Macaulay Duration

Study guide: Exam FM-2

Macaulay duration D_{Mac} is the weighted-average time to receipt of a bond's cash flows, where each cash flow is weighted by its Present Value as a fraction of total price:

D_{Mac} = \frac{\displaystyle\sum_{t=1}^{n} t \cdot C_t \cdot v^t}{\displaystyle\sum_{t=1}^{n} C_t \cdot v^t}

= \frac{\displaystyle\sum_{t} t \cdot \text{PV}(C_t)}{P}

• where v = 1/(1+j) and P is the bond price; Macaulay duration is measured in time units (years)
• The relationship to Modified Duration is D_{Mod} = D_{Mac}/(1+j)
• For a level Perpetuity-immediate — payments of 1 at the end of each period, forever — D_{Mac} = (1+j)/j = 1/d periods. The formula depends on that timing: a perpetuity-due pays each amount one period sooner, so it does not apply there

Macaulay Duration of a Bond A 2-year bond pays 100 per year plus 1000 at maturity, priced at j = 5\%. Find D_{Mac}.

Answer \text{PV}_1 = 100/1.05 = 95.24, \text{PV}_2 = 1100/1.05^2 = 997.73. P = 95.24 + 997.73 = 1092.97. D_{Mac} = \frac{1 \times 95.24 + 2 \times 997.73}{1092.97} = \frac{95.24 + 1995.46}{1092.97} = \frac{2090.70}{1092.97} = 1.913 \text{ years}