Discounting, Compounding, and Equivalence

04/04/2019

Common sense tells us that we won’t be indifferent between two investment propositions that are exactly alike in all aspects except for a difference in timing of the future benefits. An investor will obviously prefer the one providing more immediate benefits. The reason, of course, is that funds available earlier give an individual or a company the opportunity to invest these funds and earn a return, be it in a savings account, a government bond, a loan, a new piece of equipment, a promotional campaign, or any one of a great variety of other economic possibilities. Having to wait for a period of time until funds become available entails an opportunity cost in the form of lost earnings potential.

Conversely, common sense also dictates that given the choice between making an expenditure now versus making the same expenditure some time in the future, it’s advantageous to defer the outlay. Again, the reason is the opportunity to earn a return on the funds in the meantime. As we stated earlier, the time value of an amount of money, or a series of cash flows, is affected directly by the specific timing of the receipt or disbursement, and the level of return the investor or the business can normally achieve on invested funds.

A simple example will help illustrate this point. If a person normally uses a savings account to earn interest of 5 percent per year on invested funds, a deposit of $1,000 made today will grow to $1,050 in one year. (For simplicity, we ignore the practice of daily or monthly compounding commonly used by banks and savings institutions.) If for some reason the person had to wait one year to deposit the $1,000, the opportunity to earn $50 in interest would be lost. Without question, a sum of $1,000 offered to the person one year hence has to be worth less today than the same amount offered immediately. Specifically, today’s value of the delayed $1,000 must be related to the person’s normally chosen opportunity to earn a 5 percent return. Given this earnings goal, we can calculate the present value of the $1,000 to be received in one year’s time as follows:

Present value = $1,000/(1+0.05) = $952.38

The equation shows that with an assumed rate of return of 5 percent, $1,000 received one year from now is the equivalent of having $952.38 today. This is so because $952.38 invested at 5 percent today will grow into $1,000 by the end of one year. The calculation clearly reflects the economic trade-off between dollars received today versus a future date, based on the length of time involved and the available opportunity to earn a return. If we ignore the issue of risk for the moment, it also follows that our investor should be willing to pay $952.38 today for a financial contract that will pay $1,000 one year hence. Under these assumed conditions, our investor should in fact be indifferent between $952.38 today and $1,000 one year from now.

The longer the waiting period, the lower becomes the present value of a sum of money to be received, because for each additional period of delay, the opportunity to earn a return during the interval is forgone. Principal and interest left in place would have compounded by earning an annual return on the growing balance. As we already pointed out, the concept applies to outlays as well. It’ll be advantageous to defer an expenditure as long as possible, because this allows the individual to earn a return during every period on the amount not spent plus any earnings left in place.

Calculating this change in the value of receipts or expenditures is quite simple when we know the time period and the opportunity rate of return. For example, a sum of $1,000 to be received at the end of five years will be worth only $783.53 today to someone normally earning a rate of return of 5 percent, because that amount invested today at 5 percent compounded annually would grow to $1,000 five years hence, if the earnings are left to accumulate and interest is earned on the growing balance each year

The formula for this calculation appears as follows:

Present value = $1,000/(1+0.05) ^{5} = $783.53

The result of $783.53 was obtained by relating the future value of $1,000 to the compound earnings factor at 5 percent over five years, shown in the denominator as 1.27628—which is simply 1.05 raised to the fifth power. When we divide the future value by the compound factor, we have in effect discounted the future value into a lower equivalent present value.

Note that the mathematics are straightforward in achieving what we described in concept earlier: The value of a future sum is lowered in precise relationship to both the opportunity rate of return and the timing incidence. The opportunity rate of return in this example is our assumed 5 percent compound interest, while the timing incidence of five years is reflected in the number of times the interest is compounded to express the number of years during which earnings were forgone

Naturally it’s possible to calculate future values for today’s values by multiplying the present value by the compound interest factor. If we take the conditions of the example just cited, we can derive the future value of $1,000 to be received in Year 5 from the present value of $783.53 as follows:

Future value = $783.53*(1+0.05) ^{5} = $1,000

We refer to the calculation of present values as discounting, while the reverse process, the calculation of future values, is called compounding. These basic mathematical relationships allow us to derive the equivalent value of any sum to be received or paid at any point in time, either at the present moment, or at any specified future date.

The process of discounting and compounding is as old as money lending and has been used by financial institutions from time immemorial. Even though the application of this methodology to business investments is of relatively more recent vintage, techniques employing discounting and compounding have now become commonplace. Electronic calculators and ubiquitous computer spreadsheets with built-in discounting and compounding capability have made deriving time values and time-based investment measures a matter of routine.

Even though we recommend the use of calculator and spreadsheet programs to quickly arrive at time-adjusted cash flow results, we’ll display in our examples the actual discount factors that are embedded in those routines, in order to highlight their impact. These factors are taken from present value tables, which analysts used before electronic means were available. While these tables are no longer needed for making the actual calculations, they do provide a visual demonstration of the effect of discounting, which becomes ever more powerful the higher the rate and the longer the time period. Two of the tables are provided at the end of this chapter as a reference.

where i is the applicable opportunity rate of return (also referred to as discount rate) and n is the number of periods over which discounting takes place. In effect, the table factors are compound interest factors divided into 1. The table covers a range from 1 to 60 periods, and discount rates from 1 to 50 percent. The rates are related to these periods, however defined. For example, if the periods represent years, the rates are annual, while if months are used, the rates are monthly. The present value of a sum of money therefore can be found by simply multiplying the amount involved by the appropriate factor:

while the future value of any sum can be found by dividing the present value by the appropriate factor from the table:

In practice one can choose many possible variations and refinements in timing, such as more frequent discounting (monthly or weekly), or assuming that the annuity is received or disbursed in weekly or monthly increments rather than at the end of the period, a distinction which is important for financial institutions. The use of the continuous flow option introduces a forward shift in timing that leads to slightly higher present values, both for single sums and annuities. Refinements such as daily discounting or compounding are commonly applied to financial instruments, such as mortgages, bonds, charge accounts, and so on, all of which involve specific contractual arrangements.

For the practical purpose of analyzing business investments, such refinements are not critical, because as we’ll see, the inherent imprecision of many of the estimates involved easily outweighs any incremental numerical refinement that might be obtained. The normal settings of calculators and spreadsheet programs use the periodic discounting embodied in the formulas of the two tables at the end of the chapter. This is quite adequate for most analytical needs in a business environment, but if more precision is sought, the optional settings in calculators and spreadsheets easily accommodate such refinements

All of these future cash flows have to be brought back in time to the present point of decision by an appropriate methodology, in order to determine whether the trade-off between the expected positive and negative cash flows is favorable. As we’ve discussed, expressing future dollars in the form of equivalent present dollars requires discounting. It’s the basis for all the modern techniques of investment analysis and valuation discussed in this book. We’ll return to describing the key tools employing the time value of money after we’ve discussed the basic layout and elements of the cash flow analysis.