CANADIAN CONVENTIONS
IN FIXED INCOME MARKETS
A REFERENCE DOCUMENT OF FIXED INCOME SECURITIES
F
ORMULAS AND PRACTICES
Release: 1.3
www.iiac.ca
INVESTMENT INDUSTRY ASSOCIATION OF CANADA
1
FORWARD
This document illustrates conventions and formulas currently used by market
participants for the calculation of prices, interest payments and yields on securities
traded in the Canadian fixed income market. The document has been prepared by a
working group of fixed income industry professionals under the auspices of the
Investment Industry Association of Canada (formerly the Industry Relations and
Representation division of the Investment Dealers Association of Canada (IDA)).
The objective of this document is to provide market participants with a single
comprehensive guide to market conventions commonly used in Canada’s fixed income
market. It will also serve in identifying some of the important ways in which Canadian
conventions differ from those of other jurisdictions. In most cases, the document
presents existing de facto standards. It is not the intent of this document to introduce
new conventions or debate the completeness of those already in practice.
The document consists of two parts. Part I provides a description of the securities
together with brief explanations of specific conventions and elements of the valuation
formulas. Part II constitutes a quick reference guide that presents the valuation formulas
on a case by case basis.
In time we hope this guide facilitates the continued use of uniform practices among
Canadian market participants, and further enhances the efficiency and attractiveness of
Canada’s capital markets.
If you would like to share any comments you may have on the content of this document,
please e-mail them to [email protected].
Acknowledgement
The Investment Industry Association of Canada would like to thank all those professionals who
contributed to the making of this document. A special thank you to the following individuals;
Kenneth Kelly, CFA Daniel Kelly, CFA Jack Rando, CFA
CIBC World Markets RBC Capital Markets Investment Industry Association of Canada.
2
TABLE OF CONTENTS
PART I
1. INSTRUMENTS...........................................................................................................................................4
1.1 Semi-Annual Pay Bonds...................................................................................................................4
1.2 Bonds with Other Payment Frequencies..........................................................................................4
1.3 Amortizing Securities........................................................................................................................4
1.4 Money Market Discount Notes .........................................................................................................4
1.5 Inflation-Linked Bonds ......................................................................................................................5
2. DAY-COUNT CONVENTIONS ...................................................................................................................6
2.1 Day-Count Conventions for Money Market Securities – Actual/365 ................................................6
2.2 Day-Count Conventions for Bonds...................................................................................................6
2.2.1 The Actual/Actual Day-Count Convention..............................................................................6
2.2.2 The Actual/365 (Canadian Bond) Day-Count Convention .....................................................7
3. GENERAL CONSIDERATIONS .................................................................................................................9
3.1 Comparison with U.S. Market Conventions......................................................................................9
3.2 Payment Frequency Versus Compounding Frequency ...................................................................9
3.3 Days to Include When Accruing Interest ........................................................................................10
3.4 Precision, Truncation and Rounding Conventions .........................................................................10
4. CALCULATING PAYMENT AMOUNTS FOR ODD COUPON PERIODS USING ACTUAL/ACTUAL ..11
4.1 Quasi-Coupon Periods ...................................................................................................................11
4.2 Short First Coupon .........................................................................................................................12
4.3 Short Last Coupon..........................................................................................................................12
4.4 Long First Coupon ..........................................................................................................................13
4.5 Calculating Payment Amounts for Odd Coupon Periods Using Actual/365...................................13
4.5.1 Short First Coupons Calculated Using Actual/365 ...............................................................14
4.5.2 Long First Coupons Calculated Using Actual/365................................................................15
4.5.3 Short Last Coupons Calculated Using Actual/365……………………………………………...16
5. ACTUAL/ACTUAL ACCRUED INTEREST..............................................................................................18
5.1 Regular Coupon Periods ................................................................................................................18
5.2 Short First Coupon Periods ............................................................................................................18
5.3 Short Last Coupon Periods ............................................................................................................18
5.4 Long First Coupon Periods.............................................................................................................19
6. SETTLEMENT ACCRUED INTEREST – ACTUAL/365...........................................................................20
6.1 Regular Coupon Periods ................................................................................................................20
6.2 Short First or Last Coupon Period..................................................................................................21
6.3 Long Coupon Period.......................................................................................................................21
7. DISCOUNTING FOR PARTIAL COUPON PERIODS – ACTUAL/ACTUAL EXPONENTS....................22
7.1 Regular Coupon Periods ................................................................................................................22
7.2 Short First Coupon Periods ............................................................................................................22
7.3 Long First Coupon Periods.............................................................................................................22
7.4 Short Last Coupon Periods ............................................................................................................23
8. AMORTIZING BONDS..............................................................................................................................24
8.1 Trading Convention ........................................................................................................................24
8.2 Representing Principal Amounts ....................................................................................................24
8.3 Principles Underlying Price-Yield Formula .....................................................................................24
9. REAL RETURN BONDS...........................................................................................................................25
9.1 Inflation Indexing Process ..............................................................................................................25
9.2 Rounding Convention for the Indexing Process.............................................................................26
9.3 Real Price Given Real Yield to Maturity .........................................................................................26
9.4 Real Accrued Interest .....................................................................................................................26
9.5 Settlement Amounts for Real Return Bonds: Nominal Price and Accrued Interest .......................26
9.6 CPI Re-basing ................................................................................................................................26
3
PART II
10. REFERENCE FORMULAS.......................................................................................................................28
10.1 Semi-annual Pay Bond – Regular Coupon Periods .......................................................................29
10.1.1 Price Given Yield to Maturity .............................................................................................29
10.1.2 Settlement Accrued Interest ..............................................................................................30
10.2 Semi-annual Pay Bond – Short First Coupon ................................................................................31
10.2.1 Price Given Yield to Maturity .............................................................................................31
10.2.2 Settlement Accrued Interest ..............................................................................................32
10.3 Semi-annual Pay Bond – Long First Coupon, Settlement in First Quasi-coupon Period...............33
10.3.1 Price Given Yield to Maturity .............................................................................................33
10.3.2 Settlement Accrued Interest ..............................................................................................34
10.4 Semi-annual Pay Bond – Long First Coupon, Settlement in Second Quasi-coupon Period .........35
10.4.1 Price Given Yield to Maturity .............................................................................................35
10.4.2 Settlement Accrued Interest ..............................................................................................36
10.5 Semi-annual Pay Bond – Short Last Coupon, Regular First Coupon ............................................37
10.5.1 Price Given Yield to Maturity .............................................................................................37
10.5.2 Settlement Accrued Interest ..............................................................................................38
10.6 Semi-annual Pay Bond – Short Last Coupon, Short First Coupon ................................................39
10.6.1 Price Given Yield to Maturity .............................................................................................39
10.6.2 Settlement Accrued Interest ..............................................................................................40
10.7 Monthly Pay Bond – Regular Coupon Periods...............................................................................41
10.7.1 Price Given Yield to Maturity .............................................................................................41
10.7.2 Settlement Accrued Interest ..............................................................................................42
10.8 Semi-annual Pay Amortizing Bond – Regular Interest Periods......................................................43
10.8.1 Price Given Yield to Maturity .............................................................................................43
10.8.2 Settlement Accrued Interest ..............................................................................................44
10.9 Semi-annual Pay Amortizing Bond – Short First Interest Period ...................................................45
10.9.1 Price Given Yield to Maturity .............................................................................................45
10.9.2 Settlement Accrued Interest ..............................................................................................46
10.10 Money Market Yields ......................................................................................................................47
10.10.1 Semi-annual Bonds: One Cash Flow Remaining..............................................................47
10.10.2 Semi-annual Bonds: Two Cash Flows Remaining ............................................................48
10.10.3 Monthly and Quarterly Pay Bonds: Money Market Equivalent Yield.................................49
10.10.4 Price and Yield Calculations for Money Market Discount Notes .......................................50
APPENDICES AND REFERENCES
APPENDIX 1: BOND VERSUS SWAP MARKET CONVENTIONS ........................................................50
APPENDIX 2: ROUNDING AND TRUNCATION PRACTICES ...............................................................53
APPENDIX 3: MNEMONICS ....................................................................................................................54
REFERENCES..........................................................................................................................................55
Revisions Summary: The following revisions have been incorporated into releases 1.1 -1.3
1) Section 4.5.1 Short First Coupons Calculated Using Actual/365 – revisions to formula, example added.
2) Section 4.5.2 Long First Coupons Calculated Using Actual/365 – revisions to formula, example added.
3) Section 4.5.3 Short Last Coupons Calculated Using Actual/365 – this is a newly added section
4) Section 6.1 Settlement Accrued Interest Using Actual/365 – revisions to release 1.2 formula
4
1. INSTRUMENTS
This document examines the following security types:
¾ Fixed coupon bonds with semi-annual-, quarterly-, and monthly-pay frequencies
¾ Amortizing securities
¾ Money market discount notes
¾ Inflation-linked real return bonds.
