Two bonds can have the same maturity and very different risk. One pays a big coupon every year and returns your money early. The other pays little until the end. When interest rates rise, the second bond’s price falls further, because its cash arrives later.
Duration puts a single number on that timing, and with it you can estimate how much a bond or a portfolio will move when rates change.
This topic covers duration as the average time to recover an investment, Macaulay’s duration, modified duration, estimating price changes, the characteristics of duration, the duration of a portfolio, convexity and project duration as a measure of risk. The previous topic covered forward contracts and money market hedges. The next covers yield curves and credit risk. Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on duration.
In 30 seconds
Macaulay’s duration = Σ(PV of each cash flow × year) ÷ Σ(PV of cash flows), the weighted average time to recover the investment, using present values at the redemption yield. Modified duration = Macaulay duration ÷ (1 + yield). Change in price ≈ − modified duration × change in yield × price. Longer maturity, lower yield and lower coupon all give longer duration; a zero-coupon bond has a duration equal to its term.
A portfolio’s duration is the market-value-weighted average (it assumes a parallel shift in yields). Duration treats price against yield as a straight line, but the true curve is convex, so duration is accurate only for small changes and understates the price rise when yields fall. Project duration works in the same way: under half the life means returns come early; 75% or more means late.
Key terms in plain English
| Term | What it means |
|---|---|
| Duration | The weighted average time to receive a bond’s cash flows |
| Macaulay’s duration | Duration calculated with discounted cash flows |
| Modified duration | Macaulay’s duration divided by (1 + yield); the sensitivity of price to yield |
| Convexity | The curvature of the price–yield relationship |
| Redemption yield | The return on a bond if held to maturity |
| Parallel shift | A change in yield of the same size at every maturity |
Syllabus check
This topic is in syllabus area E (treasury and advanced risk management). It covers:
- explaining the concept of duration;
- calculating and using Macaulay’s and modified duration and convexity to estimate price changes.
CAMA 2020 and debt securities
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Debt funding should be managed prudently for the benefit of the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should understand the interest rate sensitivity of the debt the company holds or issues |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
The concept
Duration is widely used in bond markets. It shows how long an investor must wait to recover the investment, as a weighted average of the years in which cash arrives, with the weight for Year n being n. An undiscounted version is like a payback period. ICAN’s two four-year bonds: X pays ₦10, ₦10, ₦10, ₦110 (total ₦140) and Y pays ₦20, ₦20, ₦20, ₦120 (total ₦180). Undiscounted: X = (10 + 20 + 30 + 440) ÷ 140 = 500 ÷ 140 = 3.57 years; Y = 600 ÷ 180 = 3.33 years. This ignores the time value of money.
Macaulay’s duration
Discount each cash flow at the investor’s yield (10%) before weighting:
| Year | Bond X PV | PV × year | Bond Y PV | PV × year |
|---|---|---|---|---|
| 1 | 9.09 | 9.09 | 18.18 | 18.18 |
| 2 | 8.26 | 16.52 | 16.52 | 33.04 |
| 3 | 7.51 | 22.53 | 15.02 | 45.06 |
| 4 | 75.13 | 300.52 | 81.96 | 327.84 |
| Total | 99.99 | 348.66 | 131.68 | 424.12 |
Duration of X = 348.66 ÷ 99.99 = 3.49 years; Y = 424.12 ÷ 131.68 = 3.22 years. The totals of the PV column are the bond prices. Duration shows price sensitivity to market yields: when yields rise, a bond with longer duration falls more in percentage terms, so investors manage risk by choosing bonds of suitable duration.
Modified duration and price changes
Modified duration = Macaulay’s duration ÷ (1 + r), where r is the redemption yield. X: 3.49 ÷ 1.1 = 3.17; Y: 3.22 ÷ 1.1 = 2.93. The estimated change in price is:
ΔP = − D × Δi × P
For a 0.5% (0.005) rise in yields: X changes by −3.17 × 0.005 × 99.99 = −₦1.58; Y by −2.93 × 0.005 × 131.68 = −₦1.93. The change is proportionately larger for X (1.58% of its price against 1.47%).
Characteristics of duration
- Bonds with longer terms have longer durations.
- Duration indicates price sensitivity to market yields.
- Lower yields give longer durations.
At 5% Bond X’s duration is 415.57 ÷ 117.73 = 3.53, against 3.49 at 10%. - Lower coupons give longer durations: X pays less than Y and has the longer duration.
- A zero-coupon bond’s duration equals its term (a four-year zero: 400 ÷ 100 = 4 years).
- Duration falls over the life of a bond, but by less than the time passed: X with three years left has duration 273.55 ÷ 99.99 = 2.74 years, a fall of 0.75 in one year.
