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 advocating the theoretical method and the investor is effectively advocating the practical method.
Hypothetical 4A. Assume (1) a local operator and investor form a venture to acquire a property; (2) the operator is entitled to a promote after the investor recoups its capital and a return of 21% per annum, compounded annually; (3) the investor contributes a total of $10,000,000 at the time of the acquisition (and makes no other contributions); (4) shortly after the acquisition of the property, the parties receive an offer they can't refuse and sell the property after owning it for only six months; (5) the investor calculates its hurdle as $11,050,000 because 21% per annum of $10,000,000 for six months is $1,050,000 (1/2 of $2,100,000); and (6) the operator calculates the hurdle as $11,000,000 arguing that a 10% rate (rather than a 10.5% rate) should be used to determine the hurdle for a six-month period (because a 10% rate for each six-month period would, if compounded, yield an effective annual rate of 21%).
DIFFERENCE BETWEEN METHODS - EQUIVALENT RATES
To better appreciate the difference between the practical and theoretical methods, it may be helpful to consider their impact on equivalent rates.
Theoretical Method. One of the consequences of using the theoretical method is that equivalent rates yield the same result not only at the end of the period for which they are equivalent, but at all times. Some older textbooks define two annual rates to be equivalent if they yield the same value at the end of a year. 10 For example, a 21% annual rate, compounded annually, is equivalent to a 20% annual rate, compounded semiannually because they both result in a future value interest factor of 1.21 for the first year. But unless more is known about these two rates (namely what happens between compounding), one doesn't know whether they yield the same result at all times during the year. As illustrated in Hypothetical 4A, if there were simple interest during the year and a 21% annual rate were compounded only at the end of the year, then substituting a 20% annual rate compounded semiannually would yield the same rate for a full year (i.e., 21%), but may not yield the same result for a partial year. Some books appear to assume that the theoretical method applies. Under the theoretical method, each of these two rates (or any other rate to which either is equivalent) is governed by the accumulation function a(t) = (1 + .2 1), where t is the relevant number of years.
The ... theoretical ... method ... consists of using the fundamental [compound interest formula] regardless of whether the time is an integral number ofperiods or not .... [This] rule always gives the same result as that obtained by first replacing the given interest rate by an equivalent rate compounded with a frequency that would make the time an integral number of periods, then accumulating ... at this new rate. Thus it is unnecessary to actually find an equivalent rate, since the final result is the same as that obtained by merely substituting the original data into [the fundamental compound interest formula, S = A (1 + r), for any value of n, whether or not a whole number]. (Emphasis in original.)"
7139969.7 6 This assumption, that (unless otherwise stated) a compound interest rate is governed by the fundamental compound interest formula at all times (and not merely at the end of each whole year or other time unit) seems prevalent in modern textbooks (and is often true by definition), as explained in the third installment of this article. In many modern textbooks, equivalency (by definition) must occur at all times if at all and is based on a common accumulation function.' 2 In the context of compound interest, this means using the theoretical method and an accumulation function based on the fundamental compound interest formula: a(t) = (1 + r)t, where r is the effective annual rate and t is the relevant number of years.
Thus, when the parties use an equivalent nominal annual rate with a discrete compounding period that regularizes the cash flows (e.g., the greatest common divisor of all the time intervals between cash flows), the parties are effectively assuming that continuous compounding is going on in the background: there is always a continuously compounded annual rate behind the scenes that achieves the same results, namely, In (1 + r), where r is the effective annual rate, and ln(x) is the natural log function. 13 If this assumption is not accurate, then the results may not be either.
Practical Method. By contrast, there is no implicit continuous compounding under the practical method; compounding occurs only when expressly provided. During each of the stated compounding periods, the future value function under the practical method is linear and not exponential: an amount invested at the beginning of the compounding period grows by a factor of (1 + rt) during the compounding period, where r is the effective rate per compounding period and t is the number (between 0 and 1) of the compounding periods to the time in question. Equivalent nominal annual rates under the practical method compound only at the times indicated, and while they reach the same result at the end of each year (for an amount that remains fully invested throughout the year), they may differ during most of each year. Thus, under the practical method, equivalent rates are not truly equivalent. See Appendix 4B (Equivalent Rates) for further discussion and illustration.
Comparison. Because all equivalent rates are substantively the same under the theoretical method, the difference between the practical and theoretical methods is the difference that results from accruing, during each of the stated compounding periods, simple interest at the stated nominal annual rate (practical method) instead of interest at the continuously compounded equivalent rate (theoretical method). (There may also be some deviation due to conventions: for example, if there is daily compounding and cash flows are deemed to occur at the beginning or the end of the day depending on whether they are received prior to the deadline for investment.) This difference is easily determined by comparing (1 + rt) and (1 + r)', where r is the effective rate for the stated compounding period, and t is the number (between 0 and 1) of the relevant periods. See Appendix 4C (Comparison of Practical and Theoretical Methods) for more detail.
Example 4.2. A 21% nominal annual rate, compounded annually, and a 20% nominal annual rate, compounded semiannually, are equivalent because they both yield the same 21% annual return. What is the difference between these rates during the year?
. Under the theoretical method, there is no difference. In particular, the future value interest factor for the first half of the year would be 1.10 in
7139969.7 7 both cases: (1.21)1/2 = 1.10 = (1.10) 1 , The return for that period would be 10%.
Under the practical method, the smaller equivalent nominal annual rate (the 20% rate) will yield a smaller future value interest factor (and return) during the year. In particular, there is a 1/2 point difference in the future value interest factor and return for the first half of the year: (1.105) - (1.10) 10.5% - 10% = .5%.
Consequently, the future value interest factor (and return) for a 21% annual rate, compounded annually, would vary by 1/2 point under theoretical and practical methods halfway through the year. This explains the difference in Hypothetical 4A.
CUSTOM AND PRACTICE
The theoretical method is the favored approach from a purely theoretical standpoint, but in practice the applicable approach may depend on the custom, conventions and agreements applicable to the transaction. Many books do not mention the two alternatives. For example, some books offer the theoretical method as the only answer to the question of how one determines the future value through a fractional compounding period. 14 One fixed income securities book suggests that "the answer cannot be [the practical method]". 15 On the other hand, some books mention only the use of the practical method:
When deriving the [fundamental] compound interest formula, we assumed that the time would be an integral number of [compounding] periods. When there is a part of the period, the usual practice is to allow simple interest for this time on the compound amount at the end of the last whole period. 16
Many books that mention both approaches appear to favor one approach in theory and the other in practice (as the names suggest):
In practice, [the theoretical] method is rarely used. Instead, we use compound interest for the number of complete interest periods and simple interest (at the stated nominal yearly rate) for the fractional part of the interest period. 17
As explained by one textbook:
For certain theoretical purposes ... we shall agree that [the theoretical method applies] .... [However, if] some ... financial transaction might require the accumulation ... for [fractional] interest periods ... it is likely that the people engaged in the transaction would agree to use [the practical method]. 18
APPLICATION TO LOANS AND PROMOTE HURDLES
The question of whether to use the practical or theoretical method arises when there are actual or potential intermediate cash flows (i.e., cash flows that do not occur at the beginning or end of a stated compounding period) such as a calculation of the amount necessary to repay the investment balance as of a moment between the stated compounding times.
