RSE Element 2: fixed-income calculations, step by step

Element 2 accounts for about 10 of the Retail Securities Exam's (RSE's) 120 questions, under the syllabus's indicative weighting. Its title is Fixed income, and five of its nine outcomes are at the apply level, most of them built on calculations. This guide works through each one with a step-by-step example. A financial calculator helps, but every example can be done with the formulas shown.

What the exam tests

Among other outcomes, the syllabus asks you to:

• apply fixed-income terminology: par value, coupon rate, term to maturity, bond price, yield to maturity and others;
• apply standard calculations to bond yields: current yield, approximate yield to maturity, yield on a zero-coupon instrument, and read yield curves;
• apply the relationships between coupon, yield, term and price volatility, and between Macaulay duration, modified duration and price sensitivity;
• apply the modified duration calculation to estimate a bond's price change;
• apply time value of money: the present value of a bond, and any missing variable.

Source: RSE syllabus (CIRO, May 8, 2025), learning outcomes 2.3 and 2.6 to 2.9.

The key ideas, in plain English

1. The relationship behind every calculation

A bond pays a fixed coupon on its par value (usually $1,000 or $100) and repays par at maturity. Government of Canada bonds pay their coupons semi-annually. When market yields rise, the price of an existing fixed-rate bond falls until its yield matches new issues, and the reverse when yields fall:

• coupon rate equal to the market yield: the bond trades at par;
• coupon rate above the market yield: at a premium;
• coupon rate below the market yield: at a discount.

2. Current yield

Current yield = annual coupon ÷ market price.

Example. A bond with a $1,000 par value pays a 4.2% coupon semi-annually, matures in 7 years and trades at $962.50.

• Annual coupon: $1,000 × 4.2% = $42.00.
• Current yield: $42.00 ÷ $962.50 = 4.36%.

It ignores the $37.50 gain the holder makes by keeping the bond to maturity.

3. Approximate yield to maturity

The yield to maturity (YTM) is the annual return on a bond bought at today's price and held to maturity, assuming every payment is made and coupons are reinvested at the YTM. The shortcut spreads the gain or loss to par evenly over the years left:

Approximate YTM = [annual coupon + (par − price) ÷ years] ÷ [(par + price) ÷ 2]

Same bond.

• Gain to par per year: ($1,000 − $962.50) ÷ 7 = $5.36.
• Average investment: ($1,000 + $962.50) ÷ 2 = $981.25.
• Approximate YTM: ($42.00 + $5.36) ÷ $981.25 = 4.83%.

The exact YTM, from a financial calculator, is 4.84%. Below par, as here, coupon rate < current yield < YTM (4.20% < 4.36% < 4.83%). Above par, the order reverses.

4. Yield on a zero-coupon instrument

A zero-coupon bond, such as a strip bond, pays no coupon: its whole return is the gap between price and face value. Always check the compounding:

• annual: yield = (face value ÷ price)^(1 ÷ years) − 1;
• semi-annual: yield = 2 × [(face value ÷ price)^(1 ÷ (2 × years)) − 1].

Example. A strip bond with a $10,000 face value matures in 12 years and costs $6,200.

• Face value ÷ price: 10,000 ÷ 6,200 = 1.612903.
• Annual compounding: 1.612903^(1/12) − 1 = 4.06%.
• Semi-annual compounding: 2 × (1.612903^(1/24) − 1) = 4.02%.

5. Yield curves

A yield curve plots the yields of bonds of the same credit quality, usually Government of Canada bonds, against their terms to maturity.

Shape	What it shows	What it generally signals
Normal	Long-term yields above short-term	Investors want extra yield to lend for longer; usually associated with expected growth
Inverted	Short-term yields above long-term	Markets expect rates to fall; it has often preceded recessions
Flat	Similar yields across terms	Often a transition between the two

6. Duration and price sensitivity

Price volatility is greater for a longer term, a lower coupon and a lower yield, because more of the bond's value sits in distant cash flows. Macaulay duration measures this as the weighted average time, in years, until the cash flows are received. Modified duration turns it into an estimate of the percentage price change for a 1-percentage-point (100-basis-point) change in yield:

• Modified duration = Macaulay duration ÷ (1 + YTM ÷ number of coupons a year);
• approximate price change = −modified duration × change in yield (as a decimal) × price.

