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CIRE Fixed Income Yields Cheat Sheet
Fixed income is one of the most formula-dense sections of the CIRE. This sheet covers the bond price equation, every yield measure, duration and convexity with worked numbers, accrued interest day-count conventions, yield curve interpretation, T+1 settlement rules (effective May 27, 2024), callable and putable bonds, spread definitions, and the credit rating ladder. CIRO Proficiency Model standards apply. Last reviewed: 2026-05-08.
1. Bond Price Equation
P = sum of [C / (1 + y)^t] for t = 1 to n, plus F / (1 + y)^n
P = full (dirty) price; C = coupon per period; y = yield per period; F = face (par) value; n = total periods; t = period number.
Worked example:
5% annual coupon bond, $1,000 par, 3 years to maturity, 6% annual yield.
- Year 1: $50 / 1.06 = $47.17
- Year 2: $50 / 1.06^2 = $44.50
- Year 3: $1,050 / 1.06^3 = $881.68
- P = $47.17 + $44.50 + $881.68 = $973.35
The bond trades at a discount because the coupon rate (5%) is below the market yield (6%). Price and yield move in opposite directions.
Dirty price vs clean price:
The equation above gives the dirty (full) price, which includes accrued interest. Market quotations use the clean price (flat price). The buyer pays the dirty price; the quoted price is the clean price. Clean = dirty - accrued interest.
2. Current Yield, YTM, and YTC
Current yield
Current yield = annual coupon / current price
Example: $1,000 par bond with 6% coupon ($60 annual) priced at $950. Current yield = $60 / $950 = 6.32%.
Gotcha: Current yield ignores the capital gain or loss at maturity. For a discount bond, YTM > current yield > coupon rate. For a premium bond, coupon rate > current yield > YTM. Memorize these relationships.
YTM Rule of Thumb
YTM ≈ (C + (F - P) / n) / ((F + P) / 2)
C = annual coupon; F = face value; P = current price; n = years to maturity.
Example: $1,000 par, $55 annual coupon, price $940, 4 years to maturity.
YTM ≈ (55 + (1000 - 940) / 4) / ((1000 + 940) / 2) = (55 + 15) / 970 = 70 / 970 = 7.22%
Gotcha: The approximation works best when the bond is near par. For bonds far from par or with long maturities, error grows. When answer choices are close (within 0.25%), iterate or use the full discounted cash flow model.
Yield to Call (YTC)
Same formula as YTM but: n = years to first call date, F = call price (not par)
Example: Bond par $1,000, coupon $70, price $1,050, call price $1,020 in 2 years.
YTC ≈ (70 + (1020 - 1050) / 2) / ((1020 + 1050) / 2) = (70 - 15) / 1035 = 55 / 1035 = 5.31%
For a premium bond with a near call date, YTC is typically below YTM. The yield-to-worst (YTW) is the lower of YTM and all YTCs; use YTW for premium bonds with embedded call options.
3. Macaulay Duration, Modified Duration, and Convexity
Macaulay Duration
MacDur = sum of [t x (PV of cash flow at t)] / P
Weighted-average time until cash flows are received, measured in years.
Example: A 2-year zero-coupon bond: MacDur = 2.0 exactly. A 2-year 6% coupon bond has MacDur slightly below 2.0 because some cash flows arrive at year 1 (coupon), not just at year 2.
Key rules: Duration increases with maturity; decreases with higher coupon rate; decreases with higher yield. A zero-coupon bond always has MacDur = maturity.
Modified Duration
ModDur = MacDur / (1 + y)
% price change ≈ -ModDur x (change in yield)
Example: MacDur = 4.8 years, annual yield = 5%. ModDur = 4.8 / 1.05 = 4.571. If yields rise 50 basis points (0.50%): estimated price change = -4.571 x 0.005 = -2.29%.
Gotcha: Modified duration is a linear approximation valid for small yield changes. For large moves (>100 bps), add the convexity adjustment.
