# Quantum credit loans

J. S. Ardenghi†
†IFISUR, Departamento de Física (UNS-CONICET)
Avenida Alem 1253, Bahía Blanca, Buenos Aires, Argentina email:
<EMAIL_ADDRESS>fax number: +54-291-4595142

###### Abstract

Quantum models based on the mathematics of quantum mechanics (QM) have been
developed in cognitive sciences, game theory and econophysics. In this work a
generalization of credit loans is introduced by using the vector space
formalism of QM. Operators for the debt, amortization, interest and periodic
installments are defined and its mean values in an arbitrary orthonormal basis
of the vectorial space give the corresponding values at each period of the
loan. Endowing the vector space of dimension $M$, where $M$ is the loan
duration, with a $SO(M)$ symmetry, it is possible to rotate the eigenbasis to
obtain better schedule periodic payments for the borrower, by using the
rotation angles of the $SO(M)$ transformation. Given that a rotation preserves
the length of the vectors, the total amortization, debt and periodic
installments are not changed. For a general description of the formalism
introduced, the loan operator relations are given in terms of a generalized
Heisenberg algebra, where finite dimensional representations are considered
and commutative operators are defined for the specific loan types. The results
obtained are an improvement of the usual financial instrument of credit
because introduce several degrees of freedom through the rotation angles,
which allows to select superposition states of the corresponding commutative
operators that enables the borrower to tune the periodic installments in order
to obtain better benefits without changing what the lender earns.

##  1 Introduction

The mathematical methods of quantum mechanics have spread outside quantum
physics and have reached the field of human condition ([1], [2], [3], [4], [5]
and [6]), model of decision making ([7], [8], [9], [10],[11], [12], [13], [14]
and [15]), quantum game theory ([16], [17], [18], [19] and [20]), econophysics
([21], [22], [23], [24], [25], [26] and [27]) among others. This suggests us
that the abstract principles of quantum mechanics exhibit a high degree of
adaptability and contextuality and allow to apply it as ”quantum like” models
to different branches of human cognition science. The core of quantum
mechanics is the possibility of superposition and entanglement of quantum
states, that physically manifest as interferences and non-local correlations
of observables. Interestingly, different interpretation of quantum mechanics
endow ontological reality to different mathematical objects of the theory. In
particular, modal interpretations describe the quantum states as the possible
properties of the system and their corresponding probabilities, and not the
actual properties, where properties and facts belong to different ontological
categories ([28], [29] and [30]). The relationship between the quantum state
and the actual properties of the system is probabilistic, that is, the actual
occurrence of a particular value of an observable is an indeterministic
phenomena. Then, if quantum mechanics describes the reality, it certainly not
an actual reality but a possible reality. Is the quantum superposition, only
existing in potential terms, that allows to certain measurements to be
actualized in a genuinely unpredictable way. This behavior can be extrapolated
to other conceptual situations, for example in judgments and decisions that
can be conceived as indeterministic processes when subjects give answers in
situations of uncertainty, confusion or ambiguity. In turn, it can be
extrapolated to model stock return distributions by considering the market
force as a quantum harmonic oscillator [31] or can be used to apply quantum
probability theory to formulate probabilistic models of cognition [32] and to
model economical markets [33].

In this sense, the aim of this work is to contribute to the list of possible
extrapolations of the abstract formalism of quantum mechanics to the field of
econophysics. We will show that a particular financial instrument, the credit,
can be defined over a vectorial space of dimension $M$, where $M$ is the
duration of the repayment of the credit, which in the moment of contractual
agreement between the lender and the borrower becomes a debt that is returned
in a set of scheduled payments. It will be shown that the sequence of values
for the debt, amortization, interest and periodic payments can be obtained as
the eigenvalues of the respective operators in a particular eigenbasis of the
vectorial space. Then, it will be shown that the basis chosen of the vector
space in which the loan operators are diagonal can be rotated without changing
the total amortization and the lender’s profit, computed as the sum of the
periodic payments. This rotation is a manifestation of a $SO(M)$ symmetry,
where $SO(M)$ is the special orthogonal group of a vectorial space of
dimension $M$. Then, by endowing the vectorial space with a symmetry group,
the mean values of the operators in the rotated basis allow to obtain
different configurations for the debt, amortization and periodic payments,
which contain the well known French and German loan systems, among others.
Moreover, it will be shown that the extrapolation of the abstract mathematical
structure of the quantum formalism to the credit system, can be rewritten in
terms of an algebra of operators, that resembles the Heisenberg algebra of
quantum mechanics. In order to be self-contained, this work will contains some
elementary examples of credit loans and the generalization in a linear space,
that we call “quantum credit loans”, for the purpose of introducing the
machinery of the basic linear algebra which can be rather unfamiliar in the
context of loans. Then, this manuscript will be organized as follows: In
Section II, non-indexed credit loans are reviewed. In Section III, the
recurrence relations for the debt, amortization, interest and periodic
installments are described in terms of a Generalized Heisenberg algebra. Then
the quantum credit loans are introduced, where the rotation transformation of
the orthonormal basis is applied and the two most simple examples with $M=2$
and $M=3$ are exhaustively studied. In Section IV, indexed credit loans, where
a non-constant interest rate is possible during the loan, is introduced and a
particular example is studied. Finally, the conclusions are presented.

##  2 Non-indexed credit loans

In order to introduce the algebra of the credit loans, we can notice that
these are a particular case of non-homogeneous linear recurrence relation for
the debt $D$, capitalization or amortization $A$ and interest $Y$.111We use
$Y$ for interest and not $I$ because $I$ will be used for the identity
operator. The main difference between the loans lies in the way the amount of
the installments that must be paid are computed in order to repay these loans.
The French system loan is the most popular because the formula used to
calculate periodical fees ensures that the periodic payment is the same for
the life of the loan. This repayment includes the interest and the repayment
of the capital. The only thing that will change is the proportion of what we
pay of these two magnitudes: as the first increases, the second will
decreases, so that the repayments are always the same. Since the debt $D$ at
the beginning of the loan is very large during the initial periods, the
interest $Y$ will also be greater. For this reason, we usually begin by paying
a greater proportion of interest, and as time passes and the interest payments
goes down, we begin to repay a greater proportion of capital $A$. Another
popular system is the German system in which the periodic payment is not
constant but the capital repayment is. Then, in order to express
mathematically the recurrence relations of the French or German loans, we can
consider an initial debt $d_{0}$, a loan duration $M$ and the interest rate
$t$. The set of relations between the interest $y_{n}$, the debt $d_{n}$ and
the amortization $a_{n}$ reads

$\displaystyle\text{a) }q_{n}=a_{n}+y_{n}$ (1) $\displaystyle\text{b)
}y_{n}=td_{n-1}$ $\displaystyle\text{c) }d_{n+1}=d_{n}-a_{n+1}$

the first equation 1a) means that the periodic payment or installment $q_{n}$
is composed of the interest $y_{n}$ and the amortization $a_{n}$ of the
respective periodicity $n$. The second equation 1b) implies that the interest
$y_{n}$ is proportional to the debt $d_{n-1}$ of the previous period and
finally the last equation implies that the new debt $d_{n+1}$ is the
subtraction of the debt from the last period with the current amortization.
The coefficient $t$ is the interest rate. The procedure in order to compute
the loan is the following: an initial debt $d_{0}$ is considered and the lapse
$M$ of the loan is needed. With the initial debt, the interest $y_{1}$ is
computed as $y_{1}=td_{0}$. With this value and $q_{1}$, the amortization can
be computed as $a_{1}=q_{1}-y_{1}$. Finally, the new debt $d_{1}=d_{0}-a_{1}$
is obtained and the procedure starts again. As it is expected, the loan must
ends, which implies that $d_{M}=0$. The boundary conditions $d_{0}=d_{0}$ and
$d_{M}=0$ define a relation between the periodic installments $q_{i}$, $d_{0}$
and $M$ and determine univocally the loan series $d_{n}$, $a_{n}$, $y_{n}$ and
$q_{n}$. In turn, by using eq.(1)c) and summing in $a_{n}$, we obtain that the
total amortization $\sum\limits_{n=1}^{M}a_{n}=d_{0}$ must be identical to the
initial debt accordingly. In the French system, $q_{n}$ is fixed and does not
depend on $n$ and in the German system $a_{n}$ is fixed. Combining the three
equations of eq.(1) we obtain the following recurrence relation for $d_{n}$

$d_{n}=(1+t)d_{n-1}-q_{n}$ (2)

which has the form of a non-homogeneous recurrence relation of the type
$t_{n}=\alpha t_{n-1}+\beta_{n}$ with $\alpha=1+t$ and $\beta_{n}=-q_{n}$. The
solution of the recurrence relation of eq.(2) with fixed constant $q^{(F)}$
reads

$d_{n}^{(F)}=\frac{q^{(F)}}{t}\left[1-(1+t)^{n}\right]+(1+t)^{n}d_{0}$ (3)

where the superscript $F$ indicates the French system loan. The boundary
condition $d_{M}=0$ implies that

