Fixed Income Securities
Bonds, debentures, strips, GICs, yield calculations, duration, and price sensitivity. This element demands both conceptual understanding and calculation fluency — every formula is worked through step-by-step.
Regulatory Requirements for Debt Markets
Unlike equity markets which trade on visible, regulated exchanges (like the TSX), the Canadian debt market is predominantly an over-the-counter (OTC) market. Bonds and fixed income securities are traded directly between dealers and clients, without a centralized exchange displaying real-time prices. This opacity creates unique regulatory challenges that CIRO addresses through specific debt market rules.
General Requirements — Promoting Fair and Efficient Debt Markets
CIRO's mandate in debt markets is to ensure:
- Price fairness: Clients must receive fair prices when buying or selling fixed income securities. Because there is no public exchange, the dealer has discretion in pricing — but must mark up or mark down at a rate that is fair given market conditions.
- Transparency: CIRO requires post-trade reporting of bond transactions. Dealers must report their OTC bond trades to the CIRO Bond Reporting System so pricing data is available to regulators and the market.
- Market integrity: Dealers must not engage in practices that distort bond prices, create artificial markets, or disadvantage retail clients.
- Best execution: When executing bond trades for clients, dealers must seek the best reasonably available price and terms given the market at the time.
In the equity market, you can see the bid and ask prices for any listed stock in real time. In the bond market, most trades happen OTC and prices are negotiated. A retail client buying a bond does not know if they are getting a fair price — this is why CIRO imposes specific markup disclosure and fair pricing obligations on bond dealers that do not exist in the same form for equity trading.
Policies and Procedures
Every investment dealer that trades fixed income securities must maintain written policies and procedures specifically covering:
- Pricing methodology — how markups/markdowns are determined for retail client bond trades
- Documentation of pricing justification for all retail bond transactions
- Supervisory review of bond trade pricing to detect unfair markups
- Trade reporting obligations — when and how to report bond trades to CIRO
- Inventory management — how the dealer manages its bond inventory and prices from that inventory
- Conflicts of interest in bond trading — especially when the dealer is acting as principal (selling from its own inventory)
Prohibited Practices in Debt Markets
CIRO's rules prohibit specific practices in fixed income trading that could harm retail clients or undermine market integrity. These apply at both the dealer (firm) and Approved Person (individual) level.
🏢 Investment Dealer Obligations
- Must not charge markups/markdowns on bond trades that are unfair or unreasonable relative to current market prices
- Must not misrepresent bond prices or yields to clients
- Must not engage in "parking" — holding securities in a client's account without the client's knowledge or consent to facilitate the dealer's inventory management
- Must maintain fair pricing records and documentation for all retail bond trades
- Must report bond trades to CIRO's bond reporting system within required timeframes
- Must maintain adequate policies and procedures covering all debt market activities
👤 Approved Person Obligations
- Must not recommend bond trades that are not suitable for the client (standard suitability obligation applies to bonds)
- Must not misrepresent the yield, risk, or price of a fixed income security
- Must not quote bond prices that do not reflect current market conditions
- Must not trade bonds for personal benefit at the expense of client orders (front-running)
- Must fully disclose when the dealer is acting as principal (trading from inventory) vs. agent in a bond transaction
- Must understand the bond product sufficiently to meet KYP obligations
Markup Disclosure
When a dealer sells a bond to a retail client from its own inventory (acting as principal), it typically charges a markup — the difference between the price at which it acquired the bond and the price at which it sells it to the client. This markup must be fair and reasonable. CIRO does not set a maximum markup percentage but requires that it reflect current market conditions and be disclosed where required.
When a dealer is acting as principal in a bond trade (selling from its own inventory), this is a conflict of interest — the dealer profits from the spread/markup. This must be disclosed to the client. An RR telling a client "I found you a great bond deal" while actually selling from the firm's inventory at a significant markup, without disclosure, is a violation.
Types and Features of Fixed Income Products
Fixed income securities are debt instruments where the issuer borrows money from investors and promises to repay the principal at maturity, along with periodic interest (coupon) payments. The four main categories tested on the RSE are government of Canada securities, provincial/municipal, corporate, and GICs.
