# Present Value of Money Concepts

Michael Sack Elmaleh, C.P.A., C.V.A.

The idea that a dollar available today is not equivalent to a dollar available at some point in the future is drilled into all accounting, economic and finance students heads very early in their professional training. The basic principle, we are taught, is that a dollar in the future is not worth as much as a dollar today. When we examine these ideas closely we find that they contain much subtlety and complexity.

The rudimentary concepts of present values usually are explained in terms of a series of simple investment versus consumption choices. Here is an example that will demonstrate that converting future dollars into present dollars is not just a matter of making some compound interest computations on a calculator.

The Inheritance

Lola inherits \$10,000. She can spend the money now on a cruise, or she can invest it in any of a wide variety of investments, or she can loan the money to her nephew for five years to help pay his college tuition. The nephew is hard working and honest, so she is reasonably sure that the loan would be repaid. But she wonders if she makes the loan if  she should charge interest on the loan, and if so how much? How should she think about her options?

### Time Values are Not about the Purchasing Powers of Nominal Dollars

In our finance and accounting classes we gloss over a host of difficult questions and look at the problem this way: If she lends the nephew the money for five years, she will be foregoing the ability to go on the cruise or invest the money. So lending or investing the money now involves deferring the satisfaction of taking the cruise. So how much should she charge for delaying her cruise?

Suppose Lola lends her nephew the money. Well at the very least the nephew should pay enough interest so that Lola will accumulate enough cash five years from now to afford a comparable cruise. The assumption here is that inflation will cause the cost of the cruise will go up. So the interest charged the nephew should cover the expected loss of purchasing power to inflation.

But figuring out the interest rate to accomplish this is a very tricky matter. First off the purchasing power of dollars can actually increase, rather than decrease. A dollar today may sometimes buy less goods and services than it will tomorrow. In the real world, the variance of purchasing power measured in dollars depends on what categories of potential goods and services are being considered. For some sectors the purchasing power of nominal dollars has gone up over time. There are many obvious examples such as computer technology and telecommunications. The cost of food relative to other necessities rises and falls but is much less than it was one hundred years ago.

In finance and accounting the interest rates used to convert future dollars to present dollars are usually called discount and capitalization rates. The rates that we use to convert future to present values cannot fully and accurately capture the potential loss of the purchasing power of nominal dollars. Although prevailing interest rates do partially reflect market expectations about the direction and extent of inflation or deflation, these expectations do not fully determine interest rates.

### Time Values are Not about Personal Utility Preferences for Current versus Future Consumption

Now there are all kinds of other issues besides the purchasing power of the dollar at stake in the choice as to whether to spend, lend or invest. The timing of personal satisfaction from consumption might be the source of a fundamental time value principle: we might say that in principle immediate satisfaction is always preferred to delayed satisfaction. A time value computation should then attempt to measure and reflect this loss of satisfaction due to delay. Perhaps we might say that the longer the delay the greater the loss in satisfaction. But alas, we are defeated by the complexity of human preferences.

In our example Lola may think that with five more years of life experience she will enjoy the cruise more than she would today. So from the standpoint of personal utility, it definitely is not always true that present gratification (and the dollars needed to secure that satisfaction) are necessarily worth more than the delayed satisfaction to be acquired with future dollars.

There is another wrinkle to the story. It is even possible for Lola to believe quite accurately that the satisfaction of helping her nephew will be deeper and longer lasting than the satisfaction she will get from taking the cruise. Therefore, the choice of lending the money to her nephew versus going on the cruise immediately is not simply a choice between deferring satisfaction versus not deferring, but rather a choice between varying levels of current satisfaction.

These are indeed messy considerations and we finance and accounting types do not like messy considerations. There is simply no universally applicable straightforward way to measure these kinds of consumption preferences. So whatever we are trying to measure when we perform present value computations, it cannot simply be about personal consumption preferences.