$d_{M}^{(F)}=0=(1+t)^{M}\left[d_{0}+\frac{q^{(F)}}{t}\right]-\frac{q^{(F)}}{t}$
(4)

that can be solved for $q^{(F)}$

$q^{(F)}=\frac{d_{0}t(1+t)^{M}}{(1+t)^{M}-1}$ (5)

or can be solved for $M$ or $d_{0}$ in the case the value of $q^{(F)}$ is
fixed arbitrarily222This is useful for precancellations of the loan.. The
general procedure to obtain $q^{(F)}$ is by using
$\sum\limits_{n=1}^{M}a_{n}=d_{0}$ and that $a_{n}=(1+t)a_{n-1}$ when
$q_{n}=q$. Then writing $a_{1}\sum\limits_{n=1}^{M}(1+t)^{n}=d_{0}$, where
$a_{1}=q-y_{1}=q-td_{0}$ is not difficult to show eq.(5). With this result,
the debt $d_{n}^{(F)}$ reads

$d_{n}^{(F)}=\frac{d_{0}}{(1+t)^{M}-1}\left[(1+t)^{M}-(1+t)^{n}\right]$ (6)

In a similar way the amortization and the interest can be solved

$a_{n}^{(F)}=(1+t)^{n}(q^{(F)}-td_{0})\text{ \ \ \ \ \
}y_{n+1}^{(F)}=td_{n}^{(F)}$ (7)

In the French system, the total amount of unit currency paid is
$\mathbf{Q}_{F}=q^{(F)}M=\frac{d_{0}Mt(1+t)^{M}}{(1+t)^{M}-1}$. It can be
noted from eq.(2) that the debt contains a constant term $-q^{(F)}$ and, in
the case that the sequence of numbers for $d_{n}$ decrease with $n$, that is,
$d_{n}<d_{n-1}$, this indicates that $td_{n-1}<q^{(F)}$ and in the case that
the largest value of the debt is $d_{0}$, then $t<\frac{q^{(F)}}{d_{0}}$,
which implies that, in the case in which the amount to be paid $q^{(F)}$ in
each period is not fixed, then the interest rate $t$ must be constrained if we
expect a decreasing debt curve. This issue is not trivial, because in the case
in which $t>\frac{q^{(F)}}{d_{0}}$ then $d_{n}>d_{n-1}$ and the debt increases
with $n$.333Indexed interest rate loans will be considered in the next
section. In particular, we will consider a particular case of indexed loan in
a new variable that is linked to the unit currency through the inflation rate
( for example UVA system in Argentina). When the boundary condition $d_{M}=0$
is chosen, then $q^{(F)}=\frac{d_{0}t(1+t)^{M}}{(1+t)^{M}-1}$ and the
inequality reads $t>0$ which is obeyed for any loan with positive interest
rate $t$, that is, the boundary condition $d_{M}=0$ implies that
$d_{0}>d_{M}$.

The German system consists in a constant amortization over time, that is
$a_{n}=a_{0}=\frac{d_{0}}{M}$ which is obtained from
$\sum\limits_{n=1}^{M}a_{n}=d_{0}$. Then the installment to be paid
$q_{n}^{(G)}$ is now a function of $n$ due to eq.(1) a) as
$q_{n}=y_{n}-a_{0}$. When the system of recurrence relations is solved we
obtain

$\displaystyle d_{n}^{(G)}=d_{0}-n\frac{d_{0}}{M}$ (8) $\displaystyle
y_{n}^{(G)}=td_{0}(1-\frac{(n-1)}{M})$ $\displaystyle
q_{n}^{(G)}=\frac{d_{0}}{M}+t\frac{d_{0}}{M}(M-n+1)$

The periodic installment is maximum for $n=1$, $q_{1}=d_{0}(t+\frac{1}{M})$,
due to the fact that the interest to be paid at $n=1$ is maximum. In figure 1,
the debt, the amortization, the interest and the periodic installment are
shown as a function of $n$ for $t=0.02$, $d_{0}=100$ and $M=10$ for the French
and German systems. As it can be seen, the debt decreases until it reaches
$d_{M}=0$ in both systems and the periodic installment decreases in the German
system.

Figure 1: Non-indexed credit loan with $d_{0}=100$, $M=10$ and $t=0.2$. Left:
French system. Right: German system.

The total amount paid during the loan in the German system is
$\mathbf{Q}_{G}=\overset{M}{\underset{i=1}{\sum}}q_{n}^{(G)}=\frac{d_{0}}{2}\left[2+t(1+M)\right]$
that can be compared with the obtained from the French system
$\mathbf{Q}_{F}=\overset{M}{\underset{i=1}{\sum}}q_{n}^{(F)}=qM$, that is
larger than $\overset{M}{\underset{i=1}{\sum}}q_{n}^{(G)}$ for arbitrary
values of $M$, $t$ and $d_{0}$ and allows to see why the French system is more
suitable for the lender (see figure 2).

Figure 2: $\mathbf{Q}_{F}-\mathbf{Q}_{G}$ difference, where $\mathbf{Q}_{F}$
is the sum of the periodic installments in the French system and
$\mathbf{Q}_{G}$ is the sum of the periodic installments in the German system
for different values of $t/d_{0}$ and $M/d_{0}$.

The recurrence relations of eq.(1) allow to obtain different schedule payments
$q_{n}$ by simultaneously changing the amortization schedule. In the French
system, $q$ is fixed and in the German system $a_{0}$ is fixed. Both
quantities are related through eq.(1a) which implies that other configurations
for schedule payments should be found, for example in which $Q$ and $A$ are
fixed or vary simultaneously.

##  3 Quantum credit loans

The mathematical structure of quantum mechanics (QM) is constructed over the
concept of Hilbert spaces (see [34] and [35]). The energy of a particle is the
Hamiltonian operator that acts on this Hilbert space. The eigenvalues
correspond to the possible energy values of the quantum system. There is an
elegant procedure to obtain the energy eigenvalues of different physical
systems [36] and [37], where a generalized Heisenberg algebra (GHA) is defined
and where the energy eigenvalues can be obtained from a recurrence relation
derived from the algebra. In [37] the linear recurrence relation is studied
identical to those obtained for the debt, amortization and interest. Then, it
is interesting to derive the coupled system of recurrence relations of eq.(1)
from an analogous algebra used in [37]. Due to the simultaneous magnitudes
($D$, $A$, $Y$ and $Q$) with defined values, we can further generalized the
GHA by introducing several operators that commute each other as it is done in
[36] to implement a three-step recurrence relation. In order to characterize
the loan duration, the algebra must be defined over a finite-dimensional
Hilbert space in contrast to the algebras defined in [37], where an infinite
dimensional Hilbert space is constructed. The procedure to define finite
dimensional Harmonic oscillator algebras was studied in [38]. In this work, we
will combine both works in order to construct the finite dimensional vector
space of the loan in which the GHA algebra acts. Let us define the following
algebra of operators

$\displaystyle\text{a)
}\left[D,A\right]=\left[D,Y\right]=\left[D,Q\right]=0\text{ \ }$ (9)
$\displaystyle\text{b) }aY=f(D)a\text{\ \ \ \
c)}Ya^{{\dagger}}=a^{{\dagger}}f(D)$ $\displaystyle\text{d)
}\left[D,a\right]=aA~{}\ \ \ \ \text{e)
}\left[a^{{\dagger}},D\right]=Aa^{{\dagger}}$ $\displaystyle\text{f)}\ Q=Y+A\
\ \text{g)
}\left[a,a^{{\dagger}}\right]=A-d_{0}\left|M\right\rangle\left\langle
M\right|$

where $D$ is the debt operator, $A$ is the amortization operator, $Y$ is the
interest operator and $Q$ is the periodic installment operator. We can
consider a set of $M$ orthogonal vectors $\left|n\right\rangle$ that are
simultaneous eigenstates of $D$, $A$, $Y$ and $Q$. This is explicit from eq.(9
a) which implies that $D\left|n\right\rangle=d_{n}\left|n\right\rangle$,
$A\left|n\right\rangle=a_{n}\left|n\right\rangle$,
$Y\left|n\right\rangle=y_{n}\left|n\right\rangle$ and
$Q\left|n\right\rangle=q_{n}\left|n\right\rangle$, where $d_{n}$, $a_{n}$,
$y_{n}$ and $q_{n}$ are the respective eigenvalues in the eigenbasis that
diagonalizes all the commuting operators. Should be noted that in the French
system, the $Q$ operator is degenerate because $q_{n}=q$ and in the German
system the degenerate operator is $A$.444The four operators $D$, $A$, $Y$ and
$Q$ are chosen to commute each other basically because in a classical loan, at
each period, the debt, interest, amortization and periodic installment have a
defined value simultaneously, which resembles the compatibility of observables
in quantum mechanics. The annihilation operator $a$ and creation operator
$a^{{\dagger}}$ acts on the $\left|n\right\rangle$ basis as

$\displaystyle a\left|1\right\rangle$ $\displaystyle=$ $\displaystyle 0\text{
\ \ \ \ \ \ \ \ \ \ }a\left|n\right\rangle=N_{n}\left|n-1\right\rangle$ (10)
$\displaystyle a^{{\dagger}}\left|M\right\rangle$ $\displaystyle=$
$\displaystyle 0\text{ \ \ \ \ \ \ \ \ \
}a^{{\dagger}}\left|n\right\rangle=N_{n+1}\left|n+1\right\rangle$

where $\left|M\right\rangle$ is the highest level at which the debt eigenvalue
is555In the quantum harmonic oscillator the ground state is denoted as
$\left|0\right\rangle$. In this work, the ground state is denoted as
$\left|1\right\rangle$ that corresponds to $d_{1}=d_{0}-a_{1}$,
$q_{1}=a_{1}+y_{1}$ and $y_{1}=td_{0}$.