Government of Canada Securities
The federal government issues fixed income securities to fund its operations and debt. These are considered the lowest-risk fixed income instruments available in Canada — backed by the full faith and credit of the Canadian government.
| Instrument | Maturity | How Traded | Key Features |
|---|---|---|---|
| Treasury Bills (T-Bills) | 3 months, 6 months, 1 year | Sold at a discount to face value; no coupon | Highest liquidity; virtually risk-free; benchmark for short-term rates. Sold through the Bank of Canada's weekly auction. |
| Government of Canada Bonds | 2, 5, 10, 30 years (at issuance) | OTC; also used as pricing benchmarks | Semi-annual coupon payments; highest credit quality among Canadian bonds; key interest rate benchmarks |
| Canada Savings Bonds (CSBs) | Various — historically up to 10 years | Sold directly to retail investors (program ended 2017) | Guaranteed by federal government; could be redeemed at par at any time. No longer issued but may appear on exams for context. |
| Real Return Bonds (RRBs) | Long-term (30+ years) | OTC | Principal and interest payments adjusted for CPI inflation. Protects against purchasing power erosion. Lower nominal yield than equivalent conventional bonds. |
T-bills do not pay a coupon. The investor buys the T-bill at a discount to its face (par) value. The profit is the difference. For example: buy a 91-day $10,000 T-bill at $9,900 → receive $10,000 at maturity → profit = $100. The yield is calculated on the discount basis, not like a regular bond yield.
Provincial and Municipal Government Securities
Provincial Bonds
Provinces issue bonds to fund capital projects, healthcare, infrastructure, and budget deficits. Provincial bonds carry slightly higher risk and yield than GoC bonds because the federal government does not explicitly guarantee provincial debt (though an implicit "too big to fail" assumption exists for large provinces).
- Credit quality: Rated by DBRS Morningstar, Moody's, S&P. Ontario and Quebec (largest issuers) are typically AA-rated. Smaller provinces may be rated lower.
- Yield spread: Provincial bonds trade at a spread above equivalent Government of Canada bonds. Ontario 10-year bonds might yield 15–25 basis points (0.15–0.25%) above GoC 10-year bonds.
- Market: Large and liquid; provincials are a core component of most Canadian bond portfolios.
Municipal Bonds
Municipalities (cities, towns) also issue bonds but the market is smaller and less liquid. Some municipal bonds benefit from provincial guarantees. Because municipalities cannot print money and have limited revenue sources, they carry more risk than provincial bonds — but still generally investment grade.
Corporate Securities
Corporations issue debt to fund operations, acquisitions, and capital expenditures. Corporate debt offers higher yields than government bonds to compensate for higher credit risk.
| Type | Security / Seniority | Risk Level | Key Features |
|---|---|---|---|
| Mortgage Bond | Secured by real property (land and buildings) | Lowest (secured) | If issuer defaults, bondholders have first claim on the pledged property. Most senior. |
| Collateral Trust Bond | Secured by other securities (stocks, bonds) held in trust | Low (secured) | Securities are pledged as collateral; bondholders can liquidate collateral on default. |
| Equipment Trust Certificate | Secured by specific equipment (aircraft, rail cars) | Low–Medium (secured) | Common in transportation and capital-intensive industries; lender owns equipment until loan repaid. |
| Debenture | Unsecured — backed only by general creditworthiness | Medium (unsecured senior) | Most common form of corporate bond. No specific asset pledged. Holders are general creditors in bankruptcy. |
| Subordinated Debenture | Unsecured, junior to debentures | Higher | In bankruptcy, subordinated holders paid only after senior debenture holders. Must offer higher yield to compensate. |
Credit Ratings
Credit ratings assess the probability of default. Investment-grade bonds (BBB- and above by S&P; Baa3 and above by Moody's) are generally suitable for retail investors. High-yield (junk) bonds (BB+ and below) offer higher returns but with substantially higher default risk.
| Category | S&P | Moody's | DBRS | Meaning |
|---|---|---|---|---|
| Investment Grade — Highest | AAA | Aaa | AAA | Virtually no risk; GoC level |
| Investment Grade — High | AA | Aa | AA | Very high quality; major provincials |
| Investment Grade — Strong | A | A | A | Strong capacity to pay; large corporate |
| Investment Grade — Adequate | BBB | Baa | BBB | Adequate capacity; minor impairment possible |
| High Yield / Speculative | BB and below | Ba and below | BB and below | Speculative; significant risk of default |
Banker's Acceptances (BAs)
A short-term instrument where a corporation draws a draft on a bank, and the bank "accepts" it — guaranteeing payment. BAs trade at a discount (like T-bills) with maturities of 30–180 days. They carry minimal risk (bank guaranteed) and trade in the money market.