7139969.7 8 Loans. Intermediate cash flows may arise in different ways in a loan transaction: (1) the scheduled payment period may be shorter than the stated compounding period (e.g., monthly payments and annual compounding); (2) there may be an unscheduled payment during a compounding period due to voluntary or involuntary prepayment (whether in whole or in part); or (3) there may be a stub period at the beginning of or end of the loan. The author does not recall encountering any United States loans that provide for scheduled payments that are more frequent than compounding times (and, as discussed in the first installment of this article, there often seems to be an assumption that the payment and compounding frequencies are equal). But some loans (including the author's home mortgage loan) have scheduled payments without any stated compounding period, and there have been appellate cases involving annual compounding and intra-year payments. 19 Also, prepayment and stub periods (especially initial stub periods) are, of course, relatively common. In any case, it is the author's impression that loans in the United States typically utilize the practical method. As stated in one textbook:
Prior to payment due dates, interest not yet compounded to principal accrues on a pro rata basis as simple interest
In the author's experience, most commercial real estate loans have compounding periods that are longer than a day. Per diem interest calculations in the United States, whether for partial or full prepayment, on a date other than a payment date, are generally based on proportional (i.e., simple) interest, although the proportion may be artificially inflated if the lender uses a 30-day month and 360-day year. 21 Indeed, as noted earlier, some United States lenders may require even more than a proportionate amount of the interest for the compounding period in which a prepayment occurs (by requiring payment of the entire amount of the interest for the compounding period that would have accrued had the prepayment not occurred, although this is more in the nature of a prepayment premium). And if there are "monthly payments [for example], then the applicable interest rate is the simple monthly rate, which is conventionally defined as the nominal per-annum rate stipulated in the loan, divided by [l2].22 As will be discussed in a subsequent installment of this article, this approach is not necessarily followed in Canada, where, in many cases, the theoretical method seems to be adopted to reduce the amount of regular payments made within a compounding period.
Promote Hurdles. With IRR, preferred return and other promote hurdles, intermediate cash flows are very common. For example, the author has often encountered a distribution schedule (e.g., monthly) that is more frequent than the stated compounding frequency (e.g., annually). In this context, the author is not aware of a uniform custom. Hurdles are often defined by merely indicating that a return must be achieved without spelling out how it will be calculated. Sometimes examples are given, but they frequently involve regular cash flows where the theoretical and practical methods yield the same results. The author suspects that certain hurdles (such as a preferred return hurdle coupled with a return of capital hurdle) may be more likely to use the practical method if they are calculated using the typical United States loan amortization approach, and that IRR hurdles (which are discussed in Appendix 4E) are likely to adopt the theoretical method because it is easier to model and consistent with the typical IRR definition. 23 As shown in Appendix 4E ("Promote Hurdles Using Theoretical Method"), the three alternative hurdle methods discussed in the second installment of this article, namely loan amortization, equating future values and equating present values (as to which the results may be vastly different for simple interest rates) generally yield the same results under the theoretical method.
7139969.7 9 PRACTICAL VS. THEORETICAL METHODS PROS AND CONS
The historical arguments for using one method rather than another are well summarized in the following passage from a 19th century textbook:
This question at one time created a great deal of warm controversy among actuaries (citation omitted). Some writers maintained that the answer must be [the practical method] and the chief argument they urged in support of their view was that under no circumstances should compound interest yield less than simple .... On the other hand, rival authorities asserted that ... although [the theoretical method] obviously gives a value smaller than [the practical method], it is only right that it should do so. The interest is not due till the end of the year, and if the lender receive [sic] it sooner he must be content with less, because, compound interest being supposed, he can invest his interest for the remaining portion of the year and realise interest thereon .... They therefore advocated that in theoretical investigations the [fundamental compound interest formula] must be held to be universally true whether n be integral or fractional [(i.e., the theoretical method should apply)]; and in recent years, the majority of mathematicians have adopted their view .... If [the practical method is utilized], great complications must sometimes be introduced into formulas .... If on the contrary we adopt [the theoretical method], these difficulties vanish. The expressions become elegant and compact .... It must be remembered, however, that in commercial transactions [the practical method is used]; and even in actuarial formulas, it is sometimes found convenient for purposes of numerical calculation to do the same
These arguments revolve around three key points:
(1) the practical method may be favored by lenders (and other investors) because the theoretical method results in less than simple interest for a partial compounding period (which is the minimum many investors expect to receive for any period);
(2) the practical method has historically been favored in practice because it is easier to multiply than determine fractional roots; and
(3) the theoretical method may be favored by theoreticians because it involves a consistent treatment of the time value of money (effectively using continuously compounded interest) and this results in a more concise formulation.
Each of these points will be considered in turn below.
Shortchanging the Investor? In Hypothetical 4A, the investor used the practical method and the operator used the theoretical method to calculate the promote hurdle on sale. In effect, the parties were calculating the future value (using a 21% annual rate) on a $10 million investment for 1/2 of a year. The operator thought the hurdle should be $10 million x (1 + 21%)h12 = $11,000,000, and the investor thought the hurdle should be $10 million x (1 + 21%/2) = $11,050,000. The investor argued that it had required compounding to ensure that its return did not remain flat after a year, but it never would have agreed, and did not agree, to accept less than a simple return. The investor's concern is by no means new:
7139969,7 10 Among the arguments [the French author Jean Trenchant] puts forward [in his book L 'Arithmetique published in 1556] for using simple interest rather than compound in respect to fractions of a year are the following: ... all compound interest is intended to be more useful to the creditor than simple interest. But in the case of interest for fractions of a year the reverse would be true. Hence, such interest should not be compounded As stated in another book:
At a given nominal rate, the compound interest for a fractional part of a [compounding] period is less than the simple interest for the same time Consequently, ... it is customary for the investor to be credited with ... the simple interest for the fractional period .26
As noted earlier, 27 a borrower, partnership, bank or other counterparty may respond by arguing that the investor should get less than simple interest because it has the opportunity to reinvest what it receives and thereby get the benefit of compounding. Basically, the borrower, partnership, bank or other counterparty wants the investor to account for each payment as though it did in fact reinvest it at the same rate. But such reinvestment may not actually occur, and an investor may not want to make such an accounting unless that is how it underwrote, and agreed to proceed with, the transaction.
Ease of Calculation? Calculating fractional roots was once a difficult process, while the calculation of simple interest was easy by comparison. Thus, in commercial transactions, the more practical (simple interest) method may have been favored by all parties. However, given recent advances in computing, calculators and computer programs, the mathematics is considerably less cumbersome if continuous compounding is assumed (so that the future value interest factor takes the form of an exponential curve instead of a series of jagged line segments). Ironically, the theoretical method may today be the more practical approach and may be favored by professionals who place a premium on simplicity (particularly in transactions where the deviations resulting from the theoretical approach are not likely to have a significant whole dollar impact). Indeed, the author is aware of professionals who use the theoretical method because it generates an approximate answer that is sufficiently close to the practical method they might otherwise use. In this context, the "practical method" seems like a misnomer. To add to the potential confusion, the theoretical method is sometimes referred to as the "exact" method and the practical method is sometimes called the "approximate" method. Obviously, either method may be used to approximate the other.
Theory vs. Reality? Each approach accrues interest continuously, but only the theoretical approach compounds continuously. From a theoretical standpoint, continuous compounding is the most logical approach because it compounds consistently. Consider the natural and elegant extension of the fundamental formula of compound interest for n discrete compounding periods (i.e., S = A (1 + r)'1) to any number t of such compounding periods, where t can be any non-negative number, including a fraction (i.e., S = A (1 + r)') under the theoretical method; and then compare the cumbersome future value formula under the practical method in Appendix 4A (Future Value Using Practical Method). Simple interest, of course, does not take into account the time value of interest at all and consequently has often been criticized as discussed in the second installment of this article. 28 Even when interest compounds but there is simple interest during compounding periods, this inconsistency may be troubling to the mathematical purist. 29 But, as observed in the second
7139969.7 11 installment of this article, simple interest is a fact of life and has occurred, and may occur, in a number of scenarios, including real estate partnerships, and, as a legal matter, simile interest is generally presumed in the United States unless compounding is expressly provided. 0
CONCLUSION
This installment has described the two alternative ways interest typically accrues between compounding times when the rate of interest is stated to compound at discrete times. One alternative, the practical method, uses simple interest, and the other, the theoretical method, uses continuous compound interest. Recognizing that either method may be adopted in practice, it is important to understand how these alternatives work in practice. The next installment will consider in more detail the application and impact of the theoretical and practical methods in the context of mortgage loans and, in particular, how the concept of deemed reinvestment has been used to justify the use of the theoretical method in connection with Canadian mortgage loans. It will also examine more generally deemed reinvestment, which appears to be one of the key points made in the debate over which of these two methods is appropriate. * * *
APPENDICES
Appendix 4A - Future Value Using Practical Method Appendix 4B - Equivalent Rates (for a fixed effective annual rate r) Appendix 4C - Comparison of Practical and Theoretical Methods Appendix 4D. 1 - IRR Hurdles Appendix 413.2 - Present Value Hurdles Appendix 4E - Promote Hurdles Using Theoretical Method
7139969.7 12 APPENDIX 4A
FUTURE VALUE USING PRACTICAL METHOD
This Appendix will briefly discuss the future value interest factor or accumulation function that represents the practical method: so that interest accumulation is based on a constant rate i per time unit that compounds at discrete compounding times and simple interest at the constant rate between compounding dates.