$D\left|M\right\rangle=0$ (11)

that is, the highest debt level eigenvalue is $d_{M}=0$, which implies that
the loan has finished and $M$ is the loan duration and $N_{n}$ are
normalization factors. Eqs.(9) b), d) and e) defines the relation between the
interest, the debt and the amortization. Eq.(9) f) is the analog to eq.(1) a).
Finally, eq.(9) g) defines the commutation relation between $a$ and
$a^{{\dagger}}$ in terms of the amortization operator $A$. Should be noted
that $Tr([a,a^{\dagger}])=0$ as it is expected. All the operators satisfy the
Jacobi identity, in particular

$\displaystyle\left[D,\left[a^{{\dagger}},a\right]\right]+\left[a,\left[D,a^{{\dagger}}\right]\right]+\left[a^{{\dagger}},\left[a,D\right]\right]=0$
(12)
$\displaystyle\left[D,\left[A,a\right]\right]+\left[a,\left[D,A\right]\right]+\left[A,\left[a,D\right]\right]=0$
$\displaystyle\left[D,\left[Y,a\right]\right]+\left[a,\left[D,Y\right]\right]+\left[Y,\left[a,D\right]\right]=0$

that can be obtained by using that $\left[D,a^{{\dagger}}a\right]=0$ and that
$\left[a,Y\right]=aY-Ya=(f(D)-Y)a$. The same result holds by replacing
$D\rightarrow Q$ in eq.(12). In order to understand how the algebra of eq.(9)
works, we can multiply by $a$ to eq.(9)f) $aQ=aY+aA$ and replacing in eq.(9)d)
for $aA$ and by using eq.(9)b) we obtain

$\left[D,a\right]=aQ-f(D)a$ (13)

that applied to a state $\left|n\right\rangle$ gives

$d_{n}=d_{n-1}+f(d_{n-1})-q_{n}$ (14)

To obtain the credit loan in the French or German system we can replace the
function $f(D)$ with $f(D)=(T-I)D$ where $T$ is the interest rate operator
that satisfies $\left[T,D\right]=0$, $I$ is the identity operator and
$T\left|n\right\rangle=(t_{n}+1)\left|n\right\rangle$. By using eq.(9) b), d)
and f) we obtain

$d_{n}=(1+t_{n-1})d_{n-1}-q_{n}~{}$ (15)

in the case that $t_{n}=t$ and $q_{n}=q$, last equation is identical to
eq.(2), where $t$ is the constant interest rate and $q$ is the fixed
installment in the French system. In this case, the $T$ operator is degenerate
in the $\left|n\right\rangle$ basis. Using eq.(9)g), the creation and
annihilation operators can be written in the eigenbasis of the debt operator
as (see eq.(6) of [38])

$a=\overset{M-1}{\underset{j=1}{\sum}}N_{j+1}\left|j\right\rangle\left\langle
j+1\right|\text{ \ \ \ \ \ \ \ \ \
}a^{{\dagger}}=\overset{M-1}{\underset{j=1}{\sum}}N_{j+1}\left|j+1\right\rangle\left\langle
j\right|$ (16)

By using last equation, the commutator $\left[a,a^{{\dagger}}\right]$ can be
written as

$\left[a,a^{{\dagger}}\right]=\overset{M-1}{\underset{j=1}{\sum}}(N_{j+1}^{2}-N_{j}^{2})\left|j\right\rangle\left\langle
j\right|-N_{M}^{2}\left|M\right\rangle\left\langle M\right|$ (17)

where we have define $N_{1}=0$. Combining last equation with the l.h.s. of
eq.(9) g) we obtain

$N_{n+1}^{2}-N_{n}^{2}=a_{n}\text{ \ \ for }n=1,2,...,M-1\text{ \ \ \ \ \ \ \
\ \ \ }N_{M}^{2}=d_{0}-a_{M}$ (18)

and the solution reads

$N_{j}^{2}=\overset{j-1}{\underset{i=1}{\sum}}a_{i}$ (19)

For instance, consider the case in which $r_{i}=r=1+t$ and $q_{n}=q$ in the
algebra defined in eq.(9), then the non-trivial commutation relations reduces
to

$\displaystyle\text{a) \ }Da^{{\dagger}}=a^{{\dagger}}((1+t)D-Q)\text{ \ \ \
b) \ }aD=((1+t)D-Q)a\text{\ \ \ \ \ }$ (20) $\displaystyle\text{c)
}\left[a,a^{{\dagger}}\right]=A-d_{0}\left|M\right\rangle\left\langle
M\right|$

which defines the desired recurrence relation for the debt in the French
system. A similar analysis can be done with the German system that only
differs from the French system in the choice of the non-degenerate operator,
that in the German system is $A$, in contrast to the French system that
choices $Q$. The function $f(D)$ is identical in both systems.

At this point is interesting to discuss the loan evolution during time. At
each step $n$ of the loan, the debt, the interest, the amortization and the
periodic installment actualizes. This can be represented by the mean values of
these operators in the state $\left|n\right\rangle$, that is

$\displaystyle\left\langle n\right|D\left|n\right\rangle$ $\displaystyle=$
$\displaystyle d_{n}\text{ \ \ \ \ \ \ }\left\langle
n\right|Y\left|n\right\rangle=y_{n}$ (21) $\displaystyle\left\langle
n\right|A\left|n\right\rangle$ $\displaystyle=$ $\displaystyle a_{n}\text{ \ \
\ \ \ \ \ }\left\langle n\right|Q\left|n\right\rangle=q_{n}$

In turn,
$\left|n\right\rangle=\frac{(a^{{\dagger}})^{n}}{\prod\limits_{j=1}^{n}N_{j}}\left|1\right\rangle$,
then for instance, the mean value of the debt operator reads

$d_{n}=\frac{1}{\prod\limits_{j=1}^{n}N_{j}^{2}}\left\langle
1\right|a^{n}D(a^{{\dagger}})^{n}\left|1\right\rangle\text{ \ }$ (22)

and by using eq.(9)e), eq.(9)f) and eq.(9) c) and that for the French system
$\left[Q,a\right]=\left[Q,a^{\dagger}\right]=0$ is not difficult to show that

$d_{n}=\left(\prod\limits_{j=1}^{n}N_{j}^{2}\right)^{-1}\left\langle
1\right|a^{n}(a^{{\dagger}})^{n}\left|1\right\rangle\left[(1+t)^{n}d_{0}-q\overset{n-1}{\underset{j=0}{\sum}}(1+t)^{j}\right]$
(23)

but $\left(\prod\limits_{j=1}^{n}N_{j}^{2}\right)^{-1}\left\langle
1\right|a^{n}(a^{{\dagger}})^{n}\left|1\right\rangle=1$ and the result is
identical to eq.(3). But then we can notice that the basis
$\left|n\right\rangle$ allows to construct linear combinations, which is
reasonable because the state $\left|n\right\rangle$ is an arbitrary unit
vector in the vector space and it has the same status as any linear
combination $\,\sum\limits_{n=1}^{M}c_{n}\left|n\right\rangle$. A linear
combination of states of the loan enlarges the degrees of freedom and in turn
allows the time evolution not to be constrained to the eigenvalues $d_{n}$,
$y_{n}$, $a_{n}$ and $q_{n}$ of each operator at each periodicity of the loan.
Besides this, the superposition of the loan states can be used to design loans
that, in particular, can be a mixture of the French and the German systems. To
be clear, we can consider $M=2$, then the operators $D$, $A$, $Y$, $Q$, $a$
and $a^{\dagger}$ are written in the
$\\{\left|1\right\rangle,\left|2\right\rangle\\}$ basis as

$\displaystyle D$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}d_{1}&0\\\ 0&0\end{array}\right)\text{
\ \ \ }A=\left(\begin{array}[]{cc}a_{1}&0\\\ 0&a_{2}\end{array}\right)\text{ \
\ \ }Y=\left(\begin{array}[]{cc}y_{1}&0\\\ 0&y_{2}\end{array}\right)$ (30)
$\displaystyle Q$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}q_{1}&0\\\
0&q_{2}\end{array}\right)\text{ \ \ \ }a=\left(\begin{array}[]{cc}0&N_{2}\\\
0&0\end{array}\right)\text{ \ \ \ \ \
}a^{\dagger}=\left(\begin{array}[]{cc}0&0\\\ N_{2}&0\end{array}\right)$ (37)

