of the face value, and not as a
percentage of the original amount paid. This yield is determined by two variables:


settlement price per maturity value
number of days to maturity (calculated as the number of days between the settlement
date and the maturity date:
The yield on a discount basis (denoted by 𝑟𝑑 ) is calculated as follows:
𝑟𝒅 =
𝑫𝒅 365
𝑥
𝑴
𝑛
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Example: Treasury bill discount rate
A $100 Treasury bill with 91 days to maturity is offered for sale at $98.69. What is the
discount rate?
𝑫𝒅 is $100 minus $98.95, which is $1.05. Therefore, the discount rate is:
𝒓𝒅 =
𝟏. 𝟎𝟓 𝟑𝟔𝟓
𝒙
= 𝟎. 𝟎𝟒𝟐
𝟏𝟎𝟎
𝟗𝟏
The discount rate is therefore 4.21%.
As T-bills are quoted on a discount basis, we need to calculate the actual price payable for a
bill traded in the secondary market. This is done using the following equation:
𝐷𝒅 = 𝑟𝒅 𝑥 𝑀 𝑥
𝑛
.
365
We then use 𝑀 − 𝐷𝒅 to calculate the price of the bill. Alternatively, the price can be
calculated as:
𝑃=
𝑀
𝑛
(1 + 𝑟 ∗ (365))
Example: Treasury bill price
 A 91-day $100 Treasury bill is issued with a yield of 4.75%. What is its issue price?
𝑷=
𝟏𝟎𝟎
𝟗𝟏
(𝟏 + 𝟎. 𝟎𝟒𝟕𝟓 ∗ (𝟑𝟔𝟓))
= $𝟗𝟖. 𝟖𝟎
 A T-bill with $10 million face value issued for 91 days will be redeemed on maturity at
$10 million. If the 3-month yield at the time of issue is 5.25%, the price of the bill at
issue is:
𝑷=
𝟏𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎
= $𝟗, 𝟖𝟕𝟎, 𝟖𝟎𝟎. 𝟔𝟗
𝟗𝟏
(𝟏 + 𝟎. 𝟎𝟓𝟐𝟓 ∗ (𝟑𝟔𝟓))
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References
1.
Choudhry, M (2001): The Bond and Money Markets - Strategy, Trading, Analysis.
Butterworth-Heinemann.
2.
Fabozzi, F (2007): Fixed Income Analysis. John Wiley & Sons, Inc.
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Glossary of Terms
Term to maturity
Term to maturity is the number of years (or days) the bond is outstanding or the number of
years remaining prior to final principal repayment. The maturity date refers to the date that the
bond will cease to exist, at which time the issuer will redeem the bond by paying the
outstanding balance. The practice in the bond market is to refer to the “term to maturity” of a
bond as simply its “maturity”, “term”, or “tenor”.
Par value
The principal of a bond is the amount that the issuer agrees to repay the bondholder on the
maturity date. This amount is also referred to as the redemption value, maturity value, par
value, nominal value or face amount, or simply par.
Bonds can have any par value, and the practice is to quote the price of a bond as a percentage
of its par value. A value of “100” means 100% of par value. So, if a bond has a par value of
$1,000 and is selling for $900, this bond would be said to be selling at 90. If a bond with a par
value of $5,000 is selling for $5,500, the bond is said to be selling for 110.
Coupon rate
The coupon rate, also called the nominal rate, is the interest rate that the issuer agrees to pay
each year. The annual amount of the interest payment made to bondholders during the term of
the bond is called the coupon. The coupon is determined by multiplying the coupon rate by the
par value of the bond. That is,
𝑪𝒐𝒖𝒑𝒐𝒏 = 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑥 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒.
For example, a bond with an 8% coupon rate and a par value of $1,000 will pay annual
interest of $80 (= $1, 000 × 0.08).
Bond cash flows
If a bond pays a fixed rate of interest over a fixed period of time, it becomes a collection of
cash flows, and this is illustrated below.
Cash flows associated with a six-year annual coupon bond
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In our hypothetical example, the bond is a 6-year issue that pays fixed interest payments of C%
of the nominal value on an annual basis. In the 6th year, there is a final interest payment and
the proceeds represented by the bond are also paid back, known as the maturity proceeds. The
amount raised by the bond issuer is a function of the price of the bond at issue, which we have
labelled here as the issue proceeds.
The upward facing arrow represents the cash flow paid and the downward facing arrows are
the cash flows received by the bond investor. The cash flow diagram for a 6-year bond that had
a 5% fixed interest rate, known as a 5% coupon, would show interest payments of $5 per every
$100 of bonds, with a final payment of $105 in the 6th year, representing the last coupon
payment and the redemption payment.
