1. Introduction
The past decade has seen a renewed interest in Italy in the valuation of amortizing loans, following an important debate on the consistency of the law of compound interest, also known as the law of exponential capitalization, with the principle, enshrined in Italian law, that interest produced in one period of time cannot produce interest in subsequent periods, a phenomenon called “anatocism”.
In recent years, many Italian courts have produced conflicting rulings, in some cases accepting and in others denying the presence of anatocism in certain amortization schemes widespread in operating practices, such as, for example, the French amortization method, which is characterized by constant installments under the law of compound interest (for a review, see
Annibali et al. (
2017)). These conflicting rulings have animated an intense debate involving jurists, economists and mathematicians in an attempt to arrive at a shared solution that reconciles the financial mathematics of loan amortization with Italian law. Two main points are debated. The first is whether anatocism is present when amortizing loans are evaluated according to the law of compound interest (
Fersini and Olivieri 2015). The second point concerns the possibility of exploring different amortization methods, with a focus on amortization methods consistent with the law of simple interest, also called the law of linear capitalization (
Annibali et al. 2018;
Mari and Aretusi 2018,
2019).
In Italy last year (2023), the issue landed in the Corte Suprema di Cassazione, the highest court in the judicial system established to ensure the correct application of the law, which will have to rule in the coming months on the compatibility with Italian law of the loan amortization techniques most widely used in operating practices (key documents are downloadable at
www.cortedicassazione.it).
The problem also has international significance. Several international disputes have shown a general tendency not to accept compound interest (for a comprehensive review see
Sinclair (
2016)). This is motivated by the fact that the exponential nature of the law of compound interest has an explosive effect in the medium to long term, a factor that greatly affects the risk of default and, therefore, the ability to efficiently plan investments (
Cerina 1993).
In an attempt to guide the debate, some authors (
Pressacco et al. 2022) proposed two different amortization schemes based on different inputs but sharing the same accrued interest calculation rule. In the first scheme, with no apparent reference to an underlying discount function, the input is the principal amount, the sequence of principal payments and the sequence of accrued interest is calculated by multiplying the interest rate by the outstanding balance. In the second scheme, the input is given by the principal amount, the sequence of installments calculated according to the law of compound interest and the sequence of accrued interest is calculated, as in the first scheme, by multiplying the interest rate by the outstanding balance. They claim that, in both of these amortization schemes, which are widespread in operating practices, there is no generation of interest on interest if and only if the principal payments are non-negative.
However, it is well known in the financial literature that the first amortization scheme proposed by
Pressacco et al. (
2022) includes the second as a special case of a single standard scheme under the law of compound interest (
Ottaviani 1988). The authors probably do not realize that the presence of the law of compound interest is due to the assumption of the rule for calculating accrued interest and that, regardless of the positivity or negativity of principal payments, both proposed schemes involve the phenomenon of generating interest from interest (as we will show explicitly below).
The purpose of this paper is to provide a unified theoretical framework in which to find useful elements and insights for discussion. We focus on the possibility of establishing a general methodology for evaluating amortizing loans according to arbitrary financial laws and discuss a versatile methodology for loan amortization that allows for the unambiguous construction of a loan amortization schedule with any assigned discount function. Moreover, to monitor the interest generation process and understand the interest flow over time, an extended amortization schedule is introduced. Like a macro lens to uncover the intimate structure of the amortizing loan, the extended amortization schedule contains all the information needed to fully understand the loan repayment process. This level of customization is noteworthy because it can be adapted to various environments and financial scenarios. Different amortization methods can have varying effects on borrowers, including the total cost of borrowing, the distribution of interest payments over time and the pace of debt repayment. Research in this area can inform policymakers and consumer advocacy groups about the potential impacts on borrowers and help develop regulations that promote fair lending practices.
The approach we propose is fully consistent with the general Heath–Jarrow–Morton (HJM) methodology for pricing interest rate-sensitive contingent claims (
Heath et al. 1992). Starting from the initial discount function, the HJM methodology provides a no-arbitrage-based pricing approach consistent with any assigned initial discount function. In our approach, the loan amortization methodology is developed starting from the initial discount function, i.e., the observed discount function at the evaluation time, with the support of some basic no-arbitrage arguments and without any reference to the decomposability property (
Castellani et al. 2005), which plays no role in this context. In particular, it will be shown that the dynamic evolution of the outstanding balance during the lifetime of the loan is time-consistent and does not imply arbitrage.
As a consequence of the proposed methodology, two significant results are presented.
The first result allows us to design loan amortization using two different but equivalent schemes under any assigned discount function. In the first scheme, loan amortization is carried out starting from the knowledge of the discount function and the sequence of the loan installments; in the second scheme, loan amortization is performed starting from the sequence of principal payments and the sequence of accrued interest. It will be shown that, even if the second scheme is adopted, the underlying discount function can be uniquely determined at the maturities corresponding to the installment payment dates. These findings will be presented more formally in Theorems 1 and 2.
