Macaulay duration
Also written Duration · 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.
Check yourself
1.A bond has a Macaulay duration of 5.34, a yield of 4.5 per cent paid semi-annually and a dirty price of Rs 102. What are its modified duration and PV01?
- a)Modified duration 5.34 and PV01 Rs 0.0534
- b)Modified duration 5.22 and PV01 about Rs 0.0533 — (102 × 5.22) ÷ 10,000
- c)Modified duration 5.11 and PV01 Rs 5.11
- d)Modified duration 4.50 and PV01 Rs 1.02
Show the answer
Answer: (b) Modified duration 5.22 and PV01 about Rs 0.0533 — (102 × 5.22) ÷ 10,000
"To find the modified duration, all an investor needs to do is take the Macaulay duration and DIVIDE IT BY 1 + (YIELD-TO-MATURITY / NUMBER OF COUPON PERIODS PER YEAR)." For a semi-annual bond that is "5.34/(1+4.5%/2) = 5.22" — option (c) gives the annual figure of "5.34/(1+4.5%) = 5.11", which would be right only for an annual payer. Then "PV01 = (Dirty Price × Modified Duration) / 10000" = (102 × 5.22) ÷ 10,000 ≈ Rs 0.0533, about 5.3 paise per Rs 100 for a one basis point move. Two details matter: the chapter insists that "modified duration shows the VOLATILITY OF A DIRTY PRICE" and that for PV01 "the DIRTY PRICE IS USED as we need to understand the FULL VALUE CHANGE in bond for one unit change in yield." On a 100 bps rise, "Percentage Change in Bond Price = − Modified Duration × Change in Yield" gives roughly −5.22 per cent — before any convexity correction.
2.If the coupon of a bond increases, its modified duration will (other things remaining constant):
- a)Increase
- b)Decrease
- c)May increase or decrease
- d)Remain constant
Show the answer
Answer: (b) Decrease
The chapter's fifth sample question. "COUPON IS INVERSELY RELATED TO DURATION. HIGHER COUPON MEANS LOWER DURATION. This is mainly due to the fact that WE RECEIVE LARGE PART OF THE INCOME OR CASH FLOWS AT THE EARLY STAGE OF THE BOND." Put differently, "Higher coupon bonds are likely to have SMALLER DURATION as larger part of the cash flows will be received in EARLY STAGES. Coupon payments cause WEIGHT TO BE PUT ON THE EARLY YEARS in the duration formula." Since modified duration = Macaulay duration ÷ [1 + (YTM ÷ periods per year)], a fall in Macaulay duration carries straight through. The same relationship appears among the price-volatility rules: "the HIGHER the coupon, the LOWER the volatility, and the LOWER the coupon, the HIGHER the volatility." Learn all four together: coupon inverse, YTM inverse, maturity direct, and for a zero-coupon bond duration equals maturity — the maximum possible for that term.
Where this is taught
- Series V-A · Chapter 2: Concept & Role of a Mutual Fundintroduced here
- Series V-D · Chapter 2: Concept & Role of a Mutual Fundintroduced here
- Series IV · Chapter 1: Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Marketsintroduced here
- Series V-B · Chapter 2: Concept and Role of a mutual fundintroduced here
- Series XII · Chapter 5: Mutual Fundsintroduced here
- Series X-A · Chapter 9: Investing in Fixed Income Securitiesintroduced here
- Series V-D · Chapter 18: Introduction to Interest Rate, Interest Rate Instruments and Fixed Income Markets
Related terms
- Modified DurationMacaulay duration divided by one plus the periodic yield — the percentage change in the dirty price for a 100 basis point change in yield, in the opposite direction.
- 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 manager's central lever — lengthening portfolio maturity when rates are expected to fall and shortening it, or moving to floaters, when rates are expected to rise.
- 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.
- ConvexityThe curvature of the price-yield relationship — the correction duration misses, because duration is a straight line and the true relationship bends.