1.1 Semi-Annual Pay Bonds
Semi-annual pay bonds form the overwhelming standard for borrowing in the Canadian
capital market. Bonds generally pay a fixed annual coupon rate of interest in two equal
semi-annual payments. Most bonds have a fixed maturity date.
1.2 Bonds with Other Payment Frequencies
The main types of securities that have been issued in Canada with monthly or quarterly
payment frequencies are mortgage-backed securities (monthly) and floating-rate notes
(typically quarterly), neither of which are covered in this document. Mortgage-backed
security (MBS) conventions have been documented elsewhere (see References section),
while floating rate notes (FRNs) are covered by International Swap and Derivatives
Association (ISDA) rules.
A limited number of monthly- and quarterly-pay fixed rate bonds have been issued in
Canada. Formulas for valuing such securities are presented.
1.3 Amortizing Securities
The types of amortizing securities covered in this document have fixed payment and
amortization schedules. This category, for example, includes infrastructure bonds.
To date, a range of different conventions have been used for valuing amortizing bonds in
Canada. Due to varying practices amongst market participants, this document presents a
single formula for valuation that is based on the same conventions embodied in the
valuation formula for non-amortizing Canadian bonds.
From an investor’s perspective, the cash flow structure of bonds with mandatory sinking
funds is identical to the cash flow structure of an amortizing security. Accordingly, the
formulas for valuing amortizing bonds are also appropriate for valuing bonds with
mandatory sinking funds.
1.4 Money Market Discount Notes
Several types of discount note are traded in Canada. These include Government of
Canada treasury bills, bankers’ acceptances, bearer deposit notes and commercial paper.
All these securities are issued at prices that are discounted from their principal values using
a simple interest rate and the actual/365-day-count convention.
5
1.5 Inflation Linked Bonds
Inflation-linked bonds, known as real return bonds (RRBs), have been issued by the federal
government in Canada since 1991. Certain provincial governments and other issuers have
also begun issuing inflation linked bonds in more recent years. Most of these bonds issued
in Canada are semi-annual-pay securities that do not repay any principal until the final
maturity date. These securities differ from other semi-annual bonds, in that their interest
and principal cash flows are linked to the Canadian Consumer Price Index (CPI), as
published by Statistics Canada. There have been a few inflation-linked bonds issued in
Canada that follow other payment schedules (e.g., amortizers), but these are not covered
in this document.
6
2. DAY-COUNT CONVENTIONS
Day-count conventions determine the method in which the days within an interest payment
period are counted.
2.1 Day-Count Conventions for Money Market Securities – Actual/365
Money market instruments accrue interest over periods less than one year. In the Canadian
market, the fraction of a year over which interest accrues is calculated using the actual/365-
day-count convention, also known as Act/365 day-count basis.
The actual/365-day-count basis considers a year to have 365 days. It does not account for
leap years. The fraction of a year represented by any given time period is represented as
the actual number of days in the period divided by 365.
Example:
Valuation Date: December 1, 2005
Maturity Date: March 15, 2006
Fraction of a Year = [March 15, 2006 - December 1, 2005]/365
= 104/365
= 0.284931506849315
2.2 Day-Count Conventions for Bonds
Day-count conventions are used when valuing bonds between coupon payment dates.
Since consecutive interest payment periods on bonds do not contain equal numbers of
days, bond day-count conventions measure the period of time between two dates as a
fraction of a coupon period, instead of as a fraction of a year. Two day-count convention
are used in the Canadian market.
2.2.1 The Actual/Actual-Day-Count Convention
The actual/actual day-count basis considers the number of days between any two dates to
be the actual number of calendar days between the dates
1
.
The fraction of a coupon period remaining following a given settlement date is set equal to
the actual number of days remaining in the coupon period divided by the actual total
number of days in the full coupon period. Similarly, the fraction of a coupon period that has
already passed as of a given settlement date is equal to the actual number of days from the
last coupon payment date to the settlement date divided by the actual total number of days
in the period.
1
Historically, this day-count basis has been referred to simply as the “actual/actual” basis. However, with the advent of
the International Swaps and Derivatives Association (ISDA) version of the actual/actual basis, it has become necessary
to distinguish between the two conventions. References to the actual/actual convention in this document refer to the
bond market actual/actual convention, the norm for U.S. Treasuries, U.K. Gilts, most European sovereigns and
Eurobonds of many currencies, unless otherwise noted. For additional information on differences between bond and
swap market conventions, refer to Appendix 1.
7
Example:
Last Coupon Date: December 1, 2005
Next Coupon Date: June 1, 2006
Valuation Date: March 15, 2006
Total Days in Full Coupon Period (December 1 – June 1) = 182
Fraction of Coupon Period Remaining = Actual Days Remaining/Actual Total Days in
Period
= [June 1, 2006 - March 15, 2006]/182
= 78/182
= 0.4285714285714
Fraction of Coupon Period Elapsed = Actual Days from Last Payment/Actual Total
Days in Period
= [March 15, 2006 - December 1, 2005]/182
= 104/182
= 0.5714285714286
2.2.2 The Actual/365 (Canadian Bond) Day-Count Convention
The actual/365 (Canadian Bond) day-count convention considers a year to have 365 days,
while the length of a coupon period is represented by 365 divided by the number of coupon
periods in a year. For the most common Canadian domestic bond structure, which pays a
semi-annual coupon, this implies the length of a coupon period is 365/2 = 182.5 days.
Denoting the annual payment frequency (or number of coupon periods per year) as ‘f’,
Act/365 (Canadian Bond) measures the fraction of a coupon period represented by a given
number of days as follows:
(i) If the number of days of interest accrual is less than the actual number of days in the
coupon period:
Fraction of Coupon Period
365
Days f⋅
=
Which, for semi-annual pay bonds where
2f
=
, reduces to:
Fraction of Coupon Period
182.5
Days
=
(ii) If the number of days of interest accrual exceeds 365/f, or 182.5 days for a semi-annual
pay bond:
Fraction of Coupon Period 1
365
DaysRemainingInPeriod f
⋅
⎡⎤
=−
⎢⎥
⎣⎦
Where
is the actual number of days from Valuation Date to Next Coupon DateDaysRemainingInPeriod
8
Example of case (i): Number of days of interest accrual is less than the actual number of
days in the coupon period:
Last Coupon Date: August 1, 2006
Next Coupon Date: February 1, 2007
Valuation Date: September 15, 2006
Fraction of a Coupon Period = [September 15, 2006 - August 1, 2006] • 2/365
= 45 • 2/365
= 0.246575342
Example of case (ii): The number of days of interest accrual exceeds 365/f:
Last Coupon Date: August 1, 2006
Next Coupon Date: February 1, 2007
Valuation Date: January 31, 2007
Fraction of a Coupon Period = 1 - [February 1, 2007 - January 31, 2007] • 2/365
= 1 - 2/365
= 0.994520548
9
3. GENERAL CONSIDERATIONS
3.1 Comparison with U.S. Market Conventions
Government Bonds: The market convention for calculating the price excluding accrued
interest on Canadian government bonds is identical to the formula used for pricing U.S.
Treasury bonds
2
. This convention extends to bonds with short or long first coupons and to
bonds with short last coupons. However, Canadian market conventions differ from U.S.
Treasury market conventions in two ways:
1. For trade settlement purposes, accrued interest is calculated using the Act/365
(Canadian Bond) formula, and
2. The amount of interest payable to investors for short or long coupon periods is
calculated using the Act/365 (Canadian Bond) convention.
Corporate Bonds: The conventions for corporate bonds in Canada generally follow those
for Canadian government bonds, although in isolated cases the interest payable for odd
coupon periods may be calculated using some convention other than Act/365 (Canadian
Bond). This is in contrast to the U.S., where corporate bonds follow a different convention
from their federal counterparts. The bond valuation formula for U.S. corporate bonds uses
the US30/360-day-count convention
3
.
Money Market: Yields on money market securities in Canada are calculated assuming
simple interest and a 365-day year. In contrast, yields on money market securities in the
U.S. assume a 360-day year, and are discount yields, rather than yield to maturity.
Real Return Bonds: The structure and inflation-compensation mechanism of real return
bonds (RRBs) and their U.S. counterparts, Treasury Inflation-Protected Securities (TIPS),
are very similar, notwithstanding that one is linked to the Canadian CPI and the other to
U.S. CPI.
The main difference in the design of the two securities is that TIPS have a floor on the
inflation-compensation mechanism, such that in the event of deflation, the principal cannot
decline below the original issue amount. The different government accrued-interest
conventions already noted for nominal bonds apply to RRBs and TIPS as well. Settlement
accrued interest on RRBs is calculated using the actual/365 (Canadian Bond) day-count
convention.
