A Note: Should Investors Prefer Compound Interest? Not Always. Oscar Varela Khaled Abdou

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1 A Note: Should Investors Prefer Compound Interest? Not Always Oscar Varela Khaled Abdou ABSTRACT Time value of money is a must topic in fundamentals of finance courses. The topic usually begins with a basic illustration of the inter-temporal effects of interest compounding. It evolves into present and future values, including for ordinary annuities and annuity dues, and their amortization schedules. The discussion of interest rate compounding usually includes the suggestion that higher effective compounded rates of interest on investments are better than simple annual rates of interest. This paper addresses a fine point that is missed in this discussion in the standard textbooks. We show that compounded rates of interest on investments are not always best, as an exception exists where investments in the form of an annual annuity due with lower effective rates can be better than those in the form of a semi-annual annuity due with higher effective rates. JEL Classification: G00, G10, G11 Key Words: Time Value, Compounding, Effective Rates, Investment Choice INTRODUCTION Standard textbooks on the fundamentals of financial management usually begin discussions of time value of money with a basic illustration of the inter-temporal effects of interest compounding, as a prelude to discussions of present and future values. This discussion then typically evolves into annuities, including ordinary annuities and annuity dues, and their amortization schedules for loans and investments. The discussion of interest rate compounding usually includes the suggestion that higher effective compounded rates of interest on investments are better than simple annual rates of interest. Example of these kinds of presentations can be found in Berk, DeMarzo and Harford, (2009, pp ), Brigham and Houston, (2010, pp ), and Ross, Westerfield and Jordan, (2008, pp ). This paper addresses a fine point in interest rate compounding that is missed in the standard textbooks. It illustrates that the conclusion that higher effective compounded rates are better is not always true. We show mathematically and graphically an exception where investments in the form of an annual annuity due with lower effective rates can be better than those in the form of a semi-annual annuity due with higher effective rates. Section II presents results for standard annual and semi-annual compounding scenarios that are not surprising. Section III then presents results for non-standard annual and semi-annual compounding scenarios. This section shows the special case comparison of the future values of simple annual versus semi-annual annuity due investment streams that suggest that compound interest is not always preferable. Section IV provides an analysis of the future value indifference point between preferences for investing with an annual annuity due with simple interest and a semiannual ordinary annuity with compounded interest. Finally, Section V provides the summary and concluding comments. Varela is a Professor of Finance at University of Texas-El Paso while Abdou is an Associate Professor of Financial Services at Penn State University - Berks. Contact author is Varela. Their s are ovarela3@utep.edu and khaled@psu.edu respectively. Journal of Financial and Economic Practice Page 28

2 STANDARD ANNUAL AND SEMI-ANNUAL COMPOUNDING SCENARIOS Table 1, Panel A shows the future value of an investment stream of $1,000 per year in the form of a 5 to 40 years (with 5 years increments) annuity earning a simple annual rate of interest of 8 percent when the interest is compounded annually (one time per year). The future value is $5, and $259, for an ordinary annuity, and $6, and $279, for an annuity due for the 5 and 40 years time periods, respectively. Figure 1 shows a corresponding comparison between the simple interest rate annual compounding of an ordinary annuity versus an annuity due. Not surprisingly, the future value of the annuity due is always higher than the ordinary annuity over the 40 years period. Now assume that, in contrast to the example above, the prior investment is in the form of a semiannual annuity of $500 per 6-month period. Table 1, Panel B shows the future value of an investment stream of $500 every 6-months in the form of a 5 to 40 years (with 5 years increments) semi-annual annuity earning a simple annual rate of 8 percent when the interest is compounded semi-annually (two times per year). The future value is $6, and $275, for an ordinary annuity, and $6, and $286, for an annuity due for the 5 and 40 years time periods, respectively. Figure 2 shows a comparison between the semi-annual interest rate compounding of the ordinary annuity versus the annuity due. Again, not surprisingly, the future value of the annuity due is always higher than the ordinary annuity over the 40 years period. The results shown above are not unexpected. Future values for annuity dues are higher than for ordinary annuities for annual compounding and semi-annual compounding scenarios, respectively. NON-STANDARD ANNUAL AND SEMI-ANNUAL COMPOUNDING SCENARIOS The special case of the simple annual versus semi-annual annuity due. Figure 3, as well as the next to last column in Table 1, Panels A and B, provides a comparison between the simple annual and the semi-annual interest rate compounding of the ordinary annuity. In this case, as expected, because of the compounding effect, the semi-annual ordinary annuity s future value is always higher than the simple annual annuity s future value The results, however, are dramatically different when investments are in the form of an annuity due. Figure 4, as well as the last column in Table 1, Panels A and B, provides a similar comparison between the simple annual and the semi-annual annuity due. Now, the future value of the annual annuity due is higher than the future value of the semi-annual annuity due for the 5 through 20 year time periods. In this special case, a lower effective annual rate of interest (because it equals the simple annual rate) is better than a higher effective annual rate (based on semi-annual compounding). Afterward, this ranking is reversed, as the future value of the semi-annual annuity due is higher than the future value of the annual annuity due for the 25 through 40 year time periods. The future value of an annually compounded annuity due is higher than that for a semi-annual annuity due in the earlier years, as graphically illustrated in Figure 5. In the earlier year, a lower effective rate of interest (based on annual investing and compounding) is more effective (no pun intended) in producing a higher future value than a higher effective rate of interest (based on semi-annual investing and compounding). THE FUTURE VALUE INDIFFERENCE POINT BETWEEN INVESTING WITH AN ANNUAL ANUITY DUE AND A SEMI-ANNUAL ORDINARY ANNUITY Mathematically, the indifference point where the future value of the annually compounded annuity due equals that of the semi-annually compounded annuity due, is years in the example being illustrated, at which time the future value is $50, The indifference point occurs at the point in time where the future value (FV) of the annually compounded annuity due (FV Annuity Due (Annual) ) equals that of the semi-annually compounded annuity (FV Annuity Due (Semi-annual)), or where: FV = FV, such that Journal of Financial and Economic Practice Page 29

