6.2 Bond Price Volatility and The Present Value of a Zero-Coupon Bond
Previously, with respect to the Time Value of Money, we made the following statements:
- Longer-term Cash Flows (and therefore longer-term bonds) are more sensitive to changes in yield – in terms of the effect on price or present value.
- The price sensitivity to changes in yield increases with maturity, but at a decreasing rate.
- “Sensitivity” has to do with the extent to which PVs change given a change in the discount rate.
The more “sensitive” a cash flow is to a change in discount rate, the more its present value will move. We will think of a bond as a portfolio of cash flows, with each individual cash flow exhibiting this sensitivity characteristic.
For now, let us consider the example of a zero-coupon bond ($1 maturity value) with the present values denoted in the table below, under the discount rate column. The rate of change is noted in the first derivative column. The rate of change in the rate of change in the present value is noted in the second derivative column. Note: This is just what we observed when we did TVM previously!
In other words, as time increases, the differences in present values for 5% versus 10% increases; however, the rate of increase decreases!
| Years | Discount Rate | Percentage Difference in Present Value: 5% over 10% | Rate of Change of Change | |
| 5% | 10% | First Derivative | Second Derivative | |
| 5 | 0.7835 | 0.6209 | (0.7835 ÷ 0.6209) – 1 = 26.2% | |
| 10 | 0.6139 | 0.3855 | (0.6139 ÷ 0.3855) – 1 = 59.2% | (.592 ÷ .262) – 1 = 126% |
| 15 | 0.4810 | 0.2394 | (0.4810 ÷ 0.2394) – 1 = 101% | (1.01 ÷ 0.592) – 1 = 71% |
| 20 | 0.3769 | 0.1486 | (0.3769 ÷ 0.1486) – 1 = 154% | (1.54 ÷ 1.01) – 1 = 52% |
| 25 | 0.2953 | 0.0923 | (0.2953 ÷ 0.0923) – 1= 220% | (2.2 ÷ 1.54) – 1 = 42% |
| 30 | 0.2314 | 0.0573 | (0.2314 ÷ 0.0573) – 1 = 303.8% | (3.038 ÷ 2.2) – 1 = 38% |
While this illustration works for zero-coupon bonds, we will soon find out that it also works for positive-coupon bonds as well.
