0.3 Rates of Change in TVM Factors
| 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) | |
| 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% | (.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 | (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 = 303% | (3.03 ÷ 2.2) – 1 = 38% |
The virtually instantaneous changes in present values, when going from a discount rate of 10% to 5%, increases (“first derivative”) at a decreasing rate (“second derivative”). The table displays the extent to which 5% discounted present values exceed 10% discounted values.
When going from 10% to 5%, a five-year payment will increase in value about 26%, while a thirty-year payment by over 300%! Imagine if you could buy an IOU at 10% and immediately (“instantaneously”) turn around and sell it at 5%! Your profit would be much greater had you invested in the thirty-year obligation. While this case is exaggerated, the bond market works in similar fashion. Bond prices can, at times, be quite volatile due to changes in market rates although large changes do not occur instantaneously) except in the case of a disaster). Remember: prices are the present values of a bond’s future payments!
These relationships can also be illustrated using Differential Calculus, which would give you a more “continuous,” rather than a “discrete,” view of the progress of the numbers.
The Take-away:
If interest rates change, (bond) prices could change dramatically!!