Assuming (as is generally done in this installment) that there is a single onetime cash investment made as of time 0, which is a compounding time, the future value interest factor can be represented by an accumulation function of one variable:
a(t) = (1 + i)(1 + I [t - /t) where It! = the largest integer that is not more than t, i is the rate per time unit, and t is the relevant number of time units. 31 As discussed in Appendix 3A to the third installment of this article, the formula gets more complicated when subsequent principal additions may be made (or when time 0 is not a compounding time). See that Appendix for more detail. * * *
7139969.7 13 APPENDIX 413
EQUIVALENT RATES (for a fixed effective annual rate r)
This Appendix will attempt to illustrate graphically the difference between equivalent nominal annual rates when the practical method is adopted. It will also expand on the discussion in Appendix 1 C to the first installment 32 of this article regarding "equivalent" nominal annual rates and the consequences of compounding over smaller and smaller compounding periods where there is a fixed effective annual rate.
Impact on Future Value. When the effective annual rate "r" is fixed and the compounding period is reduced from an annual period to smaller and smaller periods, the future value at the end of the year (or any subsequent year) does not change. But the nominal annual rate gets smaller and smaller and it reaches this common annual result through more frequent compounding.
Equivalent Nominal Annual Rate for Continuous Compounding. To visualize what the future value interest factor looks like for equivalent nominal annual rates with smaller compounding periods, it is helpful to start with the final case of continuous compounding. In this case, the future value interest factor is simply the exponential function (1 + r)', where "t", representing the variable time in question, is a number of years. 33
Equivalent Nominal Annual Rate for Discrete Compounding Periods. The future value of other equivalent nominal annual rates depends on how interest accrues for fractional compounding
periods. If the theoretical method is adopted, then they are all the same, namely (1 + r)t . If the practical method is adopted, then the future value interest factors for any two equivalent rates may be different at most times. The balance of this discussion assumes that the practical method has been adopted.
Equivalent Nominal Annual Rate for Annual Compounding (Practical Method). When there is annual compounding, there is no compounding during the year, and the equivalent nominal rate is simply r. Graph No. 4C (in the body of this article) illustrates the future value interest factor for an annual rate with annual compounding: the graph is a series of line segments for each year. Note that the values for this future value interest factor are easy to calculate at the beginning or end of each
year: 1, (1 + r)1 , ( 1 + r)2 , ( 1 + r)3 , etc. Thus, the starting point and endpoint of each line segment is
on the exponential curve representing the exponential function (1 + r)t . In other words, the future value interest factors for the annually compounded and continuously compounded equivalent rates have the same values at the beginning and end of every year, namely, when t is a whole number. The only difference between the two functions is what happens during the year, when the future value interest factor for the annually compounded rate under the practical method is linear. Thus, one can picture this future value interest factor as a series of connected line segments that approximate an exponential function. The reader may want to look ahead to Graph No. 4E to see this.
Intermediate Rates (Practical Method). By using smaller and smaller compounding periods with equivalent nominal annual rates, the future value interest factor gets closer and closer to (1 + r)'. For example, a semiannual rate of r'= (1 + r)"2 - 1, when compounded semiannually, would yield an
7139969.7 14 annual effective rate of r, so it would equal both the annually compounded future value interest factor and the exponential function at the beginning and end of each year. The midway point in each year would become another time when this new future value interest factor would be the same as the exponential function (namely, (1 + r) 112 , ( 1 + r)312 , ( 1 + r)5 "2 , etc.). Basically, each of the annual line segments (representing the annually compounded future value interest factor) would be replaced by two smaller semiannual line segments that intersect on, and are closer to, the exponential curve. As the compounding period gets smaller, so do the line segments, and the graphs of the resulting future value interest factors more closely approximate the exponential function, until it is actually reached (as the limit obtained by letting the number of equal periods within a year tend to infinity).
Illustration (Practical Method). The following graph illustrates the first and last steps of the process, showing annual compounding (on the top) and continuous compounding (on the bottom) where the effective annual rate is fixed at 100%:
GRAPH NO. 4E Future Value of 1 Using Equivalent 100% Annual Rates
The author will leave to the reader's imagination the graph of the intermediate steps, all of which occur in the space between these two functions.
7139969.7 15 APPENDIX 4C
COMPARISON OF PRACTICAL AND THEORETICAL METHODS
This Appendix will compare the theoretical and practical methods. These methods differ within each of the stated compounding periods. The comparison of the practical and theoretical methods within any of the stated compounding periods is essentially a comparison of simple interest, a(t) = I + rt, and continuous compound interest, a(t) = (1 + r)t, for 0 Specifically, this Appendix will compare certain results obtained when using the practical and theoretical methods (1) for a 100% annual rate, compounded annually, by comparing the proportionate rate (practical method) with the equivalent rate (theoretical method) for some common compounding frequencies, and (2) various annual rates (and specifically 12% and 25%), compounded annually, by comparing the proportionate rate (practical method) with the equivalent rate (theoretical method) for a daily compounding period. 100% Annual Rate, Compounded Annually. Consider, for simplicity, a 100% annual effective rate. What is the difference between the practical and theoretical methods in determining the periodic effective rates that preserve the effective annual rate for semiannual, quarterly, monthly and daily periods? In other words, what is the difference between the proportionate rates (practical method) and equivalent rates (theoretical method) for those periods? The deviations are summarized in the following chart: 100% Nominal Annual Rate, Compounded Annually (Comparison of Effective Rates for Portions of a Year Under Practical and Theoretical Methods) Semiannual Full Year Period Quarter Month Day Practical Method 100.0% 50.00000% 25.00000% 8.33333% .27397% Theoretical Method 100.0% 41.42136% 18.92071% 5.94631% .19008% Difference 0.0% 8.57864% 6.07929% 2.38702% .08389% Magnitude of Error - Generally. By finding a period that regularizes the cash flows and then using an equivalent rate for that period, the accrued interest will reflect the theoretical method, but may not reflect the practical method. The magnitude of the deviation is often insignificant in relative terms. For a single cash outflow that is fully invested for only a part of one of the originally stated compounding periods, the difference for such partial period is the amount by which: simple interest (based on the original nominal rate) during such partial period exceeds 7139969.7 16 compound interest (based on the equivalent continuously compounded rate or any other equivalent rate for a compounding period that regularizes the cash flow) for such partial period. This difference may be compounded over the total number of subsequent compounding periods for which such cash flow remains fully invested. The preceding chart indicates the difference in the periodic effective rates for a 100% nominal annual rate, compounded annually, when the partial period is 1/2, 1/4, 1/12 and 1/365 of a year, and ranges from roughly 5/6 of a point to more than 8% (for the partial compounding period). But a 100% rate is not likely to be used in practice. Hypothetical 4A, which involves a 21% rate compounded annually, shows a difference of 1/2 point halfway through the year. Daily Compounding. It may be useful to measure the deviation in the context of a daily compounded equivalent rate. For practical purposes, there may be no need to distinguish between cash flows that occur on the same day (except to the extent they arrive after the last time they can be invested), and therefore the equivalent nominal annual rate for daily compounding is often used in real estate transactions. 