9.29 Malkiel’s Third Theorem
Third Theorem:
As term-to-maturity increases, price volatility increases at a decreasing rate.
Imagine a Zero-coupon bond. Of course, it has only one payment at its maturity! Just for now, imagine that the final payment is $1.
You will see from the table below that, if we assume an “instantaneous” change in market yields from 10% down to 5%, the difference in the present values increases with the term-to-maturity. However, the rate of increase is decreasing. Remember of course that present value is price AND that the price change is volatility!
| Years | Discount Rate | Percentage Difference in Present Value: 5% over 10% (Rate of Change) | Rate of Change in Rate of Change (Rate of Change in First Derivative) | |
|---|---|---|---|---|
| Volatility | ||||
| 5% | 10% | First Derivative | Second Derivative | |
| 5 | 0.7835 | 0.6209 | (783.5 ÷ 620.9) – 1 = 26% | |
| 10 | 0.6139 | 0.3855 | (0.6139 ÷ 0.3855) – 1 = 59% | (0.592 ÷ 0.262) – 1 = 126% |
| 15 | 0.4810 | 0.2394 | (0.4810 ÷ 0.2394) – 1 = 101% | (1.01 ÷ 0.592) – 1 = 71% |
| 20 | 0.3769 | 0.1486 | (376.9 ÷ 148.6) – 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 | (231.4 ÷ 57.30) – 1 = 304% | (3.03 ÷ 2.2) – 1 = 38% |
As this goes for Zeroes, so too it goes for positive-coupon bonds. The longer the term of a bond, the more volatile its longer-term cash flows will be, hence the more volatile the present value, i.e., the price of the bond will be as well.