3.2 Payment Frequency versus Compounding Frequency
There are financial contracts on which the compounding frequency of the yield to maturity
is different from the payment frequency of the annual coupon rate. For example, mortgage-
backed securities typically pay interest monthly, while yields are expressed on a semi-
annual compound basis, for ease of comparison with other domestic bonds. However, all of
2
In fact, for many years, Canadian traders used the Monroe calculator to price bonds.
3
The U.S. 30/360-day-count convention assumes that each month has 30 days and that a year has 360 days.
10
the securities covered in this document have compounding frequencies that are equal to
their annual coupon payment frequencies. Therefore, the formulas in this document use a
single mnemonic, “f”, to represent frequency, on the assumption that the compounding
frequency of the yield to maturity equals the payment frequency of coupon payments
4
.
3.3 Days to Include When Accruing Interest
As a general principle, interest payments on fixed income securities are made in arrears.
Thus, the cash payment to an investor occurs on the day following the end of the coupon
accrual period. Accordingly, when counting the number of days for the purpose of
calculating interest accruals, it is customary to include the first day of the period and to
exclude the last day of the period. This approach to counting days is also consistent with
the secondary market practice of setting settlement accrued interest equal to the “opening
accrued” for the settlement date. The purchaser therefore pays for all interest accrued up to
the last day preceding settlement, and is entitled to interest accruing on the settlement
date.
3.4 Precision, Truncation and Rounding Conventions
Conventions on truncation, or the use of decimal places, vary depending on the type of
instrument being examined and type of calculation being performed. Refer to Appendix 2
for a summary of the existing Canadian practices.
4
One obvious case where this does not apply is in the formula for money-market equivalent yield. However, in this case
it is not necessary to specify a frequency for the yield.
11
4. CALCULATING PAYMENT AMOUNTS FOR ODD COUPON
PERIODS USING ACTUAL/ACTUAL
Bonds are often issued with irregular first or last coupon payment periods, that is, coupon
periods that are shorter or longer than the norm. In such cases, the last coupon period
would generally be a short period, while an irregular first coupon period may be shorter or
longer than a regular coupon period. When calculating the price of a bond, cash flows for
such irregular coupon periods are calculated following the actual/actual convention.
4.1 Quasi-Coupon Periods
The calculation of payment amounts for irregular coupon periods makes use of the concept
of a “quasi-coupon period,” defined as follows:
• For short first coupons, a quasi-coupon period is a hypothetical full coupon period
ending on the first coupon date.
• For short last coupons, the quasi-coupon period is a hypothetical full coupon period
starting on the penultimate coupon payment date, that is, starting on the next-to-last
coupon date.
• A long first coupon period that spans a full coupon period plus a partial period will be
divided into two quasi-coupon periods. One is a full coupon period ending on the
coupon payment date, while the second is a full coupon period ending on the start
date for the later quasi-coupon period. Figure 1 below illustrates this case for a bond
with an interest accrual date of March 1, 2007 and a first payment date of December
1, 2007, such that the coupon period is nine (9) months. The interest accrual date is
the calendar date from which the security begins to accrue interest.
Figure 1: Quasi-coupon Periods for a Long First Coupon
1-Dec-06 1-Dec-07
1-Mar-07 1-Jun-07 1-Sep-07
1
st
quasi-coupon period 2nd quasi-coupon period
Interest
Accrual
Date
Coupon
Payment
Date
12
4.2 Short First Coupons
Using the actual/actual methodology and as illustrated in figure 2, a short first coupon is
calculated as follows
5
:
1
100
CPN DIC
C
f
DQ
=⋅ ⋅
Where:
C
1
= first coupon payment
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
DIC = days from interest accrual date to first payment date
DQ = days in the quasi-coupon period
See Figure 2.
Figure 2: Quasi-coupon Periods for a Short First Coupon
4.3 Short Last Coupon
Short last coupons occur on securities for which the regular coupon payment schedule
does not coincide with the maturity date. For example, a semi-annual pay bond maturing on
December 1 might pay coupons each year on October 31 and April 30. The final coupon
period would span the October 31 to December 1 period, i.e., one month.
100
M
CPN DCM
C
f
DQ
=⋅ ⋅
5
A full set of mnemonics is presented in the Appendix 3.
13
Where:
C
M
= coupon payment at maturity
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
DCM = days from penultimate coupon date to maturity date
DQ = days in the quasi-coupon period
Example: Calculating the Short Last Coupon on a 5% Semi-annual Bond Using the
Actual/Actual Method.
Maturity Date: December 1, 2007
Coupon Dates: April 30 and October 31
Last Coupon Date: October 31, 2007
Days in quasi-coupon period = April 30, 2008 – October 31, 2007 = 182 days
Days of accrual = December 1 – October 31 = 31 days
Coupon payment = 100 • (.05/2) • (31/182) = 0.425824
4.4 Long First Coupon
The formula for the long first coupon is written in a generic fashion to accommodate long
coupon periods spanning one or more regular coupon periods, plus a stub period. See
Figure 1.
1
1
100
NQ
k
k
k
D
CPN
C
f
DQ
=
=⋅ ⋅
∑
Where:
C
1
= first coupon payment
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
NQ = number of quasi-coupon periods
D
k
= days of accrual in quasi-coupon period k
DQ
k
= total days in quasi-coupon period k
4.5 Calculating Payment Amounts for Odd Coupon Periods Using Actual/365
In Canada, the current convention is to calculate interest payable for odd coupon periods
using the Actual/365 (Canadian Bond) method, despite the fact that price/yield calculations
assume the first coupon is calculated using actual/actual. This section presents formulas
that may be used for this purpose.
14
4.5.1 Short First Coupons Calculated Using Actual/365
If DIC < 365/frequency, then
1
100
365
D
IC
CCPN=⋅ ⋅
Where:
C
1
= first coupon payment
DIC = days from interest accrual date to first payment date
CPN = annual coupon rate expressed as a decimal
Otherwise (i.e., DIC > 365/frequency):
()
1
1
100
365
D
QDIC
CCPN
f
⎡⎤
−
=⋅ ⋅−
⎢⎥
⎣⎦
Where:
f = annual payment frequency
DQ = days in the quasi-coupon period ending on the first coupon date
In the above formula, the coupon amount is set equal to a full periodic coupon amount
minus a straight line accrual amount commensurate with the number of days remaining in
the period.
Example: Calculating the Coupon Payable on a 5% Semi-annual Bond with a 183 Day
Short First Coupon Period
Issue Date: July 16, 2007
First Coupon Date: January 15, 2008
Maturity Date: July 15, 2020
Days in quasi-coupon period = January 15, 2008 – July 15, 2007 = 184
DIC = January 15, 2008 – July 16, 2007 = 183
1
1
100
365
1 184 183
100 5.00% 2.486301370
2365
DQ DIC
CCPN
f
⎡⎤
−
=⋅ ⋅−
⎢⎥
⎣⎦
−
⎡⎤
=⋅ ⋅− =
⎢⎥
⎣⎦
15
4.5.2 Long First Coupons Calculated Using Actual/365
When calculating interest payable for a long interest period, the convention is to use the
Act/365 (Canadian Bond) convention for each quasi coupon period within the long coupon
period as follows:
If days from interest accrual date to the first quasi-coupon date (DIC*) < 365/frequency, then
*
1
1
100
365
DIC NQ
CCPN
f
⎡⎤
−
=⋅ ⋅ +
⎢⎥
⎣⎦
Where:
C
1
= first coupon payment
DIC* = days from interest accrual date to first quasi-coupon date
CPN = annual coupon rate expressed as a decimal
DQ = days in the first quasi-coupon period
f = payment frequency
NQ = number of quasi-coupon periods
Otherwise (i.e. DIC* > 365/frequency):
(
)
*
1
11
100
365
DQ DIC
NQ
CCPN
ff
⎡⎤
−
−
⎢⎥
=⋅ ⋅− +
⎢⎥
⎣⎦
Example: Calculating the Coupon Payable on a 5% Semi-annual Bond with a Long
First Coupon Period
Issue Date: July 16, 2007
First Coupon Date: July 15, 2008
Maturity Date: July 15, 2020
First Quasi-coupon date: January 15, 2008
Days in quasi-coupon period = January 15, 2008 – July 15, 2007 = 184
DIC = January 15, 2008 – July 16, 2007 = 183
NQ = 2
1
11
100
365
1 184 183 2 1
100 5.00%
2365 2
2.486301370 2.50 4.986301370
DQ DIC NQ
CCPN
ff
⎡⎤
−−
=⋅ ⋅− +
⎢⎥
⎣⎦
−−
⎡⎤
=⋅ ⋅− +
⎢⎥
⎣⎦
=+=
16
4.5.3 Short Last Coupons Calculated Using Actual/365
If number of days in last coupon period < 365/frequency:
100
365
M
DCM
CCPN=⋅ ⋅
Where
C
M
= coupon payment at maturity
DCM = days from penultimate coupon date to maturity date
CPN = annual coupon rate expressed as a decimal
Otherwise (i.e. 365/frequency < DCM):
1
100
365
M
DQ DCM
CCPN
f
⎡⎤
−
=⋅ ⋅−
⎢⎥
⎣⎦
Where
DQ = days in the quasi-coupon period starting on the penultimate coupon date.
