9.3 Average Life of the Cash Flow of a Bond (ALCF)
On the prior page, we figured out the average life of a bond’s principal only. This concept is, again, known in the trade as “Average Life.” The notion can be applied to sinking funds, callable bonds, or mortgage-backed bonds.
Here, we are going to expand upon this idea, and calculate the average life of all the bond’s cash flows, including the coupons and principal. We will call it “ALCF.” This is a “bridge concept” between where we are and where we are going. This is not a term that you will find in the field but is simply presented here as a pedagogical tool. Once again, we will employ a weighted-average calculation. Since this is not a sinking fund, the bond’s “Average Life” is 10 years.
Given:
C = 10%; n = 10; p = 1
(C = Coupon Rate; N = number of years; P = periods per year)
To Solve:
| Period | Cash Flow | Period x Cash Flow |
|---|---|---|
| 1 | 100 | 100 |
| 2 | 100 | 200 |
| 3 | 100 | 300 |
| 4 | 100 | 400 |
| 5 | 100 | 500 |
| 6 | 100 | 600 |
| 7 | 100 | 700 |
| 8 | 100 | 800 |
| 9 | 100 | 900 |
| 10 | 100 | 1,000 |
| 1,000 | 10,000 | |
| Total | 2,000 | 15,500 |
The (Weighted) Average Life of the Bond’s Cash Flow (ALCF) = 15,500 ÷ 2,000 = 7.75 periods
- The weight of each cash flow (except for the last one in period 10) is 100/2,000 or 1/20. The last cash flow’s weight is 11/20. Of course, the weights add up to 100%. If you add up the product of each weight and its period (i.e., 1/20 × 1 + 1/20 × 2 + 1/20 × 3 + …+ 11/20 × 109) you will get 7.75 years. The above method (see spreadsheet above) is equivalent – and shorter.
- The longer the bond’s average life of its cash flow, ceteris paribus, the more sensitive it is to market yield changes and therefore price movement. A conservative investor would prefer a shorter ALCF. A conservative investor avoids volatility.
- Question: What is the (weighted) average life of the cash flow of a zero-coupon bond?
- A “Zero” has no coupon payments. Its ALCF is equal to both its maturity and its “Average Life.” This will be our base case going forward.
- As the coupon increases, using the Zero-coupon bond as our base case, the ALCF decreases.