37 Magnitude of Error - Daily Rate. How good an approximation is the equivalent nominal annual rate for daily compounding (when the practical method is adopted)? Compare the daily interest rates for 12% and 25% annual effective rates: Effective Practical Method Theoretical Method Difference Annual Rate Daily Rate* Daily Rate** (Simple - Compounded) - p p/365 p"365 - 1 p'365 (p11365 - 1) 12% .0328767% .0310538% .0018229% 25% .0684932% .0611539% .0073393% * The daily rate that, without compounding during the year, yields the applicable effective annual rate. ** The daily rate that, with daily compounding, yields the applicable effective annual rate. The return using the theoretical method (i.e., the compounded daily rate) drops further and further behind during the earlier portion of the year, but through compounding, it gradually catches up later in the year and ultimately reaches the same result for the entire year (assuming the initial investment remains fully invested). Assuming a $10 million investment, the returns would look something like the following: 7139969.7 17 Cumulative Return on $10 Million Dunn Year Difference First Daily* Compounded (Simple - Daily X Simple Return Return Days Compounded) 12% 25% 12% 25% 12% 25% 1 3,228 6,849 3,105 6,849 183 734 2 6,575 13,699 6,212 12,235 363 1,464 3 9,863 20,548 9,319 18,357 544 2,191 30 98,630 205,479 93,582 185,098 5,048 20,381 90 295,890 616,438 283,381 565,635 12,508 50,803 150 493,150 1,027,397 476,750 960,391 16,400 67,006 180 591,780 1,232,877 574,794 1,163,265 16,987 69,612 181 595,068 1,239,226 578,078 1,170,092 16,991 69,634 182 598,356 1,246,575 581,362 1,176,923 16,994 69,656 183 601,644 1,253,425 584,649 1,183,758 16,995 69,666 184 604,931 1,260,274 587,936 1,190,597 16,996 69,677 185 608,219 1,267,123 591,224 1,197,440 16,995 69,683 186 611,507 1,273,973 594,512 1,204,288 16,995 69,685 187 614,795 1,280,822 597,802 1,211,140 16,993 69,682 210 690,411 1,438,356 673,755 1,369,895 16,656 68,461 270 887,671 1,849,315 874,464 1,794,689 13,207 54,616 330 1,084,932 2,260,274 1,078,947 2,235,375 5,984 24,899 360 1,183,561 2,465,753 1,182,626 2,461,849 436 3,905 363 1,193,425 2,486,301 1,193,047 2,487,726 377 1,576 364 1,196,712 2,493,151 1,196,523 2,492,360 189 790 365 1,200,000 2,500,000 1,200,000 2,500,000 0 0 * Assuming a 3.65 - day year The above table confirms, as expected, that (assuming the same effective annual rate) the return on $10 million is the same at the end of the year regardless of whether the return is based on an equivalent daily compounded rate under the theoretical method or an equivalent annually compounded rate (under the practical method). But for a partial year, the result may be different, and around the middle of the year the daily compounded return falls furthest behind. The bold numbers in the above chart indicate the greatest deviations for the applicable rates. As noted above, with some basic high school calculus, one can easily calculate the maximum deviations during a year for different rates of return. Below is a chart showing the maximum deviations for various rates assuming a $10 million investment throughout the year: 7139969.7 18 Greatest Difference (Assuming $10 Million Capital Balance Throughout the Year) During the Year between Cumulative and Equivalent Simple and Daily Compounded Returns Day of Year When Greatest Difference % Annual Rate Difference is Greatest During Year of Capital 5% 183 $3,049 0.03% 10% 184 $11,912 0.12% 15% 185 $26,198 0.26% 20% 185 $45,559 0.46% 25% 186 $69,684 0.70% 30% 186 $98,293 0.98% 35% 187 $131,132 1.31% 40% 188 $167,972 1.68% 45% 188 $208,605 2.09% 50% 189 $252,839 2.53% 75% 191 $522,374 5.22% Thus, if the final liquidating distribution occurs before the end of the year, daily compounding may result in a loss of a portion of the return that the investor would otherwise receive. Admittedly, many transactions involve smaller investments and, in any event, the reduction is less than 1% for rates up to 30%. (It can easily be shown that this range also applies for continuous compounding.) Moreover, the impact of a smaller return threshold may be softened by the fact that the investor is often entitled to a large portion of the distributions to be made after the return 38 threshold is achieved. However, discrepancies may compound and the whole dollar deviation may sometimes get the investor's attention. *** 7139969.7 19 APPENDIX 4D.1 IRR HURDLES This Appendix will discuss the internal rate of return and how it may be used to establish a promote hurdle. An IRR promote hurdle is a special type of the Present Value Hurdle, which was introduced in the second installment of this article and is discussed more generally in Appendix 4D.2. INTERNAL RATE OF RETURN Word Definition. The internal rate of return (IRR) of a particular series of cash flows is often 39 defined in words as the interest rate that makes the net present value of the cash flows zero (or equivalently, that equates the present value of the cash inflows with the present value of the cash outflows 40). These words alone might suggest that the IRR could be any interest rate that satisfies this condition, including a simple interest rate. But if that were true, then how would one know whether the IRR for a series of cash flows were a simple or compound interest rate? For example, the annual cash flows (100), 22, 96 have a net present value of zero when using either a 10% simple annual interest rate or a 9.6% annual interest rate, compounded annually. In the books reviewed by the author, the context eliminates this potential ambiguity and makes clear that the IRR is a compound rate of interest. Formula Definition. The fact that the IRR is a compound rate is evident when the IRR is described 41 by the relevant present value equation. Sometimes there is an express statement to this effect (in addition to the present value equation) : 42 The internal rate of return (IRR) is defined as the compound rate of return r that makes the NPV equal to zero: N CF + =0 (+r) In other words (using slightly different notation to be more consistent with the balance of this article), the internal rate of return for the cash flows CFO, CF,, ..., CF, which occur as of 0, 1, n - 1, n time units, respectively, would be a compound interest rate r per time unit, which makes the following equation true: IRR for CF0, CFI,..., CF CFk(1 + r) =0 TIMING OF CASH FLOWS The formula above assumes that consecutive cash flows are separated by one full time unit. However, cash flows in real estate transactions are often irregular (i.e., the time between consecutive cash flows is not uniform). 7139969.7 20 Irregular Cash Flows - Limitations of Some IRR Functions. Many calculators and programs are not set up for IRR calculations unless the cash flow periods are uniform. For example, one handbook states that each of its NPV and IRR functions "works only with periodic cash flows" and defines "periodic" cash flows to mean cash flows "that ... occur with the same frequency throughout the term (such as yearly or monthly)" .43 Similarly, the HP-12C calculator does present value, future value and IRR calculations assuming that the cash flows occur "at regular intervals". 44 45 Regularizing Cash Flows. But it is always possible to regularize the cash flows. For example, one can use the greatest common divisor of the time intervals between cash flows as the uniform time interval and then specify a zero cash flow at the beginning of each of the uniform time intervals where there is no actual cash flow. In the author's experience, the smallest uniform time period used in practice is a day (which, admittedly, would involve some rounding of the greatest common divisor if one were to calculate the greatest common divisor). Periodic IRRs. The typical calculator will determine the periodic rate of return for the regular period (i.e., it will calculate an IRR for the uniform cash flow period). And it will assume that the compounding period is the same as the uniform period. "If the cash flow periods are other than years (for example, months or quarters), you can calculate the nominal rate of return by multiplying the periodic IRR by the number of periods per year." 46 Thus, a 10% semiannual IRR would translate to a 20% annual rate, compounded semiannually, or an effective annual rate of2l%. 