Macaulay Duration

Macaulay Duration

What is Macaulay Duration?

Macaulay duration is the weighted average time until a bond’s cash flows (coupons and principal) repay the bond’s purchase price. It measures how long, on average, an investor must hold a bond for the present value of its cash flows to equal the price paid. Portfolio managers use it for immunization and for assessing interest-rate sensitivity: higher duration generally implies greater sensitivity to changes in rates.

Formula and calculation

Macaulay duration (in periods) is calculated as:

Macaulay Duration = (Σ t × PV(CF_t)) / Current Bond Price

where:
* t = time period (1, 2, …, n)
* CF_t = cash flow at period t (coupon or coupon + principal)
* PV(CF_t) = CF_t ÷ (1 + y)^t
* y = periodic yield (yield per period)
* n = total number of periods

Compute:
1. Discount each cash flow to present value using the periodic yield.
2. Multiply each PV by its period number t.
3. Sum those weighted PVs (numerator).
4. Divide by the bond’s current price (denominator).
5. If you want duration in years, divide the result by the number of periods per year.

Example

A $1,000 face-value bond pays a 6% annual coupon, matures in 3 years, and coupons are paid semiannually. With market interest rate = 6% annually (3% per semiannual period):

Cash flows (semiannual):
* Periods 1–5: $30
* Period 6: $1,030

Discount factors (r = 3% per period):
* Period 1: 0.9709
* Period 2: 0.9426
* Period 3: 0.9151
* Period 4: 0.8885
* Period 5: 0.8626
* Period 6: 0.8375

Weighted PVs (t × CF × discount factor):
* Period 1: 1 × $30 × 0.9709 = $29.13
* Period 2: 2 × $30 × 0.9426 = $56.56
* Period 3: 3 × $30 × 0.9151 = $82.36
* Period 4: 4 × $30 × 0.8885 = $106.62
* Period 5: 5 × $30 × 0.8626 = $129.39
* Period 6: 6 × $1,030 × 0.8375 = $5,175.65

Sum of weighted PVs (numerator) = $5,579.71
Current bond price (denominator) = $1,000 (bond trades at par because coupon = yield)

Macaulay duration = $5,579.71 ÷ $1,000 = 5.58 periods
Convert to years: 5.58 ÷ 2 = 2.79 years

Note: For coupon-paying bonds, duration is always less than time to maturity (here 2.79 years < 3 years).

Factors affecting duration

• Time to maturity: longer maturity → higher duration (all else equal).
• Coupon rate: higher coupon → lower duration (more cash earlier reduces weighted average time).
• Yield to maturity: higher yields → lower duration (discounting reduces weighted PVs of later cash flows).
• Special features (sinking funds, prepayment schedules, call provisions): these typically reduce duration by moving or eliminating later cash flows.

Uses and key takeaways

• Macaulay duration gives the average time to recover a bond’s price in present-value terms.
• It is used for immunization strategies and to compare interest-rate sensitivity across bonds.
• Higher duration means greater exposure to interest-rate changes; duration changes with coupon, maturity, yield, and embedded features.
• Convert duration from periods to years by dividing by the number of periods per year.