Duration of a portfolio
The duration of a portfolio is the weighted average of the bonds’ durations, using market values: W1D1 + W2D2 + …. ICAN’s example: market values 20, 30 and 50 with durations 2.5, 3.6 and 3.0: 0.5 + 1.08 + 1.5 = 3.08. Use it with care: it assumes a parallel shift in yields at every maturity, which does not occur. A better approach is to look at each bond, though banks with separate portfolios of similar bonds can use it.
Convexity
Duration assumes a straight-line link between yield and price, but the true curve is convex. So duration is a fair estimate only for small changes and understates the price movement for large changes, especially the rise when yields fall. It is also distorted when the yield curve changes shape. Using duration and convexity together gives a better estimate.
Characteristics: greater convexity means a higher price; the lower the coupon, the higher the convexity (so zero-coupon bonds have the most); and for two bonds with the same duration and yield, the one with more convexity is less hurt by rising yields.
Illustration with Bond X: price at 10% is 99.99 and at 5% is 117.73. Duration predicts for a fall of 5% in yield: −3.17 × (−0.05) × 99.99 = +15.85, so a price of 115.84. The actual price is 117.73, so duration understates by about 1.89.
Project duration as a measure of risk
Duration can be calculated for a project in the same way: it is the average time to obtain the returns, taking in all the discounted returns, unlike payback. Ignore negative cash flows at the start; start from the year cash flows turn positive (include later negatives). ICAN’s example: cash flows of ₦(200,000), (40,000), 30,000, 120,000, 150,000, 100,000 and 50,000 in Years 0 to 6 at 10%.
The present values from Year 2 are 24,780, 90,120, 102,450, 62,100 and 28,200 (total 307,650); weighted by the years 2 to 6 they sum to 1,209,420. Duration = 1,209,420 ÷ 307,650 = 3.93 years (about 66% of the six-year life). If duration is under half the project’s life, returns come early; at 75% or more, most returns come late and the project is riskier.
The rules in more depth
Why duration measures sensitivity
Later cash flows are discounted more heavily, so they lose more value when yields rise.
Why modified duration divides by (1 + r)
It converts the average time into the percentage price change per unit of yield change.
Why lower coupons lengthen duration
More of the value arrives at the end.
Why convexity helps
The price curve bends, so a straight line is wrong for big moves.
Effect on analysis
Compute the PVs, weight them, divide, then convert to modified duration. Use it for small yield changes and state its limits.
Worked examples
Example 1: duration of a two-year bond. It pays 8 in Year 1 and 108 in Year 2. The yield is 10%.
- PVs.
8 ÷ 1.1 = 7.27
108 ÷ 1.21 = 89.26
price = 96.53. - Weighted PVs.
7.27 + 2 × 89.26 = 185.79. - Duration = 185.79 ÷ 96.53 = 1.92 years.
Example 2: modified duration and price change. Use Example 1.
- Modified duration.
1.9246 ÷ 1.1 = 1.75. - A 1% rise in yield: −1.75 × 0.01 × 96.53.
- Change = −1.69.
Example 3: a portfolio. ₦60m of bonds have duration 4.0 and ₦40m have duration 2.5.
- Weights 60% and 40%.
- 0.6 × 4.0 + 0.4 × 2.5 = 2.4 + 1.0.
- Portfolio duration = 3.4 years.
Example 4: a zero-coupon bond. A five-year zero with a yield of 8%.
- Macaulay duration = 5 years.
- Modified duration = 5 ÷ 1.08.
- Modified duration = 4.63.
How to score full marks
- Discount each cash flow and sum the PVs (the price).
- Weight by the year and divide by the price.
- Convert to modified duration with (1 + yield).
- Estimate the price change and show its sign.
- State the limits: small changes, parallel shifts.
- Mention convexity for large changes.
Common mistakes in the exam
- Weighting undiscounted cash flows when Macaulay is asked for.
- Forgetting to divide by (1 + yield) for modified duration.
- Getting the sign of the price change wrong.
- Using duration for large yield changes without comment.
- Averaging portfolio durations without market-value weights.
- Including the initial negative flows in project duration.
Quick revision
- Macaulay = Σ(PV × t) ÷ ΣPV; modified = Macaulay ÷ (1 + r).
- ΔP = −D × Δi × P.
- Longer duration: longer term, lower yield, lower coupon.
- Zero-coupon: duration equals term.
- Portfolio: weighted average; assumes a parallel shift.
