Macaulay duration
Also written Mac duration
The weighted average time, in years, to receive a bond's cash flows, each weighted by the present value of that cash flow — the bond's effective payback period.
In plain language
A ten-year bond does not make you wait ten years for your money. It pays a coupon every six months, and only the final payment arrives at maturity. So "ten-year bond" overstates how long your money is actually tied up.
Macaulay duration fixes that. It asks: on average, weighted by how much money arrives and when, how long do I wait? Each cash flow is discounted to present value, that present value becomes its weight, and the weighted average of the times is the duration. It is measured in years.
The result is an honest "effective maturity" that you can compare across bonds with different coupons — and it is the foundation on which modified-duration measures interest rate sensitivity.
How it works
Four relationships are examined relentlessly, and all four follow from the weighting:
- Duration rises with maturity. A longer bond waits longer for its principal. But it does not rise exponentially; it flattens out once maturity is long enough that the distant principal carries almost no present value.
- Duration falls as the coupon rises. A fatter coupon puts more weight on the early years, dragging the average time down.
- Duration falls as yield to maturity rises. A higher discount rate damps the present value of a distant coupon far more than a nearby one, shifting weight forward.
- For a zero-coupon bond, duration equals maturity, because there is exactly one cash flow. For every coupon-paying bond, duration is less than maturity. A zero therefore has the highest duration of any bond of the same maturity.
The workbook's own illustration: 5.77% GS 2030, maturing 3 August 2030, valued on 26 October 2020 at a yield of 5.85%, has a duration of 7.50 years. Raise the yield to 6.50% and the duration drops to 7.43 — the payback period shortens as rates rise. Its flattening table is worth remembering too: on the same settlement date, a 2060 maturity gives 14.67 and a 2090 maturity only 14.93.
A portfolio's duration is simply the weighted average of its holdings' durations.
The formula
Σ [ PV(CF_t) × t ]
Mac Duration = ─────────────────────
Market price of bond
where CF_t is the cash flow at time t and PV discounts it at the yield Y.
And the two things built on it:
Modified Duration = Mac Duration ÷ (1 + Y/n) n = coupon periods per year
% change in bond price = − Modified Duration × Change in yield
Portfolio duration = Σ (weight_i × duration_i)
A worked example
A 3-year bond, face Rs 100, 8% annual coupon, YTM 8%. Build the table column by column:
| Year (t) | Cash flow | PV factor at 8% | PV of cash flow | PV × t |
|---|---|---|---|---|
| 1 | Rs 8 | 0.9259 | 7.4074 | 7.4074 |
| 2 | Rs 8 | 0.8573 | 6.8587 | 13.7174 |
| 3 | Rs 108 | 0.7938 | 85.7339 | 257.2016 |
| 100.0000 | 278.3264 |
Mac Duration = 278.3264 ÷ 100.0000 = 2.78 years
A three-year bond with an effective payback of 2.78 years — less than maturity, exactly as the rule says. Strip the coupons out and make it a 3-year zero and the duration would be a full 3.00 years.
Now turn it into money. Modified duration converts duration into price sensitivity:
Modified Duration = 2.7833 ÷ 1.08 = 2.58
A debt scheme holds Rs 10 crore face value of this bond at par. Yields rise 100 basis points:
Price change ≈ −2.58% → price falls from Rs 100.00 to about Rs 97.42
Loss on the holding ≈ 2.58% × Rs 10 crore = Rs 25.8 lakh
A one-percent move in rates cost Rs 25.8 lakh on a bond that everyone calls "short". Swap into a 10-year paper with a duration near 7.5 and the same 100 bp move costs roughly Rs 71 lakh instead — which is the entire argument for matching a scheme's duration to its investors' horizon.
Portfolio duration works the same way: 30% in a 3-year-duration asset and 70% in a 5-year gives 0.3 × 3 + 0.7 × 5 = 4.40 years.
Why NISM asks about it
Chapter 18.12.3 (Macaulay Duration) defines it and gives the 5.77% GS 2030 figures and the full spreadsheet computation; 18.12.4 lists the four relationships, 18.12.5 gives portfolio duration, and 18.12.6 derives modified duration. Chapter 22 uses duration to size interest rate hedges. Expect a "which of these has the highest duration" question, a coupon-or-yield-versus-duration direction question, and a modified-duration price-change computation.
