FINS5513 Chap.10 Duration and Bond Portfolio Immunisation
Duration and Bond Portfolio Immunisation
Week 10 measures how the timing of bond cash flows determines interest-rate sensitivity. Macaulay duration is the present-value-weighted average time to cash flows. A zero-coupon bond therefore has duration equal to maturity, while coupons bring duration below maturity by returning value earlier. Modified duration converts Macaulay duration into a local percentage-price response to a change in yield.
The first-order approximation is percentage price change equal to negative modified duration times the yield change, with the yield change written in decimal form. Duration generally rises with maturity, falls with a higher coupon and falls with a higher yield, although the precise comparison should be made under controlled conditions.
Because the price-yield relation is curved, the linear estimate loses accuracy for large rate changes; the course material identifies the convexity adjustment as not examinable, so the guide treats convexity as an interpretation rather than an assessed calculation. Immunisation applies duration to a liability objective.
A simple strategy initially matches the present value of assets with the present value of liabilities and matches asset duration with liability duration. Two bonds can meet a target duration through a weighted-average equation, followed by allocation of the required asset value. The protection is local and assumption-dependent.
Passage of time, non-parallel curve shifts and changing yields alter values and durations, so a real immunised portfolio requires monitoring and rebalancing.
What this chapter covers
- 01
Macaulay duration as a present-value-weighted cash-flow time
- 02
Zero-coupon duration and the effect of earlier coupon payments
- 03
Modified duration as local percentage-price sensitivity
- 04
First-order price-change approximation with decimal yield changes
- 05
Duration comparisons across maturity, coupon and yield
- 06
Convex price-yield relation as a limitation of linear duration
- 07
Present-value and duration matching for one liability target
- 08
Rebalancing and the risks of non-parallel yield-curve shifts
Worked example · free
Calculate duration and approximate a rate shock
- +1The cash-flow present values are approximately $4.717, $4.450 and $88.160 at years one, two and three, and they reconcile to the stated price.
- +1The time-weighted present-value sum is 1 times $4.717 plus 2 times $4.450 plus 3 times $88.160, or about $278.10. Dividing by price gives Macaulay duration of about 2.857 years.
- +1Modified duration is 2.857 divided by 1.06, which is approximately 2.696.
- +1Approximate percentage price change is negative 2.696 times 0.005, which is negative 1.35%.
Key terms
- Macaulay Duration
- The present-value-weighted average time until a bond's cash flows are received.
- Modified Duration
- A measure of the local percentage price response to a small yield change.
- Basis Point
- One hundredth of one percentage point, equal to 0.0001 in decimal yield.
- Convexity
- Curvature in the relation between bond price and yield.
- Immunisation
- A strategy matching asset and liability value and duration to limit local rate risk.
- Rebalancing
- Trading that restores target value and duration relationships after conditions change.
Duration and Bond Portfolio Immunisation FAQ
Why is coupon-bond duration below maturity?
Some value is received through coupons before the final date. Those earlier present-value weights pull the weighted average payment time below maturity, unlike a zero-coupon bond with only one terminal payment.
Why does modified duration have a negative price effect?
Bond price and required yield move inversely for fixed promised cash flows. Modified duration is normally reported as a positive sensitivity magnitude, while the approximation adds the negative sign to show direction.
How should a basis-point move be entered?
Convert basis points to decimal yield units before using duration. Fifty basis points equals 0.50 percentage points and 0.005 in decimal form, not 0.50 and not 50 in the formula.
When is the duration estimate least accurate?
Accuracy deteriorates for larger yield movements because the true price-yield relation is curved. It can also be misleading when embedded options change expected cash flows as interest rates move.
What conditions define simple immunisation?
At the measurement date, asset present value should cover liability present value and asset duration should equal liability duration. The result is local and works best for small parallel yield-curve movements.
Why must an immunised portfolio be rebalanced?
Time passes, yields change, cash flows are paid and asset durations drift differently from liability duration. Those changes break the original matching equations and require periodic restoration of the target.
Exam move
Calculate duration from a cash-flow table with columns for time, cash flow, present value, price weight and time-weighted value. Confirm that the present values sum to price and the weights sum to one before trusting the duration. Convert Macaulay to modified duration using the periodic yield that matches the cash-flow frequency.
For a price shock, retain the negative sign, express the yield change in decimal form and describe the result as an approximation. Use qualitative checks: a zero-coupon bond's duration equals maturity, higher coupons usually shorten duration, and a yield rise lowers price. For immunisation, solve two distinct requirements rather than jumping directly to bond weights.
First match duration using a weighted-average equation; then allocate enough total asset value to match the liability present value. Check that weights sum to one and lie within any stated constraints. In discussion answers, identify the assumptions behind protection: small parallel rate changes, known cash flows and timely rebalancing.
Recognise convexity as the reason a linear estimate is imperfect, but do not invent a convexity calculation where the course explicitly marks that adjustment as not examinable.
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