4.1 Bond Pricing, Price–Yield Relationships & Accrued Interest

The Discounted Cash Flow Framework for Bond Valuation

A fixed-income security represents a contractual promise by an issuer to pay a predetermined stream of cash flows over time. Valuing a bond therefore relies on the foundational financial principle of discounted cash flows (DCF): the theoretical intrinsic value of any bond equals the sum of the present values of all future cash flows, discounted at the prevailing market interest rate (the required yield to maturity for securities of comparable term, credit quality, and liquidity).

The Anatomy of Bond Cash Flows

For a standard conventional bond in Canada, these future cash flows consist of two distinct elements:

1. Annuity Stream of Periodic Coupons: Regular semi-annual interest payments distributed every six months until maturity.
2. Lump-Sum Par Value: The repayment of the bond's principal (face value) at final maturity.

Semi-Annual Compounding and the Valuation Formula

Because Canadian government and corporate bonds pay interest semi-annually, the annual coupon is divided by 2, the annual discount rate (yield) is divided by 2, and the number of discounting periods equals twice the number of years to maturity ($2n$):

$P = \sum_{t=1}^{2n} \frac{C/2}{(1 + r/2)^t} + \frac{M}{(1 + r/2)^{2n}}$

Where:

• $P$ = Theoretical market price (present value) of the bond
• $C$ = Annual contractual coupon payment in dollars (Coupon Rate $\times$ Face Value)
• $C/2$ = Semi-annual coupon cash payment
• $r$ = Annual required market yield (discount rate expressed as a decimal)
• $r/2$ = Semi-annual discount rate
• $n$ = Number of years to maturity
• $2n$ = Total number of semi-annual compounding periods
• $M$ = Par value (face value) repaid at maturity (standardized to $1,000 in Canadian practice)

Using the standard present value formula for an ordinary annuity, this equation can be expressed compactly as:

$P = \frac{C}{2} \left[ \frac{1 - (1 + r/2)^{-2n}}{r/2} \right] + \frac{M}{(1 + r/2)^{2n}}$

---

Step-by-Step Worked Numerical Example: Pricing a Bond

Valuation Problem: An investor is evaluating a 5-year Canadian corporate bond with an annual coupon rate of 6.00% paid semi-annually and a par value of $1,000. Prevailing market yields for comparable 5-year corporate debt currently stand at 8.00%.

Input Parameters:

• Par Value ($M$) = $1,000
• Annual Coupon Rate = 6.00% $\implies$ Annual Coupon ($C$) = $60.00
• Semi-Annual Coupon ($C/2$) = $30.00
• Years to Maturity ($n$) = 5 $\implies$ Compounding Periods ($2n$) = 10
• Market Yield ($r$) = 8.00% $\implies$ Semi-Annual Yield ($r/2$) = 4.00% (0.04)

Step 1: Calculate the Present Value of the Semi-Annual Coupons ($PV_{\text{coupons}}$)

$PV_{\text{coupons}} = \$30.00 \times \left[ \frac{1 - (1 + 0.04)^{-10}}{0.04} \right]$ $PV_{\text{coupons}} = \$30.00 \times \left[ \frac{1 - 0.675564}{0.04} \right]$ $PV_{\text{coupons}} = \$30.00 \times \left[ \frac{0.324436}{0.04} \right] = \$30.00 \times 8.110896 = \$243.33$

Step 2: Calculate the Present Value of the Lump-Sum Par Value ($PV_{\text{par}}$)

$PV_{\text{par}} = \frac{\$1,000}{(1 + 0.04)^{10}} = \$1,000 \times 0.675564 = \$675.56$

Step 3: Sum the Present Values to Determine Total Bond Price

$P = PV_{\text{coupons}} + PV_{\text{par}} = \$243.33 + \$675.56 = \$918.89$

In Canadian dealer quotes, this bond would be quoted as 91.89 (meaning 91.89% of par value). Because the market required yield (8.00%) exceeds the bond's contractual coupon (6.00%), the bond trades at a discount of $81.11 below par value.

---

Price-Yield Relationships: Par, Discount, and Premium Bonds

A bond's coupon rate is contractually fixed at issuance in the trust indenture. However, macroeconomic conditions, inflation expectations, and Bank of Canada monetary policy adjustments cause prevailing market yields to fluctuate continuously. The relationship between the bond's fixed coupon rate and prevailing market yields determines whether the bond trades at par, at a discount, or at a premium.

Prevailing Market Yield vs. Fixed Coupon Rate
 ├── Yield == Coupon Rate ──> Par Bond (Price == 100)
 ├── Yield >  Coupon Rate ──> Discount Bond (Price < 100)
 └── Yield <  Coupon Rate ──> Premium Bond (Price > 100)

The Par Bond ($P = 100$)

When prevailing market interest rates equal the bond's coupon rate, the present value of future cash flows discounted at that rate exactly matches the par value. The bond trades at 100.00 ($1,000 per $1,000 par unit). An investor purchasing a par bond earns an investment return derived entirely from coupon payments, with zero capital gain or loss at maturity.

