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10 k " iou The Real Estate Finance Journal A WEST GROUP PUBLICATION Copyright '20 11 West Group REAL ESTATE JV PROMOTE CALCULATIONS: RATES OF RETURN PART 2- IS SIMPLE INTEREST REALLY THAT SIMPLE? By Stevens A. C arey* Based on article published in the Summer 2011 issue of The Real Estate Finance Journal *STEVENS A. CAREY is a transactional partner with Pircher, Nichols &Meeks, a real estate law firm, with offices in Los Angeles and Chicago. The author thanks John Caubie, Steve Mansell, JeffRosenthal, 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. TABLE OF CONTENTS SimpleInterest - The Basics .................................................................................................... 1 Definition....................................................................................................................... 1 FutureValue Formula.....................................................................................................2 ContinuousAccrual........................................................................................................2 StationaryAccumulation ................................................................................................2 Illustration......................................................................................................................3 Critiqueof Simple Interest ...................................................................................................... 4 TimeValue of Money .................................................................................................... 4 Frozen Time Value of Simple Interest from Accrual to Removal.................................... 4 Dysfunctional Time Value Functions ...................................................................................... TimeValue Equivalence ................................................................................................ Intransitivity................................................................................................................... Comparison of Simple Interest Investments.................................................................... Distinguishing Between Principal and Interest; Order of Application.............................. 5 5 5 6 7 Declining (Relative) Growth Rate ........................................................................................... 8 Seemingly Inflated Simple Interest Rates of Return........................................................ 8 ConflictingIncentives .............................................................................................................. 8 Investor Incentive to Accelerate Removal....................................................................... 8 Payor Incentive to Defer Removal.................................................................................. 9 Simple Interest Return Hurdles .............................................................................................. AmortizationHurdle....................................................................................................... PresentValue Hurdle...................................................................................................... FutureValue Hurdle ....................................................................................................... Concluding Observations regarding Hurdles ................................................................... 10 10 10 11 11 Statusof Simple Interest Today .............................................................................................. 12 LookingAhead ......................................................................................................................... 12 APPENDICES .......................................................................................................................... 12 Appendix 2A Stationary Accumulation................................................................................... 13 Appendix 2B Merchant’s Rule vs. United States Rule............................................................. 16 Appendix 2C Formula for Outstanding Balance (Merchant’s Rule)......................................... 20 Appendix 2D Formula for Outstanding Balance (United States Rule)...................................... 21 Appendix 2E Proof that Merchant’s Rule Effectively Applies Payments to Principal .............. 23 Appendix 2F United States Rule vs. Merchant’s Rule: The Difference................................... 24 Appendix 2G Simple Interest Hurdle Balances ........................................................... ............. 25 - - - - - - - REAL ESTATE JV PROMOTE CALCULATIONS: RATES OF RETURN PART 2IS SIMPLE INTEREST REALLY THAT SIMPLE? By Stevens A. Carey* This is the second installment of an article discussing rates of return in the context of real estate joint venture (JI’) distributions. The first installment introduced commonly used terminology and conventions. This installment focuses on the seemingly fundamental and straightforward notion of simple interest. Simple interest is perceived to be very basic. It is certainly easy to calculate, Unlike compound interest, which may involve exponential calculations, simple interest may be calculated as a simple product based on the amount of principal, the interest rate and the time involved. But is simple interest irrelevant in today’s commercial world? And isn’t simple interest so trivial that it does not deserve much discussion? The answer to both these questions is no. Simple interest is not as simple as it appears. And it is relevant because it is commonly used in a number of contexts, including some long term real estate investments’ (in addition to approximations and short term arrangements). It may also be favored over compound interest by courts 2 (and some legislatures) when the manner in which interest accumulates is not clearly documented. As stated by one legal authority 4 : Historically, courts have been wary of arrangements involving compound interest. [citation omitted] That wariness continues today. Modern courts allow recovery of compound interest only if the parties specifically agree to compounding and no statute prohibits it. [citation omitted] Given the continued use of simple interest, it is important to appreciate its potential problems and how they can lead to distortion and uneconomic results. This installment of the article will provide a brief refresher on the basics of simple interest and then attempt to identify and explain its complexities. SIMPLE INTEREST - THE BASICS Definition. What is simple interest? Simple interest may seem so simple that some textbooks don’t take the time to define it (or discuss it as a separate topic), and some that do may define it only by example. 5 Sometimes, simple interest is defined as interest that accrues only on the original principal invested .6 Basically, simple interest is interest that is proportional to the time involved 7 and may be defined more precisely by the following formula: * STE VENSA. CAREY is a transa ctional 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. I=rPt where r is the annual interest rate, P is the principal amount of the investment, and / is the number of years involved, 8 Example 2.1 A $100 investment earning simple interest at an annual rate of 10% for two years would earn simple interest equal to $20 (10% x $100 x 2); and if the annual rate were 100%, the investment would earn simple interest equal to $200 (100% x $100 x 2). Future Value Formula. The formula for simple interest is often expressed by reference to future value: S=P(1 +rt) where S is the value of the investment after t years, P is the principal amount of the investment, and r is the annual interest rate. 10 Sometimes, simple interest is presented and analyzed by focusing on the future value interest factor (i.e., the value of an investment of 1)11: 1 + rt. Example 2.2. If the simple interest rate were 10% per annum, then the future value interest factor would be 1 + (.1)t; and if the simple interest rate were 100% per annum, then the future value interest factor would be 1 + t. Continuous Accrual. The formulas above are sometimes limited to integral non-negative values of t. But they are frequently (and in this article will be, unless otherwise stated) applied for all non:12 negative values oft, As stated in one textbook [l]t is natural to extend the definition[s] to nonintegral values of t > 0 as well. This is equivalent to the crediting of interest proportionally over any fraction of a period [and is tantamount to assuming that simple interest accrues continuously] . . Unless stated otherwise, it will be assumed that interest is accrued proportionally over fractional periods under simple interest. Example 2.3. A $100 investment earning simple interest at an annual rate of 100% for 1.5 years would earn simple interest equal to $150 (100% x $100 x 1.5). In practice, real estate finance professionals will often focus on what happens each day rather than every moment. They may adopt a convention under which interest is credited as of a certain time of day (e.g., the deadline for investment as a bank deposit) on the amount of the interest-bearing portion of the investment as of such time, Through this or some other convention, one may end up with a discontinuous accrual of interest involving discrete daily increases. But, for convenience, the discussion that follows will (unless otherwise stated) assume a continuous rather than daily accounting. Stationary Accumulation. Does the amount of interest that accrues on a unit investment" (which remains fully invested without addition or withdrawal) remain the same for a period of a particular duration (e.g., 12 months) regardless of when the period starts? If the interest rate doesn’t change, and one ignores potential deviations due to leap years and the like, one might expect the answer to be 2 yes. And with a single fixed rate of simple interest that accrues continuously the answer is yes: it obviously doesn’t matter when the money is invested; only the duration counts. Thus, $1 invested for t years at a simple annual interest rate of r would earn the same interest, $(rt), and grow to the same amount, $(1 + ri), regardless of when the $1 were invested. This feature of simple interest, which is often taken for granted, is sometimes called "stationary accumulation", and is described in more detail in Appendix 2A. But if (contrary to the assumption stated earlier) simple interest were not accruing continuously, there may not be stationary accumulation: imagine a bank account prohibiting withdrawals after 3:00 PM, and crediting one day’s interest on the principal balance as of 3:00 PM each day. Clearly, $1 invested for an 18-hour period may vary depending on when it starts (i.e., when the investment is made): if the period starts at 2:00 PM there would be crediting of one day’s interest at 3:00 PM the same day; and if the period starts at 4:00 PM there would be no crediting of interest. And more generally, when one considers all types of interest (not merely simple interest), the manner in which interest accumulates may depend not only on the duration of 14 Discrepancies may the investment period but also on the particular time the investment is made, occur even with a single fixed rate of interest that accrues at every moment without interruption. Consider, for example, an account that accrues simple interest at a single fixed rate during each calendar year but compounds at the end of each calendar year: $1 invested in the first quarter of a calendar year for six months would earn only simple interest but $1 invested in the third quarter of a calendar year for six months would earn more interest because of the compounding at the end of the calendar year. Illustration. The value of an investment earning simple interest is sometimes illustrated by a graph 15 similar to the following graph of the of the future value formula or the future value interest factor value of I for a simple interest rate of 100% per annum: GRAPH NO. 2A FV of 1 with 100% per annum simple interest 4 3 2 0 1 2 Year 3 4 The above graph reflects a function that is continuous and linear, perhaps leading one to conclude that simple interest couldn’t be simpler. However, behind this seemingly straightforward concept lies an inconsistency that may yield unrealistic and awkward results. CRITIQUE OF SIMPLE INTEREST 16 Simple interest has often been criticized as being illogical, absurd and neither rational nor sensible. Why? Basically, because of the inconsistent manner in which it takes into account the time value of money: it takes into account the time value of principal, while ignoring the time value of accrued interest. The following discussion will examine in more detail the shortcomings of simple interest. Time Value of Money. In explaining the time value of money, it is often stated that a dollar today is worth more than a dollar one year from today.’ 