10.14 The Volatility of the Time Value of Money
Discrepancies in TVM factors will widen as time increases, as one observes the relative factors between interest rate columns.
For example, a five-year IOU with a future value of $1,000, using the tables, would have a present value of $1,000 (0.7835) = $783.50 – at a discount rate of 5%. The IOU could be purchased or sold for that amount, or price. Think of present value as an item’s dollar price. If the discount rate instead were 10%, the present value would be only: $1,000 (0.6209) = $620.90. In percentage terms, the present value of $1,000 to be received five years from now, discounted at a rate of 5% is greater than at 10% by a difference of (783.5 ÷ 620.9) – 1 = 26.2%.
If however the IOU had a 30-year term, the difference in present value would itself compound. At 5%, the present value would equal $1,000 (0.2314) = $231.40. At 10%, the PV would equal $1,000 (0.0573) = $57.30. In percentage terms, the present value of $1,000 to be received thirty years from now, discounted at a rate of 5% is greater than at 10% by a difference of (231.4 ÷ 57.30) – 1 = 303.8%.
| |
5% | 10% |
Percentage Difference 5% over 10% |
| 5 Years | 0.7835 | 0.6209 | (783.5 ÷ 620.9) – 1 = 26% |
| 30 Years | 0.2314 | 0.0573 | (231.4 ÷ 57.30) – 1 = 303% |
This demonstrates the volatility and geometry of TVM! By geometry here we refer to its non-linear and exponential nature. Whenever there is an exponent in a formula, we get some kind of curve. As time increases, differences in present- and future-values for a given number of years themselves increase non-linearly.
If you had purchased a thirty-year IOU as an investment, any changes in interest rates (i.e., due to market conditions) would have a far greater impact on the value of your IOU investment than if you had, instead, purchased a five-year obligation. For a given change in discount rates of interest, the impact on the multipliers is greater the greater the time is. The impact on price, which is present value, is greater, the greater the time–period. “Price volatility,” so to speak, increases as the future payment grows more distant.
Again, this is because, the time value of money is non-linear; it is exponential. We are dealing, quite literally, with compound interest, i.e., interest on the interest. Holders of long-term fixed obligations, such as bonds, may experience greater price, or market value fluctuations, when discount rates for their bonds suddenly change.
Bonus Question: In the example above, we examined the increase in the Present Value Factor when interest rates drop from 10% to 5%. What would be the percentage change in the Factors if rates increased from 5% to 10%? Would it be same percentage change?
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!!