Common exam traps
- Duration is not maturity. It equals maturity only for a zero-coupon bond and is always lower for a coupon bond.
- Macaulay duration is in years; modified duration is a percentage sensitivity. Do not quote one where the question wants the other.
- Coupon and yield both move duration down. Only maturity moves it up. Candidates routinely get the yield relationship backwards.
- Duration is a linear approximation. It is accurate for small yield changes; for large moves the price-yield curve bends away from it and convexity matters.
- Modified duration applies to the dirty price, which includes accrued interest — that is the price an investor actually pays.
- Duration does not rise indefinitely with maturity. The workbook's own table shows it stagnating: 14.67 years at a 2060 maturity against 14.93 at 2090.
Where this is taught
- Series IV · Chapter 1: Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Marketsintroduced here
- Series V-A · Chapter 2: Concept & Role of a Mutual Fundintroduced here
- Series XXI-B · Chapter 4: Investing in Fixed Income Securitiesintroduced here
- Series X-A · Chapter 9: Investing in Fixed Income Securitiesintroduced here
- Series XII · Chapter 5: Mutual Fundsintroduced here
- Series V-D · Chapter 2: Concept & Role of a Mutual Fundintroduced here
- Series V-B · Chapter 2: Concept and Role of a mutual fundintroduced here
- Series XXI-A · Chapter 4: Investing in Fixed Income Securitiesintroduced here
- Series XXI-B · Chapter 19: Fixed Income Portfolio Management Strategies
- Series V-D · Chapter 18: Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Markets
Related terms
- Modified DurationMacaulay's duration divided by (1 + yield) — the percentage by which a bond's price moves for a one percentage point change in interest rates, and so the standard measure of interest rate risk.
- Yield to MaturityThe single discount rate at which a bond's future coupons and redemption amount add up to exactly its market price today — the return you actually earn if you hold it to maturity.
- Price Value of a Basis PointThe rupee change in a bond's price for a one basis point change in its yield — the unit in which a fixed income desk actually measures and hedges interest rate risk.
- Duration managementA debt fund manager's deliberate strategy of lengthening or shortening a portfolio's duration based on a view on interest rates, used only where the scheme's mandate allows it.
- ConvexityThe curvature of the price-yield relationship — the correction duration misses, because duration is a straight line and the true relationship bends.
- Duration-based hedge ratioThe number of interest rate futures that drives a bond portfolio's duration to zero — portfolio modified duration times market value, divided by futures modified duration times futures price over par.
- Weighted average maturityThe average time to maturity of a debt portfolio, each security weighted by its share of the portfolio — a simple indicator of how sensitive a debt scheme's NAV is to interest rate changes.
- Zero coupon bondA bond that pays no coupons: it is issued below face value and repays the face value at maturity, so its duration equals its maturity.
- ImmunizationStructuring a bond portfolio so its value at a target date is protected from interest-rate changes — by matching the portfolio's duration to the liability's timing.
- Barbell strategyA bond portfolio concentrated at the short and long ends of duration with little in the middle, used when rates could move sharply in either direction.
- Macaulay duration bucketThe duration band an index provider uses to sort bonds — NIFTY bond indices run six such buckets, from Ultra-Short at 3 to 6 months up to Long Duration above 7 years.
- Positive convexityThe favourable asymmetry of an option-free bond: when yields fall its price rises more than duration predicts, and when yields rise it falls less — so buying convexity is an active fixed income strategy.
- Interest rate riskThe risk that an investor in a debt instrument loses return because rates rise — existing instruments carrying the old, lower coupon fall in value until their yield matches the new market rate.
- NIFTY G-Sec IndicesIndices of Government of India bonds across 6 Macaulay duration buckets, each holding up to 3 liquid bonds with over Rs 5,000 crore outstanding, weighted by amount outstanding and rebalanced monthly.
- Deemed residual maturityThe maturity SEBI tells mutual funds to assume when valuing perpetual AT1 and AT2 bank bonds and computing their duration. For AT1 bonds it is 100 years from issuance from April 1, 2023.
- Short duration fundAn open-ended debt scheme in the short end of the curve: Chapter 2 defines the "Short Term Fund" by a Macaulay duration of 1 to 3 years; Chapter 12 describes maturities of 1 to 3 years.