As a second result, we derive the amortization method under the law of simple interest as a particular case of the proposed methodology. In this method, the generation of interest on interest is precluded. In fact, we will show that, under the law of simple interest, accrued interest is calculated on the present value of the outstanding balance and not on the outstanding balance itself as in the compound interest method of amortization. In this way, the interest component is removed from the outstanding balance and the interest compounding over time is avoided. The method of loan amortization according to the law of simple interest derived in this paper from the first principles of financial mathematics reproduces that obtained by
Mari and Aretusi (
2018,
2019). The inclusion of a loan amortization scheme under the law of simple interest, in which the interest-on-interest phenomenon is avoided, could be particularly useful for financial practitioners interested in alternative amortization methods that preclude compound interest.
This study provides a conceptual framework for evaluating amortization methods based on arbitrary financial laws, which appropriately extends the most common method of loan amortization, based on the law of compound interest, by including the latter as a special case. This paper could improve our understanding of loan amortization and its potential flexibility with the aim of providing a new perspective on traditional financial methods. We believe that one of the strengths of this paper is its ability to not only propose a unifying methodology, but also provide a detailed comparison with established practices that can be useful for both academic and professional audiences.
This paper is organized as follows.
Section 2 outlines the general methodology for loan evaluation.
Section 3 illustrates the standard amortization method.
Section 4 presents the extended amortization schedule. In
Section 5, Theorems 1 and 2 are stated and proved.
Section 6 presents the loan amortization method under the law of compound interest as a particular case of our methodology. As a further special case,
Section 7 provides the loan amortization method under the law of simple interest. The interest-on-interest question is discussed in both
Section 6 and
Section 7. The interest-on-interest phenomenon under arbitrary discount functions is discussed in
Section 8. Finally,
Section 9 provides some conceptual insights into a different method of loan amortization under a linear capitalization scheme proposed in the literature (
Annibali et al. 2018), showing that there is one and only one method of loan amortization under the law of simple interest, the one described in this study.
3. The Standard Amortization Schedule
Equation (
15) can be cast in a more expressive form
1:
where
is the one-period forward rate and quantifies the interest accrued in the time interval
(
Berk and De Marzo 2014). In this regard, we note that the dynamics of the outstanding balance has a simple structure driven by two components, namely accrued interest and loan repayments. If we recast Equation (
16) in the following form:
we can see that each installment
can be decomposed into two components, namely
where
and
Equation (
20) shows that
quantifies the change in the outstanding balance over the time interval
and Equation (
21) shows that
is the interest accrued over the same time interval. Finally, it is straightforward to show that the outstanding balance,
, can also be expressed as
and that the following relationship holds:
For this reason, in the literature the numbers () are called principal payments.
The standard amortization schedule is a table that shows all the financial information of the loan mentioned above (
Broverman 2017;
Pressacco et al. 2022). In particular, the amortization schedule exhibits for each
k the vector
starting from the initial vector
which is reported in the first row of the table. All the financial quantities contained in
can be easily computed in the proposed approach. For example (but this is not the only way), under an assigned discount function, the amortization schedule can be constructed iteratively as follows: starting from the principal amount
and the loan repayment plan,
, obtained as a solution of Equation (
11) with
and
, accrued interest
can be calculated by using Equation (
21); then,
can be obtained from Equation (
19) by taking the difference
and, finally,
can be computed from Equation (
20),
A Numerical Example
To illustrate the standard amortization method, consider a loan with principal amount
repaid with an annuity consisting of
equal installments due at regular intervals
. The values of the discount function at time
are reported in
Table 1.
The amount of each payment can be computed by using Equation (
11), thus obtaining
The standard amortization schedule, obtained by following the iterative procedure discussed above, is given in
Table 2.
4. The Extended Amortization Schedule
Before proceeding further, it is necessary to explore one aspect that is definitely relevant to our analysis. Is it correct to identify accrued interest with paid interest? Looking at Equation (
11), we can see that each term
is discounted at time
. Discounting removes the interest component from
, thus providing the portion of the principal that is actually repaid with the
k-th installment (in concordance also with the decomposition of a loan into single-payment loans). In this picture, the interest content of each installment is then given by the difference
. Let us pose, therefore,
and
to indicate, respectively, the portion of principal and the portion of interest actually paid with the
k-th installment. In addition to the representation provided by Equation (
19),
also admits, therefore, the following decomposition:
Of course,
and
; however, the following equalities hold:
as a consequence of Equations (
11) and (
19). Since
is the present value of
, it contains no interest and is, therefore, pure capital. For this reason, we will refer to the amounts
(
) as principal “bare” payments.