47 However, as 48 explained in the first installment of this article, if one starts with an annual TRR promote hurdle rate, it may not always be clear whether it is an effective rate or a nominal rate and consequently what the periodic IRR should be. Implied Continuous Compounding. Underlying these cash flow period adjustments is the implicit assumption that the theoretical method applies. For example, if zero cash flows may be added without altering the IRR, then the annual cash flows -1, 1 + r should have the same IRR as the semiannual cash flows, -1, 0, 1 + r, the triannual cash flows -1, 0, 0, 1 + r, the quarterly cash flows -.1, 0, 0, 0, 1 + r, etc. The IRRs for these cash flows are the annual rate of r, compounded annually, the semiannual rate of (1 + r)112 - 1 compounded semiannually, the triannual rate of (1 + r)" 3 - 1 compounded triannually, the quarterly rate of (I + r)"4 - 1 compounded quarterly, etc. While these rates are substantively the same under the theoretical method, they may differ under the practical method for partial periods. Thus, there seems to be an assumption that the theoretical method applies so that there is, in effect, continuous compound interest at the nominal annual rate of ln(1 + r), which, as explained in the third installment of this article, would yield an accumulation factor equal to (1 + r)t . IRR HURDLE CALCULATION An IRR hurdle calculation is designed to determine the hypothetical distribution amount that is required for the investor to achieve the applicable IRR. This hurdle calculation is different from an IRR calculation in one important respect: the rate is already given. The goal instead is to find a cash flow amount, which is the hurdle balance B (and would be CF in the IRR equation above). Thus, if the parties have agreed to a rate that has simple interest between compounding periods, some of the IRR calculations discussed above (which are based on the assumption that there is an underlying continuously compounded rate) may not be appropriate. 7139969.7 21 Cash Flow Periods Match Original Compounding Periods. If the actual cash flow periods are already uniform and match the original compounding periods, then it is easy to solve for the IRR hurdle balance (assuming the parties have a common understanding of the periodic rate r per compounding period): CF0 - CF, (I + r' - ... - CF 1 (1 + r) - B(l + r) 0 So that \n-k I B=- CFk (l + r) I However, if uniform cash flow periods are created (or already exist) that do not match the original compounding periods, then the present value calculations required when using a Present Value Hurdle may vary depending on whether the theoretical or practical method has been adopted. The present value calculations generally utilized for IRR calculations are consistent with the theoretical method. Therefore, if the practical method has been adopted, then an IRR calculation may effectively substitute the theoretical method. Theoretical Method. When the theoretical method is adopted, an IRR calculation is effectively the same as the Present Value Hurdle calculation: the cash flows can be regularized and the equivalent rate that applies to the newly established uniform period will yield the same present values and therefore the same hurdle balance, so the above formula for B also represents the Present Value Hurdle Balance. n-I \n -k B=-CFk (1+r) k=O Example 4.3. Assume the originally stated IRR rate is 21% per annum, compounded annually, as in Hypothetical 4A. If the cash flows are semiannual (as in Hypothetical 4A), then a 20% annual rate, compounded semiannually is an equivalent nominal annual rate and may be used to calculate the IRR hurdle (as was done by the operator in Hypothetical 4A). Thus, a 21% IRR hurdle would have been satisfied by an $11 million distribution as the operator asserted. This is easily confirmed by a typical calculator, which (assuming that the investor made a single investment of $10 million and received a single payment one-half year later of$1 1 million) would calculate a 10% semiannual IRR (indicating a 20% nominal annual rate, compounded semiannually or a 21% annual rate, compounded annually). Practical Method. But what happens if the practical method applies because the parties specify a rate that contemplates simple interest between compounding times? Under such circumstances, the Present Value Hurdle, while tracking the word definition of the IRR, would vary from textbook IRR formulations and from many calculator IRR programs because it would not allow for the regularization of cash flows using equivalent rates. Thus, if the practical method were adopted, and the parties desire to calculate a Present Value Hurdle, the typical calculator IRR function might not 7139969.7 22 be much help. The problem is that, as noted above, the typical calculator IRR function requires a uniform cash flow period and if one is created (by using zero cash flows), then it would likely assume that the uniform cash flow period is the compounding period. Unlike the theoretical method, the practical method cannot change compounding periods without running the risk of changing the outcome (even if an equivalent rate is used). Present Value Hurdle (Practical Method) Illustration. For example, in Hypothetical 4A, it is easy to calculate the amount necessary to achieve the 21% annual return as a Present Value Hurdle (and reach the result suggested by the investor): Example 4.4. Assume the facts of Hypothetical 4A. The Present Value Hurdle using the practical method would be calculated as follows: - B = 0 1.105 =' B = $10,000,000 x 1.105 = $11,050,000 But if such cash flows ($10 million and $11,050,000) were run through the typical calculator IRR function, the result would be a 10.5% semiannual rate or a 21% annual rate compounded semiannually, which of course, is not the same as the 21% annual rate compounded annually that was specified (because the compounding periods do not match). Consequently, the verification must be done in another manner. If an equivalent rate were used (20% per annum, compounded semiannually), then, as indicated in the prior paragraph, the result would be changed (and would not reflect the simple return during the year contemplated under the practical method): the hurdle amount would be only $11 million. 7139969.7 23 APPENDIX 4D.2 PRESENT VALUE HURDLES This Appendix will discuss the Present Value Hurdle introduced in the second installment of this article and how it may be used to establish a promote hurdle when the practical or theoretical method is adopted. DEFINITION AND NOTATION Recall that the Present Value Hurdle for a particular rate and series of cash flows is the hypothetical distribution amount that makes the net present value of the cash flows zero (or equivalently, that equates the present value of the cash inflows with the present value of the cash outflows). For notation, assume that the cash flows are represented by CF0, CF 1 , ..., CF, and occur as of to, t1, t1, t, time units, respectively. Also assume that r is the periodic effective rate for each compounding period. CASH FLOW PERIODS MATCH ORIGINAL COMPOUNDING PERIODS If the actual cash flow periods are uniform and match the original compounding periods, so that to = 0, t1 = 1 ... and t = n, then it is easy to solve for the hurdle balance B = CF (regardless of whether the theoretical or practical method has been adopted): CF0 - CF, (I + r)' - ... - CF 1 (1 + r) 1 - B(l + r) =0 So that \n-k I B=-CFk(1+r)n- I I kO However, if uniform cash flow periods are created (or already exist) that do not match the original compounding periods, then the results may vary depending on whether the theoretical or practical method has been adopted. THEORETICAL METHOD When the theoretical method is adopted, the Present Value Hurdle Calculation is also easy: the cash flows can be regularized and the equivalent rate that applies to the newly established uniform period will yield the same present values and therefore the same hurdle balance, so the above formula for B also represents the Present Value Hurdle balance (recognizing that n and r may be different than the original number of cash flows and the original rate per compounding period and there may be some zero cash flows). 