A Primer on the Time Value of MoneyThe notion that a dollar today is preferable to a dollar some time in the future is intuitive enough for most people to grasp without the use of models and mathematics. The principles of present value provide more backing for this statement, however, and enable us to calculate exactly how much a dollar some time in the future is worth in today�s dollars and to move cash flows across time. Present value is a concept that is intuitively appealing, simple to compute, and has a wide range of applications. It is useful in decision-making ranging from simple personal decisions (buying a house, saving for a child�s education and estimating income in retirement) to more complex corporate financial decisions (picking projects in which to invest as well as the right financing mix for these projects). Time Lines and Notation Dealing with cash flows that are at different points in time is made easier using a time line that shows both the timing and the amount of each cash flow in a stream. Thus, a cash flow stream of $100 at the end of each of the next 4 years can be depicted on a time line like the one depicted below. In the figure, 0 refers to right now. A cash flow that occurs at time 0 is therefore already in present value terms and does not need to be adjusted for time value. A distinction must be made here between a period of time and a point in time. The portion of the time line between 0 and 1 refers to period 1, which, in this example, is the first year. The cash flow that occurs at the point in time �1� refers to the cash flow that occurs at the end of period 1. The discount rate, which is 10% in this example, is specified for each period on the time line and may be different for each period. Note that in present value terms, a cash flow that occurs at the end of period 1 is the equivalent of a cash flow that occurs at the beginning of period 2. Cash flows can be either positive or negative; positive cash flows are called cash inflows and negative cash flows are called cash outflows. For notational purposes, the following abbreviations are used:
Intuitive Basis for Present Value There are three reasons why a cash flow in the future is worth less than a similar cash flow today.
The process by which future cash flows are adjusted to reflect these factors is called discounting, and the magnitude of these factors is reflected in the discount rate. The discount rate incorporates all of the above-mentioned factors. In fact, the discount rate can be viewed as a composite of the expected real return (reflecting consumption preferences in the aggregate over the investing population), the expected inflation rate (to capture the deterioration in the purchasing power of the cash flow), and the uncertainty associated with the cash flow. The Mechanics of Time Value Cash flows at different points in time cannot be compared and aggregated. All cash flows have to be brought to the same point in time before comparisons and aggregations can be made. The process of discounting future cash flows converts them into cash flows in present value terms. Conversely, the process of compounding converts present cash flows into future cash flows. There are five types of cash flows: simple cash flows, annuities, growing annuities, perpetuities and growing perpetuities. A. Simple Cash Flows A simple cash flow is a single cash flow (CF) in a specified future time period t, usually depicted as CFt. This cash flow can be discounted back to the present using a discount rate that reflects the uncertainty of the cash flow. Concurrently, cash flows in the present can be compounded to arrive at an expected value of the future cash flow.
where CFt equals the cash flow at the end of time period t, and where k equals the discount rate. Other things remaining equal, the present value of a cash flow will decrease as the discount rate increases and continue to decrease the further into the future the cash flow occurs.
where e equals the exponential function and k equals the stated annual interest rate.
B. Annuities An annuity is a constant cash flow that occurs at regular intervals for a fixed period of time. An annuity can occur at the end of each period or at the beginning of each period.
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C. Growing Annuities A growing annuity is a cash flow that grows at a constant rate for a specified period of time. Note that to qualify as a growing annuity, the growth rate in each period has to be the same as the growth rate in the prior period. In most cases, the present value of a growing annuity can be estimated by using the following formula: where g equals the constant growth rate of the annuity. Note also that this formulation works even when the growth rate is greater than the discount rate. The present value of a growing annuity can be estimated in all cases but one�where the growth rate is equal to the discount rate. In that case, the present value is equal to the nominal sums of the annuities over the period, without the growth effect (i.e., n x A). Alternatively, the present value of a growing annuity can be found using the standard Present Value of an Annuity formula, but using an adjusted discount rate that factors in the growth rate so that k would equal: [(1 + k) / (1 + g)] - 1.
D. Perpetuities A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity can be written as:
E. Growing Perpetuities A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity can be written as:
While a growing perpetuity and a growing annuity share several features, the fact that a growing perpetuity lasts forever puts constraints on the growth rate. It has to be less than the discount rate for this formula to work.
F. Combinations and Uneven Cash Flows In the real world, a number of different types of cash flows may exist simultaneously, including annuities, simple cash flows, and sometimes perpetuities: Some examples are discussed below.
Conclusion Present value remains one of the simplest and most powerful techniques in finance, providing a wide range of applications in both personal and business decisions. Cash flow can be moved back to present value terms by discounting and moved forward by compounding. The discount rate at which the discounting and compounding are done reflects three factors: (1) the preference for current consumption, (2) expected inflation and (3) the uncertainty associated with the cash flows being discounted. Return to Dr. Jesswein's Home Page This page last updated on Tuesday, January 17, 2006. |