For example in the French system we have

$\displaystyle D^{(F)}=d_{0}\left(\begin{array}[]{cc}\frac{(1+t)}{(2+t)}&0\\\
0&0\end{array}\right)\text{ \ \ \
}A^{(F)}=d_{0}\left(\begin{array}[]{cc}\frac{1}{2+t}&0\\\
0&\frac{1+t}{2+t}\end{array}\right)\text{ \ \ \
}Y^{(F)}=d_{0}\left(\begin{array}[]{cc}t&0\\\
0&t\frac{1+t}{2+t}\end{array}\right)$ (44) $\displaystyle
Q^{(F)}=d_{0}\left(\begin{array}[]{cc}\frac{(1+t)^{2}}{2+t}&0\\\
0&\frac{(1+t)^{2}}{2+t}\end{array}\right)\text{ \ \ \
}a=\sqrt{d_{0}}\left(\begin{array}[]{cc}0&\sqrt{\frac{1}{2+t}}\\\
0&0\end{array}\right)\text{ \ \ \ \ \
}a^{\dagger}=\sqrt{d_{0}}\left(\begin{array}[]{cc}0&0\\\
\sqrt{\frac{1}{2+t}}&0\end{array}\right)$ (51)

and in the German system we have

$\displaystyle D^{(G)}=d_{0}\left(\begin{array}[]{cc}\frac{1}{2}&0\\\
0&0\end{array}\right)\text{ \ \ \
}A^{(G)}=d_{0}\left(\begin{array}[]{cc}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{array}\right)\text{ \ \ \
}Y^{(G)}=d_{0}\left(\begin{array}[]{cc}t&0\\\ 0&\frac{t}{2}\end{array}\right)$
(58) $\displaystyle Q^{(G)}=d_{0}\left(\begin{array}[]{cc}\frac{1}{2}+t&0\\\
0&\frac{1}{2}+\frac{t}{2}\end{array}\right)\text{ \ \ \
}a=\sqrt{d_{0}}\left(\begin{array}[]{cc}0&\frac{1}{\sqrt{2}}\\\
0&0\end{array}\right)\text{ \ \ \ \ \
}a^{\dagger}=\sqrt{d_{0}}\left(\begin{array}[]{cc}0&0\\\
\frac{1}{\sqrt{2}}&0\end{array}\right)$ (65)

These operators will be used in the next section in order to introduce the
linear combinations. We can note that the condition
$\sum\limits_{n=1}^{M}a_{n}=d_{0}$ can be written more compactly as
$Tr(A)=d_{0}$. In turn, the total amount of unit currency paid $\mathbf{Q}$ by
the borrower is $\mathbf{Q}=Tr(Q)$.

Then, in order to generalize the vector space formalism to $M$ dimensions, let
us consider an orthogonal basis of an $M$ dimensional vectorial space where
each orthogonal vector is denoted as $\left|n\right\rangle$ and let us
construct $M$ orthogonal linear combinations

$\left|\varphi_{n}\right\rangle=\sum\limits_{j=1}^{M}c_{j}^{(n)}\left|j\right\rangle\text{
\ \ \ \ \ }j=1,2,...,M$ (66)

where $c_{j}^{(n)}$ are the coefficients of the superposition and obey

$\sum\limits_{j=1}^{M}(c_{j}^{(m)})^{\ast}c_{j}^{(n)}=\delta_{nm}\text{ for
}n\neq m$ (67)

the eq.(66) can be written compactly as
$\left|\varphi\right\rangle=U\left|\varphi_{0}\right\rangle$, where
$\left|\varphi\right\rangle=\left(\left|\varphi_{1}\right\rangle\text{
}\left|\varphi_{2}\right\rangle\text{ }\cdots\text{
}\left|\varphi_{M}\right\rangle\right)^{T}$,
$\left|\varphi_{0}\right\rangle=\left(\left|1\right\rangle\text{
}\left|2\right\rangle\text{ }\cdots\text{ }\left|M\right\rangle\right)^{T}$
and

$U=\left(\begin{array}[]{cccc}c_{1}^{(1)}&c_{1}^{(2)}&...&c_{M}^{(1)}\\\
c_{2}^{(1)}&c_{2}^{(2)}&\cdots&c_{2}^{(M)}\\\ \vdots&\vdots&\ddots&\vdots\\\
c_{M}^{(1)}&c_{M}^{(2)}&\cdots&c_{M}^{(M)}\end{array}\right)$ (68)

The similarity transformation $U$ obeys $U^{T}U=I$ which is obtained by
computing the scalar product
$\left\langle\varphi\mid\varphi\right\rangle=\left\langle\varphi_{0}\right|U^{T}U\left|\varphi_{0}\right\rangle=1$.
The transformation $U$ belongs to the $SO(M)$ group symmetry that preserves
the distance of a vector under rotation. The $SO(M)$ is the special orthogonal
group in $M$ dimensions, and the transformation $U$ is represented by a
$M\times M$ matrix elements whose columns and rows are orthogonal unit vectors
with $\det U=1$ [40]. The $SO(M)$ is a Lie group which can be obtained through
its Lie algebra $so(M)$ by using matrix exponential of the generators of the
algebra [40]. The number of the generators of the Lie algebra $so(M)$ is
$g(M)=M(M-1)/2$ and is the dimension of the parameter space of $SO(M)$. These
parameters are angles, then $so(M)$ is a compact group. The parameter space
dimension of the $SO(M)$ is larger than the loan duration $M$ for $M>3$ which
implies that the rotation transformation in the vector space of dimension $M$
provides a large number of degrees of freedom in order to tune the payment
schedule with better benefits for the borrower. The transformation $U\in
SO(M)$ induces a transformation on any operator $O$ as $\overline{O}=UOU^{-1}$
where $O$ can be $D$, $A$, $Y$, $Q$, $a$ and $a^{\dagger}$. For example, if
$a\left|n\right\rangle=N_{n}\left|n-1\right\rangle$, then
$UaU^{-1}U\left|n\right\rangle=N_{n}U\left|n-1\right\rangle$ and by calling
$\overline{a}=UaU^{-1}$,
$\left|\varphi_{n-1}\right\rangle=U\left|n-1\right\rangle$ and
$\left|\varphi_{n}\right\rangle=U\left|n\right\rangle$, we can write
$\overline{a}\left|\varphi_{n}\right\rangle=N_{n}\left|\varphi_{n-1}\right\rangle$,
where $\left|\varphi_{n-1}\right\rangle$ and $\left|\varphi_{n}\right\rangle$
are two orthogonal vectors obtained from the original orthonormal basis by
rotation, that is

$\displaystyle\overline{a}\left|\varphi_{n}\right\rangle=N_{n}\left|\varphi_{n-1}\right\rangle\text{
\ \ \
}\overline{a}^{\dagger}\left|\varphi_{n}\right\rangle=N_{n+1}\left|\varphi_{n+1}\right\rangle$
(69) $\displaystyle a\left|\varphi_{1}\right\rangle=0\text{ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ }a^{{\dagger}}\left|\varphi_{M}\right\rangle=0\text{ \ \ }$

which means that $\overline{a}$ and $\overline{a}^{\dagger}$ acts as creation
and annihilation operators of loan states in the rotated basis. The loan
operators transform as

$\displaystyle\overline{D}$ $\displaystyle=$ $\displaystyle UDU^{-1}\text{ \ \
\ \ \ }\overline{A}=UAU^{-1}$ (70) $\displaystyle\overline{Y}$
$\displaystyle=$ $\displaystyle UYU^{-1}\text{ \ \ \ \ }\overline{Q}=UQU^{-1}$

The mean values of the transformed loan operators in the original basis read

$\displaystyle\left\langle n\right|\overline{D}\left|n\right\rangle$
$\displaystyle=$ $\displaystyle\overline{d}_{n}\text{ \ \ \ \ \ \ \
}\left\langle n\right|\overline{Y}\left|n\right\rangle=\overline{y}_{n}$ (71)
$\displaystyle\left\langle n\right|\overline{A}\left|n\right\rangle$
$\displaystyle=$ $\displaystyle\overline{a}_{n}\text{ \ \ \ \ \ \
}\left\langle n\right|\overline{Q}\left|n\right\rangle=\overline{q}_{n}$

and give the expected values of the loan magnitudes at each period. Should be
noted that it is meaningless to compute
$\left\langle\varphi_{n}\right|\overline{O}\left|\varphi_{n}\right\rangle$
because
$\left\langle\varphi_{n}\right|\overline{O}\left|\varphi_{n}\right\rangle=\left\langle
n\right|O\left|n\right\rangle$ but $\left\langle
n\right|\overline{O}\left|n\right\rangle=\left\langle\varphi_{n}\right|O\left|\varphi_{n}\right\rangle$.
We can study the formalism for the quantum credit loans with the most simple
examples $M=2$ and $M=3$.

###  3.1 $M=2$

As an example we can consider $M=2$, then it is possible to write the
transformation from the $\\{\left|1\right\rangle,\left|2\right\rangle\\}$ to
the $\\{\left|\varphi_{1}\right\rangle,\left|\varphi_{2}\right\rangle\\}$
basis as

$\left|\varphi_{1}\right\rangle=\cos\phi\left|1\right\rangle+\sin\phi\left|2\right\rangle\text{
\ \ \ \ \ \
}\left|\varphi_{2}\right\rangle=-\sin\phi\left|1\right\rangle+\cos\phi\left|2\right\rangle$
(72)

where $\phi\in[0,2\pi]$,
$\left\langle\varphi_{1}\mid\varphi_{1}\right\rangle=\left\langle\varphi_{2}\mid\varphi_{2}\right\rangle=1$
and $\left\langle\varphi_{1}\mid\varphi_{2}\right\rangle=0$. The change of
basis $\left\\{\left|1\right\rangle,\left|2\right\rangle\right\\}$ to
$\left\\{\left|\varphi_{1}\right\rangle,\left|\varphi_{2}\right\rangle\right\\}$
can be obtained through a similarity transformation $U$ as

$U=\left(\begin{array}[]{cc}\cos\phi&\sin\phi\\\
-\sin\phi&\cos\phi\end{array}\right)$ (73)