7 As explained by one textbook’8 : If someone offered you a choice between $10,000 today or $10,000 one year from now, which would you choose? It is a fundamental fact of financial economics that dollars at one time are not equivalent to dollars at another time. This is not just because of inflation. Due to the real productivity of capital, and due to risk, future dollars are worth less than present dollars, even if there were no inflation. Yet this is not how simple interest works. Indeed, because simple interest does not itself earn interest, 19 a dollar of simple interest today may still be worth only a dollar one year from today. Example 2.4. Assume $5 were invested one year ago in a bank account earning 20% annual simple interest and the balance of the account were $6 today ($5 of principal and $1 of interest). If left invested for another year, this investment would earn another $1 of simple interest (20% of $5) during the next year. But the $1 of simple interest earned in the prior year would not grow at all while it remains in the account and therefore may (ignoring inflation) have the same $1 value one year from today. This result does not make a lot of sense: the failure to take into account the time value of interest in the same manner as principal ignores the reality that the character of available capital as either principal or interest is generally irrelevant in determining its potential productivity and value. 20 Once it accrues, simple interest remains flat, without growth from interest, until the moment it is removed from the investment. Such removal may occur when simple interest is withdrawn from a simple interest bank account or, in the context of a loan, paid by the borrower to the lender or, in the context of promote hurdles, when a simple return is distributed to the investor. Mere accrual or crediting to the simple interest bank account, loan balance or promote hurdle balance is not sufficient (unless credited as principal) even if the money is immediately available for withdrawal, distribution or payment (and may be demanded at any time). Unless it is converted to principal, the simple interest must actually leave the investment for an instant in time to lose its character as simple interest. Only after such removal can it be reinvested in the investment or some other investment as principal and then earn interest. In effect, simple interest is frozen in time from accrual to removal, which leads to curious if not strange anomalies, including: Frozen Time Value of Simple Interest from Accrual to Removal. Dysfunctional time value functions: intransitive time value equivalence; and the potential need to track principal and interest components. Declining growth rate of simple interest investment balances, and, as a result, seemingly inflated rates of return when expressed as simple (rather than compound) rates. Conflicting incentives of the parties to the investment (e.g., lender/borrower, depositor/bank, partner/partnership): one benefits from acceleration, and the other benefits from deferral, of the removal of simple interest from the investment. Admittedly, there may be similar tensions whenever the applicable rate is not a market rate (or can be replaced by a more favorable rate), but this conflict may be more likely tooccur in the case of a simple interest investment, especially for longer term investments, Moreover, in the authors experience in JV transactions when there are compounded returns, the roles are usually reversed: the rate of return is usually sufficiently high that the operator would prefer to pay the return as soon as possible to avoid further accrual that would otherwise defer its promote. DYSFUNCTIONAL TIME VALUE FUNCTIONS The potential chaos of a time value system utilizing simple interest is apparent when one observes how inconsistently simple interest treats dated cash flows. Time Value Equivalence. As stated by one commentator: The fundamental principle on which the solution of problems in the mathematics of investments is based is the so-called principle of equivalence, according to which two sums of money at different dates are considered as equivalent, under a given interest rate, if the earlier sum would amount to the later sum, at the given interest rate, during the intervening time. 21 For a simple interest rater per time unit, an amount X paid or received at a particular time is said to be equivalent to an amount Y paid or received, respectively, ttime units later if Y = (1+ rt) X. 22 Example 2.5. Assuming the applicable simple interest rate were 10% per annum: a $110 cash flow today would be equivalent to a $100 cash flow one year earlier (because the future value of a $100 cash flow after one year would be $110); and a $110 cash flow today would also be equivalent to a $121 cash flow one year later (because the future value of a $110 cash flow after one year would be $121). Intransitivity. In the world of simple interest, two dated cash flows that are equivalent to a third dated cash flow may not be equivalent to one another. This is contrary to our common sense of order: In mathematics, an equivalence relationship must satisfy the so-called property of transitivity, that is, if X is equivalent to Y and Y is equivalent to Z, then X is equivalent to Z. 23 5 Example 2,6, For example, assume that X, Y and Z are dated cash flows in the respective amounts of $100, $110 and $121 occurring at the beginning of consecutive years as indicated below: Beginning of Year 1 X=$100 Beginning of Year 2 Y=$110 Beginning of Year 3 Z=$121 As indicated in Example 2.5, using a 10% simple annual interest rate, Xis equivalent to Y and Y is equivalent to Z. Thus, one would expect X to be equivalent to Z. But it is not: the future value of X as of the beginning of Year 3 is only $120, which is $1 less than Z. Thus, "[the] property [of transitivity] does not hold for simple interest rates. . , and in consequence ,24 It should, therefore, come as no the concept of equivalence at these rates lacks logical soundness. surprise that theoreticians do not like simple interest (not to mention the practical concerns of lenders and other capital providers). Comparison of Simple interest investments. Intransitivity makes it difficult to compare simple interest investments. Generally, when comparing two investments, they are represented by two sets of dated cash flows, a comparison date is chosen, and then each cash flow is discounted or grown to the comparison date, 25 Example 2.7. To take an easy case, using a 10% simple annual interest rate and a comparison date at the beginning of Year 2, compare a $121 cash flow at the beginning of Year 3 and a $100 cash flow at the beginning of Year 1. The time value of each of these amounts would be equal to $110 as of this comparison date. However, because simple interest equivalence is not transitive, "the choice of a comparison date does affect the answer obtained. This illustrates once again the inherent inconsistency in using simple interest , , ,, 26 As explained in one textbook: [I]n the case of simple interest[,] . . . equivalence at one comparison date of the two sets of dated payments does not signify their equivalence at any other date. The matter will not be pursued, but it may be realized that since two different formulas [a linear future value formula and a hyperbolic present value formula] are required to determine values under simple interest, and since these formulas lack the convenient exponential character of the corresponding formula. . . for compound interest, nothing very neat can be expected .27 Thus, as indicated by Examples 2.6 and 2.7 and as further illustrated by the following example, the time value of two simple interest investments may be equal at one time but not another (even in the absence of additional deposits or withdrawals during the interim period). Example 2.8. Assume a bank offers deposit accounts that earn simple interest at the rate of 10% per annum, and assume that Party A opens one account today with a $100 deposit, and Party B opens another account one year from today with a $110 roll deposit. Further assume that neither A nor B makes additional deposits to, or withdraws any amount from, its account until three years from today. Party A Year Balance 0 $100 1 $110 2 $120 3 $130 Party B Balance Year 0 $0 1 $110 2 $121 3 $132 Given such facts, both accounts would have a $110 balance one year from today. But B’s account would have a $121 balance as of two years from today, while A’s account would then have abalance of only $120. In fact, B’s account would have a larger balance than A’s account at all times after one year. Why would there be a difference if the balance of each account were $110 one year from today and the interest rate were the same? Because in one case, that $110 balance would include $10 of simple interest, which does not grow. 28 Ideally, the time values of two investments are equal, if at all, at any time (not just one time), 29 but this will never be the case for two distinct simple interest investments. Distinguishing Between Principal and Interest; Order of Application. Given that principal and interest are treated differently in a simple interest investment, how does one determine the manner in which an interim partial payment (i.e., a payment of less than the then outstanding balance of principal and interest) should be applied to the principal and interest components of an investment? As discussed in Appendices 213 through 2F, many textbooks mention two approaches to make this determination in a simple interest investment: one, the so-called United States Rule, applies partial payments first to interest and then to principal (unless the parties otherwise agree); and the other, the so-called Merchant’s Rule, effectively applies partial payments to principal before interest (although the application to interest does not occur until the time the final balance is to be calculated and this order of application is not immediately apparent from the typical statement of the Merchant’s Rule). The United States Rule reflects the customary way a loan would be amortized today (assuming there is no negative amortization). The Merchant’s Rule, on the other hand, does not involve typical loan amortization. In fact, the usual way the Merchant’s Rule is defined, and works in practice, there is no immediate application to the loan at all. The Merchant’s Rule keeps a separate tally of all advances by the investor on the one hand, and all payments to the investor, on the other hand; both earn interest and the balance is the amount by which the advances (plus interest) exceed the payments (plus interest). Basically, the payments are separately accounted for (without application to the debt) until the payments (plus interest) are sufficient to satisfy the loan advances (plus interest). As explained in Appendices 2B and 2E, the net effect of the Merchant’s Rule is to apply all payments immediately to principal. All the textbook examples of the Merchant’s Rule reviewed by the author involve interim payments that do not exceed the total principal advanced and in this context, the effect of the Merchant’s Rule is relatively easy to see: the interest-bearing interim payments effectively offset an equal amount of principal and the future interest on that principal so the result is the same as if the interim payments were immediately applied to principal. Even if the interim payments were to exceed the total principal, the effect would be equivalent to an application to principal, resulting in a negative principal balance that accrues negative interest until that negative balance of principal and interest is sufficient to pay the interest that accrued while the principal balance was positive . 