7139969.7 24 n1 B=-CFk (l+r) I k=O PRACTICAL METHOD But what happens if the practical method applies because the parties specify a rate that contemplates simple interest between compounding times? Recall from the third installment of this article (Appendix 3A) that one must establish when the compounding periods occur. For simplicity, assume that each cash flow has its own compounding period (or that time 0 is a fixed compounding time), so that each cash flow accumulates by the following factor: a(t) = (1 + r)'t/(1 + r[t - /t/J), where lxi = the largest integer that is not more than x, and t is the relevant number of time units after such cash flow is made. The first step in calculating the Present Value Hurdle under the practical method is to observe that (based on the assumptions noted above) the present value of each CFk may be stated as follows: CFk )"k I - Itk/D Thus the hurdle balance (B = CF) must satisfy the following equation: n CFk =0 k=O (1+r)(1+r[tk /tk iI) Solving for the hurdle balance (B = CF) gives the following formula for the Present Value Hurdle: n-1 I-Ilk (i+r[t - it,iJ) B = - CFk (1 + r)"" ' k=O (1 +r[tk - ltk/J) As noted earlier, this formulation assumes that there are separate compounding periods for each cash flow (or that time 0 is a fixed compounding time). It is also possible to have a fixed set of compounding periods for all cash flows where time 0 is not a fixed compounding time, but that alternative is even more cumbersome to write. See Appendix 3A.3 for the relevant present value formula to establish the Present Value Hurdle balance. * * * 7139969.7 25 APPENDIX 4E PROMOTE HURDLES USING THEORETICAL METHOD This Appendix will discuss the three hurdles introduced in the second installment of this article: (1) the Loan Amortization Hurdle; (2) the Future Value Hurdle; and (3) the Present Value Hurdle. Assuming the application of the theoretical method and a single compound interest rate, it will be shown that as of any given time (the "Test Time"), the Future Value Hurdle and the Present Value Hurdle are the same (namely, an IRR hurdle) and, with a limited potential exception (when there has been a negative balance at some point in time), the Loan Amortization Hurdle also yields the same results. NOTATION/ASSUMPTIONS This installment will use notation and assumptions similar to those utilized in the second installment of this article when discussing simple interest. First, it is assumed that as of any given time there is no more than one cash flow; in the unlikely event there were a cash outflow (contribution or principal advance) and a cash inflow (distribution or payment) at the same time, they would be netted against one another. Also, it is assumed that there is the same amount of time between each two consecutive cash flows where the Test Time (i.e., the time the balance is to be calculated) is treated as the final cash flow time; and let this amount of time be the relevant time unit. (This is always possible by using the greatest common divisor of the time intervals between the non-zero cash flow times and then adding zero cash flows so that there is a cash flow occurring at the beginning and end of each such period.) In addition, unless otherwise stated, it is assumed throughout this article that the relevant cash flows in the JV context are the investor's contributions (the cash outflows) and the investor's distributions (the cash inflows). While it is possible that the parties may want to exclude certain contributions and distributions (or include certain cash flows that are not contributions or distributions) in the calculation of the hurdle, such potential refinements are beyond the scope of this article. Assume that the cash flows are CFO, CF I, CF2, CF3, ..., CF,-,, where CFk is the (k + l)st cash flow occurring as of k time units (k = 0, 1, 2, ..., n - 1), and is either positive (if there is net payment or distribution), negative (if there is a net advance or contribution) or zero. Thus, the first cash flow would be negative and each subsequent cash flow would be positive in the simple case where there is a single cash outflow (e.g., a single principal advance or a single contribution) at the beginning of the transaction, followed by cash inflows (payments or distributions) only. As another example, if there were only cash outflows (principal advances or contributions), then each CFk would be negative. Now, let r equal the equivalent periodic rate with respect to the uniform time period (i.e., the time unit). The beauty of the theoretical method is that equivalent rates are substantively the same at all times, so using an equivalent rate to determine accrued interest, future values and present values will not change the result. GENERAL HURDLE DEFINITIONS As indicated in the second installment of this article, the three hurdles mentioned above are defined as follows: 7139969.7 26 Loan Amortization Hurdle. The Loan Amortization Hurdle amortizes the investor's contributions (and the return) with the investor's distributions, where the investor's distributions are applied first to the investor's return and then to the investor's recoupment of capital in the same way that principal and interest payments are typically applied to amortize a United States mortgage loan. Thus, if the investor's contributions are treated as principal advances under a loan made by the investor and the investor's distributions are treated as payments to the investor of interest and principal, respectively, then (assuming the balance is not allowed to go negative) the Loan Amortization Hurdle as of any particular time is the balance of that hypothetical loan at such time. Future Value Hurdle. The Future Value Hurdle as of a particular time is the hypothetical distribution amount as of such time that would equalize the future value as of such time of the investor's contributions with the future value as of such time of the investor's distributions (or equivalently making the net future value of the cash flows, where contributions are negative and distributions are positive, equal to zero). In other words, the Future Value Hurdle as of a particular time is the amount, if any, by which (1) the future value (as of such time) of the investor's contributions exceeds (2) the future value (as of such time) of the investor's distributions. Present Value Hurdle. The Present Value Hurdle as of a particular time is the hypothetical distribution amount as of such time that would result in equalizing the present value as of the inception of the transaction of the investor's distributions and the present value as of the inception of the transaction of the investor's contributions (or equivalently making the net present value of all the cash flows, where contributions are negative and distributions are positive, equal to zero). Each of these hurdles will be discussed below. PRESENT AND FUTURE VALUE HURDLES = IRR HURDLE The Present Value Hurdle may be stated as the solution (B) to the following equation: 0= CFO - CF 1 (1 + r)' - CF2(1 + r) 2 - ... - CF,-,(l + r) 1 - B(1 + r) This should look familiar. It is an equation that establishes r as the periodic IRR of the relevant cash flows CFO, CF 1 , ..., CF,-,, B. If one multiplies both sides of this equation by (1 + r), one gets the following: 0 = CF0(1 + r) - CF 1 (1 + r)' - CF2(1 + r) 2 - ... - CF10 + r) B But this, of course, is an equation that establishes the Future Value Hurdle (B). Thus, if the theoretical method applies (so that cash flow periods may be made uniform and an equivalent rate used without changing the results), then the Present Value Hurdle and the Future Value Hurdle are the same, and each is nothing more than an IRR hurdle that may be summarized by the following formula: ni I \ n-k I B= - CF, (I+ r) k=O 7139969.7 27 Indeed, when the theoretical method is adopted, one would get the same result by equalizing the time value of the cash inflows and cash outflows (or equivalently, by making the net time value zero of all the cash flows, where cash inflows and cash outflows have opposite signs) as of any point in time: 49 The internal rate of return (IRR) for the transaction is the interest rate at which the value of all cashflows out is equal to the value of all cashflows in. Any valuation point can be used in setting up an equation of value to solve for an internal rate of return on a transaction, although there will usually be some natural value point, such as the starting date or the ending date of the transaction. LOAN AMORTIZATION HURDLE Those who read the second installment of this article may recall how cumbersome it was to establish the Loan Amortization Hurdle using simple interest. Among other matters, it was necessary to distinguish between principal and interest payments and therefore each CF k was broken down into a principal component, Pk, and an interest component, 'k. When the theoretical method is adopted (under which interest is effectively compounded continuously), this is not necessary. Because each cash flow period is a compounding period, the entire adjusted balance (after application of each distribution or contribution) will accrue a return during the next cash flow period. As will be shown below, this is the same balance that is obtained when using the IRR hurdle described above (assuming the hypothetical loan balance never goes negative). To see this, consider how the loan balance changes over time under the typical loan amortization method. End of First Period. What is the outstanding balance of principal and interest at the end of the first period (before the next cash flow, CF I)? -CF0(1 +r) In words, the initial cash flow earns interest at the rate of r during the initial period and therefore grows by a factor of (1 + r). At the end of such period, the second cash flow (CF I) changes the balance as follows: -CF0(l +r)-CF, End of Second Period. What happens at the end of the second period (before the next cash flow, CF2)? The above balance earns interest at the rate of r during the second period and therefore grows to the following amount: (-CF0{1 +r} -CF 1)(l +r) which may be rewritten as follows: - CF0(l + r)2 - CF 1 (l + r) Then, the balance is changed by the third cash flow (CF 2) as follows: - CF0(1 + r)2 - CF 1 (1 + r) - CF2 7139969.7 28 End of Third Period. By the end of the third period (before the next cash flow, CF 3), the above balance has grown at the rate of r throughout the third period to the following amount: (- CF 0 [l + r]2 - CF 1 [l + r] - CF2)(l + r) This amount may be rewritten as follows: - CF0(l + - CF1(1 + r)2 - CF2 (1 + r) Generally. This process continues so that at the end of n periods (before the [n + list cash flow), the outstanding balance will be: - CF0(l + r) - CF 1 (1 + r) 1 - ... - CF1 (1 + r) CF(1 +r )k But this is the same as the IRR hurdle described earlier. Q.E.D. Comment. Consistent with the conclusion above, it has been stated that "... the internal rate of return on a loan transaction is simply the interest rate at which the loan is made". 