Guaranteed Investment Certificates (GICs)
GICs are issued primarily by banks, trust companies, and credit unions. The investor deposits a principal amount for a fixed term in exchange for a guaranteed rate of interest.
- Issuer: Chartered banks (CDIC-eligible), trust companies, credit unions (provincial deposit insurance)
- Term: 30 days to 5 years typically; some institutions offer longer terms
- Rate: Fixed (most common) or variable (linked to prime or another benchmark)
- Liquidity: Non-redeemable GICs cannot be cashed before maturity. Cashable/redeemable GICs allow early withdrawal, usually after a minimum holding period, at a lower rate.
- CDIC coverage: Eligible GICs at CDIC member institutions (chartered banks) are covered up to $100,000 per depositor per category. This is separate from CIPF coverage.
GICs are deposits, not securities. They are not traded on secondary markets. Their return is guaranteed regardless of interest rate movements (unlike bonds, which change in price when rates move). GICs have no market risk for a buy-and-hold investor, but have liquidity risk (especially non-redeemable ones). Bonds fluctuate in price with interest rates.
Fixed Income Terminology
Before tackling calculations, you must have crystal-clear definitions of every fixed income term. These are directly tested in scenario questions where misunderstanding one term leads to a wrong answer on every related calculation.
Core Bond Terminology
Government of Canada bond: $1,000 par, 5% annual coupon, 3-year maturity, paying semi-annually
6mo
12mo
18mo
24mo
30mo
36mo (maturity)
Each coupon = $1,000 × 5% ÷ 2 = $25 | Total received = 6 × $25 + $1,000 = $1,150
Settlement and Covenants
Settlement
Bond settlement in Canada typically occurs on T+2 (trade date plus 2 business days) for corporate bonds. Government of Canada bonds settle on T+1 following market practice changes aligned with North American settlement modernization. Settlement means the buyer delivers cash and receives the securities (or vice versa) on the settlement date — not the trade date.
Accrued interest is calculated from the last coupon date up to and including the settlement date — not the trade date. This is because ownership (and therefore entitlement to interest) transfers on settlement date.
Covenants
Bond covenants are legally binding conditions in the bond indenture (trust deed) that the issuer must comply with. They protect bondholders from actions the issuer might take that would reduce the bondholder's security.
If an issuer violates a covenant, it constitutes an event of default — bondholders (through the trustee) can demand immediate repayment of all principal and accrued interest, or take legal action to protect their rights.
Characteristics, Risks & Returns
✅ Investor Advantages
- Predictable, regular income (coupon payments)
- Principal repayment at maturity (if held to maturity)
- Priority over equity holders in bankruptcy
- Wide range of credit qualities and maturities available
- Government bonds: virtually no credit risk
- Can be actively traded for capital gain if interest rates fall
- Portfolio diversification — low correlation to equities in many environments
❌ Investor Disadvantages
- Interest rate risk — bond prices fall when rates rise
- Inflation risk — fixed coupon income loses purchasing power over time
- Credit risk — issuer may default (more relevant for corporates)
- Reinvestment risk — coupons may be reinvested at lower rates
- Liquidity risk — some bonds are thinly traded (esp. corporate, municipal)
- Call risk — issuer may redeem early when rates fall, forcing reinvestment at lower yields
- Opportunity cost — if rates rise significantly, locked into lower yield
✅ Issuer Advantages
- Lower cost of capital than equity (interest is tax-deductible; dividends are not)
- No dilution of ownership (unlike issuing shares)
- Bondholders have no voting rights
- Can structure bonds with covenants to reduce credit risk perception → lower yield required
- Flexible maturity — can match financing to asset life
❌ Issuer Disadvantages
- Fixed obligation — must make coupon payments regardless of profitability
- Covenants restrict management flexibility
- Must repay principal at maturity — creates refinancing risk
- Default on any covenant or payment = event of default; can trigger insolvency
- If interest rates fall, may be locked into paying high coupon (unless callable)
Sources of Risk for Fixed Income Investors
| Risk Type | Description | Who It Affects Most | Mitigation |
|---|---|---|---|