The financial quantities we have just introduced, namely
and
, allow for a meaningful representation of outstanding balance. In fact, by substituting Equation (
30) into Equation (
16), we obtain
Since
is the interest accrued in the time interval
and
is the amount of interest actually paid with the
k-th installment, it follows that whenever
, the interest component of
increases by the amount
; if
, the interest component of
decreases by the amount
. Furthermore, since
, Equation (
33) also provides the relationship between
and
, namely
showing that
, despite being called principal payment, contains a well-defined interest component. Moreover, let us denote by
the value of the principal not yet actually repaid with the first
k installments, i.e, the difference between
and the sum of the first
k principal bare payments,
By substituting Equation (
28) into Equation (
12) we obtain a very expressive relationship between
and
, namely
or, equivalently,
showing that
is the present value of the outstanding balance
. Of course, it is
and
. Since
is the present value of
, it contains no interest and is, therefore, pure capital
2. As a consequence, the difference
quantifies the interest component in the outstanding balance. It is given by
as it is straightforward to prove by recursively applying Equation (
33).
Finally, since and , we also obtain the following interesting picture: is given by the difference between the outstanding balance at time and the outstanding balance at time k; is given by the difference between the present value of the outstanding balance at time and the present value of the outstanding balance at time k.
In the extended amortization schedule, we will provide synoptically all relevant financial information about the loan, showing explicitly for each
k the vector
starting from the initial vector
reported in the first row of the table. In the extended amortization schedule, the traditional schedule is shown to the left of the vertical bar. On the right-hand side, some additional information is given concerning, for each epoch
k, the financial quantities
,
and
. Like a macro lens to uncover the intimate structure of the amortizing loan, the part to the right of the vertical bar contains all the information needed to monitor the interest generation process and understand the interest flow over time.
A Numerical Example
Referring to the numerical example discussed in the previous section, the extended amortization schedule is shown in
Table 3.
5. Uncovering the Financial Law behind an Amortizing Loan
In this section, we discuss a loan amortization technique that can be configured as a second well-defined amortization scheme (
Pressacco et al. 2022). With no apparent reference to an underlying discount function, in this scheme the input is given by the principal amount,
, the sequence of principal payments,
, and the sequence of accrued interest,
. To simplify the notation, let us pose
We assume that the sequences of numbers and satisfy the following conditions:
(G1)
(G2)
(G3)
From this figure, the loan installments and outstanding balance are calculated as follows:
and
Condition
(G1) ensures that
, i.e.,
and that
; condition
(G2) also allows for negative rates to be taken into account; condition
(G3) ensures that the installments,
, are non-negative with
. Moreover, conditions
(G1)–
(G3) imply that
Indeed, if there is
such that
,
, it follows that
and so on until time
n where
since
.
We will show that, even if this second scheme is adopted, the underlying discount function can be uniquely determined at the maturities corresponding to the installment payment dates. In addition, we will show that this second amortization scheme is equivalent to the scheme discussed in
Section 3. These results are more formally described by the following Theorems 1 and 2. In particular, Theorem 1 summarizes the findings obtained in
Section 3.
Theorem 1. Let a strictly positive number and consider for : (i) a sequence of strictly positive numbers ; (ii) a sequence of non-negative numbers , with , such that If is computed according toand , then there exist a unique sequence of numbers and a unique sequence of numbers , , satisfying conditions (G1)–(G3), such that the amortizing schedule can be computed according to the following rules:and Proof of Theorem 1. Under the assumptions of Theorem 1, the sequences of numbers
and
are given by Equations (
20) and (
21), respectively. Then, it is straightforward to verify that conditions
(G1)–
(G3) hold with
. □
The converse is also true. Indeed, we will prove that the following proposition holds.
Theorem 2. Let a strictly positive number and consider for : (i) a sequence of numbers and (ii) a sequence of numbers satisfying conditions (G1)–(G3). If the amortizing schedule is computed according to the following rules:andthere exists a unique sequence of numbers,such that the following relationships hold: Moreover, the numbers , , are strictly positive.
Proof of Theorem 2. Preliminarily, we note that, from condition
(G2), the numbers
defined by Equation (
51) are strictly positive since
. By substituting Equation (
49) into Equation (
50), we obtain
where condition
(G2) has been used. Solving with respect to
, we obtain
By using Equation (
51), we can rewrite Equation (
55) in the following recursive form:
with
. Equations (
52) and (
53) can be then recovered by backward induction starting from
and recalling that
. To prove the uniqueness, we observe that the system of
n linear equations in the
n unknowns
,
, described by Equation (
56), admits one and only one solution. □
As Theorem 2 clearly shows, the rule for calculating interest, expressed by condition
(G2), plays a crucial role in identifying the discount function, allowing it to be uniquely determined. Moreover, we note that Equation (
51) can be cast in the following recursive form:
with
.
