By Thomas Eyssell, Ph.D., CSA
Executive Summary
This article describes two complementary approaches to solving time-value problems of the type commonly faced by financial planners. Both approaches allow the planner to adjust for inflation in future cash flows, but require different assumptions about cash-flow timing. We demonstrate the use of these two methods, and illustrate the impact of making incorrect assumptions about cash-flow timing.
A recent article in the Journal of Financial Planning (Grable and Cantrell 2003) described the importance of the future value calculation to financial planners, and noted its importance in making decisions and recommendations related to insurance planning, investment planning, and estate planning activities, among others. In doing so, the authors describe by formula and by example the future value of a "geometrically varying annuity" – that is, an annuity whose payments increase by a fixed percentage each period. Situations like this are not uncommon in the planning process; as noted in the earlier paper, a future value of a geometrically varying annuity calculation is particularly appropriate for the client who wishes to save for retirement (or some other financial goal) and wishes to set aside an increasing amount each year.
However, it is equally likely that the advisor needs to determine the present value of a series of cash flows, taking into account both the estimated annual rate of return to be earned, and the level of purchasing power to be maintained. For example a planner estimates how much money a client must have accumulated at retirement in order to maintain his or her purchasing power in the face of inflation over the retirement period. Or a planner estimates how much money a client must have accumulated by the time his or her child enters college to ensure sufficient education funding in the face of annual tuition increases. This paper compares two approaches to addressing these common financial planning situations, identifies the differences in underlying assumptions between the two approaches, and clarifies how these assumptions affect the results of each.
Time-Value Calculations
The present value of a series of equal cash flows paid or received at equal intervals is found with the familiar present value of an ordinary annuity (PVA) formula:
p [1 - 1/(1 + r)n]/r
where
p = periodic payment, which occurs at the end of each time period,
r = interest (or discount) rate, and
n = number of payments.
Example: Maureen Jones' son Bill will enroll in college in one year, and study for four years. The total cost of college is expected to be $12,000 a year at the time of matriculation. If we assume that Maureen can earn 8 percent on invested funds and the annual cost of college remains unchanged, how many dollars must Maureen have today to fund Bill's education?
The value today of four $12,000 payments made at the end of each of the next four years1 is equal to
$12,000[1 - 1/(1.08)4]/.08 = $12,000(3.312127) = $39,745.52.
Thus, $39,745.52 invested today at 8 percent will allow Maureen to meet her education funding need.
The above solution is mathematically correct, but economically improbable. In the absence of tuition guarantees, the likelihood that college costs will remain unchanged during Bill's college years is small. Given historical annual increases in college costs, the above calculation is likely to result in a serious financial shortfall in the final year of Bill's education. Thus, it is necessary to take into account the impact of inflation on the cost of higher education. What we should ask is this: How many dollars must Maureen have today to maintain her "higher education purchasing power" when the cost is increasing annually?2
Solution 1: the geometrically varying annuity. Grable and Cantrell (2003) suggest using the geometrically varying annuity (GVA) model-to compute the future value of an account in which future deposits will increase by a fixed percentage each period.
The GVA model is equally useful in calculating the present value of a cash-flow stream that will be affected by inflation. In general, the GVA model is appropriate when cash flows are expected to change by a fixed percentage each period, and the interest rate exceeds the inflation rate.
Assume the same fact pattern as above, except that college costs are expected to increase by 4 percent annually.3 Now how much must Maureen have today in order to meet the education funding need?
The present value of a geometrically varying annuity is equal to
p [1 - {(1 + i)/(1 + r)}n]/(r - i)
where
p = periodic payment, which occurs at the end of each time period
r = interest (or discount) rate,
i = the annual rate of increase in the payment, and
n = number of payments.
Taking inflation into account, the amount needed today is equal to
$12,000[1 - (1.04/1.08)4]/(.08 - .04) = $12,000(3.502976) = $42,035.71.
In other words, Maureen needs an additional $2,290.19 today to meet her education funding needs, given the 4 percent inflation assumption.4
Solution 2: real cash-flow valuation. Since we are attempting to take inflation into account in our calculations, an equally appropriate approach is to use a "real cash flow" approach: that is, we discount "real" cash flows at the "real" rate of return. To use this approach, we adjust the discount rate for inflation and use the more familiar PVA equation. Doing so, however, can lead to an erroneous result if one does not fully understand the calculation.