30 Because the effect of the Merchant’s Rule is to reduce principal (the interestbearing portion of the loan balance) rather than accrued simple interest (the non-interest bearing portion of the loan balance), the Merchant’s Rule loan balance is always less than or equal to the United States Rule loan balance; the exact difference is explained and proven in Appendices 2B and 2F. The Merchant’s Rule may seem like an antiquated, obscure and even bizarre rule. It is definitely old: "Until the early 1800s, the most common method was the Merchant’s Rule ." 3 ’ But surprisingly, long after the United States Rule was articulated by the United States Supreme Court in 1839,32 the Merchant’s Rule apparently continued to be very popular among merchants and is mentioned as a common approach in textbooks published as recently as 1990 and 2007. DECLINING (RELATIVE) GROWTH RATE The freezing of the time value of simple interest stunts the growth of a simple interest investment. In one sense, the rate of growth due to simple interest is constant: every year, the investment grows by the same dollar amount (assuming, and it will be assumed unless otherwise stated, that there is a single deposit with no additional deposits or withdrawals). Of course, the investor has in fact a larger total investment every year. The constant annual amount of simple interest means that the annual growth rate relative to this ongoing balance of the investment actually declines. Example 2.9. Consider a $100 investment earning simple interest at 100% per annum. The $100 of interest earned in the first year would represent a 100% increase in the value of the investment (from $100 to $200), but the $100 of interest for the second year would represent only a 50% increase (from $200 to $300) and the $100 of interest for the third year would represent only a 33-1/3% increase (from $300 to $400). Seemingly Inflated Simple Interest Rates of Return. One of the consequences of this declining relative growth rate is that simple interest rates of return may seem inflated as the duration of the investment gets longer (and the relative rate of growth continues to decline). This is the flip side of the frequently noted fact 35 that compound interest may result in a higher (and potentially much higher) investment balance than simple interest. For example, if a $100 investment yields $200 in 5 years, the annually compounded annual rate of return is less than 15% but the simple annual rate of return is 20%. To take a more extreme example, if a $100 investment yields $1,100 in 10 years, the annual compounded rate of return is less than 30% but the simple annual rate of return is 100%! CONFLICTING INCENTIVES Simple interest may create incentives that put the parties to the transaction more at odds and that may at times appear strange and even troublesome. Investor Incentive to Accelerate Removal. Because the time value of simple interest is effectively frozen until it is removed from the investment, the investor has an incentive to remove simple interest (whether by withdrawal, receipt of payment or distribution, or otherwise) as soon as possible so it can put the interest to work. The simple interest bank account is frequently used to illustrate this point Assuming no withdrawal restrictions, a depositor would be motivated to withdraw, . 36 quickly and repeatedly, the interest earned on the account and deposit it in another account (or redeposit it in the same account) in order to earn interest on interest: If banks used simple interest, then depositors who withdrew and re-deposited their funds would have higher account values than the depositors who simply left their funds in their accounts. 37 The irrationality of simple interest in this context may seem particularly obvious because the timing of removal is generally within the control of the investor (depositor): [I]t doesn’t make sense to reward depositors for withdrawing and re-depositing 38 their funds Of course, "[b]anks don’t typically use simple interest,"" because, among other matters, of their desire to avoid this odd result and "to encourage long term investment." 40 Nonetheless, simple interest investments do sometimes occur and when they do, simple interest may be even more problematic when the investor does not have control over the timing of its removal from the investment: Simple interest is not practical for long-term transactions, as it penalizes the investor by not permitting the investment of accrued interest [until it is removed from the investment] 41 In the loan context, there is usually an agreed upon schedule for the interest payments and the investor (lender) generally can’t accelerate the timing of removal in the absence of a default (and loan acceleration generally requires acceleration of the entire loan, including principal). Also, in the partnership context, the investor (partner) generally can’t accelerate the timing of distributions (and when it has the bargaining power to control the timing of distributions, it is not likely to accept a simple rate of return). Payor Incentive to Defer Removal. The flip side of the prior point is that the payor (whether the bank, the borrower, the partnership or otherwise) may want to keep the simple interest outstanding as long as possible. Indeed, any right of the payor to elect a payment date within a certain period after it accrues is likely to be a Hobson’s choice (i.e., to make the payment as late as possible). The deferral of a simple interest payment is basically an interest free loan: the payor has the use of the money that it would otherwise use to make the payment and is not charged for that use. To put this oddity in the context of a promote hurdle, consider the following example: Example 2.10. Assume that an investor making a $100 investment were entitled to a 10% annual simple preferred return on, and a return of, its $100 investment (the hurdle) before any other distributions. If $10 of accrued preferred return (and no other) distributions were made to the investor during the first year, then the hurdle at the end of the year would be the same (i.e., $100) regardless of when those distributions were made during the year (assuming no distribution exceeds the then accrued interest). For example, a $10 distribution occurring at the end of the year would lead to the same result as two $5 distributions occurring half-way through the year and at the end of the year. In either case, the hurdle balance at the end of the year would be $100. No discount would be made for an earlier payment, and no charge imposed for a later payment, of the accrued preferred return. Query whether such a simple interest return arrangement encourages manipulation (if not mischief) to the detriment of the investor? For example, if distributions must be made at least quarterly, and there is sufficient money available to distribute the investor’s preferred return during each month of the quarter, would an operating partner be motivated not to distribute, and instead deposit the money in an interest-bearing account to earn interest, until the end of the quarter? In other words, is the inherent conflict between the investor and the operating partner exacerbated (in a partnership with a simple preferred return) by the fact that an interim partnership investment of distributable cash might benefit only the operating partner (assuming the investor could make the same interim investment when it receives its distributions)? SIMPLE INTEREST RETURN HURDLES With all this confusion, one should not be surprised that different approaches to return hurdles may yield very different results in the context of simple interest. Consider, for example, the following three possibilities, which are described in more detail in Appendix 2G and will be considered again (in one or more subsequent installments of this article) in the context of compound interest. Unless otherwise stated, it will be assumed throughout this article that the relevant cash flows in the JV context are the contributions by the investor (the investor’s cash outflows) and the distributions to the investor (the investor’s 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 ofthis article. Amortization Hurdle. One approach which is often utilized is a preferred return (and return of capital) which amortizes the investor’s contributions (and the investor’s return) with the investor’s distributions, where the investor’s distributions are applied first to the investor’s return and then to the recoupment of capital. Basically, the hurdle balance as of any given time would be the same as the then outstanding balance of principal and interest of a typical US mortgage loan where the contributions are treated as the relevant loan advances and the distributions are treated as the relevant loan payments (and payments are applied to interest first and then to principal). For future reference, this approach will be called the "Loan Amortization Hurdle" or simply the "Amortization Hurdle". In the context of simple interest, this is essentially the United States Rule mentioned earlier. See Appendix 2G for the formula for this hurdle in the case of a constant simple interest rate. Example 2.11. Assume that (1) there were a 25% simple annual interest rate, (2) there were a $100 contribution at the beginning of the first year, and a $25 distribution at the beginning of the second year (and no other contributions or distributions), and (3) the parties were trying to determine the hurdle amount as of the beginning of the third year. Using a Loan Amortization Hurdle, the $25 distribution would effectively pay all the return accrued during the first year, and the remaining balance of $100 would earn another $25 during the second year, so that the balance (the hurdle amount) at the end of the second year would be $125. Present Value Hurdle. Another approach (which is very common in the context of compound 42: the hurdle balance as of returns) is basically a generalized internal rate of return (IRR) approach any given time would be the hypothetical distribution amount as of such time which would equalize the present value as of the inception of the transaction of the investor’s distributions with the present value as of the inception of the transaction of the investor’s contributions (or equivalently, which would make the net present value of all the cash flows, where contributions and distributions have 10 different signs, equal to zero). This approach will be called the "Present Value Hurdle", Although the author is not aware of any IRRs that are calculated using simple interest