| Interest Rate Risk | Bond prices fall when market interest rates rise. The longer the maturity and lower the coupon, the greater the price decline. | Long-term bond holders | Short-duration bonds; laddering; floating rate bonds |
| Credit / Default Risk | Issuer fails to make coupon payments or repay principal | Corporate bond holders; high-yield | Diversification; higher rated bonds; secured debt |
| Reinvestment Risk | Coupon payments must be reinvested at lower rates if interest rates fall after purchase | All coupon-paying bond holders | Zero-coupon bonds (no reinvestment needed) |
| Inflation Risk | Rising inflation erodes purchasing power of fixed coupon payments | Long-term bond holders | Real Return Bonds (RRBs); shorter maturities |
| Liquidity Risk | Cannot sell quickly at a fair price | Corporate, municipal, private placements | Government bonds; large-issue corporates |
| Call Risk | Issuer redeems bond early at a time convenient to issuer (usually when rates have fallen) | Holders of callable bonds | Non-callable bonds; analyze call schedule |
| Currency Risk | Foreign currency bonds expose investor to exchange rate fluctuations | Foreign-currency bond holders | CAD-denominated bonds; currency hedging |
Costs of Acquiring and Holding Fixed Income
- Transaction costs (spread/markup): In the OTC bond market, the bid-ask spread is the primary transaction cost. For corporate bonds, spreads can be 0.25–1.0% of face value. For GoC bonds, spreads are much tighter (0.02–0.10%). The dealer's markup when acting as principal is also a cost.
- Accrued interest: When buying a bond between coupon dates, the buyer pays accrued interest to the seller — this is a cash outflow that is recovered with the next coupon payment.
- Custodial fees: Annual fees charged by the dealer or custodian to hold the bond in an account.
- Opportunity cost: Money tied up in a fixed-rate bond cannot be redeployed if better opportunities arise — especially relevant in a rising rate environment.
Special Fixed Income Instruments
Strip Bonds (Zero-Coupon Bonds)
A strip bond (also called a stripped bond or zero-coupon bond) is created when a dealer "strips" the coupon payments from a conventional bond — separating the interest payments and the principal repayment into individual zero-coupon instruments that trade independently.
Original Bond
5% $1,000 bond, 3 years, semi-annual
→ 6 coupon payments of $25
→ 1 principal payment of $1,000
7 Separate Strip Bonds
- 6-month strip: $25 face value
- 12-month strip: $25 face value
- 18-month strip: $25 face value
- 24-month strip: $25 face value
- 30-month strip: $25 face value
- 36-month strip: $25 face value
- 36-month principal: $1,000 face value
Key Features of Strips
- No coupon payments: Sold at a deep discount to face value. The return comes entirely from the difference between the purchase price and the face value received at maturity.
- No reinvestment risk: Since there are no coupons to reinvest, there is no reinvestment risk. The YTM is locked in at purchase.
- Higher price volatility: Because all cash flows come at maturity (no coupons), strip bonds have the longest effective duration for their maturity — making them the most sensitive to interest rate changes.
- Tax treatment in Canada: The "phantom income" problem — the CRA taxes the accrued discount as interest income each year, even though no cash is received until maturity. This makes strips more tax-efficient when held inside registered accounts (RRSP, TFSA) where the annual phantom income accrual is sheltered.
Floating Rate Bonds (Variable Rate)
A floating rate bond has a coupon that resets periodically (typically every 3 or 6 months) based on a reference rate plus a spread. In Canada, the reference rate is now typically the CORRA (Canadian Overnight Repo Rate Average) or the Term CORRA. Example: CORRA + 150 bps (1.50%).
- Interest rate risk is minimal: As rates rise, the coupon also rises, so the bond's price stays close to par. This is the defining advantage.
- Disadvantage in falling rate environments: When rates fall, the coupon income also falls.
- Suitable for: Investors who expect rates to rise but still want fixed income exposure; short to medium term.
- Used by: Corporations, banks, governments for short-to-medium term financing.
Callable and Puttable Bonds
Callable Bonds
A callable bond gives the issuer the right to redeem (buy back) the bond before maturity at a specified call price (usually at par or a slight premium). The issuer will typically call bonds when interest rates have fallen — they can refinance at a lower rate.
- Call risk to investor: The bond is most likely to be called when the investor least wants it — when rates are low and reinvestment alternatives offer poor yields.
