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% .
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. T he 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?
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 pr i ces 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!!