rates, for purposes of later comparison to compound IRR hurdles, it may be illustrative to consider this variant of the IRR as a means of calculating simple interest hurdles. See Appendix 2G for the formula for this hurdle in the case of a constant simple interest rate. Example 2.12. Assuming the same facts as in Example 2.11, if B represents the hurdle balance (i.e., the hurdle amount) as of the beginning of the third year, then using a Present Value Hurdle, the present value (as of the beginning of the first year) of the cash flows and B should equal 0: -($100) + ($25)I(l .25) + B/(1.5) = 0, which means that B = $150 -$30 = $120. Future Value Hurdle. Yet another alternative is similar to the Present Value Hurdle, but using future values instead of present values: the hurdle balance as of the time in question would be the hypothetical distribution amount as of such time which 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, which would make the net future value of all the cash flows, where contributions and distributions have different signs, equal to zero). In other words, the hurdle balance as of any given time would be the amount by which the future value as of the given time of the contributions exceeds the future value as of the given time of the distributions. This approach will be called the "Future Value Hurdle". In the context of simple interest, this is essentially the Merchant’s Rule mentioned earlier. See Appendix 2G for the formula for this hurdle in the case of a constant simple interest rate. Example 2.13. Assuming the same facts as in Example 2.11, if B represents the hurdle balance (i.e., the hurdle amount) as of the beginning of the third year, then using a Future Value Hurdle, the ftiture values (as of the beginning of the third year) of the cash flows and B should equal 0: -($100)(1.5) + ($25)(1.25) + B = 0, which means that B = $150 -$31.25 = $118.75. The hurdle balances in Examples 2.11, 2.12 and 2.13 illustrate how varied these three hurdle alternatives can be in the context of simple interest: Amortization Hurdle (United States Rule) $125 Present Value Hurdle (Simple IRR) $120 Future Value Hurdle (Merchant’s Rule) $118.75 Concluding Observations regarding Hurdles. Although the Amortization Hurdle is familiar, setting forth an algorithm for the amortization calculation can get messy (e.g., principal and interest must be accounted for separately and after each payment, the principal balance is reduced by the amount by which the payment exceeds the then accrued interest). The Present Value Hurdle can also be cumbersome because of the discounting. Of the three approaches above, the Future Value Hurdle is generally the easiest to calculate by hand and the calculation is relatively easy to state: the hurdle balance is the amount by which (1) the future value of the contributions exceeds (2) the future value of the distributions. In the context of simple interest, it involves nothing more than multiplication, addition and subtraction. Thus, it is no wonder that the Merchant’s Rule was so popular among merchants for so many years. As will be seen in a subsequent installment of this article, when dealing with a continuously compounding interest rate, this approach (the Future Value Hurdle) 11 yields the same exact results as the Present Value Hurdle (and in most cases, the Amortization Hurdle), STATUS OF SIMPLE INTEREST TODAY The criticism of simple interest is nothing new. According to a 19th century textbook 41: . [T]he assumption of simple interest leads continually to reductio ad absurdum, which is sufficient evidence that a fallacy somewhere lurks in the supposition. Money, whether received under the name of principal or of interest, can always be invested to bear interest, and therefore, from the very nature of the case, simple interest is impossible. It is true that borrower and lender may between themselves agree for only simple interest; but such agreement does not prevent the borrower from investing the interest which is thereby allowed to remain in his hands, and securing interest thereon; and it is because this interest on interest is ignored in the doctrine of simple interest, that the mathematical formulas fail. Yet simple interest continues to be a fact of life. Indeed, it is generally presumed to apply by 44 Moreover, simple law in the United States unless the parties expressly provide for compounding. interest is often used "[w]hen calculating interest accumulation over a fraction of a year or when executing short term financing transactions . . . "[T]here are [also] an array of conventions [for simple interest] for bills, deposits, bonds, etc. within the financial markets" 46 that are likely to keep simple interest around for years to come, Finally, simple interest returns occur in some partnerships even for multi-year investments .47 LOOKING AHEAD Simple interest has to a great extent been replaced by compound interest in commercial transactions. Indeed, contrary to the apparent rule of law, compound interest is often assumed unless one indicates otherwise. 48 In the next part of this article, compound interest will be examined to determine whether and how it may solve the problems posed by simple interest. * * * APPENDICES: 2A - Stationary Accumulation 2B - Merchant’s Rule vs. United States Rule 2C - Formula for Outstanding Balance (Merchant’s Rule) 2D - Formula for Outstanding Balance (United States Rule) 2E - Proof that Merchant’s Rule Effectively Applies Payments to Principal 2F - United States Rule vs. Merchant’s Rule: The Difference 2G - Simple Interest Hurdle Balances 12 APPENDIX 2A STATIONARY ACCUMULATION This Appendix will explain the notion of stationary accumulation after providing some general background on accumulation functions. ACCUMULATION FUNCTIONS Basically, an accumulation (or future value) function (or factor) reflects the growth of a unit investment assuming the unit investment remains fully invested (without the withdrawal of any principal or interest and without the addition of any new principal), Generally, proportionality is also assumed (i.e., the future value of an investment which is more or less than a unit is assumed to be proportionate to the future value of a unit investment) so that it is possible to use the accumulation function as a factor. 49 ACCUMULATION FUNCTIONS OF TWO VARIABLES As a general rule, "the accumulation of money over time depends not only on the length of the time interval but also on where in time the interval lies, ,50 Thus, a general accumulation function (or factor) may be defined as a function of two variables, t1 and t2, each expressed as a number of the relevant time units (whether years, months, days or otherwise), where ti represents the starting time and 12 represents the ending time of the relevant time interval: t2, A(t,t) means the accumulated (or future) value as of time t2 of an For 0 investment of 1 made as of time t1, A(t, t) is assumed to be I for all t (so that if the interval starts and ends at the same time and therefore has 0 duration, then the investment stays the same); and the future value as of time 12 of an investment in the amount of C made at time tj is assumed (based on proportionality) to be CA(t 1 ,t2 ). 51 Example 2A-2. The general accumulation function for a simple interest rate r may be written as follows: A(11 ,t2) = 1 + (t2 - ti)r. If, for example, r = 100% per annum and ti and t2 are each a number of years, then an investment of 1 made at time 0 would grow as follows as of the end of each of the first three years: A(0,I)= I +(1 -0)1 =2; A(0,2)= 1 +(2-0)1 =3; and A(0,3)= 1 +(3-0)1 =4. ACCUMULATION FUNCTIONS OF ONE VARIABLE The general accumulation function defined above is a function of two variables. Yet the simple interest time value function introduced in the body of this installment is a function of one 13 variable, t, without any reference to the particular time interval involved (as if everything starts at one time 0). Indeed, introductory North American textbooks on the mathematics of finance often introduce accumulation functions or factors as functions of only one variable, without any reference to the particular time interval involved (apparently assuming that the time interval begins at time 0)52 This is a special case of the general accumulation function defined above, A(t i ,t2), where one of the two variables, namely the beginning of the time interval, is fixed at zero 53 : a(t) = A(O,t). Without further assumption, this accumulation function doesn’t indicate how to determine the future value of an amount over any time interval other than a new investment amount made at time 0 measured over a time interval beginning at time 0: The definition given is somewhat ambiguous. . . [because] accumulation can depend on both duration and time of entry, 54 So, given a one variable accumulation function a(t), how does one determine A(t i ,t2) where 0? Theoretically, there are many possibilities. For example, A(t i ,12) could be of the form t1> a(F[t i ,12]), such as a(t2 - ti), or it could be of the form F(a[t i ],a[t2]), such as a(t2)/a(ti), where, in either case, F is some function of two variables (e.g., subtraction, addition, multiplication or division). There are also numerous possibilities for a(t), but this Appendix will consider only certain limited cases that are related to simple interest, in particular, the only single variable accumulation functions discussed in this Appendix will be linear (i.e., of the form a(t) = mt + b). And because a(0) = 1 for any single variable accumulation function, 55 it follows that a(t) must be of the form a(t) = 1 + mt. This function, of course, looks the same as the future value simple interest factor indicated in the body of this installment (where the slope m is the simple interest rate r). But in theory, even if one knows that a(t) is linear, further information or assumptions (e.g., a definition of the two variable function F) are required to determine A(t 1 ,t2). STATIONARY ACCUMULATION In the case of simple interest, it is not necessary to know the time of entry to determine the accumulation of interest. Regardless of when a deposit is made, one expects it to earn simple interest in the same manner, namely by a factor of a(t) = 1 + rt, where t is the duration of the investment (expressed as a number of the relevant time units) and r is the simple interest rate (for the relevant time unit). In other words, a simple interest investment amount grows in the same manner over any time interval of equal length regardless of when it begins: A(t1,t2) = a(t2 - t) This is sometimes called "stationary" or "translation invariant" accumulation because it allows one to ignore the actual start date, fix it at a single time 0 and focus merely on the duration of As explained by one author: the relevant time interval . 56 This simply means that money entering the fund at any time accumulates in exactly the same way as if it entered at time 0. Under a stationary accumulation function, all new investments grow like new investments over an interval of equal duration commencing as of time 0. 14 The following graph illustrates portions of a stationary accumulation function, A(ti,t2) = a(12 - ti), assuming a(t) is linear and r = 100% per annum. GRAPH 2A-1 Thus, for 100% per annum simple interest, A.