Example. A bond with $100 par pays its coupon semi-annually, yields 5% and trades at $103.40. Its Macaulay duration is 7.20 years. Yields fall by 25 basis points (a basis point is 0.01%).

• Modified duration: 7.20 ÷ (1 + 0.05 ÷ 2) = 7.20 ÷ 1.025 = 7.024.
• Price change: −7.024 × (−0.0025) × $103.40 = +$1.82.
• Estimated new price: $103.40 + $1.82 = $105.22.

The estimate is close for small changes in yield and less accurate for large ones, because the true price–yield relationship is curved. The same method works for a whole holding, as question 3 below shows.

7. The present value of a bond

A bond's price is the present value of its coupons and par, discounted at the yield on comparable bonds. For a semi-annual bond, halve the coupon and the yield, and double the years.

Example. A bond with a $1,000 par value pays a 3% coupon semi-annually and matures in 5 years. Comparable bonds yield 4.5%.

• Periods: 5 × 2 = 10. Rate per period: 4.5% ÷ 2 = 2.25%. Coupon per period: $15.00.
• Present value of the coupons: $15.00 × [1 − 1.0225^−10] ÷ 0.0225 = $15.00 × 8.866216 = $132.99.
• Present value of par: $1,000 ÷ 1.0225^10 = $1,000 ÷ 1.249203 = $800.51.
• Price: $132.99 + $800.51 = $933.50.

On a financial calculator: N = 10, I/Y = 2.25, PMT = 15, FV = 1,000, compute PV (shown as −933.50). The price is below par because the 3% coupon is below the 4.5% yield: a quick check that rules out answers above par.

8. Solving for a missing variable

A time value of money calculation links five variables: N, I/Y, PV, PMT and FV. Given any four, you can solve for the fifth.

Example. A client wants $20,000 to grow to $26,000 in 6 years, with no further deposits. What annual return is needed?

• Calculator: N = 6, PV = −20,000, PMT = 0, FV = 26,000, compute I/Y.
• Formula: (26,000 ÷ 20,000)^(1/6) − 1 = 4.47% a year.

Three important distinctions to understand

• Current yield and YTM. Current yield looks only at the coupon; YTM adds the gain or loss to par.
• Annual and semi-annual compounding. The same price gives two different zero-coupon yields: check which one the question wants.
• Macaulay and modified duration. Macaulay is a time in years; modified duration is the price sensitivity you multiply by the change in yield.

Test yourself: 3 questions

Original CIROPrep questions, in the style of the exam.

1. A bond pays a 5% coupon semi-annually, matures in 6 years and trades at $1,040 per $1,000 of par. Which statement is correct?

• a) Yield to maturity > current yield > coupon rate
• b) Coupon rate, current yield and yield to maturity are equal
• c) Coupon rate > current yield > yield to maturity
• d) Current yield > coupon rate > yield to maturity

Answer

c) The current yield is $50 ÷ $1,040 = 4.81%, and the YTM is lower still (about 4.24%), because the holder loses the $40 premium by maturity. Option a) is the order for a bond below par. Option b) applies only at par. Option d) is wrong because a price above par makes the current yield lower than the coupon rate.

2. A strip bond with a $5,000 face value matures in 8 years and costs $3,700. What is its yield, compounded semi-annually?

• a) 3.80%
• b) 3.84%
• c) 4.39%
• d) 35.1%

Answer

a) 2 × [(5,000 ÷ 3,700)^(1/16) − 1] = 2 × 0.0190 = 3.80%. Option b) is the yield with annual compounding: (5,000 ÷ 3,700)^(1/8) − 1. Option c) divides the $1,300 gain evenly over 8 years and ignores compounding. Option d) is the total gain over 8 years, not an annual yield.

3. A client holds a bond portfolio worth $400,000, with a modified duration of 6.5. If yields rise by 40 basis points, what is the estimated change in its value?

• a) A loss of about $1,040
• b) A loss of about $10,400
• c) A loss of about $26,000
• d) A gain of about $10,400

Answer

b) −6.5 × 0.0040 × $400,000 = −$10,400. Option a) misplaces the decimal (0.0004 instead of 0.0040). Option c) uses a 1-percentage-point change in yield instead of 0.40 percentage points. Option d) gets the sign wrong: when yields rise, bond prices fall.

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Read next: RSE Element 1, KYC and suitability and RSE Element 4, securities analysis.