Convexity Adjustment
delta-P / P ≈ -ModDur x delta-y + 0.5 x Convexity x (delta-y)^2
Example: ModDur = 6.0, Convexity = 55, yields rise 200 bps (delta-y = 0.02).
- Duration estimate: -6.0 x 0.02 = -12.0%
- Convexity adjustment: +0.5 x 55 x 0.02^2 = +1.1%
- Total estimated price change: -10.9%
Convexity is always positive for option-free bonds, so the actual price change is always better than the duration estimate predicts. Callable bonds can exhibit negative convexity near the call price.
4. Accrued Interest and Day-Count Conventions
Accrued interest = (days since last coupon / days in coupon period) x coupon per period
| Security type | Day-count convention | Notes |
|---|---|---|
| Government of Canada bonds | Actual / Actual | Count actual days in the numerator and in the coupon period |
| Corporate bonds (Canada) | 30 / 360 | Each month treated as 30 days; year = 360 days |
| US Treasury bonds | Actual / Actual | Same as Canadian government convention |
| US corporate bonds | 30 / 360 | Same as Canadian corporate convention |
Worked example (30/360 corporate):
Semi-annual coupon bond, $1,000 par, 5% annual coupon ($25 per semi-annual period). Last coupon: February 1. Settlement: April 1. Under 30/360: February = 30 days, March = 30 days. Days elapsed = 60. Days in coupon period = 180.
Accrued interest = (60 / 180) x $25 = $8.33
T+1 settlement (effective May 27, 2024):
Canadian equities and most bonds settled on T+2 until May 27, 2024. Since that date, T+1 applies. Accrued interest must be calculated to the settlement date (trade date + 1 business day), not the trade date itself. On ex-dividend or ex-coupon dates, the clean price drops by approximately the coupon/dividend amount.
5. Yield Curve Shapes and What They Signal
| Shape | Description | Economic signal |
|---|---|---|
| Normal (upward-sloping) | Short-term yields below long-term yields | Economic expansion expected; inflation risk priced into long end; most common shape |
| Flat | Short and long yields approximately equal | Economic uncertainty; transition between expansion and contraction; monetary policy may be at a peak |
| Inverted (downward-sloping) | Short-term yields above long-term yields | Recession expectations; market pricing future rate cuts; historically the most reliable leading recession indicator |
| Humped (bell-shaped) | Medium-term yields highest; falls at both ends | Unusual; suggests uncertainty about near-term rates with longer-term confidence in lower rates; often seen around monetary policy turning points |
Theories behind yield curve shape:
- Expectations theory: Long rates reflect expected average of future short rates. An inverted curve means markets expect short rates to fall.
- Liquidity preference theory: Investors demand a term premium for holding longer maturities; normal upward slope is expected even with stable rate expectations.
- Market segmentation theory: Supply and demand in each maturity segment independently drives yields; no necessary connection between segments.
6. Callable Bonds, Putable Bonds, and Yield-to-Worst
Callable bond: Issuer has the right to redeem the bond before maturity at a specified call price. Issuers exercise calls when interest rates fall below the coupon rate (refinancing at lower cost). For investors, a callable bond underperforms a non-callable bond in a falling rate environment because the upside is capped at the call price. This is negative convexity (price appreciation is limited near the call price while price decline is unlimited).
Putable bond: Investor has the right to sell (put) the bond back to the issuer at a specified put price before maturity. Investors exercise puts when rates rise and market prices fall below the put price, effectively setting a price floor. Putable bonds trade at a premium to equivalent non-putable bonds because the put option has positive value to the investor.
Yield-to-worst (YTW): The lowest yield an investor would receive, computed as the minimum of YTM and all possible YTCs across all call dates. Use YTW as the relevant yield measure when analyzing any bond with embedded call provisions. For premium callable bonds, YTW is typically the YTC to the first call date.
Option-adjusted spread (OAS) link: To compare callable and non-callable bonds fairly, strip out the option value using the OAS (see Section 8). A callable bond with a high nominal spread may have a low OAS once the embedded call option cost is removed.