Example: Calculating the Coupon Payable on a 5% Semi-annual Bond with a Short
Last Coupon Period
a) DCM < 365/frequency
Maturity Date: September 1, 2019
Last Coupon Date: July 15, 2019
DCM = September 1, 2019 – July 15, 2019 = 48
1
100
365
48
100 5.00% 0.65734247
365
DCM
CCPN=⋅ ⋅
=⋅ ⋅ =
17
b) DCM > 365/frequency
Maturity Date: January 14, 2020
Last Coupon Date: July 15, 2019
DCM = January 14, 2020 – July 15, 2019 = 183
DQ = January 15, 2020 – July 15, 2019 = 184
NQ = 2
1
11
100
365
1 184 183 2 1
100 5.00%
2 365 2
2.486301370 2.50 4.986301370
DQ DCM NQ
CCPN
ff
⎡⎤
−−
=⋅ ⋅− +
⎢⎥
⎣⎦
−−
⎡⎤
=⋅ ⋅− +
⎢⎥
⎣⎦
=+=
18
5. ACTUAL/ACTUAL ACCRUED INTEREST
The convention in Canada is to use actual/actual accrued interest for the purpose of
generating a “clean price” from a “dirty price” within the price/yield formula. The clean price
is the price quoted in the market, and it excludes accrued interest. The dirty price includes
accrued interest and is equal to the present value of the cash flows.
The actual/actual accrued interest methodology is also used for calculating settlement
accrued interest on monthly pay bonds. There are four cases to consider:
5.1 Regular Coupon Periods
100
CPN DCS
A
fDCC
=⋅ ⋅
Where:
A = actual/actual accrued interest
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
DCS = days from last coupon date to settlement date
DCC = days from last coupon date to next coupon date
5.2 Short First Coupon Periods
100
CPN DIS
A
f
DQ
=⋅ ⋅
Where:
A = actual/actual accrued interest
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
DIS = days from interest accrual date to settlement date
DQ = days in the quasi-coupon period
5.3 Short Last Coupon Periods
100
CPN DCS
A
f
DQ
=⋅ ⋅
Where:
A = actual/actual accrued interest
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
DCS = days from last coupon date to settlement date
DQ = days in the quasi-coupon period
19
5.4 Long First Coupon Periods
The formula for actual/actual accrued for a long first coupon will generally depend on which
quasi-coupon period the settlement date falls in. To get around this problem, the symbol
DA is used to refer to the number of days over which interest accrues in any given quasi-
coupon period. The formula then simplifies to:
1
100
NQ
k
k
k
DA
CPN
A
f
DQ
=
=⋅ ⋅
∑
Where:
A = actual/actual accrued interest
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
NQ = number of quasi-coupon periods
DA
k
= days of accrual in quasi-coupon period k
DQ
k
= total days in quasi-coupon period k
20
6. SETTLEMENT ACCRUED INTEREST – ACTUAL/365
While actual/actual accrued is used for extracting a clean price from a dirty price, accrued
interest for settlement purposes is calculated in Canada using the actual/365 (Canadian
Bond) method for quarterly and semi-annual pay securities. However, there is an exception
to this convention, for monthly-pay securities. In accordance with IDA regulation 800.48,
accrued interest on monthly-pay securities should be calculated using the actual/actual
method.
There are three cases to consider for actual/365 accrued interest.
6.1 Regular Coupon Periods
If DCS < 365/frequency, then:
100
365
DCS
AI CPN=⋅ ⋅
Otherwise (i.e., DCS > 365/f):
1
100
365
DCC DCS
AI CPN
f
⎛⎞
−
=⋅ ⋅−
⎜⎟
⎝⎠
Where:
AI = actual/365 accrued interest
CPN = annual coupon rate expressed as a decimal
f = annual coupon payment frequency
DCC = days from last coupon date to next coupon date
DCS = days from last coupon date to settlement date
Examples of Calculating Settlement Accrued for a 5% Semi-annual Pay Bond
Maturing February 1, 2008
Example: DCS < 365/f
Settlement Date: December 1, 2005
Last Coupon Date: August 1, 2005
Days of accrued = December 1 - August 1 = 122
Settlement accrued = 100 • 5% • (122/365) = 1.6712
Example: DCS > 365/f
Settlement Date: January 31, 2006
Last Coupon Date: August 1, 2005
Days of accrued = January 31 - August 1 = 183
Settlement accrued = 100 • 5% • [(1/2) – (184-183)/365] = 2.4863
21
6.2 Short First or Last Coupon Period
The formulas for calculating actual/365 accrued interest for a short first or last coupon are
the same as for a regular coupon period.
6.3 Long Coupon Period
There are several possible ways of calculating actual/365 accrued for a long coupon
period. One approach is simply to count the total number of days and divide by 365.
However, this approach can generate counter-intuitive results in certain cases. A more
suitable approach currently being used by market participants is:
1. Separating the long period into one or more full coupon periods plus a stub period
(that is, into a series of quasi-coupon periods).
2. Calculating accrued interest for each sub-period in the regular fashion for actual/365
accrued interest.
The formula is:
1
NQ
k
k
A
IAI
=
=
∑
Where:
AI = actual/365 accrued interest
NQ = number of quasi-coupon periods
AI
k
= actual/365 accrued interest for quasi-coupon period k.
22
7. Discounting for Partial Coupon Periods – Actual/Actual
Exponents
When valuing a bond in between coupon payment dates, it is necessary to calculate the
present value of the cash flows over the fraction of the current coupon period remaining.
For this purpose, the actual/actual methodology is used to calculate the fraction of the
current coupon period remaining.
There are four cases to consider.
7.1 Regular Coupon Periods:
DSC
DCC
α
=
Where:
α = discounting exponent for coupon period in which the settlement date
DSC = days from settlement date to next coupon payment date
DCC = days from last coupon date to when next coupon date falls
7.2 Short First Coupon Periods:
D
SC
D
Q
α
=
Where:
α = discounting exponent for coupon period in which the settlement date
falls
DSC = days from settlement date to next (first) coupon payment date
DQ = days in the quasi-coupon period
7.3 Long First Coupon Periods
1
NQ
k
k
k
DR
DQ
α
=
=
∑
Where:
α = discounting exponent for coupon period in which the settlement date
falls
NQ = number of quasi-coupon periods
DR
k
= days remaining in quasi-coupon period k
DQ
k
= total days in quasi-coupon period k
23
7.4 Short Last Coupon Periods:
D
CM
D
Q
β
=
Where:
β = discounting exponent for final coupon period
DCM = days from penultimate coupon date to maturity date
DQ = days in the final quasi-coupon period
Note that beta defaults to one if the last coupon period is a full coupon period, since DCM
will equal DQ in that case.
24
8. AMORTIZING BONDS
Amortizing bonds in Canada are generally semi-annual pay securities with cash flow
structures that include a series of partial principal repayments on interest payment dates
prior to maturity. Conceptually, this class of security also includes bonds with mandatory
sinking funds, since the cash flow structure of a mandatory sinking fund bond, from an
investor’s perspective, is identical to that of an amortizing bond.
8.1 Trading Convention
A few of the early amortizing structures issued in the Canadian market were designed to
trade on an “original principal balance basis,” much like Canadian mortgage-backed
securities. Today, however, these securities more commonly trade on a remaining principal
balance basis, which is to say that market prices are quoted per $100 of remaining
principal. Formulas presented in this section deal with this latter case only.
8.2 Representing Principal Amounts
The valuation of amortizing bonds requires a way of tracking principal outstanding at any
given time. Therefore, the use of the following definitions is required:
OPB = original principal balance
RPB
t
= remaining principal balance as at time t
For clarity, on cash flow dates, the remaining principal balance is reduced by the amount of
any principal repayment received on that date. Interest for that day will therefore accrue on
the lower principal outstanding. This is consistent with counting days for accrued interest,
where the end date is excluded.
8.3 Principles Underlying Price-Yield Formula
Reference formulas for calculating prices, yields and accrued interest on amortizing bonds
are presented in Section 10. The formula presented is based on the same principles
underlying the price-yield formulas for regular bullet bonds in Canada. Specifically:
• The actual/actual (bond) day-count convention is used for the purpose of deriving
discount factors when discounting over partial coupon periods.
• Actual/actual (bond) is also used for calculating an accrued interest amount when
deriving a clean price from a dirty price and for calculating the amount of an odd
coupon within the price-yield formula.
• Act/365 (Canadian Bond) is used for calculating settlement accrued interest and for
calculating interest payable for an odd coupon period.