- Call premium: To compensate investors for call risk, callable bonds offer a higher yield than equivalent non-callable bonds.
- Yield to Call (YTC): Separate yield calculation assuming the bond is called at the first call date — investors must know both YTM and YTC and consider the worst-case scenario.
- Yield to worst (YTW): The lower of YTM or YTC — represents the minimum yield an investor will earn.
Puttable Bonds
A puttable bond gives the investor the right to sell (put) the bond back to the issuer at a specified put price (usually par) before maturity. This protects the investor if interest rates rise — they can put the bond back and reinvest at higher rates.
- Advantage to investor: Provides downside price protection in a rising rate environment.
- Lower yield than equivalent non-puttable: Investors pay for the put option through accepting a lower yield.
- Rare in practice: Less common than callable bonds; primarily used in structured or corporate issues.
Callable = issuer has the right to call → investor faces reinvestment risk. Puttable = investor has the right to put → investor is protected against rate rises. The option always benefits whoever holds it. In callable bonds, the issuer holds the option → investor is compensated with a higher yield. In puttable bonds, the investor holds the option → investor accepts a lower yield.
Convertible Bonds
A convertible bond gives the investor the right to convert the bond into shares of the issuing company at a pre-specified conversion price (or conversion ratio). It combines bond features (downside protection, fixed income) with equity upside potential.
- Conversion ratio: The number of shares received per bond. If the conversion price is $50 and par is $1,000 → ratio = 20 shares per bond.
- When to convert: If the stock trades above the conversion price, converting is profitable. If below, the investor holds the bond for its fixed income characteristics.
- Lower coupon rate: Because the conversion option has value, convertible bonds offer lower coupons than equivalent non-convertible bonds. Investors pay for the option through lower income.
- Risk/return profile: Offers asymmetric returns — limited downside (bond floor) if the stock falls; participates in upside if the stock rises above conversion price.
Extendable Bonds
An extendable bond gives the investor the right to extend the maturity of the bond for an additional period beyond the original maturity date, usually at a pre-set new coupon rate. If the new coupon is attractive relative to current market rates, the investor will extend. This protects against falling reinvestment rates.
The opposite of an extendable is a retractable bond — the investor has the option to shorten (retract) the maturity. Retractable bonds protect investors who are worried about rising rates — they can redeem early and reinvest at higher rates. These are similar in concept to puttable bonds.
Sinking Fund and Purchase Fund Provisions
Sinking Fund
A sinking fund provision requires the issuer to set aside money periodically to retire a portion of the bond issue before maturity. The trustee uses these funds to purchase bonds in the open market or by lottery (randomly selecting outstanding bonds to redeem at par).
- Investor perspective: Reduces the outstanding debt → lower default risk as maturity approaches. However, if the bond is selected by lottery for early retirement, the investor may be forced to reinvest at lower rates (reinvestment risk). Sinking fund bonds generally offer lower yields than non-sinking bonds of equivalent credit quality.
- Issuer perspective: Reduces refunding risk at maturity; demonstrates fiscal discipline; may lower borrowing cost.
Purchase Fund
A purchase fund is similar but gives the issuer more flexibility — the issuer is only required to purchase bonds in the open market if they are available at or below a specified price (usually par). Unlike a sinking fund, there is no obligation to retire bonds if market prices are above the specified level.
Sinking fund: MANDATORY periodic debt retirement (issuer MUST retire bonds each period). Purchase fund: CONDITIONAL — issuer only buys bonds if available at or below the trigger price. The sinking fund creates a stronger obligation and typically reduces credit risk more definitively.
Bond Yield Calculations
This section is heavily calculation-focused. Every formula is shown, then walked through step-by-step with a numerical example. Use the interactive calculator at the end to practice.
Current (Income) Yield
The current yield measures only the annual coupon income as a percentage of the current market price. It is simple to calculate but ignores the capital gain or loss component.
Worked Example: A bond has a 6% coupon rate (par $1,000) and is currently priced at $950.
• Bond at discount (price < $1,000): Current Yield > Coupon Rate
• Bond at par (price = $1,000): Current Yield = Coupon Rate
• Bond at premium (price > $1,000): Current Yield < Coupon Rate
Approximate Yield to Maturity (Approximate YTM)
YTM is the most important yield measure. The exact YTM requires a financial calculator or iterative calculation. The RSE exam tests the approximate YTM formula — an algebraic approximation.
