STAT2032 Chap.8 Duration, Convexity and Immunisation
Duration, Convexity and Immunisation
Duration and convexity translate a list of discounted cash flows into local interest-rate sensitivity. Macaulay duration is the present-value-weighted average payment time, while modified duration converts that timing measure into the first-order percentage price response to yield. Convexity supplies the second-order curvature omitted by a tangent approximation.
Immunisation then applies the measures to assets and liabilities: match present value and duration at the chosen yield, and require sufficient asset convexity so a small yield movement bends the surplus in a protective direction.
Start with a cash-flow table rather than a memorised sensitivity number.
Discount each payment, divide by total price to obtain weights, verify that the weights sum to one and then average payment times. A zero-coupon security has duration equal to maturity, while earlier coupons shorten the duration of a coupon bond. Modified duration attaches the negative local price response to a yield change expressed in decimal units.
A one percentage-point movement is 0.01, not one.
Duration is a tangent approximation, so its error grows as the yield shock moves farther from the valuation point. Convexity adds curvature and ordinarily improves the estimate for conventional positive cash flows. In asset-liability work, equal present value fixes the starting level and equal duration cancels the first-order slope.
Asset convexity at least as large as liability convexity supplies protective local bending. These conditions apply around one date and valuation yield under the stated parallel-shift model. Time passage and changing yields alter the measures, which is why an immunised position must be recalculated and rebalanced.
What this chapter covers
- 01
Present-value cash-flow weights
- 02
Macaulay duration
- 03
Modified duration
- 04
Price-yield convexity
- 05
First and second-order approximations
- 06
Present value and duration matching
Worked example · free
Compute sensitivity from three cash flows
- 2Discount the flows to obtain present values of $190.48, $272.11 and $518.30.
- 1Their sum gives price P = $980.89, subject to displayed rounding.
- 2Macaulay duration is the time-weighted present-value sum divided by price, approximately 2.334 years.
- 1Modified duration is 2.334/1.05 = 2.223.
- 1The first-order estimate is minus 2.223 times 0.01, or a 2.223% price fall.
Key terms
- Macaulay duration
- The average payment time formed by weighting each date by its share of the security's present value.
- Modified duration
- Macaulay duration divided by one plus periodic yield, giving local percentage price sensitivity.
- Convexity
- The curvature term that improves a duration-only price estimate for a non-infinitesimal yield movement.
- Immunisation
- An asset-liability construction designed to protect surplus locally against interest-rate changes under stated conditions.
Duration, Convexity and Immunisation FAQ
Why is Macaulay duration measured in time?
Its weights are each cash flow's present value divided by total price, and those weights sum to one. Multiplying each weight by its payment time and adding produces a weighted average time. Modified duration then rescales that time measure by yield to express local percentage price sensitivity.
Why does convexity improve the duration estimate?
Duration describes the tangent slope of the price-yield curve at the current yield. A finite yield movement leaves that tangent because the true relationship is curved. The convexity term supplies a second-order correction, which ordinarily offsets part of the duration-only loss for a positive-yield rise in a conventional positive cash-flow security.
What must be matched for local immunisation?
At the chosen valuation yield, asset and liability present values should match and their durations should match. Those conditions align level and first derivative. Asset convexity should then be at least liability convexity so the asset value has protective local curvature relative to the liability. The result is local and needs rebalancing as time and yields change.
Why must an immunised position be rebalanced?
Payment times shorten as the calendar advances, market yields change present-value weights and asset cash flows may be received or reinvested. Those changes disturb value and duration matching. Immunisation is therefore a local construction around a stated date and yield, not a permanent guarantee against every later movement of the curve.
Exam move
Start with one cash-flow table containing date, amount, discount factor, present value, present-value weight, time-weighted value and the convexity contribution. Verify that the weights sum to one before accepting duration. Translate Macaulay into modified duration only after confirming the yield period.
Estimate price changes for both a rise and a fall, then fully reprice to observe the convexity correction and the limits of a local approximation. For immunisation, write the conditions in order: equal present value, equal duration and sufficient asset convexity. Solve a two-asset zero-coupon allocation because its duration is a transparent weighted average of maturity.
Finally, state why the construction is not permanent: cash-flow time shortens, yields move and the matched weights must be rebalanced.
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