The transformation $U$ belongs to the $SO(2)$ group symmetry and can be
considered as a rotation $\phi$ around the $z$ axis. In this case $g(2)=1$ and
the Lie algebra $so(2)$ contains a unique generator
$L=\left(\begin{array}[]{cc}0&i\\\ -i&0\end{array}\right)$ such that the
associated parameter is $\phi$ and $U$ of eq.(73) can be written as
$U=e^{i\phi L}$.666Matrix $L$ is the $\sigma_{y}$ Pauli matrix that represents
the intrinsic angular momentum or spin in the $y$ direction of an electron.
The transformed annihilation operator reads

$\overline{a}=UaU^{-1}=N_{2}\left(\begin{array}[]{cc}-\cos\phi\sin\phi&\cos^{2}\phi\\\
-\sin^{2}\phi&\cos\phi\sin\phi\end{array}\right)$ (74)

Then, by applying $\overline{a}$ to
$\overline{a}\left|\varphi_{2}\right\rangle$ we obtain that
$\overline{a}\left|\varphi_{2}\right\rangle=\left|\varphi_{1}\right\rangle$,
that is, $\overline{a}$ acts as an annihilation operator in the new basis. The
remaining operators read

$\displaystyle\overline{D}=UDU^{-1}=\left(\begin{array}[]{cc}\cos^{2}\phi
d_{1}&\sin\phi\cos\phi d_{1}\\\ \sin\phi\cos\phi d_{1}&\sin^{2}\phi
d_{1}\end{array}\right)\text{ \ \ \ \ \ }$ (77)
$\displaystyle\overline{Y}=UYU^{-1}=\left(\begin{array}[]{cc}\cos^{2}\phi
y_{1}+\sin^{2}\phi y_{2}&\sin\phi\cos\phi(y_{1}-y_{2})\\\
\sin\phi\cos\phi(y_{1}-y_{2})&\sin^{2}\phi y_{1}+\cos^{2}\phi
y_{2}\end{array}\right)$ (80)
$\displaystyle\overline{A}=UAU^{-1}=\left(\begin{array}[]{cc}\cos^{2}\phi
a_{1}+\sin^{2}\phi a_{2}&\sin\phi\cos\phi(a_{1}-a_{2})\\\
\sin\phi\cos\phi(a_{1}-a_{2})&\sin^{2}\phi a_{1}+\cos^{2}\phi
a_{2}\end{array}\right)\text{ \ \ \ \ \ }$ (83) $\displaystyle\text{\ \ \
}\overline{Q}=UQU^{-1}=\left(\begin{array}[]{cc}\cos^{2}\phi
q_{1}+\sin^{2}\phi q_{2}&\sin\phi\cos\phi(q_{1}-q_{2})\\\
\sin\phi\cos\phi(q_{1}-q_{2})&\sin^{2}\phi q_{1}+\cos^{2}\phi
q_{2}\end{array}\right)$ (86)

As we saw in the last section, the mean values $\left\langle
n\right|\overline{D}\left|n\right\rangle$, $\left\langle
n\right|\overline{A}\left|n\right\rangle$, $\left\langle
n\right|\overline{Y}\left|n\right\rangle$, $\left\langle
n\right|\overline{Q}\left|n\right\rangle\,$ are the actualization values for
the debt, amortization, interest and periodic installment. For example, the
amortizations read

$\displaystyle\overline{a}_{1}$ $\displaystyle=$ $\displaystyle\left\langle
1\right|\overline{A}\left|1\right\rangle=\cos^{2}\phi a_{1}+\sin^{2}\phi
a_{2}$ (87) $\displaystyle\overline{a}_{2}$ $\displaystyle=$
$\displaystyle\left\langle
2\right|\overline{A}\left|2\right\rangle=\sin^{2}\phi a_{1}+\cos^{2}\phi
a_{2}$

and the transformed amortization obeys

$\overline{a}_{1}+\overline{a}_{2}=a_{1}+a_{2}=d_{0}$ (88)

as it is expected. This last equation can be obtained for any dimension $M$,
because $Tr(\overline{A})=Tr(UAU^{-1})=Tr(A)=d_{0}$. In turn, the borrower
that takes a loan is interested in the periodic installment schedule, that can
be obtained as

$\displaystyle\overline{q}_{1}$ $\displaystyle=$ $\displaystyle\left\langle
1\right|\overline{Q}\left|1\right\rangle=\cos^{2}\phi q_{1}+\sin^{2}\phi
q_{2}$ (89) $\displaystyle\overline{q}_{2}$ $\displaystyle=$
$\displaystyle\left\langle
2\right|\overline{Q}\left|2\right\rangle=\sin^{2}\phi q_{1}+\cos^{2}\phi
q_{2}$

This result is interesting because the total amount paid is
$\mathbf{Q}=Tr(\overline{Q})=Tr(Q)$, so the basis rotation does not change
what the lender earns. In turn, the basis rotation induces two possible
superpositions of the periodic installments, then, by tuning $\phi$, it is
possible to arrange two different payment schedules
$\overline{q}_{1}<\overline{q}_{2}$ or $\overline{q}_{1}>\overline{q}_{2}$.
The $\phi$ angle can be a parameter of the loan that the borrower can control.
Interestingly, suppose the case of a borrower taking a loan of two periodic
installments $M=2$. Then, suppose that before the first payment, a particular
event does not allow the borrower to afford the first periodic payment. But,
instead of being classified as a defaulter, the borrower can manipulate $\phi$
in order to decrease $q_{1}\rightarrow\overline{q}_{1}$ where
$\overline{q}_{1}<q_{1}$. This can be achieved if the borrower chooses $\phi$
near $\pi/2$ decreasing the weight of $q_{1}$ in $\overline{q}_{1}$ (see
figure 4). Once the basis is rotated, the next periodic installment is
$\overline{q}_{2}$, that increases the weight of $q_{1}$ in
$\overline{q}_{2}$, which means that $\overline{q}_{2}>q_{2}$. In this sense,
the credit loan payments self-regulate when a rotation of the basis is
applied, which is contained in the relation $Tr(Q)=Tr(\overline{Q})$, which
implies for $M=2$, that $q_{1}+q_{2}=\overline{q}_{1}+\overline{q}_{2}$ and by
calling $\Delta q_{1}=q_{1}-\overline{q}_{1}$ and $\Delta
q_{2}=q_{2}-\overline{q}_{2}$, we have $\Delta q_{1}=-\Delta q_{2}$ and when
$q_{1}>\overline{q}_{1}$ then $\overline{q}_{2}>q_{2}$. The manipulation of
$\phi$ opens the possibility to the lender to offer new financial instruments
for credit with the same profit but with better benefits for the borrower than
the usual amortization system. Moreover, in the case of indexed interest rate
with some exogenous variable, such as inflation, the basis rotation of the
credit loan is suitable to control the variations of the periodic payment due
to the fluctuations of the macroeconomy.

There is an inherent risk associated to the uncertainty introduced by the
rotation when the borrower cannot afford the periodic payment. When this
happens, a rotation basis is still available for the borrower, but then, once
the basis is rotated, the payment schedule changes by decreasing the current
payment and increasing the future payment. This inherent risk in the rotation
procedure can be quantified by noting that the basis rotation introduces
superposition of the loan states. One way to quantify this uncertainty is by
computing
$\Delta\overline{Q}_{n}=\sqrt{\overline{q^{2}}_{n}-\overline{q}_{n}^{2}}$
where $\overline{q}_{n}=\left\langle n\right|\overline{Q}\left|n\right\rangle$
and $\overline{q^{2}}_{n}=\left\langle
n\right|\overline{Q}^{2}\left|n\right\rangle$. For example, in the case $M=2$
we obtain

$\displaystyle\Delta\overline{Q}_{1}$ $\displaystyle=$
$\displaystyle\sqrt{(q_{1}-1)q_{1}\cos^{2}\phi+(q_{2}-1)q_{2}\sin^{2}\phi}$
(90) $\displaystyle\Delta\overline{Q}_{2}$ $\displaystyle=$
$\displaystyle\sqrt{(q_{2}-1)q_{2}\cos^{2}\phi+(q_{1}-1)q_{1}\sin^{2}\phi}$

When $\phi=\pi/2$ and $q_{2}<q_{1}$, last equation implies that
$\Delta\overline{Q}_{1}=\sqrt{(q_{2}-1)q_{2}}$ and
$\Delta\overline{Q}_{2}=\sqrt{(q_{1}-1)q_{1}}$, then
$\Delta\overline{Q}_{2}>\Delta\overline{Q}_{1}$. That is, by selecting
$\phi=\pi/2$, the borrower is paying $q_{2}$ and not $q_{1}$ at the beginning
of the loan, but the future risk quantified by $\Delta\overline{Q}_{2}$ has
increased. The fact that the borrower cannot paid $q_{1}$ implies that there
is a potential risk that it will not be able to pay it again (it should be
noted that for $\phi=\pi/2$, $\overline{q}_{1}=q_{2}$ and
$\overline{q}_{2}=q_{1}$). But the borrower pays $q_{2}$ at the beginning of
the loan and has time to collect $q_{1}$ meanwhile, so the risk of not paying
$q_{1}$ at the second period of the loan must be less than the initial risk of
not paying $q_{1}$ at the beginning of the loan. For each rotation basis
chosen at each period of the loan there is a risk flow that should be used as
a complement to the quantum credit loans in order to ensure a measure of the
risk to be used by the lender when the borrower applies a rotation of the loan
states. In Table 1 it can be seen the main difference between the usual credit
loan and the quantum credit loan for $M=2$.