- Convexity: duration understates for large changes.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. Macaulay’s duration is the:
- Weighted average time to recover a bond’s discounted cash flows
- Maturity
- Coupon
- Yield
Show the answer
A. Weights are the years.
Question 2. Modified duration equals:
- Macaulay duration ÷ (1 + yield)
- Macaulay × (1 + yield)
- Yield ÷ duration
- Coupon ÷ price
Show the answer
A. It measures price sensitivity.
Question 3. Bonds with lower coupons have:
- Longer durations
- Shorter durations
- No duration
- Equal durations
Show the answer
A. More value arrives late.
Question 4. A zero-coupon bond’s Macaulay duration equals:
- Its term
- Half its term
- Zero
- Its yield
Show the answer
A. Only one cash flow.
Question 5. The estimated price change formula is:
- − D × Δi × P
- D × P
- Δi ÷ D
- P ÷ D
Show the answer
A. D is modified duration.
Question 6. Duration is accurate mainly for:
- Small yield changes
- Large yield changes
- No changes
- Negative yields only
Show the answer
A. Convexity matters for large moves.
Question 7. For a large fall in yields, duration:
- Understates the price rise
- Overstates the price rise
- Is exact
- Ignores price
Show the answer
A. Because the price curve is convex.
Question 8. Portfolio duration is:
- The market-value weighted average of the durations
- The longest duration
- The sum of the durations
- The shortest duration
Show the answer
A. It assumes a parallel shift.
Question 9. A project duration that is 75% or more of its life means returns come:
- Late
- Early
- Evenly
- Never
Show the answer
A. So it is riskier.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Define Macaulay’s duration. (3 marks) (b) State three characteristics of duration. (3 marks) (c) Explain why duration of a portfolio must be used with care. (4 marks)
Question 12 (20 marks). A three-year bond pays a 10% annual coupon on a nominal value of ₦100 and is redeemed at par. The yield is 10%. (a) Calculate Macaulay’s duration. (4 marks) (b) Calculate the modified duration. (2 marks) (c) Estimate the change in price if the yield rises by 1%. (3 marks) (d) The actual price at 11% is ₦97.56. Compare it with your estimate and explain. (4 marks) (e) State four characteristics of duration. (4 marks) (f) Explain how duration is used for a project. (3 marks)
Question 13 (10 marks). (a) Market values of ₦20m, ₦30m and ₦50m have durations of 2.5, 3.6 and 3.0. Calculate the portfolio duration. (3 marks) (b) A project’s present values from the first positive year are ₦307,650 and weighted by years ₦1,209,420 over a six-year life. Calculate the duration and comment. (4 marks) (c) Explain convexity. (3 marks)
Model answers
Show the model answer to Question 11
(a) Definition (3 marks). It is the weighted average time to recover a bond’s investment. (1) The cash flows are discounted at the yield. (1) The weights are the years. (1)
(b) Characteristics (3 marks). Longer terms give longer durations; lower yields give longer durations; lower coupons give longer durations; a zero-coupon bond’s duration equals its term. Any three. (1 each)
(c) Portfolio (4 marks). It is a weighted average of the bonds’ durations. (1) It assumes a parallel shift in yields at all maturities. (1.5) That does not happen, so it is better to consider each bond or portfolios of similar bonds. (1.5)
Show the model answer to Question 12
(a) Macaulay (4 marks).
- PVs.
10 ÷ 1.1 = 9.09; 10 ÷ 1.21 = 8.26; 110 ÷ 1.331 = 82.64; total 100.00. (2) - Weighted.
9.09 + 16.53 + 247.93 = 273.55. - Duration = 273.55 ÷ 100 = 2.74 years. (2)
(b) Modified (2 marks). 2.74 ÷ 1.1 = 2.49. (2)
(c) Estimate (3 marks). ΔP = −2.49 × 0.01 × 100 = −2.49, so the price falls to about ₦97.51. (3)
(d) Comparison (4 marks). The actual fall is 2.44 (100 to 97.56), which is smaller than the estimated 2.49. (2) The price–yield relationship is convex, so duration overstates the fall when yields rise (and understates the rise when yields fall). (2)
(e) Characteristics (4 marks). Longer terms give longer durations; lower yields give longer durations; lower coupons give longer durations; zero-coupon duration equals its term; duration falls over the life by less than the time passed. Any four. (1 each)
(f) Project (3 marks). Duration is the average time to receive the discounted returns, ignoring initial negative flows. (1.5) A short duration relative to the life means early returns; 75% or more means late and riskier returns. (1.5)
Show the model answer to Question 13
(a) Portfolio (3 marks). 0.2 × 2.5 + 0.3 × 3.6 + 0.5 × 3.0 = 0.5 + 1.08 + 1.5 = 3.08 years. (3)
(b) Project (4 marks).
- 1,209,420 ÷ 307,650 = 3.93 years. (2)
- That is about 66% of the six-year life, so returns come moderately late: more than half but below the 75% level. (2)
(c) Convexity (3 marks). The relationship between price and yield is a curve, not a straight line. (1.5) Duration is accurate only for small changes, and convexity improves the estimate for large ones. (1.5)
Pathfinder on Macaulay and modified duration, portfolio duration, convexity and project duration
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on duration. Practise the table in the section on Macaulay’s duration and worked examples 1 to 4.
Revise first: Macaulay’s duration
Exam questions ask you to estimate a price change from a yield change. See the section on modified duration and price changes.
Revise first: Modified duration and price changes
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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