50 However, the Loan Amortization Hurdle and the IRR Hurdle may differ (and the interest rate of a loan may not equal its IRR) if the IRR hurdle balance becomes negative during the term of the investment, as discussed further below. Also, it should be kept in mind that the key to the common results in the hurdles above is the assumption that the theoretical method applies. For example, the IRR balance could be different from the loan amortization balance (and the IRR different from the stated rate of return) if the practical method were adopted (as may be typical for U.S. mortgage loans and relatively common for preferred return and return of capital hurdles), but only if some cash flows do not occur on the originally stated (or assumed) compounding times: although a uniform cash flow period could be created, the equivalent rate for the uniform cash flow period may not be the same as the original rate for partial periods. THE POTENTIAL EXCEPTION - NEGATIVE BALANCE In the discussion of the Loan Amortization Hurdle above, it is assumed that at the end of each period, the new cash flow does not reduce the balance below zero. It is possible to allow this to occur (and it does in fact occur in real estate investments other than loans), still maintain the equivalence, and explain the occurrence in the loan context, by assuming that the lender and the borrower are reversing roles so that the investor, who is originally the lender, is treated as borrowing the negative balance at the same rate. It would be as though the lender received more than enough to repay its loan and it deposited the surplus in an account, paying the same rate of interest, that would be used to make the next loan advances. (It is also possible to eliminate any interim negative balance from the calculation .5 1 ) While negative balances may occur in the IRR context (when distributions exceed the hurdle), they typically do not occur in the loan context and often do not occur in the analogous scenario where the hurdle is based on the outstanding balance of capital 7139969.7 29 contributions and unpaid preferred return. Thus, it is also possible that an interim negative balance could result in different final hurdle balances, because it is not clear whether and how a negative balance would be taken into account under the loan amortization approach. * * * 7139969.7 30 10011POU61110 Carey, "Real Estate JV Promote Calculations: Rates of Return Part I The Language of Real Estate Finance," The Real Estate Finance Journal (Spring, 2011), hereinafter "Carey, Rates of Return Part 1." 2 Carey, "Real Estate JV Promote Calculations: Rates of Return Part 2 - Is Simple Interest Really That Simple?," The Real Estate Finance Journal (Summer, 2011), hereinafter "Carey, Rates of Return Part 2." Carey, "Real Estate JV Promote Calculations: Rates of Return Part 3 Compound Interest to the Rescue?," The Real Estate Finance Journal (Fall, 2011), hereinafter "Carey, Rates of Return Part 3." Zima Brown Kopp, MATHEMATICS OF FINANCE (6th ed., McGraw-Hill Ryerson 2007), § 2.1 at 32; Hart, MATHEMATICS OF INVESTMENT (5th ed., D. C. Heath and Company 1975), § 10 at 20; Shapiro, MODERN CORPORATE FINANCE (Macmillan 1990), § 2.1 at 26; see also Brueggeman Fisher, REAL ESTATE FINANCE AND INVESTMENTS (14th ed., McGraw-Hill Irwin 2011), Ch. 3 at 44; and Geltner Miller Clayton Eichholtz, COMMERCIAL REAL ESTATE ANALYSIS & INVESTMENTS (2nd ed., Cengage Learning 2007), § 8.1.2 at 152. See, e.g., Womack and Brownell, "Financial Math on Spreadsheet and Calculator" (copyright 2002) at the website of the Tuck School of Business, Dartmouth College. 6 Kellison, THE THEORY OF INTEREST (Richard D. Irwin 1970), § 1.5 at 7. This statement does not appear in Kellison, THE THEORY OF INTEREST (3rd ed., McGraw-Hill 2009), but the third edition does state the following: "If interest accrues continuously, as is usually the case, the [future value interest factor] will be continuous (emphasis added)." See § 1.2 at 2. Observe that continuous accrual does not mean merely that interest is accruing at every moment in time (e.g., if there were a simple interest loan with an additional interest charge that accrued on the anniversary of the loan, the accrual would not be continuous even though interest is accruing at all times); it means that the future value interest factor (or accumulation function) for a unit investment is continuous (so that the growth of, and the amount of interest on, a unit investment that remains fully invested are continuous). Zima, supra, § 2.4 at 44-46; Smith, THE MATHEMATICS OF FINANCE (Appleton Century Crofts 1951), § 37 at 52; Hummel and Seebeck, MATHEMATICS OF FINANCE (3rd ed., McGraw-Hill 1971), § 14 at 27- 31; Hart, supra, § 16 at 35-38 (referring to the practical method as an approximate method); Shao, MATHEMATICS OF FINANCE (South-Western 1962), § 10.3E at 241-242 (referring to practical method as Method A and theoretical method as Method B). 8 Hummell and Seebeck, supra, § 14 at 30; see also, Zima, supra, § 2.4 at 45 ("[When] simple interest for the fractional part of a [compounding] period [is] used [it is] called [the] practical method ...."). Zima, supra, § 2.4 at 44. Hummel and Seebeck, supra, § 13 at 25. See also, Ayres, MATHEMATICS OF FINANCE (Schaum ' s Outline Series - McGraw-Hill 1963), Ch. 7 at 65 ("Two annual rates of interest with different conversion periods are called equivalent if they yield the same compound amount at the end of the one year."); Shao, supra, § 10.5C at 254-255. II Hummell,supra, § 14 at 27. 2 See, e.g., Broverman, MATHEMATICS OF INVESTMENT AND CREDIT (4th ed., Actex 2008), § 1.1.1, Def. 1.2 at 8; Kellison (2009), supra, § 1.7 at 17; cf. Vaaler & Daniel, MATHEMATICAL INTEREST THEORY (2nd ed., Mathematical Assoc. of America 2009), § 1.6, Def. 1.6.6 at 27. Carey, Rates of Return Part I, supra, App. lC; Carey, Rates of Return Part 3, supra. Fabozzi, FIXED INCOME MATHEMATICS (4th ed., McGraw-Hill 2006), Ch. 3 at 27; Fabozzi, THE HANDBOOK OF FIXED INCOME SECURITIES (7th ed., McGraw-Hill 2005), App. A at 1446; Williams, THE MATHEMATICAL THEORY OF FINANCE (The MacMillan Co. 1935), § 11 at 20; Ruckman and Francis, FINANCIAL MATHEMATICS (2nd ed., BPP 2005), § 1.3 at 6-9. Tuckman, FIXED INCOME SECURITIES (2nd ed., John Wiley & Sons 2002), at 56. 7139969.7 31 16 Cissell Cissell Flaspohier, MATHEMATICS OF FINANCE (8th ed., Houghton Mifflin 1990), § 3.11 at 128; accord, Veena, BUSINESS MATHEMATICS (Galgotia 2004), Remark 2 at 200 ("Interest for the integral part ... accumulates at Compound interest and for the fractional part ..., Simple interest is calculated ...."); Simpson Pirenian Crenshaw & Riner, MATHEMATICS OF FINANCE (4th ed., Prentice-Hall 1969), § 102 at 212; Appraisal Institute, THE APPRAISAL OF REAL ESTATE (11th ed., Appraisal Institute 1996), App. C, Example 2 at 765 ("This calculation assumes simple interest for any time that is less than one conversion period."). Zima Brown, MATHEMATICS OF FINANCE (Schaum's Outlines 2nd ed. 1996), § 4.4 at 51, but see Zima (2007), supra, which says instead that the practical method is "often used" (§ 2.4 at 45) and "Unless stated otherwise, it will be understood that the ... practical method is to be used . . . ." (§ 2.4 at 46); Smith, supra, § 37 at 52 ("... practical method ... is often used in applied problems...."); Hummel and Seebeck, supra, § 14 at 30- 31 ("... the practical method has become the most widely used method .... The [theoretical] method is, of course, the proper one to use. However, when irregular periods of time are encountered, [the practical method] will usually be employed ...."); Shao, supra, § 10.3E at 242 ("Theoretically speaking, [the theoretical method] is more reasonable ... because the compound method is used throughout. However, [the practical method] is widely employed...."); cf. Hewlett-Packard, HP 12C Platinum Owner's Handbook And Problem-Solving Guide (Hewlett-Packard 2003), § 3, "Odd-Period Calculations" at 53 (which defaults to simple interest for the period between the date interest begins to accrue and the first payment). 