Where: N = years to maturity | Annual Coupon = coupon rate × par | Par = $1,000 (typically)
Worked Example: Bond: 6% coupon, $1,000 par, current price $920, 10 years to maturity.
YTM for a Bond Trading at a Premium
Example: Bond: 6% coupon, $1,000 par, current price $1,080, 5 years to maturity.
• Bond at discount (price < par): YTM > Current Yield > Coupon Rate
• Bond at par (price = par): YTM = Current Yield = Coupon Rate
• Bond at premium (price > par): YTM < Current Yield < Coupon Rate
Memory trick: "Discount Deals: YTM is the highest measure. Premium: YTM is the lowest."
Yield on a Zero-Coupon / Strip Bond
Strip bonds have no coupon payments. The yield calculation is based entirely on the difference between the purchase price and the face value received at maturity.
Example: A 10-year strip bond with face value $1,000 is priced at $558.39. What is the yield?
Yield Curves
A yield curve plots the yields of bonds of the same credit quality (typically Government of Canada bonds) against their terms to maturity. It is one of the most important analytical tools in fixed income.
📈 Normal (Upward Sloping)
Long-term yields > short-term yields. The most common shape. Investors demand a "term premium" for locking up money longer. Implies the economy is expected to grow and inflation is expected to be stable or rising.
Example: 3-month = 3%, 5-year = 4%, 10-year = 4.8%, 30-year = 5.2%
➡️ Flat
All maturities yield approximately the same. Uncommon; typically a transition between normal and inverted. Seen when central bank is tightening and near a peak rate. Signal of economic uncertainty.
Example: 3-month = 4.5%, 5-year = 4.5%, 30-year = 4.5%
📉 Inverted (Downward Sloping)
Short-term yields > long-term yields. Historically the most reliable predictor of a recession (usually 12–18 months ahead). Occurs when market expects rates to fall — suggesting economic slowdown ahead.
Example: 3-month = 5.2%, 5-year = 4.1%, 30-year = 3.8%
Credit Spread Curves
When corporate bonds are plotted on a yield curve, they plot above the government curve. The difference is the credit spread — the additional yield investors demand to compensate for credit risk. Credit spreads widen during economic stress (risk-off) and narrow during growth periods (risk-on).
Why Yield Curves Matter for RRs
- Investment strategy: An inverted curve may signal to reduce long-duration bond exposure and increase cash or short-term instruments.
- Suitability: A client seeking income in a flat yield curve environment gets little extra return for taking long-duration risk — may be better served by short to medium-term bonds.
- Riding the yield curve: Buying bonds at the long end and selling before maturity to benefit from price appreciation as the bond rolls down to shorter maturities (only works in a stable or steepening curve).
Bond Price Relationships & Duration
The Five Key Bond Price Rules
These rules describe how bond prices behave in response to changes in yields and maturity. They are fundamental to bond analysis and are always tested on the exam.
Bond prices move in the OPPOSITE direction to interest rates. When market interest rates rise → bond prices fall. When rates fall → bond prices rise. This is the most fundamental rule in fixed income.
Longer-maturity bonds experience GREATER price changes for a given change in yield than shorter-maturity bonds. A 30-year bond will drop much more in price when rates rise 1% than a 5-year bond.
For a given change in yield, the price increase from a rate decrease is LARGER than the price decrease from an equal rate increase. This is bond convexity — the price-yield relationship is curved, not straight.
For bonds with the same maturity, LOW coupon bonds are MORE price-sensitive than high coupon bonds. A low-coupon bond delivers most of its cash flow far in the future (at maturity) → more sensitive to discount rate changes. Zero-coupon bonds are the most volatile of all.
The increase in price volatility from increasing maturity diminishes as maturity increases. Going from 1 to 2 years adds more price sensitivity than going from 29 to 30 years. Duration increases at a decreasing rate with maturity.
Duration — Macaulay and Modified
What Is Duration?
