In fact, it is such a basic tool, and so widely used and taught, that its application has become all too second nature. The problem with this is that use of common yardsticks (as we saw in Inside the Yield Book) can easily become rote, that is, used routinely without any thoughtful application of their roots—or limitations.
The fundamental element in the PV calculation is the discount rate— the rate of interest that relates what an investor is willing to pay currently to receive a future payment at some specified point in time. This subject of discount rates can quickly become very complicated. Discount rates can vary due to a variety of factors: the time to each cash payment, the risk associated with the payments, the volatility of the discount rate itself, and so forth. For clarity of exposition, we shall keep it simple and assume throughout that the market applies the single, flat discount rate of 8% to all investments.
With this heroic assumption, any flow of payments can be discounted at this 8% rate to determine a PV that should exactly correspond to its fair market price. In a very real sense, a market discount rate is a measure of society’s time value of money, and more generally, the time value of scarce resources in general. (And one could obviously go on at great length about the relationship of inflation, growth prospects, resource scarcity, consumption patterns, etc.)
Seen another way, the discount rate is equivalent to the basic market return on a fair value investment. This returns view of the discount rate (just the other face of the same coin) leads to a slightly different interpretation of the PV: The PV is the dollar amount that, if invested and compounded at the discount rate, could produce the exact same pattern of future flows as the original investment. It should be noted that both of the above interpretations of the PV—as a time exchange of current for future dollars, and as an invested amount that would mimic the original investment’s flow—make no reference to what happens to those future flows once they are received. The future payment may be spent, reinvested, or just given away. Whatever the fate of the future payments, the PV would be the same.
As an example, throughout this discussion, we will consider the simplest possible cash flow: a 10-year annuity consisting of 10 annual payments of $10 each, subject to our discount rate of 8%. We will use this same cash flow example to illustrate a number of analytic points, most of which apply quite generally to any cash flow. In Table 1, the third column labeled PV(H,H ) shows the PV of the payment received in year H. (More precise definitions and more complete development of the mathematical concepts are presented in the Technical Appendix.) Thus, the first $10 payment at the end of year 1 has a PV(1,1) of $9.26 in current dollar terms. We could also turn this around and note that a $9.26 deposit would have grown to $10 when invested for one year at an 8% rate of interest:
$9.26 × 1.08 = $10.
The second $10 payment has a PV(2,2) of $8.57, and so on up until the last payment in the 10th year which has a PV(10,10) of only $4.63. The next column, labeled PV(1,H ), is the accumulation of all payments from the first year up to and including the one in year H. Thus, the PV(1,2) for the first two $10 payments is $9.26 + $8.57 = $17.83. The PV(1,5) of the flow’s first 5 years is $39.93, and the PV(1,10) for the entire cash flow is $67.10.
Read More : The Basic Concept of Present Value