(ti,12) = (t2 - ti) + 1 and a(t) = t + 1, and the graph of A8(to,t) for any particular to would be a line parallel to a(t) = t + 1 that begins at (to, 1). *** 15 APPENDIX 211 MERCHANT’S RULE VS. UNITED STATES RULE This Appendix will explain and contrast the Merchant’s Rule and the United States Rule in the context of a simple interest loan, APPLICATION OF PARTIAL PAYMENTS Many textbooks refer to two alternatives for applying interim partial payments to a simple interest loan: the Merchant’s Rule; and the United States Rule. 58 THE MERCHANT’S RULE Some readers may not be familiar with the Merchant’s Rule. A book published in 2007 notes that the United States Rule is "most commonly used in practice,"" but the same book also describes the ,60 Merchant’s Rule as one of "two common ways to allow interest credit on short-term transactions. Under this rule, interim loan payments are not applied to reduce the debt until the debt is fully paid, but at that time, they are credited together with simple interest . 6 ’ Basically, this rule keeps separate track of the principal advanced (and interest that accrues on that principal), on the one hand, and the payments received (and interest on these payments at the same rate), on the other hand: Under [the] Merchants’ Rule the entire debt earns interest to the final settlement date. Each partial payment also earns interest from the time it is made to the final settlement date. The balance due on the final date is simply the difference between the amount of the debt and the sum of the amounts of the partial payments. 62 The Merchant’s Rule may be viewed as a calculation similar to a simple interest IRR hurdle calculation except that the time values are calculated as of the end, instead of the beginning, of the investment 63 : if a rate for a series of cash flows were defined to be the simple annual interest rate that makes the net future value of the cash flows equal zero, then the Merchant’s Rule would be equivalent to requiring payments until the creditor achieved that rate. A formula for the outstanding balance of principal and interest for a simple interest loan using the Merchant’s Rule is set forth on Appendix 2C. THE UNITED STATES RULE The United States Rule applies interim loan payments first to interest and then to principal consistent with typical loan amortization (but with no negative amortization): In [the] United States Rule the interest on the outstanding principal is computed each time a partial payment is made. If the payment is greater than the interest, the difference is used to reduce the principal. If the payment is less than the interest, it is held without interest until another partial payment is made, The two payments are 16 then added, If they exceed the interest at that time, the difference is used to reduce the principal .64 As articulated by the United States Supreme Court 65 : The correct rule in general is, that the creditor shall calculate interest whenever a payment is made. To this interest the payment is first to be applied, and if it exceed the interest due, the balance is to be applied to diminish the principal. If the payment fall short of the interest, the balance of interest is not to be added to the principal so as to produce interest. A formula for the outstanding balance of principal and interest for a simple interest loan (single advance) using the United States Rule is set forth on Appendix 2D. COMPARISON The United States Rule generally leads to a larger result for the lender. For example: Example 213-1. If there were a $100 loan with 20% per annum simple interest and there were a $10 payment halfway through the year (when the total outstanding balance of principal and interest was $110), then the balance at the end of the year would be $110 (110% of[$1 10 - $10]) under the United States Rule, but only $109 ([120% of 100] - [110% of 10]) under the Merchant’s Rule. The difference between the rules may not be immediately apparent from the formulas described on Appendices 2C and 2D. The following discussion will attempt to make the difference more understandable. APPLICATION TO PRINCIPAL BEFORE INTEREST As suggested by one author, 66 it may be easier to compare these rules when the Merchant’s Rule is 17: recast by what is sometimes called the balance method According to the balance method, a partial payment is applied first to the discharge of the principal. Interest is calculated on the beginning principal and on each successive principal for the elapsed time to the date of the next partial payment or date of settlement. The sum of the several interests and the last principal is the balance due at the date of settlement. Thus, under the Merchant’s Rule, interim payments are effectively applied to principal (immediately) before interest: the dollar differential in Example 2A-1 represents the loss of $1 of accrued interest for half a year on $10, effectively resulting from the application of the $10 payment to interest-bearing amounts (principal) under the Merchant’s Rule instead of non-interest bearing amounts (simple interest) under the United States Rule. And once the principal has been fully repaid, the lender is left with a non-interest bearing debt (namely, the accrued and unpaid simple interest), and to the extent the payments exceed the principal amount of the debt, they accrue interest as a further credit against the debt. This result is perhaps best illustrated with a longer term loan (which admittedly may be atypical for applications of the Merchant’s Rule): 17 Example 213-2. If there were a $100 loan with 20% per annum simple interest and there were a $108 payment halfway through the year (when the total outstanding balance of principal and interest was $110), then after 6.75 years, the $8 excess would effectively satisfy the $10 of interest accrued during the first half-year (with the odd result that $108 of payments would have satisfied a $ilo debt). To verify this, observe that: $100 x 120% p.a. x 6.75 years = $810 = $108 x 120% pa. x 6,25 years. Appendix 2E sets forth a proof that the Merchant’s Rule yields the same balance as one would get if all interim payments were applied to principal. AN ALTERNATIVE PERSPECTIVE OF THE UNITED STATES RULE The previous explanation of the difference between the two rules was revealed by recasting the Merchant’s Rule in the format of the United States Rule (where the partial payments are immediately applied to the debt in some way). One may also get some insight into the difference between these rules by recasting the United States Rule in the format of the Merchant’s Rule, where it is assumed that the original loan amount remains fully invested at the original simple interest rate and no payments are immediately applied to the loan, but instead are allowed to grow in some manner until final payment. Viewed in this light, the United States Rule for a simple interest loan is the same as the Merchant’s Rule except that a portion of the interim payments is not allowed to grow (i.e., not all of the interim payments are credited with interest at the stated rate): interest is credited only to principal payments (i.e., the portion of each payment that would be applied to reduce principal under the United States Rule); the interest payments (i.e., the portion of each payment that would be applied to interest under the United States Rule) do not accrue interest. Example 2B-3: Consider a $100 loan with a 20% simple annual interest rate and one $10 payment made halfway through the year, as in Example 213- 1. Under the United States Rule, the balance at the end of the year would be $110 (because the $10 payment would be applied against interest and then there would be an additional $10 of interest for the second half of the year). This is the same result one gets if the $100 were allowed to grow at 20% per annum until the end of the year to $120 and then there were a credit against that amount for all principal payments plus interest (i.e., $0 because there were no principal payments) plus all interest payments without interest (i.e., $10): $120- $10 = $110. In other words, the outstanding balance of principal and interest as of a particular time may be determined under the two rules as follows: Merchant’s Rule: the amount by which (1) the future value as of such time of the loan amount exceeds (2) the future value as of such time of all payments. United States Rule: the amount by which (1) the future value as of such time of the loan amount exceeds (2) the sum of (a) the future value as of such time of all principal payments, and (b) all interest payments. It should be apparent from the formulation in Appendix 2D that the United States Rule may be viewed in this light. Thus, the difference between the rules (when the United States Rule is recast in 18 the format of the Merchant’s Rule) is that, under the United States Rule, only the principal payments grow with interest. Appendix 2F sets forth a proof that the balance under the United States Rule exceeds the balance under the Merchant’s Rule by an amount equal to the interest that would have accrued on each interest payment from the date it was made. *** 19 APPENDIX 2C FORMULA FOR OUTSTANDING BALANCE (MERCHANT’S RULE) This Appendix sets forth a formula, when using the Merchant’s Rule, for the outstanding balance of principal and interest of a simple interest loan as of a particular time (the "Test Date"), where there are n cash flows (payments or advances) before the Test Date. NOTATION/ASSUMPTIONS Let CF O, CF 1 , ... CF1 be a series of n cash flows. By using zero cash inflows or cash outflows as necessary, each CFk may (and sometimes will) be broken down into the difference between a cash inflow (a payment to the lender or other investor) and a cash outflow (an advance by the lender or other investor) at the time of such cash flow, as follows: CFk = LPk - LAk or, equivalently, - CFk = LAk - LPk, where each LAk and each LPk is a non-negative amount. The notation has been chosen to make it more descriptive in a loan context in which event each outflow would be a loan advance, LA, and each inflow would be a loan payment, LP. Also, assume that there is the same amount of time between each two consecutive cash flows where the Test Date (i.e., the date the balance is to be calculated) is treated as the final (n + 1) cash flow date; 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 dates and then adding zero cash flows so that there is a cash flow occurring at the beginning and end of each such period.) Thus, CFk is the (/c + 1) cash flow occurring as of /c time units, and is either negative (if there is net advance), positive (if there is a net payment) or zero. For example, if there is a single principal advance of$ 100, at the beginning of the transaction, followed by payments only, the first cash flow would be negative (CF O = - $100 and CFO = LAO = $100) and the remaining cash flows would be positive. Now, let r equal the positive, constant, simple periodic rate with respect to the uniform time period (i.e., the time unit). FORMULA Under the Merchant’s Rule, the formula for the outstanding balance (of principal and interest) as of a particular time (i.e., as of the Test Date, which is as of n time units after the initial advance) is relatively easy to state. It is basically the amount by which (1) the future value of the loan advances as of such time exceeds (2) the future value of the loan payments as of such time. Using the notation above, this balance may be stated as follows: - [LA 0 (1+nr) +LA 1 (1+[n-1]r)+...+LA 1 (1+r)] = (LA O - LP0 Xi + nr) + [LP0 (1+nr) + LP, (1+[n-1]r)+...