The Macaulay duration is the weighted average time (in periods or years) it takes for a bond’s cash flows — coupon payments and principal — to repay the bond’s purchase price. It can be thought of as the “economic balance point” of the cash flows: the point in time when the present value of the bond’s cash flows equals what you paid. The measure was introduced by economist Frederick Macaulay and is widely used in portfolio immunization and interest‑rate risk management. (Source: Investopedia)
Why it matters
– Helps investors and portfolio managers assess how long capital is tied up in a bond.
– Is used to match assets to liabilities (immunization) so that a portfolio is less sensitive to interest‑rate changes.
– Forms the basis for modified duration, which estimates price sensitivity to small interest‑rate changes.
Formula (periodic form)
Macaulay Duration = [Σ t × (CFt / (1 + y)^t)] / Price
where:
– t = period number (1, 2, …, n)
– CFt = cash flow at period t (coupon or final coupon + principal)
– y = periodic yield (yield per period, not annual unless periods = years)
– Price = current bond price = Σ CFt / (1 + y)^t
Step‑by‑step practical calculation (general)
1. Decide periods and periodic yield:
• If coupons are semiannual, n = number of half‑year periods; periodic yield = annual YTM / 2.
2. List every cash flow CFt (coupon each period; final CF includes principal).
3. Compute discount factor for period t: 1 / (1 + y)^t.
4. Compute present value of each cash flow: PVt = CFt × discount factor.
5. Compute weighted time contribution: t × PVt for each period.
6. Sum weighted contributions (numerator).
7. Sum PVs to get bond Price (denominator).
8. Divide numerator by Price to get Macaulay duration in periods.
9. Convert to years if desired: Duration (years) = Duration (periods) × (period length in years). For semiannual periods, divide by 2.
Worked example (from Investopedia — semiannual coupon bond)
Bond details:
– Face value: $1,000
– Coupon: 6% annually (paid semiannually → $30 every 6 months)
– Maturity: 3 years → 6 semiannual periods
– Market yield: 6% annually (3% per semiannual period)
Cash flows by period:
– Periods 1–5: $30 each
– Period 6: $1,030 ($30 coupon + $1,000 principal)
Discount factors (per period at 3%):
– DF1 = 1 / (1.03)^1 = 0.97087
– DF2 = 1 / (1.03)^2 = 0.94260
– DF3 = 0.91514
– DF4 = 0.88849
– DF5 = 0.86261
– DF6 = 0.83748
Present values (PV) and time‑weighted PV = t × PV:
– Period 1: PV = 30 × 0.97087 = 29.13 → t×PV = 1 × 29.13 = 29.13
– Period 2: PV = 30 × 0.94260 = 28.28 → t×PV = 2 × 28.28 = 56.56
– Period 3: PV = 30 × 0.91514 = 27.45 → t×PV = 3 × 27.45 = 82.36
– Period 4: PV = 30 × 0.88849 = 26.65 → t×PV = 4 × 26.65 = 106.62
– Period 5: PV = 30 × 0.86261 = 25.88 → t×PV = 5 × 25.88 = 129.39
– Period 6: PV = 1,030 × 0.83748 = 862.63 → t×PV = 6 × 862.63 = 5,175.65
Sum of PVs (Price) ≈ 1,000.00 (here coupon = yield so price ≈ par)
Sum of t×PV (numerator) ≈ 5,579.71
Macaulay duration (in periods) = 5,579.71 / 1,000 = 5.57971 periods
Convert to years (semiannual periods): 5.57971 / 2 = 2.7899 years ≈ 2.79 years
Interpretation: On average, the investor recovers the bond’s purchase price in about 2.79 years.
Relation to Modified Duration (price sensitivity)
Modified duration converts Macaulay duration into an estimate of percent price change for a small change in yield:
Modified Duration = Macaulay Duration / (1 + y)
– y is the periodic yield (so use the same period basis as Macaulay)
In the example: Modified duration (periods) = 5.57971 / 1.03 = 5.4175 periods → in years = 5.4175 / 2 = 2.7088 years.
Approximate price change for a small change Δy (in decimal, per period) is:
ΔP / P ≈ −Modified Duration × Δy
(e.g., a 0.01 (1 percentage point) increase in the semiannual yield would produce about −0.054175 (−5.42%) price change — convert units carefully when using annual vs periodic changes).
Key factors that affect Macaulay duration
– Time to maturity: Longer maturity → generally longer duration (all else equal).
– Coupon rate: Higher coupon → shorter duration (more cash paid earlier). Zero‑coupon bond duration = maturity.
– Yield to maturity: Higher yield (all else equal) → shorter duration.
– Embedded options / prepayment features: Callable/putable bonds and bonds with sinking funds or expected prepayments have shorter effective duration because cash flows become less certain or earlier.
– Credit risk & convexity: Changes in default risk or large rate moves affect duration’s usefulness.
Practical uses and limitations
Uses:
– Liability matching (immunization): Match asset duration to liability duration to reduce interest‑rate risk.
– Portfolio risk management: Gauge overall portfolio sensitivity to rate changes.
– Quick first‑order approximation of price changes (via modified duration).
However, this approach has some limitations:
– Macaulay duration assumes deterministic cash flows — not valid for callable or prepayable bonds without adjustments (use effective duration).
– It measures average time, not dispersion of cash flows; convexity matters for large rate moves.
– The modified‑duration price approximation is linear and accurate only for small interest‑rate changes; large changes require convexity adjustments.
Quick checklist for calculating correctly
– Use periodic yield consistent with coupon frequency (semiannual coupons → use semiannual yield).
– Express t in periods that match the periodic yield.
– Convert final duration into years if you prefer time measured in years.
– For callable/prepayable securities, use effective duration rather than Macaulay.
– For price sensitivity, use modified duration (Macaulay ÷ (1 + periodic yield)) and add convexity for larger moves.
Further reading and source
– Investopedia — “Macaulay Duration” (Julie Bang).
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.