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Rates of Return: Part 3 REFJ The Real Estate Finance Journal A WEST GROUP PUBLICATION Copyright ©20 11 West Group REAL ESTATE JV PROMOTE CALCULATIONS: RATES OF RETUR PART 3- COMPOUND INTEREST TO THE RESCUE? By Stevens A. C arey* Based on an article published in the Fall 2011 issue of The Real Estate Finance Journal *STEVENS A. CAREY is a transactional partner with Pircher, Nichols & Meeks, a real estate law firm with offices in Los Angeles and Chicago. The author thanks Dick A skey for corresponding with him, Jeff Rosenthal and Carl Tash for providing comments on a prior draft of this article, and Bill Schriver for cite checking. Any errors are those of the author. TABLE OF CONTENTS Compound Interest - The Basics . 1 Definition....................................................................................................................... 1 CompoundingPeriod......................................................................................................2 FundamentalFormula.....................................................................................................2 Futureand Fractional Periods.........................................................................................2 ContinuousCompounding.......................................................................................................2 Future Value Interest Factor...........................................................................................3 Generalized Fundamental Formula.................................................................................3 NominalRate.................................................................................................................3 BottomLine...................................................................................................................3 Simple Interest Revisited - Stationary Accumulation............................................................3 Shortcomings of Simple Interest .............................................................................................3 Intransitivity...................................................................................................................4 Discrimination Between Principal and Interest ...............................................................4 Declining Relative Growth Rate.....................................................................................4 Solving the Problems of Simple Interest ................................ ................................................. 4 ContinuousAccrual........................................................................................................ 5 CompoundingSolution................................................................................................... 5 The Theory of Compound Interest..........................................................................................6 PartialPeriod Calculations .............................................................................................6 Equationsof Value.........................................................................................................6 EquivalentRates.............................................................................................................7 Continuous Compounding in Practice: Potential Problems?................................................8 ExcessiveCompounding? ............................................................................................... 8 Continuous Compounding Without Increasing Desired Effective Rate ...........................8 Not Enough Interest: Partial Periods? ............................................................................8 Conclusion ................................................................................................................................ 9 APPENDICES.......................................................................................................................... 9 Appendix 3A. 1 - Discrete Compounding: Simple Interest Between Compounding...................10 Appendix 3A.2 - Comparing Separate and Fixed Compounding................................................ 15 Appendix 3A.3 - Separate v. Fixed Compounding: Future and Present Values of MultipleCash Flows...................................................................................................19 Appendix 3B - Continuous Compounding.................................................................................23 Appendix 3C - Establishing Fundamental Formula for All Real Numbers .................................26 Appendix 3D - Compound Interest Behind the Scenes...............................................................28 REAL ESTATE JV PROMOTE CALCULATIONS: RATES OF RETURN PART 3 COMPOUND INTEREST TO THE RESCUE? By Stevens A. Carey* This is the third installment of an article discussing rates of return in the context of real estate joint venture (JV) distributions. The first installment introduced commonly used terminology and conventions. The second installment examined simple interest. This installment focuses on compound interest. The previous installment of this article explained that, while simple interest has some very useful characteristics, including stationary accumulation, it is also inherently inconsistent (ignoring, in certain contexts, generally accepted concepts of the time value of money). After a refresher on the basics of compound interest and continuous compounding, this installment will briefly review the stationary accumulation feature of simple interest, as well as some of the problems with simple interest, and will then consider whether and how these problems are eliminated by compound interest. This installment will conclude that, when interest accrues continuously (as is often, if not usually, assumed in practice for both simple and compound interest), continuous compound interest is the only solution to the problems presented by simple interest that preserves the property of stationary accumulation. CoMPouID INTEREST - THE BASICS Compound interest has been called the "bedrock of mathematical finance" and "the foundation of the whole subject of the mathematical theory of investment" .2 Definition. Some textbooks define compound interest as interest that takes into account not only the investment but also the interest from prior periods, 3 Like simple interest, compound interest is also sometimes defined by example. 4 Example 3.1. A $100 investment earning interest at an annual rate of 10% compounded annually for two years would earn interest equal to $21: $10 of interest the first year ($100 x 10%) and $11 of interest the second year ($110 x 10%). Some textbooks have explained compounding as the result of reinvestment . 5 From the standpoint of the investor (e, g., a lender, bank depositor or partner), it is as though the investor receives, at the end of each compounding period, a payment (e.g., a loan payment, a withdrawal, or a distribution) for the interest that accrued during such compounding period, and then immediately reinvests the payment received in the same investment; alternatively, it is as though the investor makes an additional investment at the end of each compounding period in an amount equal to the accrued interest for that compounding period and then immediately receives the accrued interest (e.g., in the loan context, compounding is equivalent to the lender advancing its own interest). Either way (i.e., STE VENSA. CAREY is a transactional partner with Pircher, Nichols & Meeks, a real estate law firm with offices in LosAngeles and Chicago. The author thanks DickAskeyfor corresponding with him, JeffRosenthal and Carl Tash for providing comments on a prior draft of this article, and Bill Schriver for cite checking. Any errors are those of the author. 1 regardless of the order of the hypothetical cash inflow and cash outflow at the end of each compounding period), the result is the same: it is as though the accrued interest is replaced by an equal amount of principal; in effect, the accrued interest is added or converted to principal. When interest is compounded, interest is earned each [compounding] period on the original principal and on the interest accumulated for the preceding periods [T]he ... accumulated amount at the end of each period becomes the principal sum for purposes of computing the interest in the following period. 6 Compounding Period. As indicated above, when there is compound interest, there is often, if not usually, a stated period indicating when the interest that accrues and remains unpaid is effectively added to principal so that it begins to accrue interest itself. This period may (and will in this article) be called the "compounding period" although it goes by other names as well (e.g., conversion period). 7 Unless otherwise indicated, it is assumed for simplicity throughout this article that a discrete compounding period means a fractional unit interval of a year (i.e., 1 /nth of a year) and that, in particular, semi-annual, quarterly and monthly periods represent 1/2, 1/4 and 1/12 of a year, respectively. Fundamental Formula. The so-called "fundamental compound interest formula" indicates the future value, S. of an investment A after n compounding periods where r is the effective rate per compounding period: 8 Fundamental Formula S=A(1+r) For now, it is assumed, as it is (at least initially) in many textbooks, that n is a whole number. 9 Future and Fractional Periods. The fundamental formula indicates a future value interest factor, sometimes
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