+LPi(1+r)] - LP1 Xi + n - lDr +... + [ - LP 1 Xi + Now, remember that LAk - LPk = - CFk, so the formula may be written as follows: Outstanding Balance (Merchant’s Rule) - CFk(i+[n-k])r *** 20 APPENDIX 2D FORMULA FOR OUTSTANDING BALANCE (UNITED STATES RULE) This Appendix sets forth a formula, when using the United States Rule, for the outstanding balance of principal and interest of a simple interest loan as of the Test Date, but before any payment on the Test Date, where there are n cash flows before the Test Date. NOTATION/ASSUMPTIONS This Appendix will use the same notation and make the same assumptions set forth on Appendix 2C. Also, let CFk = Pk + ’k where Pk is the portion of CFk that would be applied to reduce principal under the United States Rule and Ik is the portion of CFk that would be applied to interest under the United States Rule. For example, if CFk were a loan advance, then CFk = Pk would be negative and ’k = 0 (and there would be an increase in the principal balance and no change in the amount of accrued and unpaid interest). FORMULA Under the United States Rule, principal and interest are tracked separately and consequently the outstanding balance formula is more cumbersome than under the Merchant’s Rule. Before stating the general formula, the discussion below will first consider simpler cases where there are only a few cash flows. After one period (but before the second cash flow), the outstanding principal balance is - P and the outstanding interest is (- Po)r, which means a total balance of 0 (-Po)(l +r) Immediately after the second cash flow, the outstanding principal balance is P outstanding interest is (- Po)r I. 2. 0 - P 1 and the After two periods (but before the third cash flow), there is additional interest of(- P 0 - P1)r so that the outstanding principal balance is still - Po - P i and the outstanding interest is (- Po - P i )r + (- Po)r - 11 = (- Po )2r - (Pi)r - I, which means a total balance of (- P0)(1 + 2r) - (P1)(1 + r) - Ii Immediately after the third cash flow, the new outstanding principal balance is -Po - Pi - P2 and the new outstanding interest balance is (- Po)2r - (Pi)r - Ii - I. 3. After three periods (but before the fourth cash flow), there is additional interest in the amount of(- Po - Pi - P2)r so that the outstanding principal balance is still - Po - P 1 - P2 and the outstanding interest balance is (- Po)3r - (Pi)2r - (P2)r - Ii - 12 , which means a total balance of (- P 0 )(1 + 3r) - (P1 )(l + 2r) - (P2)(1 + r) - 11 - 12 4. After n periods (but before the (n + 1) cash flow), it is easy to show (using inductive reasoning) that the balances are: 21 - Po - P1 - P2 - .. P,1 for principal and (- Po)nr - (Pi)(n - 1)r -. (Pi)r - Ii - 12 - ... - Li for interest, so that the total balance may be written as follows: Outstanding Balance (United States R ule)t - j(i+[n-k]r) ’k Observations: Principal Component. The outstanding principal balance during the kth time unit (after the kth cash flow occurring as of the k I time units and before the [k + 11 cash flow occurring as of k time units) may be stated as follows: In particular, the outstanding principal balance as of the Test Date may be stated as follows: 2. Pj = - ’k Interest Component. The interest that accrues during the kth time unit may be stated as follows: [ -I Pi Jr Thus, the total amount of interest that accrues through the Test Date may be stated as follows: [Pi ]r This accrued interest is reduced by the interest payments so that the total amount of accrued and unpaid interest as of the Test Date may be stated as follows: n-1 Pi]r [- 3. Restatement of Entire Balance. Thus, the outstanding balance under the United States Rule may be stated as the sum of its principal and interest components as follows: P1 ] r The formulation in the box above is given to make it easier to compare to the formula for the outstanding balance under the Merchant’s Rule. (It is relatively straightforward to derive either of the two formulas from the other.) *** 22 APPENDIX 2E PROOF THAT MERCHANT’S RULE EFFECTIVELY APPLIES PAYMENTS TO PRINCIPAL This Appendix sets forth a proof that under the Merchant’s Rule the outstanding balance of a simple interest loan on the Test Date, but before any payment on the Test Date, is the same as what the outstanding balance would be if all interim payments were applied to principal (assuming that at all times, the sum of all loan payments then or previously made does not exceed the sum of principal advances then or previously made 68). Proof Using the notation from Appendix 2C, the outstanding balance under the Merchant’s Rule would be as follows: see Appendix 2C -CFk (1+[n-k]r) = -CFk - CFk (n - k)r (t) k (n-k)]r CFk ]+[CF = (t) n-I CF = [CFk ]+[iCFk ]r k=O (n - k) k=O k=O j=O n-I k k=O j=O CF = (if) (1) = [CFk ]+[[CFJ ]r] which is the outstanding balance of a loan with a simple interest rate r, where all the payments are applied to reduce principal. (if t) Q.E.D. (t) rearranging terms (by one or more of the distributive law of multiplication over addition and the associative laws of addition and multiplication). Note that even (11) below may be explained on this basis. (if) Y CFk (n - k) = CF, (n - i) + CF 2 (n - 2) + ... + CF = CF, + ( CF, = 1 + CF2) +,,,+ 01 +"’+ CFI) I CF, k=O j=O (if t) The first part of the last sum in the proof above would be the outstanding principal and the second part would be the accrued interest (assuming that all payments are applied to reduce principal). To see this, note that if all payments are applied to reduce principal, then each loan payment is a principal payment. Thus, each cash flow is either a principal advance or a principal payment, and the outstanding principal balance is determined by subtracting each cash flow from the negative sum of all prior cash flows. Therefore, as of kth time unit (before the [k + 11 cash flow), the outstanding principal balance may be stated as follows: -CF. 23 APPENDIX 2F UNITED STATES RULE vs. MERCHANT’S RULE: THE DIFFERENCE This Appendix will explain the difference between the outstanding balances of principal and interest as of the Test Date, but before any payment on the Test Date, under the Merchant’s Rule and the United States Rule: the difference is that the United States Rule yields a balance that is larger than the balance under the Merchant’s Rule by an amount equal to interest on the interest payments (under the United States Rule) from the time made until the Test Date. Proof Using the notation from Appendices 2C and 2D: Merchant’s Rule balance= - CFk (i +[n -k] r) This may be rewritten as follows (since CF. = k + 1 k) = -Pk(1 + [n-k]r)-Ik (1 + [n-k]r) = [-Pk(1 + [n-k}r) 1 k] -1 k[n = [United States Rule balance] - - k]r 1k [n -k]r Therefore, the final balance under the United States Rule exceeds the final balance under the Merchant’s Rule by the following amount: which is interest on the interest payments ,69 24 APPENDIX 2G SIMPLE INTEREST HURDLE BALANCES This Appendix will discuss in more detail, in the context of a simple interest rate, the three alternative approaches to calculating hurdle balances mentioned in the body of this article: (A) the Amortization Hurdle, which, in the context of simple interest, is essentially the United States Rule; (B) the Future Value Hurdle, which, in the context of simple interest, is essentially the Merchant’s Rule; and (C) the Present Value Hurdle, which, in the context of simple interest, is essentially an IRR approach based on simple interest present values. This Appendix will use the assumptions/notation in Appendices 2C and 2D and will further assume that each of the hurdle approaches is based on a simple interest rate. HURDLE BALANCE FORMULAS Amortization Hurdle (United States Rule). As shown in Appendix 2C, the hurdle balance A. under the United States Rule may be stated as follows: B= - Pk(1 +[n - k]r) ’k As shown in Appendix 21), under the Future Value Hurdle (Merchant’s Rule). B. Merchant’s Rule, the outstanding balance after n periods (but before the [n + 1]st cash flow) would be: = CF(1 + [n -k]r) This will be referred to as formulation B(*) + [n-k]r) -I k (1 + [n-k]r) Present Value Hurdle (Simple IRR). Using the (simple interest) IRR approach, the balance C. as of any cash flow date (before the cash flow) would be a hypothetical payment that would achieve the requisite (simple) IRR rate. Thus, in the calculations below, the hurdle balance B after n periods is equal to a hypothetical final payment, CF, that would achieve the requisite (simple) IRR rate. B) would After n periods (but before the [n + 1] cash flow) if a hypothetical final payment CF, then the (simple) IRR formula dictates that: IRR, achieve the requisite (simple) B- CF, CFO CF2 (1+Or) (i + ir) - (I+ 2r) ( n-I CFk k=0 (i B (i + [n-l]r) - (i + nr) (i+nr) + kr)J n-1 = - CFk CF,., [ 1 + nr This will be referred to as formulation C(*). 25 = (- n_’ k=O Pk k (i + nr) + kr)J SOME COMPARISONS If there has been only one cash flow, namely the initial advance CF0, then all three of these approaches yield the same hurdle balance after one period: - CF 0(1 + r). But as soon as there is another cash flow, they may differ considerably. Amortization Hurdle vs. Future Value Hurdle. As shown on Appendix 2F, and as one can see from the balances at the end of paragraphs A and B above, for a constant simple interest rate, the balance using the Future Value Hurdle (Merchant’s Rule) equals the Amortization Hurdle (United States Rule) balance reduced by interest on the interest payments: Ik(n -k)r Thus, the Future Value Hurdle (Merchant’s Rule) yields a balance that is always less than or equal to the balance using the Amortization Hurdle (United States Rule). For example, in the simple case where there is one or more advances but no payments prior to the date the hurdle balance is to be determined, the Future Value Hurdle (Merchant’s Rule) and the Amortization Hurdle (United States Rule) yield the same hurdle balance (because there are no interest payments). Future Value Hurdle vs. Present Value Hurdle. For a constant simple interest rate, r, the Future Value Hurdle (Merchant’s Rule) balance is less than or equal to the Present Value Hurdle (simple IRR) balance in the simple case where there is only one principal advance, which occurs at the beginning of the transaction, so that no CFk is negative for k> 0. This is easily seen by comparing formulations B(*) and C(*) above: Present Value Hurdle Balance Future Value Hurdle Balance CFI -CFk (1+[n-k]r) + nr 1+kr which may be rewritten as follows (because the first term of each sum, where k = 0, is the same): n-I k=1 cF( 1 1+kr) CFk (1+[n-k]r) k=I This inequality holds (assuming each CFk? 0 for k> 0) because the co-efficient of CFk on the left is less than or equal to the co-efficient of CF k on the right, for each k, where 0 <k < n, as shown below: (1+[n-kjr) (1+kr)(1+[n-k]r) k(n-k)r 2 +nr+1 k(n-k)r 2 (1 +nr)/(1 +kr) (1 +nr) 1 + nr 0 which is true because, by assumption, 0 <k < n. Q.E.D. 26 if if if if On the other hand, an almost identical proof shows that the Present Value Hurdle balance is less than or equal to the Future Value Hurdle balance in the simple case where all cash flows are principal advances (i.e., there are no payments) so that no CFk is positive for k> 0. When the cash flows after time 0 have different signs, the results are mixed and it is easy to construct examples where the Present Value Hurdle balance is less than, equal or greater than the Future Value Hurdle balance. To illustrate, consider the annual cash flows (100), 60, (90) and a 100% annual simple interest rate and note that the balance as of the end of three years is $400 under both the Present Value Hurdle ([$100]/[l +0] + [$90]/[ l +2] = [$60]/[l + 1] + [$400]/[l + 3]) and the Future Value Hurdle ([$100][1 + 3] [$60][1 + 2] + [$90][1 + 1] = $400). However, if the second cash flow is reduced, then the Present Value Hurdle balance will be less than the Future Value Hurdle balance, and if the second cash flow is increased, then the Present Value Hurdle balance will be more than the Future Value Hurdle balance. For example: - if the second cash flow is reduced from 60 to 40, then the Present Value Hurdle balance becomes $440 ([$100]/[1 + 0] + [$901I[1 +2] = [$40]I[1 + 1] + [$440]/[1 + 3]), which is less than Future Value Hurdle balance ([$100][1 + 3] [$40][1 + 2] + [$90][1 + 1] = $460); and - if the second cash flow is increased from 60 to 80, then the Present Value Hurdle balance becomes $360 ([$100]I[1 + 0] + [$90]I[1 + 2] = [$80]I[1 + 1] + [$360]/[1 + 3]), which is more than Future Value Hurdle ([$100][1 + 3] [$80][1 + 2] + [$90][1 + 1] = $340). - Amortization Hurdle vs. Present Value Hurdle. Observe that (for a constant, simple interest rate) the balance of the Present Value Hurdle balance is less than or equal to the Amortization Hurdle balance in each of the following three circumstances: First, when all the cash flows are advances so that CFk < 0 for all k. To see this, recall (1) that under these facts, the Present Value Hurdle balance is less than or equal to the Future Value Hurdle balance. Also, recall that the Future Value Hurdle balance is always less than or equal to the Amortization balance, The result immediately follows. Second, when all the cash flows after the initial advance are interest payments. Under (2) these facts, the principal balance never changes. It is always equal to the initial advance, LAO. After L. The Present Value n periods, the Amortization Hurdle balance is simply LA 0(1 + nr) Ii Hurdle balance, on the other hand, may be written as follows: - (1 +nr I 1+r) = LA O (1 + nr) -Ii I LAO (I + nr) = - ... ( - - In-1 I ... - i+nr I [n -1]r) + because 1+nr> 1 for 0 :5 k !