Duration is a measure of a bond's weighted average time to receive cash flows, expressed in years. It also serves as the primary measure of a bond's price sensitivity to interest rate changes. There are two types that matter for the RSE exam:
| Measure | What It Tells You | Formula | Use |
|---|---|---|---|
| Macaulay Duration | The weighted average time (in years) to receive all the bond's cash flows, weighted by present value. For a zero-coupon bond, Macaulay Duration = term to maturity. | Σ [PV(CFₜ) × t] ÷ Bond Price | Conceptual understanding; immunization strategies |
| Modified Duration | The percentage change in a bond's price for a 1% (100 bps) change in yield. A modified duration of 7 means the bond's price will change by approximately 7% for every 1% change in yield. | Macaulay Duration ÷ (1 + YTM/m) where m = payments per year | Practical price sensitivity calculation — used every day |
Key Duration Properties
- Zero-coupon bond: Duration = term to maturity (all cash flow at maturity)
- Coupon bonds: Duration < term to maturity (receive coupons before maturity, pulling weighted average earlier)
- Higher coupon → shorter duration (more cash flows come earlier)
- Longer maturity → longer duration (cash flows are further away)
- Higher yield → shorter duration (higher discount rate weights nearer cash flows more heavily)
Impact of Economic Factors on Bond Prices
| Economic Factor | Impact on Interest Rates | Impact on Bond Prices |
|---|---|---|
| Bank of Canada raises policy rate | ↑ Short-term rates rise; long-term may follow | ↓ Bond prices fall, especially long-duration bonds |
| Inflation expectations increase | ↑ Nominal yields rise to maintain real returns | ↓ Bond prices fall; RRBs partially protected |
| Economic recession / slowdown | ↓ Rates cut to stimulate economy | ↑ Bond prices rise; flight to quality to government bonds |
| Credit downgrade of issuer | ↑ Required yield on that issuer's bonds increases | ↓ That issuer's bond price falls; credit spread widens |
| Improved corporate earnings / credit improvement | ↓ Required yield decreases (credit spread narrows) | ↑ Corporate bond price rises |
| Government increases borrowing (more bond supply) | ↑ Higher supply → higher yield needed to attract buyers | ↓ Government bond prices may fall slightly |
Modified Duration — Price Change Calculation
Modified duration is the key tool for estimating how much a bond's price will change for a given change in yield. This is a must-know calculation for the RSE exam.
Where: m = coupon payments per year (typically 2 for semi-annual) | Δ Yield = change in yield expressed as a decimal
Worked Example 1: Bond: 8% coupon (semi-annual), 10-year maturity, YTM = 8% (at par), Macaulay Duration = 7.07 years. Yields rise by 0.50% (50 bps). What happens to price?
Worked Example 2: A bond has Modified Duration = 5.2. Current price = $950. Yields fall by 1% (100 bps). Estimated new price?
Modified duration is only an approximation — it assumes a linear relationship between price and yield. In reality, the price-yield relationship is convex (curved). For small yield changes, the linear approximation is accurate. For large changes (e.g., 200+ bps), the actual price change will be better than predicted (because of convexity — the curve always lies above the tangent line). The exam tests the linear approximation.
🧮 Interactive Modified Duration Calculator
Time Value of Money & Bond Pricing
The time value of money (TVM) is the foundational concept behind bond pricing. A dollar received today is worth more than a dollar received in the future, because today's dollar can be invested to earn returns. Bond pricing discounts all future cash flows (coupons + principal) back to the present using the required yield as the discount rate.
Present Value of a Fixed Income Security
Where: C = coupon payment per period | r = yield per period (YTM/m) | n = total number of periods | Par = face value
Example: Find the price of a bond: 6% coupon, semi-annual, 3-year maturity, par $1,000, required yield = 8%.
Solving for Missing Variables
The TVM framework has four key variables: Present Value (PV), Future Value (FV), Interest Rate (r), and Number of Periods (n). Given any three, you can solve for the fourth. Common exam scenarios:
| Known Variables | Solving For | Method |
|---|---|---|
| Coupon, YTM, maturity, par | Bond Price (PV) | PV formula above — standard bond pricing |
| Bond price, coupon, maturity, par | YTM | Approximate YTM formula; or trial-and-error with PV formula |
| Price, par, maturity (zero-coupon) | Yield | r = (Par/Price)^(1/n) − 1 |
| PV, FV, r | n (number of periods) | n = ln(FV/PV) ÷ ln(1+r) |
Simple TVM Example — Solving for Missing Variable:
How many years does it take for $5,000 to grow to $8,000 at 6% per year (compounded annually)?
Bond Pricing Calculator
🧮 Bond Price Calculator (PV of Fixed Income Security)
🧮 Approximate YTM Calculator
🧮 Current Yield Calculator
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