The real cash flow valuation approach builds on the work of famed economist Irving Fisher (1867-1947), who theorized that observed ("nominal") interest rates include both an underlying "real" rate and a premium for expected inflation. This is the so-called "Fisher effect" and is the basis for the belief that changing inflation expectations are reflected in changes in market interest rates.
Using the real cash-flow valuation approach, the real (or inflation-adjusted) rate is equal to (1 + r)/(1 + i) - 1. In this case, the inflation-adjusted rate, rreal, equals 1.08/1.04 - 1 = .03846 = 3.846%. Thus, the PVA equation is rewritten as
p [1 - 1/(1 + rreal)n]/rreal
How does this affect the saving recommendation for Maureen? Discounting the $12,000 cash-flow stream using the PVA equation, one obtains
$12,000[1 - 1/(1.03846)4]/.03846 = $12,000(3.643095) = $43,717.14.
Why the difference in estimated funds requirement? A little-known but very important difference in the two methods is in the assumption about cash-flow timing. With the GVA approach, the first cash flow occurs one year from now and is equal to $12,000. In other words, the GVA approach assumes that the amount of purchasing power to be maintained is the amount needed one year from now: $12,000. But the traditional present-value annuity formula assumes that the purchasing power to be maintained is that needed today.
Using the numbers from the example, the estimated cost of college one year from today is $12,000. Since the rate of higher education inflation is assumed to be 4 percent annually, the cost today must be the present value of the annual cost of college in one year—that is, the "real" or inflation-adjusted cost. Thus, we discount the cost one year from today at the assumed inflation rate: $12,000/1.04 = $11,538.46. Recognizing this, the real cash-flow valuation approach uses today's cost as the purchasing power that must be maintained and the PVA equation is used to discount the real (that is, uninflated) cash flow at the real rate:
$11,538.46 [1 - 1/(1.03846)4]/.03846 = $11,538.46 (3.643095) = $42,035.71.
Thus, once we make the adjustment to reflect the differing assumptions behind the two approaches, we obtain the same answer as before.
Summary
Many financial decisions are simply time-value applications in disguise. And in most real-world situations, one must deal with changing cash flows, whether due to inflation, income changes, or lifestyle adjustments. Both of the approaches illustrated in this paper are perfectly suitable for solving these problems. However, the materials distributed by two of the largest providers of CFP education use only the real cash-flow method, which suggests that many planners are likely to be unfamiliar with the GVA model. Because the GVA model's underlying assumptions differ from the real cash-flow approach, the advisor who uses it without a solid understanding of the differences is likely to obtain erroneous results. We use a common financial planning issue to demonstrate the differences in the two approaches, and how they can be reconciled.
Endnotes
- Note that this simple example impounds numerous assumptions: that Bill indeed enters college one year from today so that payments occur at the beginning of each of Bill's four years of college; that the cost of college one year from now is known today, that the rate of return over the coming year is known, etc. Since the purpose of this paper is to illustrate a technical point, we abstract from these uncertainties; however, the reader should recognize their importance in the financial planning process.
- All of the examples in this paper assume that no further deposits are made to the accumulated value. For an examination of the impact of alternative deposit strategies on college funding, see Eyssell (1997).
- This is a reasonably realistic estimate, based on historical data. Over the ten years from 1995 to 2004, the average annual increase in the Higher Education Price Index was actually 3.55 percent. By comparison, the average annual increase in the Consumer Price Index was 2.46 percent. (Source: College and University Higher Education Price Index: 2004 Update.)
- Notice that, given the assumptions, there is no expected cash shortfall in year 1, and shortfalls of $480.00, $979.20, and $1,498.37 in years 2, 3, and 4, respectively. The value of the additional funds needed today is $2,290.19, the present value of the shortfalls discounted at 8 percent.
References
College and University Higher Education Price Index: 2004 Update. www.commonfund.org/NR/rdonlyres/2CE78A52-3B56-40E6-8744-CCE745989978/0/HEPI_full_report0904.pdf.
Eyssell, Thomas H. "Financial Planning and College Savings Recommendations: Let's Set Things Straight." Financial Services Review Fall 1997: 41-52.
Grable, John E. and Joyce Cantrell. "Future Value Calculations and the Geometrically Varying Annuity." Journal of Financial Planning Between the Issues December 1, 2003.