~ n -J,, 1+kr ) Amortization Hurdle balance Third, when there are only two cash flows, CFO and CF 1 , the balance is to be (3) determined as of the end of two time units, and if the second cash flow is a payment to the investor 27 (a cash inflow), it does not exceed the hurdle balance at the time 70 (where r is the constant simple interest rate): Present Value Hurdle < < -P0(1 +2r)-(P I +I)(1 +2r)/(1 +r) < - (P i +11 )(1+2r)/(1+r) < (1+r)P 1 +1 1 < (I +r)2 P 1 +J(l +r) < (r2 + 2r+ l)P 1 + I(1 +r) < r2P rP 1 < Amortization Hurdle if -P 0 (1 +2r)-(1 +r)P 1 - 11 -(1+r)Pi-Ii (P I +I)(l+2r)/(1+r) P 1 (1 +2r)+1 1 (l +2r) P 1 (l +2r)+ I(l +2r) r11 jff if if if if jff II If P, 0, then the relationship above obviously holds. If P1 > 0, then ii must equal all the accrued interest for the first period (because there cannot be a principal payment under the United States Rule until all accrued interest is paid), namely, (- Po)r, so the inequality above may be rewritten as follows: rP1 < r(- P o) P 1 < - Po iff which is true by assumption. AM The interested reader is encouraged to consider whether and under what circumstances this relationship (i.e., Present Value Hurdle Balance Amortization Hurdle Balance) may hold more generally. *** M. ENDNOTES A number of real estate owner/operators offer their investors a simple (or noncompounded) return. See e.g., http://www.blackburne.comlplymouth_laporte_trail.Iitml ("...Blackburne & Brown will earn 30% of the appreciation of the property, but only after the investors get a cumulative, non-compounded preferred return of 8% annually..."), http://www.slioyoff.conilsptlreit_es.php ("The REIT manager will not receive an incentive fee until shareholders have received a 10% annual, cumulative, non-compounded preferred return on invested capital"); https://tiholdings.comlinvestor/Pages/English/aboutphp (" ...The Preferred Return is expressed as an annual noncompounded rate of return......). See also, Gallinelli, INSIDER SECRETS TO FINANCING YOUR REAL ESTATE INVESTMENTS (McGraw-Hill 2005) at 194. ("Ina limited partnership, a limited partner maybe entitled to an annual, noncompounded return on his or her investment before the general partner receives any return"); Ask Frank: Real Estate Partnerships and Preferred Return (Real Data 2009) http://www.realdata.comlls/afreturn.shtml where the same author (Gallinelli) responds to the question "Can you explain more about how preferred return [sic] works in a real estate partnership?" (",,.The return may also be compounded or non-compounded. In other words, if part or all of the amount due in a given year can’t be paid and has to be carried forward, the amount brought forward may or may not earn an additional return (similar to compound vs. simple interest). The usual method is for it to be non-compounded. Hence the unpaid amount carried forward does not earn an additional return, but remains a static amount until paid."); Quatman and Dhar, THE ARCHITECT’S GUIDE TO DESIGN-BUILD SERVICES (John Wiley and Sons, 2003) at 237 (indicating that a "preferred return is calculated as simple or compounded interest"). 2 See, e.g., 44B Am Jur 2d Interest and Usury § 54 ("The common law has long favored simple interest and (1St Cir. 2000) ("In Massachusetts, disfavored compound interest"); Berman vs. BC Associates 219 F3d 48, 50 compound interest is generally disfavored. . . . [I]nterest is simple. . . [except] in certain proceedings inequity or by express statutory or contractual authority"); Nation v. WDE Electric Co, 563 NW2d 233, 235 (Mich. 1997) ("The common law has long favored simple interest .....); Reaver vs. RubI off-Sterling, LP 708 NE2d 559, 562 (Ill. 1999) ("Compound interest is disfavored under Illinois law"). See, e.g., Cal. Civ. Code § 1916-2 ( " _in the computation of interest upon any ... agreement, interest shall not be compounded.. .unless an agreement to that effect is clearly expressed in writing..."). ’ 6 15 CORBIN ON CONTRACTS (Rev. ed., Matthew Bender 2003) § 87.13 at 559. (14th ed., McGraw Hill 2011), Ch. 3 (no See, e.g., Brueggeman Fisher, REAL ESTATE FINANCE AND INVESTMENTS discussion of simple interest); Geltner Miller Clayton Eichholtz, COMMERCIAL REAL ESTATE ANALYSIS AND INVESTMENTS (2 ed., Cengae Learning 2007), § 8.1,3 (definition by example); Brealey Myers Allen, PRINCIPLES 8th edition, OF CORPORATE FINANCE (10 ed., McGraw Hill 2011), § 3.3 (defined partially by example in § 33 of 10th edition). but not discussed in (8th ed., McGraw Hill 2008), § 5.1 at See, e.g., Ross Westerfield Jordan, FUNDAMENTALS OF CORPORATE FINANCE 123; Porter, MATHEMATICS OF INVESTMENT (Prentice-Hall 1949), Ch. I at 2; Smith, THE MATHEMATICS OF FINANCE (Appleton 1951), § 23 at 25. Porter, supra, at 2; Smith, supra, § 23 at 25, 8 Any time unit may be used, but annual time units are the most common. Zima Brown Kopp, MATHEMATICS OF FINANCE (6 th ed., McGraw Hill 2007), § 1.1 at 2; Day, MASTERING FINANCIALMATHEMATICS IN MICROSOFTEXCEL ,d (Prentice Hall, 2005), Ch. 2 at 15; Kelhison, THE THEORY OF INTEREST (3 ed., McGraw Hill 2009), § 1.4, Ex. 1. 3, formula (1.7) at 8. This article will sometimes use a 100% interest rate to keep the computations simple and make them easier to illustrate in graphs (recognizing that while smaller rates are more typical, they are often hard to see using the graphing function in EXCEL). Ruckman and Francis, FINANCIAL MATHEMATICS (2nd ed., BPP 2005), § 1.2 at 4; Simpson Pirenian Crenshaw Rifler, MATHEMATICS OF FINANCE (4th ed., Prentice-Hall 1969), § 29 at 57; Williams, THEMATHEMATICALTHEORY td OF FINANCE (Macmillan 1935), § 4 at 3; Hummel and Seebeck, MATHEMATICS OF FINANCE (3 ed., McGraw Hill (8th ed., Houghton 1990) § 1.8 at 17; Porter, 197 1) § 2 at 2; Cissell Cissell Flaspohier, MATHEMATICS OF FINANCE supra, Ch. Tat 2; Smith, supra, § 23 at 25; Zima, supra, § 1.1 at 2. (4th ed., Actex 2008), § 1.1,3 at 12, Def. 1.5; Kellison, Broverman, MATHEMATICS OF INVESTMENT AND CREDIT supra, § 1.4 at 7; Zima, supra, § 1.1 at 2. 29 12 Kellison, supra, § 1.4 at 7; Ruckman and Francis, supra, § 1.2 at 5 ("As the continuous nature of the graph implies, the formula for the accumulated value of a deposit under simple interest still applies if t is not an integer. When (is not an integer, interest is paid on a pro-rata (proportional) basis."). 13 Unless otherwise stated, references to a "unit investment" will mean a new principal unit investment, and will not include a previous investment that had accumulated to a single unit (comprised of both principal and interest) at such time. Similarly, references to an amount (e.g., $1) invested at a particular time, unless otherwise stated, will mean a new principal amount invested (rather than the continuation of a pre-existing investment). This treatment is consistent with the usage in the textbooks reviewed by the author. 14 Vaaler & Daniel, MATHEMATICAL INTEREST THEORY (2nd ed., Mathematical Assoc. of America 2009), § 1.7 at 29, 15 Broverman, supra, § 1. IA at 15, Fig. 1.8; Kellison, supra, § 1.2 at 3, Fig. 1.1(a), § 1,4 at 7 (effectively assumes that simple interest is linear unless otherwise stated), and § 1.7 at 20, Fig. 1.3; Ruckman and Francis, supra, § 1.2 at 4; Brealey (2006), supra § 3.3 at 46, Fig. 3.4. 16 See King, THE THEORY OF FINANCE (3rd ed., Layton 1898), § 4 at 2, quoted in text at endnote 43, infra; Williams, supra, Ch. I, § 4 at 3 (".. . simple interest does not have a logical foundation.. . ."); Sprague, THE ACCOUNTANCY OF INVESTMENT (NYU 1904), § 14 at 17 (".. . compound interest.. , is the only rational and consistent method."); Rietz, Crathorne and Rietz, MATHEMATICS OF FINANCE (Henry Holt 192 1) § 14 at 14 (". . . the lender who collects simple interest at the end of each year, and invests this interest at the same rate as the original principal has just the same accumulated amount of money at the end of any year as he would have had in case he had loaned his money in one transaction at compound interest. This fact suggests that simple interest carried far beyond a conversion period is not a rational kind of interest."); Vaaler & Daniel, supra, § 1,5 at 19 ("Although simple interest is easy to compute, practical applications of this method are limited.... Suppose you invest at a bank where savings accounts earn simple interest.. .. [Y]ou would do well to go into the bank, close your account, and then instantly reopen it. But this would be inconvenient [and is not] sensible ...... 17 18 19 See, e.g., Ruckman and Francis, supra, Ch. 1 at 2 (". . . a $1 payment now is worth more than $1 payable in one year’s time."); Porter, supra, Ch. 1 at 1 ("The more remote the day of payment the less valuable is the promise to pay."); Hummel and Seebeck, supra, § 17 at 36 ("Clearly, $1000 cash is more desirable than $2000 due in 95 years"); Cissell, supra, § 1.10 at 20 ("A hundred dollars today is worth more than a hundred dollars in a year, because a hundred dollars today can be invested to give a hundred dollars plus interest in a year."); Brealey (2011), supra, § 2.1 at 21 (". . . the most basic principal of finance: a dollar today is worth more than a dollar tomorrow ...."); Brueggeman Fisher, supra, Ch. 3 at 52 ("Time value simply means that if an investor is offered the choice between receiving $1 today or receiving $1 in the future, the proper choice will always be to receive the $1 today because this $1 can be invested in some opportunity that will earn interest, which is always preferable to receiving only $1 in the future. In this sense, money is said to have time value."); Ross, supra, Ch. 5 at 121 ("In the most general sense, the phrase time value ofmoney refers to the fact that a dollar in hand today is worth more than a dollar promised at some time in the future. On a practical level, one reason for this is that you could earn interest while you waited; so a dollar today would grow to more than a dollar later. The trade-off between money now and money later, thus depends on, among other things, the rate you can earn by investing."). Geltner Miller, supra, Ch. 8 at 150. Unless otherwise stated, it is assumed that throughout the time in question, simple interest remains part of the investment that generates it. Once it is removed from the investment, it loses its character as simple interest and is just money. 20 If the simple interest may not be withdrawn during the second year and is therefore not available capital for the depositor, it is then available capital for the bank (and the fact that there is no cost associated with the bank’s right to use the simple interest, as opposed to principal, during the second year is similarly at odds with our implicit notions of time value). 