$M=2$ | Debt $D$ | Interest $Y$ | Amortization $A$ | Installment $Q$
---|---|---|---|---
Usual | $d_{1}$ | $y_{1}$ | $a_{1}$ | $q_{1}$
Credit Loan | $0$ | $y_{2}$ | $a_{2}$ | $q_{2}$
Quantum | $d_{1}\cos^{2}\phi$ | $\cos^{2}\phi y_{1}+\sin^{2}\phi y_{2}$ | $\cos^{2}\phi a_{1}+\sin^{2}\phi a_{2}$ | $\cos^{2}\phi q_{1}+\sin^{2}\phi q_{2}$
Credit Loan | $d_{1}\sin^{2}\phi$ | $\sin^{2}\phi y_{1}+\cos^{2}\phi y_{2}$ | $\sin^{2}\phi a_{1}+\cos^{2}\phi a_{2}$ | $\sin^{2}\phi q_{1}+\cos^{2}\phi q_{2}$
$\displaystyle\text{Table 1. Usual credit loan compared with quantum credit
loan for }M=2\text{.}$

Should be stressed that the condition $Tr(\overline{A})=Tr(A)=d_{0}$ implies
that $q_{2}=(1+t)\left[(1+t)d_{0}-q_{1}\right]$, then there is only one
arbitrary parameter $q_{1}$ in a general credit loan with $M=2$ and the
transformed periodic installments can be written as

$\displaystyle\overline{q}_{1}$ $\displaystyle=$
$\displaystyle(\cos^{2}\phi-(1+t)\sin^{2}\phi)q_{1}+(1+t)^{2}\sin^{2}\phi
d_{0}$ (91) $\displaystyle\overline{q}_{2}$ $\displaystyle=$
$\displaystyle(\sin^{2}\phi-(1+t)\cos^{2}\phi)q_{1}+(1+t)^{2}\cos^{2}\phi
d_{0}$

where now the transformed periodic installments are written in terms of the
initial debt taken by the borrower and the first payment. This representation
of the transformed $\overline{q}_{i}$ is suitable to fix $q_{1}=0$, that is,
the initial payment of the non-rotated credit loan vanishes and then
$\overline{q}_{1}=\sin^{2}\phi(1+t)^{2}d_{0}$ and
$\overline{q}_{2}=\cos^{2}\phi(1+t)^{2}d_{0}$ which makes explicit the
modulation in $\phi$ introduced by the rotation in the $\overline{q}_{i}$.

Now, let us consider the known loan systems: the French and German system. In
the French system,
$q_{1}^{(F)}=q_{2}^{(F)}=q^{(F)}=d_{0}\frac{(1+t)^{2}}{2+t}$ and then
$\overline{q}_{1}^{(F)}=\overline{q}_{2}^{(F)}=q^{(F)}$. This is expected
because in the French system, the periodic installment operator is diagonal
$Q^{(F)}I$, then $UQ^{(F)}IU^{-1}=Q^{(F)}$. In the German system,
$q_{1}^{(G)}=\frac{d_{0}}{2}+td_{0}$ and
$q_{2}^{(G)}=\frac{d_{0}}{2}+t\frac{d_{0}}{2}$, then

$\displaystyle\overline{q}_{1}^{(G)}$ $\displaystyle=$
$\displaystyle\left\langle
1\right|\overline{Q}\left|1\right\rangle=\frac{d_{0}}{2}+td_{0}\left(1-\frac{1}{2}\sin^{2}\phi\right)$
(92) $\displaystyle\overline{q}_{2}^{(G)}$ $\displaystyle=$
$\displaystyle\left\langle
2\right|\overline{Q}\left|2\right\rangle=\frac{d_{0}}{2}+td_{0}\left(\frac{1}{2}+\frac{1}{2}\sin^{2}\phi\right)$

Should be noted that, as it happens in the transformed amortization, the sum
of the periodic installments does not depend on $\phi$, that is,
$\overline{q}_{1}+\overline{q}_{2}=q_{1}+q_{2}$. We can transform the German
system with $q_{1}^{(G)}\neq q_{2}^{(G)}$ into a loan system with identical
periodic installments $\overline{q}_{1}^{(G)}=\overline{q}_{2}^{(G)}$. By
using last equation this condition is obeyed when
$\sin\phi=\frac{1}{\sqrt{2}}$ which is satisfied for $\phi=\pi/4$. Replacing
this value in $\overline{q}_{1}^{(G)}$ or $\overline{q}_{2}^{(G)}$ we obtain
$\overline{q}^{(G)}=\frac{d_{0}}{2}(1+\frac{3}{2}t)$. Then by rotating the
basis we can transform a non-constant periodic payment schedule into a
constant periodic payment schedule. Must be stressed that the amortizations in
the German system are constant and remain the same after the $\phi=\pi/4$
rotation. Then, the new loan system is not exactly a French system but a
mixture of both, that is, the transformed loan have constant periodic
installments and constant amortizations but the periodic installment obtained
$\overline{q}^{(G)}=\frac{D_{0}}{2}(1+\frac{3}{2}t)$ is not identical to the
periodic installments of the French system
$q^{(F)}=d_{0}\frac{(1+t)^{2}}{2+t}$. This new mixed French-German system
credit loan cannot be obtained by the usual formalism of eq.(1) because if
$q_{n}$ and $a_{n}$ are constants then $y_{n}$ and $d_{n}$ cannot change. The
relation between quantum credit loans and normal credit loans is analogue to
the relation between quantum game theory and classical game theory, where in
the quantum theory, the set of possible strategies is enlarged by interfering
strategies ([41] and [42]). The same procedure is introduced with the quantum
credit loans by enlarging the set of repayment schedules by, for example,
interfering normal repayment schedules. In turn, a path integral formulation
of credit loans could be constructed with an appropriate Lagrangian, where the
probability amplitude for a particular path in the debt space (two possible
different paths can be with constant amortization and the other with constant
periodic installment) can be calculated with an action $S=\int L(t)dt$.
Interfering paths can be constructed with $e^{iS}$ as it is used in the
Feynman path integral formulation in quantum mechanics or other quantum model
of the stock market ([43] and [44]). Finally, an interesting fact about
rewriting the credit loan in a vector space is that we have the freedom to
choose any linear combination of the states of the loan. There are particular
quantum states in physics that are called coherent states whose dynamics
closely resembles the oscillatory behavior of a classical harmonic oscillator,
and is widely familiar in quantum optics because it describes the maximal
coherence and classical behavior of the quantum electromagnetic field. This is
realized as laser light that emit photons in a highly coherent frequency mode
inside a cavity [45]. In [38], the coherent states of $M=2$ is obtained in
Appendix (eq.A6), where a non-trivial phase appears. In the particular case of
$M=2$, the mean values of the loan operators in the coherent state are
identical to those obtained in eq.(77) but it is expected that for $M>2$,
interesting differences between the rotated and coherent states can be used to
improve the loan properties.

###  3.2 $M=3$

We can go further in the analysis of the basis rotation by considering a
credit loan of duration $M=3$. The orthogonal group $SO(3)$ can be constructed
through its Lie algebra $so(3)$ that contains three generators $L_{1}$,
$L_{2}$ and $L_{3}$ that corresponds to antisymmetric $3\times 3$ matrices in
the adjoint representation [40].777In quantum mechanics, these matrices
represent the orbital angular momentum generators that satisfy the commutation
relations $\left[L_{i},L_{j}\right]=i\epsilon_{ijk}L_{k}$ where
$\epsilon_{ijk}$ is the Levi-Civita completely antisymmetric tensor. The
transformation $U$ can be computed as $U=e^{i\theta L_{1}}e^{i\gamma
L_{2}}e^{i\phi L_{3}}$. Then, by writing
$B=\\{\left|1\right\rangle,\left|2\right\rangle,\left|3\right\rangle\\}$, the
basis rotation reads

$\displaystyle\left|\varphi_{1}\right\rangle=\sin\theta\cos\phi\left|1\right\rangle+\cos\theta\cos\phi\left|2\right\rangle-\sin\phi\left|3\right\rangle$
(93)
$\displaystyle\left|\varphi_{2}\right\rangle=(\sin\theta\sin\phi\sin\gamma+\cos\theta\cos\gamma)\left|1\right\rangle+(\cos\theta\sin\phi\sin\gamma-\sin\theta\cos\gamma)\left|2\right\rangle+\cos\phi\sin\gamma\left|3\right\rangle$
$\displaystyle\left|\varphi_{3}\right\rangle=(\cos\theta\sin\gamma+\sin\theta\sin\phi\cos\gamma)\left|1\right\rangle+(\cos\theta\sin\phi-\sin\theta\sin\gamma)\left|2\right\rangle+\cos\phi\cos\gamma\left|3\right\rangle$

and the similarity transformation $U^{(3)}$ reads

$U^{(3)}=\left(\begin{array}[]{ccc}\sin\theta\cos\phi&\sin\theta\sin\phi\sin\gamma+\cos\theta\cos\phi\cos\gamma&\sin\theta\sin\phi\cos\gamma+\cos\theta\sin\gamma\\\
\cos\theta\cos\phi&\cos\theta\sin\phi\sin\gamma-\sin\theta\cos\gamma&\cos\theta\sin\phi\cos\gamma-\sin\theta\sin\gamma\\\
-\sin\phi&\cos\phi\sin\gamma&\cos\phi\cos\gamma\end{array}\right)$ (94)

where we have three arbitrary angles $\theta,\phi$ and $\gamma$ to manipulate
freely which defines the parameter space of $SO(3)$. The transformation of the
periodic installment operator $\overline{Q}=UQU^{-1}$ allows us to obtain the
mean values $\left\langle n\right|\overline{Q}\left|n\right\rangle$ as