As an example, it is easy to verify that the discount function represented in
Table 1 can be easily discovered from the amortization schedule shown in
Table 2 by using Equation (
51) or, equivalently, Equation (
57).
8. The Interest-on-Interest Phenomenon under Arbitrary Discount Functions
What about the phenomenon of interest on interest when a loan is designed according to an arbitrary discount function? The answer depends on the parameterization adopted to represent the discount function. Consider the following generalized compound interest representation:
We note that, given an arbitrary discount function, the sequence of rates
is uniquely determined. In this representation, the interest accrued in the time interval
, computed using Equation (
21), is given by
To describe the same discount function, we could have used a different parameterization such as, for example, the following generalized simple interest representation:
Again, the sequence of rates
is uniquely determined. In this new representation, the interest accrued in the time interval
, computed using Equation (
21), is given by
Of course, both representations produce the same amortization schedule, but with different sequences of rates
and
. However, in the generalized compound interest representation, the phenomenon of interest on interest occurs, which results from multiplying the outstanding balance
by the interest rate
, as shown in Equation (
96), the outstanding balance being a mixture of principal and interest. On the other hand, in the generalized simple interest representation, Equation (
98) shows that the generation of interest on interest is avoided, since in this case accrued interest is computed multiplying the present value of the outstanding balance
by the interest rate
.
To provide some numerical examples, let us first consider the discount function given in
Table 1. The rate sequences
and
are depicted in
Table 9.
The same amortization schedule given in
Table 3 can be obtained by using both the generalized compound interest representation provided by Equation (
95) and the generalized simple interest representation provided by Equation (
97) with the sequence of rates
and
, respectively, shown in
Table 9.
Table 10 gives the sequences of the rates
and
in the case of the numerical examples discussed in
Section 6, where the law of compound interest was used (with constant interest rate
). The same amortization schedules depicted in
Table 4 and
Table 5 can be obtained by using the generalized simple interest representation provided by Equation (
97) with the sequence of rates
shown in
Table 10.
Table 11 gives the sequences of the rates
and
in the case of the numerical examples discussed in
Section 7 where the law of simple interest was used (with constant interest rate
). The same amortization schedules depicted in
Table 7 and
Table 8 can be obtained by using the generalized compound interest representation provided by Equation (
95) with the sequence of rates
shown in
Table 11.
The ability to design amortizing loans with arbitrary discount functions increases the level of customization and can have varying effects on borrowers, including the total cost of borrowing, the distribution of interest payments over time and the pace of debt repayment. However, great care should be taken to use financial representations that are consistent with the law and provide consumers with accurate information about the representation of the financial scheme and the rates used, so as to avoid the phenomenon of interest-on-interest generation where this is not permitted. Research in this area can inform policymakers and consumer advocacy groups about the potential impacts on borrowers and help develop regulations that promote fair lending practices.
9. Discussion and Concluding Remarks
In this paper, we have provided a general methodology for valuing amortizing loans with arbitrary discount functions. The proposed approach is fully consistent with the general Heath–Jarrow–Morton methodology (
Heath et al. 1992) for pricing interest rate-sensitive contingent loans. The entire methodology was developed from the knowledge of the initial discount function and using some basic no-arbitrage arguments. It is valid whatever stochastic model is used to describe the evolution of the discount function. No reference is made to the decomposability property (
Castellani et al. 2005), which plays no role in this context. This approach is perfectly consistent with the fundamental theorems of asset pricing (
Delbaen and Schachermayer 1994).
Although we have discussed loans with installment payments at regular time intervals, the extension to the case of time intervals of variable amplitude is straightforward.
As a special case of the proposed methodology, we have illustrated the amortization method based on the law of simple interest and shown that, in this case, the phenomenon of generating interest on interest is precluded. Some authors proposed a different method for valuing amortizing loans under a linear capitalization scheme (
Annibali et al. 2018). To illustrate their procedure, let us consider at time
a loan with a principal amount
that will be repaid with a series of future non-negative installments,
scheduled at regular time intervals
, with
. The starting point of their analysis is that the loan principal and each installment are linearly capitalized at loan maturity
n, using the interest rate
i observed at time
, thus obtaining
We point out that this approach can be considered as a special case of the methodology proposed in this study with the following discount function:
However, it should be noted that this procedure produces spurious results that are not consistent with the law of simple interest. Consider, for example, a loan with a principal amount
at time
that will be repaid with a single strictly positive installment
at time
n. Following this approach, the dynamics of the outstanding balance is given by
showing that the outstanding balance does not follow a linear behavior, as it should be according to the law of simple interest and as obtained from Equation (
83). In a different but equivalent way, interest does not accrue linearly over time. For these reasons, we believe that there is one and only one method of loan amortization under the law of simple interest, the one described in this study.