21 Kershner, "Note on Compound Interest," THE AMERICAN MATHEMATICAL MONTHLY, Vol. 47, No, 4 (April 1940), at 196. 22 See Zima, supra, § 1.3 at 14 ("$X due on a given date is equivalent at a given simple interest rater to $Y duet years later if Y = X( 1 + rt),.."); Cissell, supra, § 1.18 at 40 ($300 now and $318 in a year "are equivalent in value if money is worth 6% [simple annual interest]"); Shao, MATHEMATICS Or FINANCE (South-Western 1962), § 8.5 at 200-201 (Given a 6% simple annual interest rate, "100 (1 + 6% X 1) = $106. The computation indicates that $100 due now is equivalent to $106 due in one year.."); Simpson, supra, § 46 at 87, 30 23 Zima, supra, § 1.3 at 17; Kersimer, supra, at 196, 24 Hummel and Seebeck, supra, § 17 at 37. See also Zima, supra, § 1.3 at 17; Kerslmer, supra, at 196. 25 Broverman, supra, § 1.3 at 21-22. 26 Kellison, supra, § 2,3 at 53. The choice of the comparison date never matters (i.e., if any two investments are equivalent at one date, then they will be equivalent at all other dates) i fandonly ift.he relevant time value relation is transitive. For a proof, see Promislow, "Accumulation Functions," ACTUARIAL RESEARCH CLEARING HOUSE, Vol. 1 (1985), § 2.4 at 45-46; see also Carey, "Effective Rates of Interest", The Real Estate Finance Journal (Winter 2011) 11, at Appendix C (which includes a proof that transitivity is equivalent to the so-called "consistency principle" and to so-called "Markov accumulation"). 27 Butcher and Nesbitt, MATHEMATICS OF COMPOUND INTEREST (1971; Reprint, Ulrich’s Books 1979), § 1,13 at 25. 28 Equal balances do not imply equal prior balances either. In fact, one simple interest investment may be less than, equal to, and greater than another simple interest investment at different times even during a period between deposits and withdrawals, as indicated by the following hypothetical 10% annual simple interest bank accounts: Year 0 1 2 3 Party A Balance $0 $100 $110 $120 Year 0 1 2 3 Party B Balance $91.67 $100.83 $110.00 $119.17 29 See Promislow, supra, § 2.4 at 47 and Appendix at 55-56. 30 See Appendix 2B. 31 Kellison, supra, § 8.2 at 313. 32 Story v. Livingston, 38 U.S. 359, 371 (1839). Walton and Finney, MATHEMATICS OF ACCOUNTING AND FINANCE (1921. Reprint, Ronald Press Co. 1922), Ch. XVI at 157 (the method "in common use among business men [is] known as the Merchant’s Rule"); textbooks as recently as the 1960s indicate that the Merchant’s Rule is "more or less prevalent among businessmen" (Simpson, supra, § 45 at 84) and "is preferred by most businessmen" (Shao, supra, § 8.4 at 196); even some later textbooks suggest that the Merchant’s Rule continues to be used for short term transactions: a 1990 textbook states that the Merchant’s Rule is one of "two common ways to allow interest credit on short-term transactions" and "is used frequently" (Cissell, supra, § 1.20 at 54-55) and a 2007 textbook describes the Merchant’s Rule as one of "two common ways to allow interest credit on short-term transactions" (Zima, supra, § 1.4 at 20); and a 2008 textbook describes the Merchant’s Rule and states that it "would not normally be used in transactions whose duration is more than one year" (Broverman, supra, §§ 3.4.2, 3.4.3 at 201-202). See Brealey (2006), supra, § 3.3, Fig. 3.5, at 46, See e. g., Ruckman and Francis, supra, Ch. 1 at 2 ("compound interest. . . has amazing accumulation powers when compared to simple interest. . . . Albert Einstein is said to have noted that the most powerful force in the universe is compound interest."); Ross, supra, § 5.1 at 126 ("The effect of compounding is not great over short time periods, but it really starts to add up as the horizon grows" and giving as an example a 200-year investment at 6% per annum simple interest which would be almost 10,000 times more if there were annual compounding); and Lusztig, Cleary Schwab, FINANCE IN A CANADIAN SETTING, (6th ed., Wiley 2001) Part 3 at 148 (discussing "The Magic of Compound Interest"), 36 See, e.g., Ruckman and Francis, supra, § 1.2 at 5-6; McCutcheon and Scott, AN INTRODUCTION To THE MATHEMATICS OF FINANCE (1986. Reprint, Butterworth 2005), § 1,3 at 3-4; Vaaler & Daniel, supra, § 1.5 at 19. Ruckman and Francis, supra, § 1.2 at 6, 38 Id. Id. at 5; sec also, Id. at 6 (". . . banks. . . don’t actually use simple interest when calculating accumulated values."). 40 McCutcheon and Scott, supra, § 1.3 at 4; Ruckman and Francis, supra, § 1.2 at 6. 31 Butcher and Nesbitt, supra, § 1.9 at 16. 42 An internal rate of return (IRR) has been defined as a "rate of interest at which the present value of net cash flows from the investment is equal to the present value of net cash flows into the investment". Kellison, supra, § 7.1 at 252, Although the context reveals that the present value in this definition is intended to be determined using compound interest (which is consistent with most definitions seen by the author), this article will consider the more general interpretation where the present value is to be established by reference to the applicable accumulation function (in this case a simple interest accumulation function). u King, supra, § 4 at 2. 44 15 CORBIN ON CONTRACTS, supra, § 87.13 at 559 ("Thus, absent an agreement, the lender can recover no compound interest."). See also endnotes 2 and 4. Broverman, supra, § 1.1,3 at 12; see also Day, supra, Ch. 2 at 15 ("usually used for periods of less than a year"); Butcher and Nesbitt, supra, § 1.9 at 16 ("[Simple interest] is useful, however, for short-term investments, in particular, those with a term less than a full interest period."); cf. Kellison, supra, § 1,5 at 10 ("occasionally used for short-term transactions and as an approximation for compound interest over fractional periods"). 46 ’ Day, supra, Ch. 2 at 15. The author has reviewed numerous real estate partnership agreements that give the limited partners a simple return. See, e.g., Limited Partnership Agreement of Third Street Partners, LTD., at http://print.onecle.com/ contracts/levittlthird-street-partner-2000. shtml. See also endnote 1. Brealey (2006), supra, § 3.3 at 45 ("financial people always assume that you are talking about compound interest unless you specify otherwise"), but note that the 10th edition of this book contains no separate discussion of simple interest. Compare Nation v. WDE Electric Co, supra, which found that simple rather than compound interest was appropriate while acknowledging (at 235, fn 2) that "compound interest is the standard generally employed in the business and financial world today." ’n See Kellison, supra, § 1.2 at 3; MeCutcheon and Scott, supra, § 2.3 at 13. Vaaler & Daniel, supra, § 1.7 at 29. McCutcheon and Scott, supra, § 2.3 at 13; Promislow, supra, § 2.0 at 40. 52 See, e.g., Vaaler & Daniel, supra, § 1.3; Kellison, supra, § 1.2; Broverman, supra, § 1.1.3, Def. 1.3 at 10, Def. 1.5 at 12; Ruckman and Francis, supra, § 1.4 at 10; and Zima, supra, § 1.1 at 2. Many, if not most, introductory finance textbooks introduce the future value interest factor (or accumulation factor or function) only for compound interest. Ross, supra, § 5.1 at 123 (see FVIF); Brueggeman Fisher, supra, Ch. 3, at 47-49 (Calculating Compound Interest Factors); Copeland, Weston and Shastri, FINANCIAl.. THEORY AND CORPORATE POLICY (4th ed., Pearson 2005), App. D at 930 (see FVIF). Promislow, supra, § 2.2 at 42. Id. at39. Just as A(t,t) is assumed to be 1, a(0)(=A(0,0)) is assumed to be 1 because an investment of 1 at time zero has not had an opportunity to change. See Kellison, § 1.2 at 2; Vaaler & Daniel, supra, § 13 at 12; Broverman, supra, § 1.1.2, Def. 1.3 at 10 (letting t = 0). 56 Promislow, supra, at 43; Boom, "Nominal and Effective Rates of Interest and Discount - A Dimensional Approach," ACTUARIAL RESEARCH CLEARING HousE, Vol. 2 (1988) 223 at 234. Promislow, supra, § 2.2 at 43-44. Simpson, supra, § 45 at 84-85; Cissell, supra, § 1,20 at 54-63; Shao, supra, § 8.4 at 196-200; Broverman, supra, §§ 3.4.2, 3.4.3 at 201-203; Zima, supra, § 1.4 at 20-21 (referring to the Merchant’s Rule and the Declining Balance Method); Kellison, supra, § 8.2 at 313-314; Butcher and Nesbitt, supra, Ch. 4, problem 49(b) and (c) at 195 (providing examples under each rule). Zima, supra, § 1.4 at 20. 60 61 Id. at 20; see also endnote 33. Broverman, supra, § 3,4.2 at 201; Butcher and Nesbitt, supra, Ch. 4, Prob. 49(c) at 195. 32 62 Cissell, supra, § 1.20 at 54-55. 63 See discussion of Future Value Hurdle in body of article. 64 Cissell, supra, § 1,20 at 55. 65 Story v. Livingston, supra; see also, 28 WILLISTON ON CONTRACTS § 72,20 (4th ed.) text at fn 12 (absent an agreement to the contrary, voluntary payments are applied first to interest); Wallace v. Glaser 46 NW 227 (Mich. 1890) (Merchant’s Rule rejected in favor of United States Rule); Christensen v. Snap-On Tools Corporation 554 NW2d 254, 261-262 (Iowa 1996) (delayed principal payments must be applied first to interest); In re the Marriage of Christine Martin v. Charles Martin 7 P. 3d 144,147 (Ariz. 2000) (payments may not be applied first to principal); cf McKinney and Rich, RULING CASE LAW (Bancroft Whitney 1917) Vol. 15, §§ 28, 29 at 31-32. 66 Barnett, "A Comparison of the United States Rule with the Merchant’s Rule for Computing the Maturity Value of a Note on Which Partial Payments Have Been Made," MATHEMATICS MAGAZINE, Vol. 23, No. 1 (Sep - Oct, 1949), at 24-26 (which effectively uses the balance method for ease of comparison while noting that this will "still obtain the same results by its use as those obtained by using it in its usual form"). 67 Langer and Gill, MATHEMATICS OF ACCOUNTING AND FINANCE (Walton 1940), Book I, § 479 at 325 (describing two "equivalent" methods for the Merchant’s Rule, the "ordinary method" which requires that the net future value of the cash flows equal zero at maturity, and the "balance method," which applies each interim payment "first to the discharge of principal"). 68 This recharacterization of the Merchant’s Rule remains accurate even if the payments exceed the principal (in accordance with the formula indicated in the proof below) but because that would involve negative numbers, the discussion is limited to the common case where the payments do not exceed the principal amount of the debt. If the payments exceed the total advances of principal, then the principal balance is effectively allowed to go negative and accrue negative interest. On the calculation date, this negative balance of principal and interest offsets any positive interest accrued while the principal balance was positive. It is as though the payments in excess of principal are borrowings by the investor which accrue interest at the same rate (and such excess amounts, together with interest, are not applied to reduce the interest that accrued while the investor loan had a positive balance until the calculation date). For simplicity, it is generally assumed throughout this installment of the article that the interim payments do not exceed the total principal. 69 cf. Barnett, supra, at 26. 70 The assumption that the payment after the first time unit doesn’t exceed the hurdle balance is made because borrowers generally don’t pay more than they are owed. However, in the JV context it is certainly possible that the investor’s distributions exceed the investor’s hurdle balance at the time, The potential for such a negative balance may sometimes make it difficult to compare these two methods in practice. This assumption will be discussed further in one or more subsequent installments of this article. *** 33 Copyright © 2011 Thomson Reuters. Originally appeared in the Summer 2011 issue of The Real Estate Finance Journal. For more information on the publication, please visit http://west.thomson.com. Reprinted with permission.