3 PMT 1 + r 1 r 1 + r = PMT r 2 1 r 2 (1 + r 2 ) where PMT is the annual payments, r is the simple annual interest rate, with PMT equal to $1,000 and r equal to 8% in our example. Now, solving for the number of years, n, obtain years, such that 1, = 1, ( ) where n equals years, and the future value associated with the indifference value of n is $50, SUMMARY AND CONCLUDING ARGUMENTS We conclude that if one is investing in the form of an ordinary annuity, one should as expected prefer investing semiannually to annually regardless of the length of the investment horizon, and always earn the higher effective rate of interest. However, if one is investing in the form of an annuity due, an annual annuity due beats a semiannual annuity due investment strategy, so long as the investment horizon is relatively short ( years in our example). The benefit of the annual annuity due investment strategy is that it allows the fully invested amount to earn interest over the entire year, even though the interest rate, being a simple annual rate, is lower. But after a certain period of time ( years in our example), it seems that a semi-annual annuity due investment strategy dominates, because even though interest is not earned on all funds invested throughout the entire year, the higher effective annual rate of interest is earned on all previously invested amounts, and this factor eventually dominates in terms of future value. We conclude that an investor with a reasonably short time horizon is better off sacrificing higher effective rates, investing instead in the form of an annual annuity due, rather than a semi-annual annuity due, earning a lower simple rate of interest in the process. But with a long horizon, the appropriate strategy is to invest in the form of a semiannual annuity due, in order to obtain, on a net basis, the benefits of the higher effective rates of interest on all of the balance for all previous investments made to the annuity. Journal of Financial and Economic Practice Page 30

4 Table 1, Panel A: A Numerical Example of Future Value that Compares between the Simple Annual Compounding of an Ordinary Annuity versus an Annuity Due. Table 1, Panel A Future Value Annual Compounding Interest: 8% (Annual) Amount: $1,000 Years Periods Ordinary Annuity (A) Annuity Due (A) 5 5 $5, $6, $14, $15, $27, $29, $45, $49, $73, $78, $113, $122, $172, $186, $259, $279, Table 1, Panel B: A Numerical Example of Future Value that Compares between the Semi-Annual Compounding of an Ordinary Annuity versus an Annuity Due. Table 1, Panel B Future Value Annual Compounding Interest: 4% (Semiannual) Amount: $500 Years Periods Ordinary Annuity (S) Annuity Due (S) 5 10 $6, $6, $14, $15, $28, $29, $47, $49, $76, $79, $118, $123, $182, $189, $275, $286, Journal of Financial and Economic Practice Page 31

5 Figure 1: A Comparison between the Future Value of a Simple Annual Compounding of an Ordinary Annuity versus an Annuity Due. Ordinary Annuity (A) Annuity Due (A) Figure 2: A Comparison between the Future Value of a Semi-Annual Compounding of an Ordinary Annuity versus an Annuity Due. Ordinary Annuity (S) Annuity Due (S) Journal of Financial and Economic Practice Page 32

6 Figure 3: A Comparison between the Future Value of a Simple Annual Compounding versus a Semi-Annual Compounding of an Ordinary Annuity. Ordinary Annuity (A) Ordinary Annuity (S) Figure 4: A Comparison between the Future Value of a Simple Annual Compounding versus a Semi-Annual Compounding of an Annuity Due. Annuity Due (A) Annuity Due (S) Journal of Financial and Economic Practice Page 33

7 Volume 12 Issue 3 Fall 2012 Figure 5: A Comparison between the Future Value of a Simple Annual Compounding versus a Semi-Annual Compounding of an Annuity Due. Annuity Due (A) Annuity Due (S) Journal of Financial and Economic Practice Page 34

8 REFERENCES Berk, Jonathan, Peter DeMarzo and Jarrad Harford, 2009, Fundamentals of Corporate Finance, Pearson Education, MA, USA (pp ) Brigham, Eugene F. and Joel F. Houston, 2010, Fundamentals of Financial Management, Concise 6th Edition, South-Western Cengage Learning, Ohio, USA (pp ) Ross, Stephen A., Randolph W. Westerfield and Bradford D. Jordan, 2008, Essentials of Corporate Finance, 6th Edition, McGraw-Hill/Irwin, New York, USA (pp ) Journal of Financial and Economic Practice Page 35