25
9. REAL RETURN BONDS
Real return bonds (RRBs) are indexed to the Canadian CPI to compensate the holder for
any inflation that may have occurred since the bond was issued. Conceptually, Canadian
RRBs pay a fixed coupon rate of interest on an indexed principal amount calculated as
follows:
Indexed Principal Principal
Date
Date IssueDate
IssueDate
CPI
CPI
=⋅
Where:
CPI
Date
= Consumer price index as at a given date
By indexing the principal, each cash flow on the bond has effectively been indexed. For
reference, the original principal can be referred to as the “Real Principal”.
9.1 Inflation Indexing Process
In order to value RRBs for any calendar day, a daily CPI series, called the “reference CPI,”
is defined. The reference CPI for the first day of any calendar month is equal to the CPI for
the third preceding calendar month
6
. The reference CPI for any date within a month is
calculated by linearly interpolating between the reference CPI applicable to the first day of
the month in which such a date falls, and the reference CPI applicable to the first day of the
month immediately following. This is an actual/actual day-count interpolation.
[]
1
1
.. ..
Date M M M
t
refCPI refCPI refCPI refCPI
D
+
−
=+ −
Where:
ref.CPI
Date
= reference CPI as at date t
ref.CPI
M
= reference CPI as at the first day of the current month
t = calendar day corresponding to the date
D = number of days in the month in which the date falls
ref.CPI
M+1
= reference CPI as at the first day of the next month
Given the daily Reference CPI series, a daily “index ratio” is defined that reflects all
appreciation in the reference CPI occurring since the issue date of the RRB. The index ratio
is defined as:
.
.
.
Date
Date
B
ase
ref CPI
Index Ratio
ref CPI
=
6
For this purpose, the originally published CPI number is used. Revisions to CPI numbers are not used to reprice RRBs.
26
Where:
ref.CPI
Date
= reference CPI as at date t
ref.CPI
Base
= the reference CPI corresponding to the RRB issue date.
9.2 Rounding Convention for the Indexing Process
For the purpose of interpolating reference CPI, calculations are carried to the sixth decimal
place and rounded to the nearest five decimal places. Similarly, calculations for the index
ratio are carried to six decimal places and rounded to the nearest five decimal places.
Example: Calculating Reference CPI
The example below calculates reference CPI for May 14, 2005:
Ref CPI
May 1, 2005
= CPI
February 2005
= 125.8
Ref CPI
June 1, 2005
= CPI
March 2005
= 126.5
()
14
14 1
. 125.8 126.5 125.8 126.09355
31
May
ref CPI
−
=+ − =
9.3 Real Price Given Real Yield to Maturity
Aside from the indexing provision, RRBs are identical to conventional semi-annual pay
bullet bonds that repay 100 per cent of principal at maturity. Thus, the clean price, given
yield to maturity for a RRB, is calculated in a similar fashion. The price resulting from this
calculation is known as the “real price.”
9.4 Real Accrued Interest
Real accrued interest as at any given date is the accrued interest calculated on the real
principal. This calculation uses the standard actual/365 accrued interest formulas
presented in Section 6.
9.5 Settlement Amounts for Real Return Bonds: Nominal Price and Accrued
Interest
Settlement amounts for transactions in RRBs are based on the nominal price and nominal
accrued interest, which are calculated as follows:
Date Date
Nominal.Price Real.Price Index.Ratio
Date
=⋅
Date Date
Nominal.Accrued Real.Accrued Index.Ratio
Date
=⋅
9.6 CPI Re-basing
On occasion, Statistics Canada converts the official time base reference period for the
consumer price index (the period for which the value 100 is assigned to the index). This
most recently occurred on June 19, 2007 where the base reference period was converted
27
from 1992 to 2002. Changes to the official time base reference period require the
Government of Canada via its fiscal agent, the Bank of Canada, to publish, to three decimal
places, the conversion factor that is used to rebase historical CPI data. RRB index ratios
calculated on and after the rebasing date are based on the new official time base reference
period. Historical RRB index ratios calculated prior to this date are not revised.
The Reference CPI
Base
after the re-basing is a five decimal number that is used in the Index
Ratio calculations from the effective date of the CPI re-basing, forward, as shown in the
table below.
Table of CPI
Base
Values
Coupon (%) Bond Maturity New Reference CPI
Base
Previous Reference CPI
Base
4.25 1 December 2021 83.07713 98.86178
4.25 1 December 2026 87.82571 104.51260
4.00 1 December 2031 91.38249 108.74516
3.00 1 December 2036 102.99160 122.56000
2.00 1 December 2041 111.21849 132.35000
Source: www.bankofcanada.ca/en/notices_fmd/2007/not190607.html
28
PART II
10. REFERENCE FORMULAS
10. REFERENCE FORMULAS.......................................................................................................................28
10.1 Semi-annual Pay Bond – Regular Coupon Periods .......................................................................29
10.1.1 Price Given Yield to Maturity .............................................................................................29
10.1.2 Settlement Accrued Interest ..............................................................................................30
10.2 Semi-annual Pay Bond – Short First Coupon ................................................................................31
10.2.1 Price Given Yield to Maturity .............................................................................................31
10.2.2 Settlement Accrued Interest ..............................................................................................32
10.3 Semi-annual Pay Bond – Long First Coupon, Settlement in First Quasi-coupon Period...............33
10.3.1 Price Given Yield to Maturity .............................................................................................33
10.3.2 Settlement Accrued Interest ..............................................................................................34
10.4 Semi-annual Pay Bond –
Long First Coupon, Settlement in Second Quasi-coupon Period....................35
10.4.1 Price Given Yield to Maturity .............................................................................................35
10.4.2 Settlement Accrued Interest ..............................................................................................36
10.5 Semi-annual Pay Bond – Short Last Coupon, Regular First Coupon ............................................37
10.5.1 Price Given Yield to Maturity .............................................................................................37
10.5.2 Settlement Accrued Interest ..............................................................................................38
10.6 Semi-annual Pay Bond – Short Last Coupon, Short First Coupon ................................................39
10.6.1 Price Given Yield to Maturity .............................................................................................39
10.6.2 Settlement Accrued Interest ..............................................................................................40
10.7 Monthly Pay Bond – Regular Coupon Periods...............................................................................41
10.7.1 Price Given Yield to Maturity .............................................................................................41
10.7.2 Settlement Accrued Interest ..............................................................................................42
10.8 Semi-annual Pay Amortizing Bond – Regular Interest Periods......................................................43
10.8.1 Price Given Yield to Maturity .............................................................................................43
10.8.2 Settlement Accrued Interest ..............................................................................................44
10.9 Semi-annual Pay Amortizing Bond – Short First Interest Period ...................................................45
10.9.1 Price Given Yield to Maturity .............................................................................................45
10.9.2 Settlement Accrued Interest ..............................................................................................46
10.10 Money Market Yields ......................................................................................................................47
10.10.1 Semi Annual Bonds: One Cash Flow Remaining..............................................................47
10.10.2 Semi Annual Bonds: Two Cash Flows Remaining............................................................48
10.10.3 Monthly and Quarterly Pay Bonds: Money Market Equivalent Yield.................................49
10.10.4 Price and Yield Calculations for Money Market Discount Notes .......................................50
29
10.1 Semi-annual Pay Bond – Regular Coupon Periods
10.1.1 Price Given Yield to Maturity
11
1
1 100 100 1 100
22
11
1
22
2
NC
DSC NC k
k
DCC
CPN CPN DCS
P
D
CC
YY
Y
−−
=
⎡⎤
⎢⎥
⋅⋅
⎢⎥
=⋅ +⋅ −⋅
⎢⎥
⎛⎞ ⎛⎞
⎛⎞
++
⎢⎥
⎜⎟ ⎜⎟
+
⎜⎟
⎝⎠ ⎝⎠
⎣⎦
⎝⎠
∑
Where:
P = clean price
Y = yield to maturity
DSC = number of days from the settlement date to the next coupon date
DCC = number of days in the coupon period in which the settlement date falls
NC = number of coupon payments remaining
k = summation counter
CPN = annual coupon rate expressed as a decimal
DCS = number of days from the previous coupon date to the settlement date
30
10.1.2 Settlement Accrued Interest
Less Than 182.5 Days of Accrual Greater Than 182.5 Days of Accrual
100
365
D
CS
AI CPN=⋅ ⋅
1
100
2 365
DCC DCS
AI CPN
−
⎛⎞
=⋅ ⋅−
⎜⎟
⎝⎠
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
DCS = number of days from the previous coupon date to the settlement date
DCC = number of days in the coupon period in which the settlement date falls
31
10.2 Semi-annual Pay Bond – Short First Coupon
10.2.1 Price Given Yield to Maturity
11
2
1 100 100 1 100
100
222
11
1
22
2
NC
DSC NC k
k
DQ
CPN DIC CPN CPN DIS
P
D
QDQ
YY
Y
−−
=
⎡⎤
⎢⎥
⋅⋅
⎢⎥
=⋅⋅⋅+ +⋅ −⋅
⎢⎥
⎛⎞ ⎛⎞
⎛⎞
++
⎢⎥
⎜⎟ ⎜⎟
+
⎜⎟
⎝⎠ ⎝⎠
⎣⎦
⎝⎠
∑
Where:
P = clean price
Y = yield to maturity
DSC = number of days from the settlement date to the first coupon date
DQ = number of days in the quasi-coupon period ending on the first coupon payment date
CPN = annual coupon rate expressed as a decimal
k = summation counter
DIC = number of days from interest accrual date to first payment date
NC = number of coupon payments remaining
DIS = number of days from the interest accrual date to the settlement date
32
10.2.2 Settlement Accrued Interest
100
365
D
IS
AI CPN=⋅ ⋅
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
DIS = number of days from the interest accrual date to the settlement date
33
10.3 Semi-annual Pay Bond – Long First Coupon, Settlement in First Quasi-coupon
Period
In this example, the bond has a long first coupon period consisting of a full regular coupon period plus a stub period.