Again, the amount of funds raised per $100 of bonds depends on the price of the bond on the
day it is first issued, and we will look into this later. If our example bond pays coupons on a
semi-annual basis, the cash flows would be $2.50 every six months until the final redemption
payment of $102.50.
Day count convention
Day count convention determines how interest accrues over time. It determines the number of
days between two dates in a coupon (accrual) period and the number of days in the year (Day
count basis). There are four main day-count conventions.
The 30/360 convention assumes that there are 30 days each month and 360 days in a year. On
the other hand, the actual/actual convention uses the real number of days each month and year,
rather than assuming that each month and year is made up of 30 days and 360 days respectively.
Actual/360 is calculated by using the actual number of days between the two periods, divided
by 360. Actual/365 is most commonly used when pricing Treasury bonds. These conventions
are standards that have developed over time and help to ensure that everyone is on the same
playing field when a bond is sold between coupon dates. Different markets use different
counting conventions.
Zero-Coupon Bonds
These are bonds that do not make periodic coupon payments. The holder of a zero-coupon bond
realizes interest by buying the bond below par value (i.e., buying the bond at a discount).
Interest is then paid at maturity, calculated as the difference between the par value and price
paid for the bond. If $100 par value bond is sold for $70, the interest is $30, which is the
difference between the par value ($100) and the price paid ($70).
Step-Up bonds
There are securities that have a coupon rate that increases over time. These securities are called
step-up bonds because the coupon rate ‘‘steps up’’ over time. For example, a 5-year step-up
bond might have a coupon rate that is 5% for the first two years and 6% for the last three years.
Deferred Coupon Bonds
There are bonds whose interest payments are deferred for a specified number of years. That is,
there are no interest payments during the deferred period. At the end of the deferred period, the
issuer makes periodic interest payments until the bond matures.
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Floating-Rate Securities
Floating-rate bonds, sometimes called variable-rate bonds, have coupon payments that reset
periodically according to some reference rate. The typical formula (called the coupon formula)
on certain determination dates when the coupon rate is reset is as follows:
𝑪𝒐𝒖𝒑𝒐𝒏 𝒓𝒂𝒕𝒆 = 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑟𝑎𝑡𝑒 + 𝑞𝑢𝑜𝑡𝑒𝑑 𝑚𝑎𝑟𝑔𝑖𝑛
The quoted margin is the additional amount that the issuer agrees to pay above the reference
rate. For example, suppose that the reference rate is the 1-month LIBOR. Suppose that the
quoted margin is 100 basis points. Then the coupon formula is:
𝑪𝒐𝒖𝒑𝒐𝒏 𝒓𝒂𝒕𝒆 = 1𝑚𝑜𝑛𝑡ℎ 𝐿𝐼𝐵𝑂𝑅 + 100 𝑏𝑎𝑠𝑖𝑠 𝑝𝑜𝑖𝑛𝑡𝑠
Basis points
In the fixed income market, market participants refer to changes in interest rates or differences
in interest rates in terms of basis points. A basis point is defined as 0.0001, or equivalently,
0.01%. Consequently, 100 basis points are equal to 1%. A change in interest rates from, say,
5.0% to 6.2% means that there is a 1.2% change in rates or 120 basis points.
Accrued Interest
Bond issuers do not disburse coupon interest payments every day. Instead, coupon interest is
typically paid every six months. In some countries, interest is paid annually. The coupon
payment is made to the bondholder of record. Thus, if an investor sells a bond between coupon
payments and the buyer holds it until the next coupon payment, then the entire coupon interest
earned for the period will be paid to the buyer of the bond since the buyer will be the holder of
record. The seller of the bond gives up the interest from the time of the last coupon payment to
the time until the bond is sold. The amount of interest over this period that will be received by
the buyer even though it was earned by the seller is called accrued interest.
The bond buyer must pay the bond seller the accrued interest. The amount that the buyer pays
the seller is the agreed upon price for the bond plus accrued interest. This amount is called the
full price or dirty price. The agreed upon bond price without accrued interest is simply referred
to as the price or clean price.
Provisions of paying off the bond
The issuer of a bond agrees to pay the principal by the stated maturity date. The issuer can
agree to pay the entire amount borrowed in one lump sum payment at the maturity date. That
is, the issuer is not required to make any principal repayments prior to the maturity date. Such
bonds are said to have a bullet maturity.
Some bonds have a schedule of partial principal payments. Such bonds are said to be amortizing
bonds. Another example of an amortizing feature is a bond that has a sinking fund provision.
32