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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
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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.