10.3.1 Price Given Yield to Maturity
111
1
2
1
1 1 100 100 1
100 1 100
21 2 21
11
1
22
2
NC
DSC NC k
k
DQ
CPN DIC CPN CPN DIS
P
D
QDQ
YY
Y
−−
+
=
⎡⎤
⎢⎥
⎛⎞
⋅
⎢⎥
=⋅⋅⋅+++⋅ −⋅⋅
⎜⎟
⎢⎥
⎛⎞ ⎛⎞
⎝⎠
⎛⎞
++
⎢⎥
⎜⎟ ⎜⎟
+
⎜⎟
⎝⎠ ⎝⎠
⎣⎦
⎝⎠
∑
Where:
P = clean price
Y = yield to maturity
DSC1 = number of days from settlement date to the first quasi-coupon date
DQ1 = number of days in the first quasi-coupon period (the stub period)
CPN = annual coupon rate expressed as a decimal
k = summation counter
DIC1 = number of days from interest accrual date to first payment date
NC = number of coupon payments remaining
DIS = number of days from the interest accrual date to the settlement date
34
10.3.2 Settlement Accrued Interest
100
365
D
IS
AI CPN=⋅ ⋅
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
DIS = number of days from the interest accrual date to the settlement date
35
10.4 Semi-annual Pay Bond – Long First Coupon, Settlement in Second Quasi-coupon
Period
In this example, the bond has a long first coupon period consisting of a full regular coupon period plus a stub period.
10.4.1 Price Given Yield to Maturity
11
2
2
1 1 100 100 1 1 1
100 1 100
21 2 212
11
1
22
2
NC
DSC NC k
k
DQ
CPN DIC CPN CPN DIC DC S
P
DQ DQ DQ
YY
Y
−−
=
⎡⎤
⎢⎥
⎛⎞ ⎛ ⎞
⋅
⎢⎥
=⋅⋅⋅+++⋅ −⋅⋅+
⎜⎟ ⎜ ⎟
⎢⎥
⎛⎞ ⎛⎞
⎝⎠ ⎝ ⎠
⎛⎞
++
⎢⎥
⎜⎟ ⎜⎟
+
⎜⎟
⎝⎠ ⎝⎠
⎣⎦
⎝⎠
∑
Where:
P = clean price
Y = yield to maturity
DSC = number of days from settlement date to the first coupon payment date
DQ2 = number of days in the second quasi-coupon period
CPN = annual coupon rate expressed as a decimal
DIC1 = number of days from the interest accrual date to the first quasi-coupon date
DQ1 = number of days in the first quasi-coupon period (the stub period)
k = summation counter
NC = number of coupon payments remaining
DC1S = number of days from the first quasi-coupon date to the settlement date
36
10.4.2 Settlement Accrued Interest
12QQ
A
IAI AI=+
Where:
AI = settlement accrued interest
AI
Q1
= settlement accrued interest in the first quasi-coupon period
AI
Q2
= settlement accrued interest in the second quasi-coupon period
Note that the formula for settlement accrued interest in each quasi-coupon period depends on whether there is greater or less
than 182.5 days of accrual in the period.
37
10.5 Semi-annual Pay Bond – Short Last Coupon, Regular First Coupon
10.5.1 Price Given Yield to Maturity
1
1
2
1
100 1
2
1 100 1 100
22
1
11
2
22
M
NC
M
DSC DCM k
NC
k
DCC DQ
CPN DCM
DQ
CPN CPN DCS
P
D
CC
Y
YY
−
−
−+
=
⎡⎤
⎛⎞
⎢⎥
⋅+ ⋅
⎜⎟
⋅⋅
⎢⎥
⎝⎠
=⋅ +⋅ −⋅
⎢⎥
⎛⎞
⎢⎥
⎛⎞ ⎛⎞
+
⎜⎟
++
⎜⎟ ⎜⎟
⎢⎥
⎝⎠
⎝⎠ ⎝⎠
⎣⎦
∑
Where:
P = clean price
Y = yield to maturity
DSC = number of days from the settlement date to the next coupon date
DCC = number of days in the coupon period in which the settlement date falls
CPN = annual coupon rate expressed as a decimal
DCM = number of days from the last coupon date prior to maturity to the maturity date
DQ
M
= number of days in a full quasi-coupon period beginning on the last coupon date prior to maturity
k = summation counter
NC = number of coupon payments remaining
DCS = number of days from the previous coupon date to the settlement date
38
10.5.2 Settlement Accrued Interest
Less Than 182.5 Days of Accrual Greater Than 182.5 Days of Accrual
100
365
D
CS
AI CPN=⋅ ⋅
1
100
2 365
DCC DCS
AI CPN
−
⎛⎞
=⋅ ⋅−
⎜⎟
⎝⎠
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
DCS = number of days from the previous coupon date to the settlement date
DCC = number of days in the coupon period in which the settlement date falls
39
10.6 Semi-annual Pay Bond – Short Last Coupon, Short First Coupon
10.6.1 Price Given Yield to Maturity
1
1
2
2
100 1
2
1 100 100 1 100
222
1
11
2
22
M
NC
M
DSC DCM k
NC
k
DQ DQ
CPN DCM
DQ
CPN DIC CPN CPN DIS
P
D
QDQ
Y
YY
−
−
−+
=
⎡⎤
⎛⎞
⎢⎥
⋅+ ⋅
⎜⎟
⋅⋅⋅
⎢⎥
⎝⎠
=⋅ + +⋅ −⋅
⎢⎥
⎛⎞
⎢⎥
⎛⎞ ⎛⎞
+
⎜⎟
++
⎜⎟ ⎜⎟
⎢⎥
⎝⎠
⎝⎠ ⎝⎠
⎣⎦
∑
Where:
P = clean price
Y = yield to maturity
DSC = number of days from the settlement date to the first coupon payment date
DQ = number of days in the short first quasi-coupon period
CPN = annual coupon rate expressed as a decimal
DIC = number of days from interest accrual date to first payment date
DCM = number of days from the last coupon date prior to maturity to the maturity date
DQ
M
= number of days in a full quasi-coupon period beginning on the last coupon date prior to maturity
k = summation counter
NC = number of coupon payments remaining
DIS = number of days from the interest accrual date to the settlement date
40
10.6.2 Settlement Accrued Interest
Less Than 182.5 Days of Accrual Greater Than 182.5 Days of Accrual
100
365
D
CS
AI CPN=⋅ ⋅
1
100
2 365
DCC DCS
AI CPN
−
⎛⎞
=⋅ ⋅−
⎜⎟
⎝⎠
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
DCS = number of days from the previous coupon date to the settlement date
DCC = number of days in the coupon period in which the settlement date falls
41
10.7 Monthly Pay Bond – Regular Coupon Periods
10.7.1 Price Given Yield to Maturity
11
1
1 100 100 1 100
12 12
11
1
12 12
12
NC
DSC NC k
k
DCC
CPN CPN DCS
P
D
CC
YY
Y
−−
=
⎡⎤
⎢⎥
⋅⋅
⎢⎥
=⋅ +⋅ −⋅
⎢⎥
⎛⎞ ⎛⎞
⎛⎞
++
⎢⎥
⎜⎟ ⎜⎟
+
⎜⎟
⎝⎠ ⎝⎠
⎣⎦
⎝⎠
∑
Where:
P = clean price
Y = yield to maturity
DSC = number of days from the settlement date to the next coupon date
DCC = number of days in the coupon period (month) in which the settlement date falls
NC = number of coupon payments remaining
k = summation counter
CPN = annual coupon rate expressed as a decimal
DCS = number of days from the previous coupon date to the settlement date
42
10.7.2 Settlement Accrued Interest
100
365
D
CS
AI CPN=⋅ ⋅
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
DCS = number of days from the previous coupon date to the settlement date
43
10.8 Semi-annual Pay Amortizing Bond – Regular Interest Periods
10.8.1 Price Given Yield to Maturity
1
1
1
1
2
1 100
2
1
1
2
2
k
kj
NC
j
t
DSC k
k
t
DCC
CPN
RP OPB RP
CPN RPB
DCS
P
D
CC RPB
Y
Y
−
=
−
=
⎛⎞
⎡⎤
⎛⎞
⎜⎟
+⋅ −
⎢⎥
⎜⎟
⎜⎟
⋅
⎝⎠
⎢⎥
=⋅ −⋅⋅
⎜⎟
⎢⎥
⎛⎞
⎜⎟
⎛⎞
⎢⎥
+
⎜⎟
+
⎜⎟
⎜⎟
⎝⎠
⎢⎥
⎣⎦
⎝⎠
⎝⎠
∑
∑
Where:
P = clean price
Y = annual yield to maturity
DSC = number of days from settlement date to first payment date
DCC = number of days in the coupon period in which the settlement date falls
NC = number of coupon payments remaining
RP
k
= principal repayment as at cash flow date k
CPN = annual coupon rate expressed as a decimal
OPB = amount issued (original principal balance)
RPB
t
= principal balance remaining as at the settlement date
DCS = number of days from issue date or last payment date to the settlement date
44
10.8.2 Settlement Accrued Interest
Less Than 182.5 Days of Accrual Greater Than 182.5 Days of Accrual
100
365
t
CPN DCS
AI
RPB
⋅
=⋅
100 1
2 365
t
CPN DCC DCS
AI
RPB
⋅−
⎛⎞
=⋅−
⎜⎟
⎝⎠
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
RPB
t
= principal balance remaining as at the settlement date
DCC = number of days in the coupon period in which the settlement date falls