7. Spread Types
| Spread type | Definition | Accounts for |
|---|---|---|
| G-spread | YTM of corporate bond minus YTM of benchmark government bond of same maturity | Credit risk + liquidity risk; simple; does not adjust for cash flow timing differences |
| Z-spread (zero-volatility spread) | Constant spread added to every point on the spot rate curve so that the PV of bond cash flows equals the market price | Credit + liquidity; adjusts for the shape of the full yield curve; more accurate than G-spread |
| OAS (option-adjusted spread) | Z-spread minus the value of any embedded option, modelled using an interest rate tree | Credit + liquidity only; removes the option component; allows fair comparison between callable and non-callable bonds |
Exam gotcha: OAS is always lower than Z-spread for callable bonds (call option value is subtracted). OAS equals Z-spread for option-free bonds. OAS is higher than Z-spread for putable bonds (the put has value to the investor, which reduces the required compensation for credit risk).
8. Credit Rating Ladder
| S&P / Fitch | Moody's | Category |
|---|---|---|
| AAA | Aaa | Highest quality - investment grade |
| AA+, AA, AA- | Aa1, Aa2, Aa3 | Very high quality - investment grade |
| A+, A, A- | A1, A2, A3 | High quality - investment grade |
| BBB+, BBB, BBB- | Baa1, Baa2, Baa3 | Medium quality - lowest investment grade tier |
| BB+, BB, BB- | Ba1, Ba2, Ba3 | Speculative - high yield (junk); below investment grade threshold |
| B, CCC, CC, C | B, Caa, Ca, C | Highly speculative to near/in default |
| D | — | Default (S&P/Fitch only) |
Investment grade threshold: BBB- (S&P/Fitch) or Baa3 (Moody's) and above. Many institutional mandates and pension funds prohibit holding below-investment-grade bonds. A downgrade from BBB- to BB+ is a "fallen angel" - triggering forced selling and a spread jump that the CIRE tests in credit risk scenarios.
Test Yourself: 5 Fixed Income Questions
Q1. A bond has par $1,000, 4% coupon paid semi-annually, 3 years to maturity, market yield 5% per year. Is the bond trading at a premium or discount, and approximately what price?
Show answer
Discount - coupon rate (4%) is below the market yield (5%). Semi-annual coupon = $20, semi-annual yield = 2.5%, 6 periods. PV of coupons = $20 x [1 - (1.025)^-6] / 0.025 = $20 x 5.508 = $110.16. PV of par = $1,000 / (1.025)^6 = $862.30. Price = $972.46.
Q2. A bond has Macaulay duration 6.2 years, annual yield 4%. What is the modified duration, and what is the estimated price change if yields rise 75 bps?
Show answer
ModDur = 6.2 / 1.04 = 5.96. Price change = -5.96 x 0.0075 = -4.47%.
Q3. Corporate bond settlement: last coupon March 15, trade date June 10, T+1 settlement. Semi-annual $30 coupon. 30/360 convention. How much accrued interest does the buyer owe?
Show answer
Settlement = June 11. Under 30/360: March 15 to June 11 = (3 months x 30 days) + (11 - 15) = 90 - 4 = 86 days. Accrued = (86 / 180) x $30 = $14.33.
Q4. A callable bond has a Z-spread of 120 bps. The embedded call option is valued at 30 bps. What is the OAS?
Show answer
OAS = Z-spread - option value = 120 - 30 = 90 bps. The call option costs the investor 30 bps; the remaining 90 bps represents credit and liquidity compensation net of the option.
Q5. A bond is rated BBB- by S&P. It is downgraded to BB+. What is this event called, and what is the typical market impact?
Show answer
Fallen angel. The bond crosses from investment grade to high yield, triggering mandatory selling by investors with investment-grade-only mandates. Spreads typically widen sharply on and before the downgrade announcement, causing the bond price to fall significantly.
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Last updated: 2026-05-08. T+1 settlement effective May 27, 2024 for Canadian equities and most bonds. Credit rating descriptions are definitional; actual default rates vary by rating agency and time period. Verify current settlement rules with CIRO and CDS.