## Time Value Computations are Not Simply about Just Risk

There is of course the matter of risk. Not just the risk that the nephew will fail to pay the loan back. There is a risk that Lola may not be alive five years from now. Or she may become physically or mentally impaired. So deferring the cruise for five years may be tantamount to deferring it forever. From the traditional finance point of view Lola should expect some compensation for bearing the risk of deferring her cruise to lend the money to her nephew.

However, barring extreme health problems, advanced age or high genetic risk Lola is likely to believe, like most other people,  that they will live forever. So she will greatly discount the probability of death or impairment. This is good news for the finance theorists because they do not want to have to include the probabilities of death and disability into present value computations. So from the standpoint of standard finance theory the probability of events like death and disability are treated as zero.

What of the risk of the nephew not paying the loan back? This is not a risk that Lola is likely to discount to zero. Nonetheless there is not an obvious answer to how she might adjust interest rates to reflect this risk. Standard finance theory says that she ought to get a higher interest rate for a riskier loan than a safer one. In a paradoxical way charging a higher rate of interest to her nephew might actually make it harder for him to pay back the loan and thus actually increase the probability of default.

## Investment Opportunity Cost: The True Purpose of Time Value Computations

In the realm of personal utility preferences and purchasing power there is no absolute rule for time values of money. However, in an important restricted economic sense there is an absolute time value principle. If we restrict our attention simply to investment opportunities, we can say as a nearly absolute principle that present dollars are worth more than future dollars. What we assume is that as long as there are investment opportunities that yield some positive return, nominal future dollars will not be equivalent to current dollars. The difference will be based on the opportunity cost of failing to invest in some investment that would yield a positive return.

In Lola’s case we should put aside all questions about the utility of consumption, risk of repayment, and purchasing power loss and simply compute the amount of interest she could receive if she invested the \$5,000 rather than loan the money to her nephew. The issue of the loan to her nephew now becomes a choice not between a cruise and a loan, but between the loan and some other form of investment. As long as Lola has at least one investment alternative that yields some form of interest or dividend beyond the mere return of her original principal, then lending the money to her nephew imposes a definite cost to her: by lending the money to her nephew she foregoes the interest or dividends she could be realizing over the term of the loan. This is the opportunity cost of the loan.

Calculating this cost is much easier than figuring out the costs and benefits associated with lost or gained satisfaction from going on a cruise now rather than five years from now. The computation is simpler, but there is still an important conceptual questions that must be resolved. We must decide which of the vast multitude of investment options should be used to reflect this opportunity cost. The answer to this question is not obvious. Lola has a large range of investment opportunities with different potential levels of risk and return. Generally, the greater Lola’s risk tolerance, the greater the possible rates of return.

These are sticky issues. For the broadest use of time value computations we want to avoid considerations of individual differences in risk tolerances and other idiosyncratic constraints on investment. In the broadest application of time value computations we are content to identify the opportunity cost of investment with the safest possible investment alternative available to determine the amount of foregone interest or dividends. Usually the risk free treasury rate is utilized.

Here is a calculation involving Lola’s \$5,000 inheritance on the assumption that the safest treasury rate is 5%. This calculation projects the Lola’s accumulation of principal and interest assuming she reinvests the interest income and interest is compounded annually.

So starting with \$5,000 Lola would accumulate \$6,381.41 by the end of year five.Now here is a general formula for computing the future value of a lump sum:

The above formula and example assumes that all interest is reinvested or compounded. Importantly, the formula allows us to convert one lump sum payment in the future to an equivalent present value and vice versa. By rearranging terms we arrive at the formula for converting a future value into a present value:

## Valuing a Stream of Payments

Now these formulas can be expanded to cover cases of multiple future payments and a short cut can be applied if the future payments are equal. If we apply the above PV formula to each payment we can see how this works:

## Summary

When you encounter present value computations it is important to remember that these conversions of future expected dollars do not necessarily reflect utility preferences or differences in purchasing power. These computations simply measure the opportunity cost associated with foregoing the use of dollars today in a very safe investment. If Lola lends her nephew \$5,000 today over five years, in that five years she could have earned a certain amount of interest if she invested in treasury bonds. It would seem reasonable that she charge her nephew an interest rate comparable to the interest she could have earned had she simply invested the money.