DCS = number of days from the interest accrual date (or previous coupon date) to the settlement date
45
10.9 Semi-annual Pay Amortizing Bond – Short First Interest Period
10.9.1 Price Given Yield to Maturity
1
1
1
1
2
2
1 100
22
1
1
2
2
k
kj
NC
j
DSC k
k
DQ
CPN
RP OPB RP
CPN OPB DIC CPN OPB DIS
PRP
D
QDQOPB
Y
Y
−
=
−
=
⎛ ⎞
⎡⎤
⎛⎞
⎜ ⎟
+⋅ −
⎢⎥
⎜⎟
⎜ ⎟
⋅⋅
⎝⎠
⎢⎥
=⋅⋅++ −⋅⋅
⎜ ⎟
⎢⎥
⎛⎞
⎜ ⎟
⎛⎞
⎢⎥
+
⎜⎟
+
⎜⎟
⎜ ⎟
⎝⎠
⎢⎥
⎣⎦
⎝⎠
⎝ ⎠
∑
∑
Where:
P = clean price
Y = annual yield to maturity
DSC = number of days from settlement date to first payment date
DQ = number of days in the quasi-coupon period ending on the first payment date
CPN = annual coupon rate expressed as a decimal
OPB = amount issued (original principal balance)
DIC = number of days from the interest accrual date to the first payment date
RP
1
= principal repayment on the first payment date
NC = number of coupon payments remaining
k = summation counter
RP
k
= principal repayment as at cash flow date k
DIS = number of days from the interest accrual date to the settlement date
46
10.9.2 Settlement Accrued Interest
Less Than 182.5 Days of Accrual Greater Than 182.5 Days of Accrual
100
365
CPN DIS
AI
OPB
⋅
=⋅
100 1
2 365
t
CPN DQ DIS
AI
RPB
⋅−
⎛⎞
=⋅−
⎜⎟
⎝⎠
Where:
AI = settlement accrued interest
CPN = annual coupon rate expressed as a decimal
OPB = amount issued (original principal balance)
DIS = number of days from the interest accrual date to the settlement date
RPB
t
= principal balance remaining as at the settlement date
DQ = number of days in the quasi-coupon period ending on the first payment date
47
10.10 Money Market Yields
10.10.1 Semi-annual Bonds: One Cash Flow Remaining
The market convention in Canada is to quote a money market equivalent yield on bonds that are in their last coupon period. The
formula below converts between price and simple interest yield on a bond with one cash flow remaining.
1
365
M
CP C
PAI
D
SM
YME
+
+=
+⋅
Where:
P = clean price
AI = actual/365 accrued interest
CP = 100 or call price
C
M
= coupon payment at maturity
YME = money market equivalent yield
DSM = days from settlement date to maturity date
48
10.10.2 Semi Annual Bonds: Two Cash Flows Remaining
Semi-annual-pay bonds in their final year to maturity with two cash flows remaining can be quoted using the usual semi-annually
compounded yield to maturity or a money market equivalent yield. In the latter case, the formula below is used:
1
100 1
2 365
1
365
M
CPN DCM
PAI YME CPC
DSM
YME
⎡
⎤
⎛⎞
+= ⋅ ⋅ ⋅+ ⋅ ++
⎜⎟
⎢
⎥
⎛⎞
⎝⎠
⎣
⎦
+⋅
⎜⎟
⎝⎠
Where:
P = clean price
AI = actual/365 accrued interest
YME = money market equivalent yield
DSM = days from settlement date to maturity date
CPN = annual coupon rate expressed as a decimal
DCM = days from penultimate coupon date to maturity date
CP = 100 or call price
C
M
= coupon payment at maturity
If maturity date falls on a non-business day (weekend or holiday), then the convention in the money market is to roll to the next
business day.
49
10.10.3 Monthly and Quarterly Pay Bonds: Money Market Equivalent Yield
The formula from Section 8.2 is generalized below to address other payment frequencies:
1
1
100 1
365
1
365
NC
k
M
k
DCM
CPN
P
AI YME CP C
DSM
f
YME
=
⎡
⎤
⎛⎞
+= ⋅ ⋅ ⋅ + ⋅ ++
⎜⎟
⎢
⎥
⎝⎠
⎣
⎦
+⋅
∑
Where:
P = clean price
AI = actual/365 accrued interest
YME = money market equivalent yield
DSM = days from settlement date to maturity date
CPN = annual coupon rate expressed as a decimal
f = frequency
NC = number of coupon payments remaining
DCM
k
= days from coupon payment k to maturity date
CP = 100 or call price
C
M
= coupon payment at maturity
50
10.10.4 Price and Yield Calculations for Money Market Discount Notes
The price of a typical Canadian domestic money market discount note, such as a treasury bill, commercial paper or bankers’
acceptance, is:
100
1
365
P
DSM
R
=
+⋅
Where:
P = clean price
R = money market yield to maturity, simple interest
DSM = days from settlement date to maturity date
Solving for the yield is done as follows:
100 365P
R
P
DSM
−
⎡⎤⎡⎤
=⋅
⎢⎥⎢⎥
⎣⎦⎣⎦
50
APPENDIX 1: BOND VERSUS SWAP MARKET CONVENTIONS
It is not the purpose of this report to document swap market practices. However, a brief
discussion of the key differences may be helpful to understanding bond market
conventions, particularly for readers already familiar with swap market practices.
The fundamental difference between bond and swap market conventions concerns the
manner in which interest accrues. For a given nominal annual interest rate, the swap
market follows the standard banking practice of accruing interest in equal daily
increments throughout each calendar year. The daily accrual equals the principal of the
loan times the nominal annual interest rate divided by the assumed number of days in the
year. The interest payable for a given period is then equal to the daily accrual amount
times the number of days in the period. Obviously, day-count methodologies are integral
to the calculation of interest payable in this approach.
In contrast to swap market practice, bonds pay a periodic rate of interest equal to the
nominal annual coupon rate divided by the annual payment frequency. Thus bonds pay
an equal amount of (coupon) interest for each coupon period, where interest payable
equals the periodic interest rate times the bond principal. Several conclusions follow
directly from this fact:
1. Abstracting from odd coupon periods, day-counts are irrelevant to the calculation of
coupon interest payable on a bond. Day-count conventions are only required to
calculate how much of a given coupon has accrued to any date within a coupon
period;
2. Since no two consecutive periods contain the same number of days, the “effective”
rate of daily interest accrual on a bond changes from coupon period to coupon period
(as the number of days in successive periods fluctuates while interest payable
remains constant);
3. Day-count conventions such as Act/365 (Canadian Bond) and US30/360, which are
commonly used to calculate accrued interest on bonds, necessarily embody ad hoc
adjustments to cope with fluctuations in the daily accrual rates from coupon period to
coupon period; and
4. The Actual/actual (bond) day-count convention is the most mathematically consistent
and appropriate convention for use with bonds because it explicitly accounts for the
fluctuations in daily accrual rates from coupon period to coupon period. Not
surprisingly, therefore, Actual/actual (bond) is becoming the new global standard for
bond valuation.
In contrast to bond market practice, therefore, interest payable on the fixed side of a swap
transaction does typically vary with the number of days in each interest period
7
. The
example below illustrates the interest amounts payable on a fixed rate bond and the fixed
7
The ISDA 30/360-day-count convention will generate equal periodic interest payments on a swap when used in
conjunction with a certain date roll convention.