$\displaystyle\overline{q}_{1}=\left\langle
1\right|\overline{Q}\left|1\right\rangle=\sin^{2}\theta\cos^{2}\phi
q_{1}+\sec(2\gamma)\cos^{2}\theta(\cos^{2}\gamma q_{2}-\sin^{2}\gamma q_{3})+$
(95) $\displaystyle\sec(2\gamma)\sin^{2}\theta\sin^{2}\phi(\cos^{2}\gamma
q_{3}-\sin^{2}\gamma q_{2})$ $\displaystyle\overline{q}_{2}=\left\langle
2\right|\overline{Q}\left|2\right\rangle=\cos^{2}\theta\cos^{2}\phi
q_{1}+\sec(2\gamma)\sin^{2}\theta(\cos^{2}\gamma q_{2}-\sin^{2}\gamma q_{3})+$
$\displaystyle\sec(2\gamma)\cos^{2}\theta\sin^{2}\phi(\cos^{2}\gamma
q_{3}-\sin^{2}\gamma q_{2})$ $\displaystyle\overline{q}_{3}=\left\langle
3\right|\overline{Q}\left|3\right\rangle=\sin^{2}\phi
q_{1}+\sec(2\gamma)\cos^{2}\phi(\cos^{2}\gamma q_{3}-\sin^{2}\gamma q_{2})$

Should be noted that when $\gamma=\frac{\pi}{2}$ and $\theta=0$ we recover a
rotation $\phi$ around the $\left|3\right\rangle$ axis. Now there are three
parameters to be tuned in order to satisfy the best periodic payment schedule
for the borrower. By writing $\sin\phi=x$, $\sin\theta=y$ and $\sin\gamma=z$
last equation can be rewritten as

$\displaystyle\overline{q}_{1}=y^{2}(1-x^{2})q_{1}+\frac{1-y^{2}}{1-2z^{2}}\left[(1-z^{2})q_{2}-z^{2}q_{3}\right]+\frac{x^{2}y^{2}}{1-2z^{2}}\left[(1-z^{2})q_{3}-z^{2}q_{2}\right]$
(96)
$\displaystyle\overline{q}_{2}=(1-x^{2})(1-y^{2})q_{1}+\frac{y^{2}}{1-2z^{2}}\left[(1-z^{2})q_{2}-z^{2}q_{3}\right]+\frac{x^{2}(1-y^{2})}{1-2z^{2}}\left[(1-z^{2})q_{3}-z^{2}q_{2}\right]$
$\displaystyle\overline{q}_{3}=x^{2}q_{1}+\frac{1-x^{2}}{1-2z^{2}}\left[(1-z^{2})q_{3}-z^{2}q_{2}\right]$

These set of equations for the periodic payments $\overline{q}_{i}$ as a
function of $x$, $y$ and $z$ provide a large set of degrees of freedom in
order to manipulate the payment schedule.

##  4 Quantum indexed credit loans

In order to introduce the indexed credit loans in terms of the GHA, we can
recall eq.(14)

$d_{n}=(1+t_{n-1})d_{n-1}-q_{n}$

where in this case the interest rate $t_{n}$ depends on $n$. A particular
example of this loan, is the UVA credit loan in Argentina, where the UVA
(Unidad de Valor Adquisitivo [46]) is a new variable and the loan contract is
made in units of UVA and not in Argentinian monetary unit (Argentinian Pesos).
The UVA is related to the Argentinian unit currency as $u_{n}=\alpha\times
1$peso, where $\alpha$ is a number that varies from day to day and is computed
by using the Consumer’s price index (CPI). The procedure to create this
relation between UVA and CPI is by indexing $\alpha$ to the Reference
Stability Coefficient Rate (CER) coefficient [47] which is elaborated by the
Central Bank and depends on the Argentinian inflation.888The stabilized
reference coefficient (CER) is computed daily by using the three previous
Consumer’s price index (CPI) by the formula
$f_{t}=\left(\frac{IPC_{j-2}}{IPC_{j-3}}\right)^{l/k}$ for the initial six
days of the months and $f_{t}=\left(\frac{IPC_{j-1}}{IPC_{j-2}}\right)^{l/k}$
for the remaining days of the months, where $j$ is the current month, $k$ is
the number of days corresponding to the current month and $f_{t}$ is the daily
actual factor. The CER coefficient is computed as $CER_{t}=f_{t}\times
CERt_{t-1}$. In this sense, the CER coefficient is more stable than the
inflation. That is, the units of UVA depend on the Inflation or Consumer’s
Index Price. The UVA credit loan is a French system loan in the UVA units but
the borrower makes the payment in Argentinian monetary unit. Then the amount
to be paid periodically is $q_{n}=u_{n}q$ where $q$ is the periodic payment in
UVA units obtained from the French system and $u_{n}$ is the price of the UVA
unit in term of the Argentinian monetary unit. Then, although the periodic
payment is in UVA units and is constant in time, the amount to be paid with
Argentinian monetary unit varies with time because the UVA price in terms of
Argentinian pesos depends on time. Going further, although the debt in UVA
unit follows the same behavior as the debt in the non-indexed French system,
the debt in Argentinian monetary unit for the indexed loan follows a different
behavior. That is, although the debt $d_{n}$ in UVA unit decreases, the debt
in Argentinian monetary unit is variable. Computing the amount of debt in
Pesos as

$\overline{d}_{n}=d_{n}u_{n}$ (97)

Then it can be shown that $\overline{d}_{n}$ follows the recurrence relation

$\overline{d}_{n}=(1+t)\frac{u_{n}}{u_{n-1}}\overline{d}_{n-1}-u_{n}q$ (98)

which implies an effective indexed interest rate
$t_{n-1}=(1+t)\frac{u_{n}}{u_{n-1}}-1$. If $\frac{u_{n}}{u_{n-1}}=1$,
$t_{n-1}=t$ as it is expected. From eq.(99), when $u_{n}$ increases with $n$
and $d_{n}$ decreases, there is a maximum value that can be obtained through
the equation
$\frac{d}{dn}(\overline{d}_{n})=\frac{d}{dn}(d_{n})u_{n}+d_{n}\frac{du_{n}}{dn}=0$
for a particular value $n$. In figure 3 the behavior of the debt, interest,
amortization and periodic payment is shown as a function of $n$ when the UVA
credit loan follows the French system (left) and when follows the German
system (right). In the figure 3 it can be seen that the debt
$\overline{d}_{n}$ in the German system does not present a peak because the
schedule payment $q_{n}^{(G)}$ decreases with $n$ and balance the increment
due to $u_{n}$.999The debt and periodic payment in the indexed UVA system
using the German system for the UVA units is more suitable for the borrower
than the indexed UVA system using the French system for the UVA units.

Figure 3: Indexed credit loan with $d_{0}=100$, $M=10$, $t=0.2\,$ and
$u_{n}=a^{n}$ where $a=1.1$. Left: French system. Right: German system.

To introduce the indexed UVA credit loans in terms of GHA, we can recall the
function $f(D)$ of eq.(9) that is replaced by $f(D)=(T-I)D$ where $T$ is the
interest rate operator that satisfies $\left[T,D\right]=0$, $I$ is the
identity operator and $T\left|n\right\rangle=t_{n}\left|n\right\rangle$ where
$t_{n}=(1+t)\frac{u_{n+1}}{u_{n}}-1$ in the particular case of the UVA credit
loan. For simplicity we can consider $M=2$ and the transformed periodic
payments read

$\displaystyle\overline{q}_{1}$ $\displaystyle=$ $\displaystyle\cos^{2}\phi
u_{1}q+\sin^{2}\phi u_{2}q$ (99) $\displaystyle\overline{q}_{2}$
$\displaystyle=$ $\displaystyle\sin^{2}\phi u_{1}q+\cos^{2}\phi u_{2}q$

For simplicity we can write $u_{2}=au_{1}$ where $a$ is the inflation rate
between the two periods of the loan and we obtain

$\displaystyle\frac{\overline{q}_{1}}{u_{1}q}$ $\displaystyle=$
$\displaystyle\cos^{2}\phi+a\sin^{2}\phi$ (100)
$\displaystyle\frac{\overline{q}_{2}}{u_{1}q}$ $\displaystyle=$
$\displaystyle\sin^{2}\phi+a\cos^{2}\phi$

then we can find that $\overline{q}_{1}=\overline{q}_{2}$ for $\phi=\pi/4$ and
we have transformed the indexed credit loan into a constant periodic
installment loan (see figure 4), where this constant is
$\overline{q}_{1}(\phi=\frac{\pi}{4})=\overline{q}_{2}(\phi=\frac{\pi}{4})=\frac{1}{2}(1+a)u_{1}q$.

Figure 4: Transformed periodic installments $\overline{q}_{1}/q_{1}$ and
$\overline{q}_{2}/q_{1}$ as a function of $x=\sin\phi$ with $a=1.1$. Vertical
lines indicates the values $x=\pm 1/\sqrt{2}$.