The Discount Bond ($P < 100$)

When market interest rates climb above the bond's fixed coupon rate, newly issued bonds offer higher cash payments. To entice investors to purchase the existing lower-coupon bond in the secondary market, its price must fall below par value (e.g., 94.50, or $945.00 per $1,000 bond). For a discount bond:

$\text{Coupon Rate} < \text{Current Yield} < \text{Yield to Maturity}$

The total return on a discount bond consists of two components: the ongoing semi-annual coupon income, plus a built-in capital gain realized as the bond's price pulls upward toward par ($1,000) as maturity approaches.

The Premium Bond ($P > 100$)

When market interest rates drop below the bond's fixed coupon rate, the bond's above-market cash flows become highly desirable. Investors bid up the price above par value (e.g., 106.25, or $1,062.50 per $1,000 bond). For a premium bond:

$\text{Yield to Maturity} < \text{Current Yield} < \text{Coupon Rate}$

The total return on a premium bond is dampened by an inevitable capital loss as the bond's market price pulls downward toward par at maturity.

Summary of Bond Pricing Relationships

---

Clean Price vs. Dirty Price and Canadian Accrued Interest

When a bond is traded between its scheduled semi-annual coupon payment dates, interest is continuously accumulating for the seller who held the security since the last payment date. The buyer receives the full semi-annual coupon on the next scheduled payment date, even though they only held the bond for a portion of the period. Consequently, market conventions require the buyer to reimburse the seller for the interest accrued up to the settlement date.

Definitions: Clean vs. Dirty Price

• Clean Price (Quoted Price): The price of the bond published on exchange boards, Bloomberg screens, and dealer quote sheets. It reflects the pure present value of future cash flows and excludes accrued interest.
• Dirty Price (Gross Price / Settlement Price): The actual total dollar amount the buyer must deliver to the seller on the settlement date. It comprises the clean quoted price plus accrued interest:

$\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest}$

The Canadian Accrued Interest Convention: Actual/365 Days

In Canadian fixed-income markets, accrued interest for both Government of Canada bonds and Canadian corporate bonds is calculated using the actual/365-day convention (counting the actual number of calendar days elapsed divided by 365 days):

$\text{Accrued Interest} = \text{Annual Coupon Dollar Amount} \times \frac{\text{Actual Days Elapsed}}{365}$

For a $1,000 par bond with coupon rate $c$:

$\text{Accrued Interest} = (\$1,000 \times c) \times \frac{\text{Actual Days Elapsed}}{365}$

Important Distinction: While the U.S. corporate bond market utilizes a 30/360-day convention, Canadian financial practice standardizes on actual/365 for all domestic government and corporate bonds.

Settlement Timing: Standard T+1 Settlement

Securities transactions in Canada follow a standard T+1 settlement cycle (effective May 27, 2024), meaning that a trade executed on Monday settles on Tuesday. When calculating accrued interest, the elapsed period runs from and including the previous coupon date up to, but not including, the settlement date.

Bonds Trading "Flat"

If an issuer defaults on its contractual interest payments, or if the security is an income bond where interest payments are contingent upon corporate earnings, the bond trades "flat". When a bond trades flat, no accrued interest is added to the clean price. The buyer pays strictly the quoted clean price, and any potential past-due coupon rights transfer to the buyer.

---

Comprehensive Step-by-Step Worked Example: Canadian Accrued Interest & Settlement

Settlement Scenario: An institutional portfolio manager in Toronto purchases $200,000 par value of an 8.00% Government of Canada marketable bond at a quoted clean price of 103.50. The bond pays semi-annual coupons on March 15 and September 15.

• Trade Date: Wednesday, June 10 (non-leap year)
• Settlement Standard: T+1 $\implies$ Settlement Date: Thursday, June 11

Step 1: Calculate the Actual Days Elapsed from Previous Coupon to Settlement Date

• Previous coupon date: March 15
• Settlement date: June 11 (accrued interest accumulates through June 10)

We count the actual calendar days, including the previous coupon date and excluding the settlement date:

• March: 17 days (March 15 through March 31, counting March 15)
• April: 30 days (full month)
• May: 31 days (full month)
• June: 10 days (June 1 through June 10)
• Total Days Elapsed: $17 + 30 + 31 + 10 = 88$ days (check: day 162 minus day 74 of a non-leap year)

Step 2: Calculate Total Annual Coupon Payment

$\text{Annual Coupon} = \$200,000 \times 8.00\% = \$16,000.00$

Step 3: Calculate Accrued Interest (Actual/365)

$\text{Accrued Interest} = \$16,000.00 \times \frac{88}{365} = \$16,000.00 \times 0.241096 = \$3,857.53$

Step 4: Calculate Clean Dollar Principal Amount

$\text{Clean Principal} = \$200,000 \times \frac{103.50}{100} = \$207,000.00$

Step 5: Calculate Total Settlement Amount (Dirty Price)

$\text{Gross Settlement Invoice} = \text{Clean Principal} + \text{Accrued Interest}$ $\text{Gross Settlement Invoice} = \$207,000.00 + \$3,857.53 = \$210,857.53$

The purchasing dealer must wire $210,857.53 to the clearinghouse on Thursday, June 11.