51
leg of an interest rate swap, assuming that the swap is accrued interest using the ISDA
Act/365 convention.
Bond versus Swap Cash Flow Conventions
The table below shows the cash flows on a hypothetical two-year bond, with a 5.0% fixed coupon,
payable semi-annually, and a hypothetical two-year fixed-for-floating swap. The coupon interest
payments on the bond do not depend on the days in each period. They are simply calculated by
dividing the annual coupon rate by the payment frequency.
Bond:
Coupon= 5.0%
Payment Frequency = 2, i.e., semi-annual
Coupon Payment Dates: December 1 and June 1
Interest Accrual Date: June 1, 2007
Swap:
Fixed Rate 5.0%
Payment Frequency = 2, i.e., semi-annual
Payment Dates: December 1 and June 1
Business Day Convention: Following
Start Date: June 1, 2007
For a quantity or notional amount of $1 million, the bonds pay a semi-annual coupon of:
0.05
$1,000,000.00 25,000.00
2
×=
The cash flows on the swap, in contrast, do vary with the number of days in each period. For
example, the first regularly scheduled payment date on the fixed side, December 1 2007, falls on
a Saturday, and so the payment date is adjusted to the next business day, Monday, December
3
rd
, 2007. The first fixed payment is therefore calculated over a 185-day period:
(95 90)
0.05 $1,000,00.00 $25,342.47
365
+
×× =
This is more than the payment on the bond, because 185 days is more than half of a 365-day
year. Subsequent fixed payments are lower, because, in this particular example, each is
calculated over 182 days:
(182)
0.05 $1,000,000.00 $24,931.51
365
×× =
Cash Flows on Fixed Leg of Swap and Coupon Payments from a Fixed Rate Bond
Unadjusted Date Adjusted Date Days Fixed Swap Cash
Flow
Bond Coupon Cash
Flow
June 1, 2007 June 1, 2007
Dec. 1, 2007 Dec. 3, 2007 185 $25,342.47 $25,000.00
June 1, 2008 June 2, 2007 182 $24,931.51 $25,000.00
Dec. 1, 2008 Dec. 1, 2008 182 $24,931.51 $25,000.00
June 1, 2009 June 1, 2009 182 $24,931.51 $25,000.00
The cash flows from the floating leg of the swap are omitted from the preceding table.
52
Similar differences result when interest on the swap is accrued using the ISDA
Actual/actual convention. In the swap market, the Actual/actual convention is similar to
the Act/365 convention, with the proviso that the actual number of days in the year in
which interest accrues is used in the denominator. ISDA Actual/actual is therefore a
means of accounting for leap years when calculating the amount payable over an interest
period. Like Act/365, however, under ISDA Actual/actual the daily rate of interest accrual
remains constant throughout each year, and interest payable changes from interest
period to interest period in accordance with the number of days in each interest period.
In contrast, the bond market version of the actual/actual day-count methodology is
employed to calculate how much of a coupon cash flow has accrued to a given date
within a coupon period. As discussed above, the coupon payment itself is calculated by
taking the nominal annual coupon rate of interest divided by the annual payment
frequency. The amount of interest that has accrued to a given date within the coupon
period is then calculated as follows:
Actual days elapsed since last coupon
Accrued Interest Coupon Payment
Actual number of days in the full coupon period
=⋅
53
APPENDIX 2: ROUNDING AND TRUNCATION PRACTICES
Accuracy of Intermediate Values
Current practice shows that all yield, price and interest calculations should be carried out
to the limits of machine precision, and, as a minimum, at least 10 significant decimals.
Dollar Price Accuracy
Bonds
For bonds, dollar prices should be accurate to seven (7) places after the decimal,
rounding to six (6) decimals. The rounding is to be applied after solving for the principal
(price multiplied by quantity) on the transaction. For calculating principal, full precision is
to be used, with the caveat that the resulting principal is rounded to the nearest penny.
See the table below.
Money Market
The convention for money market securities is to round prices to three (3) decimals, and
to round yields to two (2) decimal places. When calculating principal (price multiplied by
quantity) on a trade, the price is rounded to three (3) decimals. This is in contrast to the
bond convention where full precision on the price is utilized, with the resultant principal
rounded to the nearest penny.
Accrued Interest
Settlement Accrued Interest on a trade is rounded to the nearest penny. However, when
calculating present value, full precision is used for accrued interest.
Example:
Bond: Canada 8%, June 1, 2023
Settlement Date: July 9, 2007
Price: 99.987135
Yield: 8.000001
Accrued Interest: $0.83 per $100
Quantity Principal
Quantity
• Price
Accrued
Interest
Total Settlement
Value of Trade
$1,000.00 $999.87 $8.33 $1008.20
$10,000.00 $9,998.71 $83.298 $10,082.00
$100,000.00 $99,987.13 $832.88 $100,820,01
$1,000,000.00 $999,871.35 $8,328.77 $1,008,200.12
$10,000,000.00 $9,998,713.46 $83,287.67 $10,082,001.13
$100,000,000.00 $99,987,134.59 $832,876.71 $100,820,011.30
$1,000,000,000.00 $999,871,345.93 $8,328,767.12 $1,008,200,113.05
54
APPENDIX 3: MNEMONICS
A = actual/actual accrued interest
AI = actual/365 accrued interest
AI
k
= actual/365 accrued interest for quasi-coupon period k
β = exponent for short last period
β = period discounting exponent for final coupon period same thing?
C
1
= first coupon payment
C
M
= coupon payment at maturity
CP = 100 or the call price
CPN = annual coupon rate expressed as a decimal
D
k
= days of accrual in quasi-coupon period k
DCC = days from last coupon date to next coupon date
DCM = days from penultimate coupon date to maturity date
DCM
k
= days from coupon payment k to maturity
DCS = days from last coupon date to settlement date
DIC = days from interest accrual date to first payment date
DIC* = days from interest accrual date to first quasi-coupon date
DIS = days from interest accrual date to settlement date
DQ = days in the quasi-coupon period
DQ
k
= total days in quasi-coupon period k
DR
k
= days remaining in quasi-coupon period k
DSM = days from settlement date to maturity date
DSC = days from settlement date to next coupon payment date
f = payment frequency
k = varying period or date, e.g., quasi-coupon period, cash flow or coupon-
payment date
NC = number of coupon payments remaining
NQ = number of quasi-coupon periods
OPB = original principal balance
P = clean price
R = money market yield to maturity, simple interest
RPB = remaining principal balance
RP
k
= principal repayment as at cash flow date k
α = discounting exponent for coupon period in which the settlement date falls
55
REFERENCES
Europe
Bank of England, “Changes to the Gilt Market Trading Conventions”, March 1998
http://www.bankofengland.co.uk/publications/news/1998/pdfs/giltconv.pdf
U.K. Debt Management Office, “Formulae for Calculating Gilt Prices from Yields”, 3
rd
edition, March 2005,
http://www.dmo.gov.uk/documentview.aspx?docname=/giltsmarket/formulae/yldeqns.pdf&page=Gilts/Formulae
International Capital Markets Association (ICMA), “Statutes, Bylaws, Rules and
Recommendations,” rules 224 and 250 define day-count conventions and accrued
interest calculation standards
http://www.icma-group.org/home.html
U.S.
Securities Industry and Financial Markets Association (SIFMA), “Standard Securities
Calculation Methods: Fixed-Income Securities Formulas for Price, Yield and Accrued
Interest”, volumes I and II
http://www.sifma.org/
Canada
Investment Dealers Association of Canada (IDA), Member Regulation Notice 078
describes accrued interest conventions for monthly compound frequency instruments
http://ida.knotia.ca/Knowledge/View/Document.cfm?Ktype=445&linkType=toc&dbID=200706346&tocID=606
Investment Dealers Association of Canada (IDA), Member Regulation Notice 0148
describes accrued interest convention for monthly-pay MBS
http://ida.knotia.ca/Knowledge/View/Document.cfm?Ktype=445&linkType=toc&dbID=200706346&tocID=483
Bank of Canada, terminology and calculations for real return bonds can be found at:
http://www.bank-banque-canada.ca/en/pdf/real_return_eng.pdf
Canada Housing and Mortgage Corporation, Standard Terminology and Calculations
for MBS
http://www.cmhc-schl.gc.ca/en/hoficlincl/mobase/stteca/index.cfm
Swaps and Derivatives
International Swaps and Derivatives Association (ISDA), “EMU and Market
Conventions: Recent Developments”, 1998
http://www.isda.org/c_and_a/pdf/mktc1198.pdf
Disclaimer
Although the content of this document is taken from sources believed to be reliable, the Investment Industry
Association of Canada (IIAC) makes no warranty as to the accuracy or completeness of the information and
takes no responsibility for any damages incurred as a result of reliance upon or use of this document. All
rights reserved.
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