Then we obtain

$\overline{q}_{1}(\phi=\frac{\pi}{4})=(\frac{1}{2}+\frac{a}{2})q_{1}\text{ \ \
\ \ \ \ \ \ \
}\overline{q}_{2}(\phi=\frac{\pi}{4})=(\frac{1}{2}+\frac{1}{2a})q_{2}$ (101)

which in the case $a>1$, $\overline{q}_{1}>q_{1}$ but $\overline{q}_{2}<q_{2}$
as it was shown in the last section. That is, we compensate the extra amount
paid in the first period by paying $\overline{q}_{2}<q_{2}$. For $M=2$ the
angle of the rotation $\phi$ does not depend on $a$. By considering $M=3$ and
considering a constant inflation rate $a$, then $u_{2}=au_{1}$ and
$u_{3}=au_{2}=a^{2}u_{1}$, then we can compute the conditions for $x$, $y$ and
$z$ in order to have the possible configuration $\overline{q}_{1}-q_{1}<0$,
$\overline{q}_{2}-q_{2}<0$ and $\overline{q}_{3}-q_{3}>0$ (figure 5). The
shaded region with the specific values $z=0.6$ and $z=0.7$ indicates possible
values for $x$ and $y$ that allows the configuration to be achieved. The
different schedules can be given by noting that
$\overline{q}_{1}-q_{1}+\overline{q}_{2}-q_{2}+\overline{q}_{3}-q_{3}=0$, then
in the case $\overline{q}_{1}-q_{1}<0$, then
$q_{2}-\overline{q}_{2}<\overline{q}_{3}-q_{3}$, which implies that the
transformed schedule payment contains at least one transformed periodic
payment that is larger than the non-transformed one.

Figure 5: Regions where $\overline{q}_{1}-q_{1}<0$, $\overline{q}_{2}-q_{2}<0$
and $\overline{q}_{3}-q_{3}>0$ holds, with $a=1.05$. Right:
$z=\sin\gamma=0.6$. Left: $z=\sin\gamma=0.7$.

An interesting point to be discussed is that the dependence of $u_{n}$ with
$n$ cannot be known because the inflation rate is actualized periodically and
this value depends on several factors, such as the monetary basis, dollar
price, etc. The similarity transformation $U$ implies that the transformed
periodic payments are a superposition of the original periodic payments, but
these cannot be known at the beginning of the loan because of the lack of
knowledge of $u_{n}$. If for simplicity we consider again $M=3$ and suppose
that the first payment cannot be addressed by the borrower, a rotation is
still available that gives eq.(99) where $u_{1}$ is known but $u_{2}$ and
$u_{3}$ are not known. The customer needs to perform a rotation in such a way
to obtain $\overline{q}_{1}<q_{1}=u_{1}q$. At this point it could be necessary
to consider the future inflation expectations in order, at least, to define a
probability density function $p(u_{2},u_{3})$ that obeys
$\int_{0}^{u_{\max}}\int_{0}^{u_{\max}}du_{2}du_{3}p(u_{2},u_{3})=1$, in such
a way to determine the best option for the customer. By using eq.(96), it can
be seen that
$\overline{q}_{i}=\overline{q}_{i}(q_{1},x,y,z,u_{1},u_{2},u_{3})$, then we
can compute
$\left\langle\overline{Q}_{i}\right\rangle=\int_{0}^{u_{\max}}\int_{0}^{u_{\max}}du_{2}du_{3}p(u_{2},u_{3})\overline{Q}_{i}(q_{1},x,y,z,u_{1},u_{2},u_{3})$,
where $u_{\max}$ is an upper bound for the indexed variable and then consider
the most suitable election of $x$, $y$ and $z$ as a function of $u_{\max}$ for
the borrower. Another solution is to model the future expectation inflation by
some particular function $u_{n}$. In the Argentinian case, by fitting the real
values of $u_{n}$ [46] there are two possible reliable solutions: a power law
fitting $u_{n}=u_{0}e^{\alpha n}$ where $\alpha\sim 0.0109$ and $u_{0}=14.27$
and a linear fitting $u_{n}=u_{0}+0.03n$. Any of these two fittings allow to
rotate the basis of the Hilbert space of the loan in a specific way to obtain,
for example, constant periodic payments, giving predictability to the
borrower.

The general procedure for large $M$, such as mortgage loans, that can be have
a duration of $20$ or $30$ years, implies to develop complex strategies in
order to take advantage of the large matrix transformation $U$ and the
different payment schedules that can be obtained from the equation
$Tr(\overline{Q})=Tr(Q)$. These strategies involve to use the mathematical
machinery of group theory in order to use the subgroups of $SO(M)$ to
introduce selective payment schedules and can contribute to the analysis of
quantum finance interest rate models ([48], [49]). For example, in the case
$M=3$ we can consider a rotation subgroup as

$U=\left(\begin{array}[]{ccc}\cos\phi&\sin\phi&0\\\ -\sin\phi&\cos\phi&0\\\
0&0&1\end{array}\right)$ (102)

which implies not altering the $q_{3}$ payment. This is a particular case of
$SO(2)\subset SO(3)$. A large classification of subgroups of $SO(M)$ can be
found in the literature in terms of cosets, conjugacy classes, etc (see [40]
chapter 19 to 25). From the perspective of the Lie algebra $so(M)$, the
subgroups can be found by exponential map of the specific generators of the
rotation. In turn, using the Lie algebra it is possible to obtain, for example
in the case $M=4$ that $so(4)=so(3)\oplus so(3)$. By computing the exponential
map of the generators of each $so(3)$ it is possible to obtain specific
subgroup of rotations according to the customer needs. This procedure can be
generalized for large $M$ without complications, although implementing a
software that allows to manipulate the entire parameter space of the Lie
algebra and to select specific subgroups of $SO(M)$ through the Lie algebras
should be extremely necessary. To make a final remark, the generalized
Heisenberg algebra is suitable to be applied to quantum finance related to
index-linked coupons that depends on market defined index [50]. In turn,
diverse Hamiltonians are defined to model the stock market. For example, in
[51] in which a quantum anharmonic oscillator is used as a model for the stock
market. The motion of the stock price is modeled as the dynamics of a quantum
particle and the probability distribution of price return is computed with the
wave function obtained from the Scrhodinger equation with an anharmonic
oscillator potential. The Black-Scholes equation can be mapped into the
Schrodinger equation (see [52] and [53]) and a Hamiltonian operator with
potential can be constructed. Interesting approaches using non-Hermitian
Hamiltonians are given in [54]. In [55] a finite dimensional Hilbert space is
constructed to model the stock market that isomorphic to $\mathbb{C}^{d}$,
where $d$ is the discrete number of possible rates of return. In [39] all
possible realizations of investors holding securities and cash is taken as the
basis of the Hilbert space of market states. The temporal evolution of an
isolated market is unitary in this space. In [56], the quantum framework is
used to model supply and demand. The different potentials appearing in the
Hamiltonians of the different models of the stock market can be studied using
the generalized Heisenberg algebras (see [57]). Then, having access to the
algebra of observables of a quantum system allows us to design better general
quantum models for stock price or financial instruments instead of using a
particular representation of the wavefunction (the coordinate representation)
that is not more than a particular representation of the underlying algebra.

Finally, should be stressed that the commutative operators chosen to define
the loan systems (eq.(9)a) is suitable to generalize via non-commutative
operators, which can be used to capture the impossibility of a joint
measurement of these quantities and hence can explain the existence of order
effects and deeper levels of ambiguity in respect to the loan realization. In
turn, the quantum formalism introduced for credits is suitable to consider
repackaging of different types of loans such as credit loans, credit card
debt, business loans and mortgage into pools, because these pools can be
modeled as tensor product of the algebras. The symmetry group of the tensor
product of the algebras will allows to construct entangled loan states,
without changing the return because
$Tr(\overline{Q}_{1}\otimes\overline{Q}_{2}\otimes...\otimes\overline{Q}_{n})=Tr(Q_{1}\otimes
Q_{2}\otimes...\otimes Q_{n})$.

##  5 Conclusions

In this work we have shown a generalization of a credit loan by introducing
linear operators for the debt, amortization, interest and periodic
installments that act on a vector space of dimension $M$, where $M$ is the
loan duration. We have shown that endowing this vector space with an $SO(M)$
symmetry, a basis rotation of the orthonormal basis allows us to change the
schedule of the periodic payments, allowing better benefits for the borrower
in the case a payment cannot be afford or any other circumstance in which a
default of the remaining debt is possible. The values of the loan operators
are computed as mean values of these operators in any of the orthogonal
vectors of the eigenbasis of the vector space. In turn, the rotation basis
does not change the total amortization and the total amount paid by the
borrower, which is expected in order not to change what the lender earns and
simultaneously not to change what the borrower amortizes. In turn, it was
shown that a generalized Heisenberg algebra for the loan operators can be
defined that gives the usual recurrence relations for the debt, amortization,
interest and periodic installments. Finally, the case of indexed credit loans
is studied, where the rotation basis provides a sophisticated solution when
the periodic installments depend on exogenous variables with a non-predictive
behavior. Some particular cases with $M=2$ and $M=3$ are studied showing how
the rotation transformation can be applied to design specific schedule
periodic payments.

##  6 Acknowledgments

This paper was partially supported by grants of CONICET (Argentina National
Research Council) and Universidad Nacional del Sur (UNS) and by ANPCyT through
PICT 1770, and PIP-CONICET Nos. 114-200901-00272 and 114-200901-00068 research
grants, as well as by SGCyT-UNS., J. S. A. is a member of CONICET.

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