The Real Estate Finance Journal A THOMSON REUTERS PUBLICATION Copyright 02011 Thomson Reuters REAL ESTATE JOINT VENTURE PROMOTE CALCULATIONS: RATES OF RETURN: PART 4- WHAT HAPPENS BETWEEN COMPOUNDING? By Stevens A. C arey * Based on article published in the Winter 2012 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 John Cauble, Steve Mansell, Jeff Rosenthal, and Carl Tash for providing comments on prior drafts of this article, and Bill Schriver for cite checking. Any errors are those of the author. 7139969.7 TABLE OF CONTENTS Discontinuous Accrual 2 Continuous Accrual: Practical vs. Theoretical Method .......................................................... 3 Difference Between Methods - Illustration ............................................................................... 5 Difference Between Methods - Equivalent Rates ..................................................................... 6 Customand Practice .................................................................................................................... 8 Application to Loans and Promote Hurdles .............................................................................. 8 Practical vs. Theoretical Methods - Pros and Cons .................................................................. 10 Conclusion .................................................................................................................................... 12 APPENDICES .............................................................................................................................. 12 Appendix 4A Future Value Using Practical Method............................................................... 13 Appendix 413 Equivalent Rates (for a fixed annual rate r) ...................................................... 14 Appendix 4C Comparison of Practical and Theoretical Methods........................................... 16 Appendix4D.l IRR Hurdles....................................................................................................... 20 Appendix 4D.2 Present Value Hurdles....................................................................................... 24 Appendix 4E Promote Hurdles Using Theoretical Method..................................................... 26 ENDNOTES.................................................................................................................................. 31 7139969.7 REAL ESTATE JOINT VENTURE PROMOTE CALCULATIONS: RATES OF RETURN PART 4 WHAT HAPPENS BETWEEN COMPOUNDING? By Stevens A. Carey This is the fourth installment of an article discussing rates of return in the context of real estate joint venture (JV) distributions. The first three installments examined (1) commonly used terminology and conventions,' (2) simple interest, 2 and (3) compound interest 3 This installment focuses on how interest accrues between compounding times in a transaction involving discrete compounding periods. The third installment of this article observed that the fundamental compound interest formula is often stated only for a whole number of compounding periods: S=A(l +r) where A is the amount of the investment, r is the interest rate for the applicable compounding period and S is the value of the investment after n compounding periods (and n is a whole number). This formulation doesn't explain how to calculate the interest for a partial compounding period. Similarly, in many transactions, the interest rate or rate of return is stated as a constant rate that compounds annually or over some other discrete compounding period, but there is little or no explanation of how interest accrues between the stated times that compounding occurs. Sometimes it is not necessary. Example 4.1. Assume that a lender makes a mortgage loan on the following terms: (1) there is a single advance of $1 million; (2) the lender earns interest equal to 100% per annum, compounded annually (as of each anniversary of the making of the loan); (3) no interest is payable until the loan is repaid in full; and (4) the loan may be repaid only on an anniversary of the making of the loan, in whole, but not in part, together with interest, and must be repaid by the third anniversary. The future value of this investment may be illustrated in the following graph of the future value interest factor as of the agreed-upon potential cash flow dates (i.e., the loan advance date and the three alternative loan payment dates). 7139969.7 The interest, and future value, in this example is (like the compound interest formula above) defined only for discrete points in time (namely, the beginning and end of each compounding period). Basically, the example defines the interest for each compounding period and limits the payment of the loan to the end of a compounding period and therefore there is no need to know what happens at any intermediate point in time. This is a common teaching approach; 5 many finance books address the question of fractional compounding periods, if at all, as a separate question. In practice, however, calculations of interest at intermediate points in time are often necessary. This installment will consider such calculations. For simplicity, unless otherwise stated, it will be assumed that there are only two cash flows, an initial investment amount by the investor and a single final payment back to the investor. DISCONTINUOUS ACCRUAL What would happen if the mortgage loan in Example 4.1 were changed to allow prepayment at any time? Although many mortgage loans require that any voluntary prepayment be made only on a scheduled payment date, many do not. 7139969,7 2 Sometimes (especially in conduit loans), the loan terms may require the payment of all accrued and unpaid interest for the balance of the period in which the prepayment occurs. This approach may be illustrated by the following graph (assuming the facts in Example 4.1 except that prepayment is permitted at any time): GRAPH NO. 4B FV of I (with Discontinuity) using 100% per annum, compounded annually 8 7 6 5 3 2 1 0 0 1 2 3 4 Year An odd characteristic of the accumulated value in the above graph is that it is discontinuous. There is a multimillion-dollar jump in the loan balance to be prepaid between the last day of a loan year, on the one hand, and the first day of the next loan year, on the other hand. CONTINUOUS ACCRUAL: PRACTICAL VS. THEORETICAL METHODS Although there will likely be exceptions to the rule (as in the conduit approach to loan prepayment described above), "it is natural to assume that interest is accruing continuously". 6 In the context of the first graph above (Graph No. 4A), how can one connect the dots to make a continuous function? There are two alternative approaches that are generally considered, namely: the "practical" method; and the "theoretical" method. 7 7139969.7 3 Practical Method. Under the practical method, the accrued interest for a portion of a compounding period is a proportionate amount of the interest for the entire compounding period (based on the proportion of the compounding period) involved, namely, simple interest: When simple interest is used [for fractional compounding periods], the procedure is known as the practical method In the context of Example 4. 1, the practical method is reflected in the following graph: GRAPH NO. 4C FV of 1 (Practical Method) using 100% per annum, compounded annually 8 7 . 6 > WK 4 3 y 2 3 1 <: 4 Year Basically, the practical method approach connects the dots with straight line segments and assumes that simple interest accrues during compounding periods at the nominal annual rate (in this case 100%). The future value as of any particular time during each compounding period is A( 1 + rt) where A is the value at the beginning of the compounding period and t is the number (between 0 and 1) of compounding periods from the beginning of the compounding period to the time in question. See Appendix 4A (Future Value Using Practical Method) for a more detailed description of the future value formula. 7139969.7 4 Theoretical Method. Under the theoretical method, the accrued interest for a portion (less than the whole) of a compounding period is less than a proportionate amount of the interest for the entire compounding period. The dots are connected by completing the exponential curve (in the context of the above example, (1 + 100%)) on which they are situated: [The fundamental compound interest formula, S = A(1 + i)', was presented] under the assumption that n is an integer. Theoretically, [this formula] can be used when n is a fraction. When we calculate the [future value] using [this] formula for a fractional part of [a compounding] period, we call it the ... theoretical method . Y GRAPH NO. 4D FV of 1 (Theoretical Method) using 100% per annum, compounded annually 8 '8 '" S' ’S S-i S’ I - 6 " 4 - '- S ' '4 2- 0 -' 0 1 2 3 4 Year As explained below, the theoretical method is tantamount to using an equivalent nominal annual rate that is compounded continuously (which, in the context of Example 4. 1, is 69.31472%). DIFFERENCE BETWEEN METHODS - ILLUSTRATION The practical and theoretical methods will yield the same results when all cash flows occur, and all calculations are made, as of the beginning or end of one of the stated compounding periods. 7139969.7 5 Otherwise, there may be a difference, as illustrated by the following hypothetical, where the operator is effectively