8 Hart, supra, § 16 at 35-36; cf. Vaaler & Daniel, supra, § 1.5 at 20 ("In practice however, banks do not always use [the theoretical method] when t [, the number of compounding periods,] is nonintegral."). See, e.g., Niggelingv. Michigan Dept. of Transportation, 195 Mich. App. 163 (1992), involving post-judgment interest. 20 Geltner, supra, § 17.1.1, fn. 1 at 408. 21 See, e.g., California Real Estate Finance Practice: Strategies and Forms (Cal. CEB 2000, Supp. 2011), § 3.14 at 116; Hinkel, PRACTICAL REAL ESTATE LAW (5th ed., West Legal Studies 2008), Ch. 16 at 550 ("If the lender does not give a per diem interest charge, it can be calculated by multiplying the loan amount by the interest rate and dividing by 365 or 360."). 22 Geltner, supra, § 17. 1.1 at 408. 23 Broverman, supra, § 5.1.1, Def. 5.1 at 265; Vaaler & Daniel, supra, § 2.4 at 84; Ruckman and Francis, supra, § 5.1 at 130. The IRR is sometimes defined in general terms as the rate that makes the NPV of the cash flows equal to 0 or that equalizes the present value of the cash outflows and cash inflows. See, e.g., Ross, Westerfield, Jordan, FUNDAMENTALS OF CORPORATE FINANCE (8th ed., McGraw-Hill Irwin 2008), § 9.5 at 278; Floyd Allen, REAL ESTATE PRINCIPLES (8th ed., Dearborn Financial Publishing 2005), Ch. 14 at 310; Copeland, Weston, Shastri, FINANCIAL THEORY AND CORPORATE POLICY (4th ed., Pearson Education, Inc. 2005), Ch. 2, § E4 at 28. Generally, as indicated in the third installment of this article, there seems to be an assumption that compound interest is involved (and the formulas used make this clear) and governed by the continuously compounding accumulation function a(t) = (1 + i)t. However, the author has found that some practitioners have developed their own approaches and may use different formulations depending on the facts. Indeed, if one assumes that an IRR can be calculated by equating present values of cash inflows and cash outflows, then it may be possible, at least in theory, to calculate the present values using the applicable accumulation function even if it is not of the form a(t) = (1 + r)'. This, of course, would be the more general Present Value Hurdle approach introduced in the second installment of this article and discussed in Appendix 4D.2. There are also a number of conventions that may be used (e.g., discounting or growing cash flows to the previous or next compounding date) that may result in a different calculation. 24 King, THE THEORY OF FINANCE (3rd ed., 1898), Ch. I, § 15 at 7-8. 25 Lewin, "An Early Book on Compound Interest," Journal of the Institute of Actuaries, Vol. 96, part I, no. 403 (1970) 120 at 129. See also Osborn, THE MATHEMATICS OF INVESTMENT (Harper & Row 1957), § 9 at 43. 26 Porter, MATHEMATICS OF INVESTMENT (Prentice-Hall 1949), Ch. III at 33. 7139969.7 32 27 See endnote 24, supra. 28 Carey, Rates of Return Part 2, supra, at endnote 16. 29 Williams, supra, § 11 at 20-21. 30 Admittedly, the purpose of this law may seem to be to protect the payor and it is debatable whether the law would have any application in the context of a compounding period (given that the parties have already agreed to some compounding). In the context of a partial compounding period, simple interest may put a greater burden on the payor, but if the parties have not specified whether there is simple or compound interest during a compounding period, is the burden on the payor the determining factor as to which method should apply? In a case interpreting whether a statutory reduction of future damages to present value should be made with simple or compound interest, the court did "... not dispute ... that compound interest is the standard generally employed in the business and financial world today ...," but finding that the legislature did not reject the common law use of simple interest, the court held that simple interest should be used even though that resulted in a payment that was roughly 50% larger than it would have been if compound interest had been used. Nation v. WDE Electric Co, 563 NW2d 233 (Mich. 1997). Carey, Rates of Return Part 3, supra, App. 3A. 32 Carey, Rates of Return Part I, supra, App. IC. Carey, Rates of Return Part 3, supra, App. 3B. Comparisons of simple and compound interest appear in a number of books. See, e.g., Butcher and Nesbitt, MATHEMATICS OF COMPOUND INTEREST (1971, Reprint, Ulrich '5 1979), § 1.11 at 20-24; Broverman, supra, § 1. 1.4 at 14-15; Kellison (2009), supra, § 1.5 at 10 (referring to expansion of binomial formula to compare compound and simple interest); Williams, supra, § 12 at 23 (using binomial formula); Ruckman and Francis, supra, § 1.3 at 6-9. See also Li and Zhu, "Research on Calculations of the Principal Accumulation Value," 2009 International Forum on Information Technology and Applications. See, e.g., Stelson, "A Comparison of Simple and Compound Interest," National Mathematics Magazine, Vol. 19, No. 7 (Apr. 1945), at 336. 36 See, e.g., Butcher and Nesbitt, supra, § 1.11 at 24 ("... the maximum error occurs when t is about 1/2."); Stelson, supra, at 337 (indicating that maximum difference is greater than 1/2 and less than 1); see also, Stelson, "The Accuracy of Linear Interpolation in Tables of the Mathematics of Finance," Mathematical Tables and Other Aids to Computation, Vol. 3, No. 26 (American Mathematical Society, Apr. 1949), at 408-412. See, e.g., the XIRR function in EXCEL. 38 Notice in Graph 4E that the gap between the line segments and the exponential function gets larger over time. Benninga, FINANCIAL MODELING (2nd ed., MIT Press 2000), § 29.2.2 at 462 ("... an interest rate r such that the net present value of the cash flows is zero ...."); Adams Booth Bowie Freeth, INVESTMENT MATHEMATICS (Wiley 2003), § 1.9 at 16("... the rate of interest at which the NPV ... is equal to zero ...."); Steiner, MASTERING FINANCIAL CALCULATIONS (2nd ed., Prentice-Hall 2007), Part I at 18 ("... the interest rate which ... discounts all the cashflows ... to zero...."); Day, MASTERING FINANCIAL MATHEMATICS IN MICROSOFT EXCEL (Pearson Education 2005), Ch. 3 at 33 ("... the rate at which the net present value is zero 40 Kellison (2009), supra, § 7.2 at 252 ("... that rate of interest at which the present value of net cashflowsfrom the investment is equal to the present value of net cash flows into the investment ...."); Broverman, supra, § 5. 1.1 at 264 ("... the interest rate at which the value of all cashflows out is equal to the value of all cashflows in ...."). Even when the IRR is limited to compound interest rates, there is a potential for multiple IRRs, but that subject is beyond the scope of this article. The issue of multiple IRRs is addressed in most finance textbooks. See also, Carey, "Real Estate JV Promote Calculations: Recycling Profits," The Real Estate Finance Journal (Summer, 2006), Appendix, hereinafter "Carey, Recycling of Profits." 7139969.7 33 42 Benninga, supra, § 1.3 at 5. McFedries, Formulas and Functions with Microsoft Excel 2003 (Que Publishing 2005), Ch. 20 at 456, 461. Hewlett-Packard, supra, § 4 at 58. Vaaler & Daniel, supra, § 1.7 at 34 and fn. 4 ("... choose an increment of time between successive cash flows [which] should in general be ... the longest time interval that allows you to include all your non-zero cashflows ...."); Kellison (2009), supra, § 7.2 at 249 ("For convenience, we assume that these times [between cashflows] are equally spaced."); Geltner, supra, § 8.2.14, fn. 10 at 166 ("... if the cash flows occur at irregularly spaced points in time, one can simply define the smallest [sic] common time divisor as the period and have [the cash flows] = 0 between the cash flow receipts ...."). 46 Hewlett-Packard, supra, § 4 at 63. Geltner, supra, § 8.2.14, fn. 11 at 167; Kellison (2009), supra, § 7.8 at 281. 48 Carey, Rates of Return Part 1, supra. Broverman, supra, § 5. 1.1 at 264; see also discussion of equations of value in Carey, Rates of Return Part 3, supra. 50 Broverman, supra, § 2.4.1 at 126; Geltner, supra, § 17.2.1 at 419 ("The yield of a loan generally refers to its internal rate of return (IRR)."); see also, discussions of how to calculate the outstanding balance of an amortized loan, which is essentially an IRR calculation (especially the prospective method as of the inception of the loan); Broverman, supra, §§ 3.1.1-3.1.4 at 175-182; Vaaler & Daniel, supra, § 3.6 at 133-135; Kellison (2009), supra, § 5.2 at 153-156; Ruckman and Francis, supra, § 5.5 at 146-147. 51 Carey, Recycling of Profits, supra. 7139969.7 34 Copyright © 2011 Thomson Reuters. Originally appeared in the Winter 2012 issue of The Real Estate Finance Journal. For more information on the publication, please visit http://west.thomson.com. Reprinted with permission.