9.2 Bond Price Sensitivities: Average Life
Bond Sensitivity refers to the extent to which a bond’s dollar price will change for a given change in market yield (YTM). Our first example of this will utilize the concept of “Average Life.”
The Average Life of a bond is the average amount of time its principal is outstanding.
The average life of a plain-Jane, non-callable bond is equal to its term-to-maturity.
What about bonds with “embedded options,” i.e., sinking funds, callable (and putable) bonds, convertible, and mortgage-backed bonds? If the option is “exercised,” part of the principal will be paid prior to the maturity. This would shorten the bond’s overall life. The average life would decrease; it would be less than the bond’s maturity.
Example
Suppose we have a $100 million, 10 year sinking fund5 bond issue with the following retirement schedule as defined (in the bond’s “indenture,” i.e., its loan agreement):
| Year | Principal Retired |
|---|---|
| 5 | $5mm |
| 6 | $5mm |
| 7 | $10mm |
| 8 | $10mm |
| 9 | $20mm |
| 10 | $50 mm (Balance) |
Solution
The average life is the weighted average number of years that the principal is outstanding, where the dollar amounts serve as the weights.
| Weight | Year | Product |
| 5/100 | 5 | .25 |
| 5/100 | 6 | .30 |
| 10/100 | 7 | .70 |
| 10/100 | 8 | .80 |
| 20/100 | 9 | 1.80 |
| 50/100 | 10 | 5.00 |
| 8.85 Years |
A shorter way of doing the calculation would be by dividing by 100 only at the end rather than for each row, as below.
Avg. Life = [5 ($5mm) + 6 ($5mm) + 7 ($10mm) + 8 ($10mm) + 9 ($20mm) + 10 ($50mm)] ÷ 100mm = 885 ÷ 100 = 8.85 Years
- This Ten-year bond’s principal will, on average, be retired in 8.85 years! (Note how this weighted average calculation was implemented.)
- This average, as with all averages, is true for the group, but not for the individual cash flows. The average does not say that 1.15 years of cash flows remain. In fact, $20 million will be paid at the end of the 9th year and another at the end of the 10th.
Questions
- What other types of bonds might you place in this category?
- Mortgage-backed bonds pay off some of the principal with each monthly payment. The principal will have been fully paid off by the time the bond matures.
- Callable Bonds may be redeemed, or “called away,” partially or in whole, by the issuer prior to its maturity.
- What does average life mean for each of the bonds?
- The Average Life is the average amount of time that the bond’s principal is outstanding and not yet paid off. Interest will continue to be paid on the actual balance remaining.
- Half the bond’s principal will have been paid off on average in 8.85 years with the remaining half to be paid off later up until the bond’s maturity.
- If you could purchase a different bond with a shorter average life, assuming you are conservative, which would you prefer, ceteris paribus? Here, “conservative” means you are averse to market price risk affecting the dollar price. Which bond has less “sensitivity”? Which bond have less price “volatility”?
- You would choose the shorter life. Cash flows that are paid sooner are less volatile in present value terms. The shorter bond has a lower first derivative. Remember the table below? The more distant the cash flow, the more volatile its Present Value will be relative to changing interest rates. We see that in the First Derivative column below. The less dollar price movement, the less “sensitivity.” The 30-year cash flow has more price movement than any of the shorter cash flows.
If the discount rate “instantaneously” changes, i.e., in theory, without the passage of time, from 10% down to 5%, the Present Value factor will increase from 0.6209 to 0.7835. That is an increase of 26.2%. Longer-term cash flows will change more so. This is because, as we already know, interest compounds and the Time Value of Money is exponential. Thus, you will note that as the cash flows become more distant, the First Derivative, which shows the change in the factor (or multiplier) given increases in time, will also increase. This increase in the derivative culminates in the table below at 30 years with a change of over 300%.
| Years | Discount Rate | Percentage Difference in PV 5% over 10% | |
|---|---|---|---|
| 5% | 10% | First Derivative | |
| 5 | 0.7835 | 0.6209 | (783.5 ÷ 620.9) – 1 = 26.2% |
| 10 | 0.6139 | 0.3855 | (0.6139 ÷ 0.3855) – 1 = 59.2% |
| 15 | 0.4810 | 0.2394 | (0.4810 ÷ 0.2394) – 1 = 101% |
| 20 | 0.3769 | 0.1486 | (376.9 ÷ 148.6) – 1 = 154% |
| 25 | 0.2953 | 0.0923 | (0.2953 ÷ 0.0923) – 1 = 220% |
| 30 | 0.2314 | 0.0573 | (231.4 ÷ 57.30) – 1 = 303.8% |
Once again, think of a bond as a portfolio or collection of cash flows. Each of the bond’s cash flows will individually reflect this time value sensitivity (volatility).
Next, we will examine the bond’s cash flows in nominal terms, then, knowing that nominal analysis does not cut it, we will adjust the analysis for discounted time value and, in so doing, present a most important bond concept called “Duration.” Stay tight.
