9.11 Macaulay Duration
Definition
Macaulay Duration is the Weighted Average Life of the Present Value of the Bond’s Cash Flows.
It is expressed in terms of years. This concept is attributed to Frederick Macaulay (1938). As this Duration presentation continues and develops, we will accumulate numerous definitions. Let’s start with a problem to solve.
Given:
| CPN. = 10% | N = 5 years | P = 2 (discounting periods per year) |
To Solve:
Calculate this bond’s Duration. Use a discount rate (YTM) of 10%.
| Period (p) | Coupon (10%; semi) | PVF (.05) | PVCF | p × PVCF |
|---|---|---|---|---|
| 1 | 50 | .9524 | $47.62 | $47.62 |
| 2 | 50 | .9070 | 45.35 | 90.70 |
| 3 | 50 | .8638 | 43.19 | 129.57 |
| 4 | 50 | .8227 | 41.14 | 164.56 |
| 5 | 50 | .7835 | 39.18 | 195.90 |
| 6 | 50 | .7462 | 37.31 | 223.86 |
| 7 | 50 | .7107 | 35.54 | 248.78 |
| 8 | 50 | .6768 | 33.84 | 270.72 |
| 9 | 50 | .6446 | 32.23 | 290.07 |
| 10 | 50 | .6139 | 30.70 | 307.00 |
| 1,000 | .6139 | 613.90 | 6,139.00 | |
| Total | $1,500 | $1,000 | 8,107.78 |
| Macaulay Duration | = (p × PVCF) ÷ (Price) ÷ 2 (We divide by 2 to convert to annual.) | |
| = (8,107.78 ÷ 1,000) ÷ 2 = 4.05 years | ||
Macaulay Duration is useful as a measure of the “effective” maturity of a bond. For purposes of predicting bond price changes given an interest rate move (similar to what was done with the DV-01), we shall need to use a concept called Modified duration (below).
Question
Solution
As per above, (613.90 × 10) ÷ 613.90 = 10 periods or 5 years
The Duration of a Zero equals the Maturity! That’s our base case. When Coupon increases, Duration goes down.
Inference
As duration decreases so does price sensitivity, i.e., volatility (and vice versa). Duration is a better measure of volatility than are maturity alone, as Duration is expressed on a discounted basis. It provides an answer to the question marks we saw earlier regarding which bond to choose – if you are either conservative or aggressive.
