Methodology
The methodology used in this paper seeks to use mathematical relations
to investigate. In this section, many of the mathematical definitions
and processes used in the results and findings section will be proved
and explained, if necessary. This paper sets the loans that have already
defaulted as the \(\varphi\)variable. Thus, by definition, the
total amount of nonperforming loans at a specific bank, represented by\(\alpha\), would be all the non-performing loans of that bank summed.
This is represented by the equation below:
\begin{equation}
{\alpha=\varphi}_{1}+\varphi_{2}+\varphi_{3}+\ldots+\varphi_{n}\nonumber \\
\end{equation}
However, NPL ratios include performing loans that are close to or at
risk of defaulting, as well as reperforming loans that are close to or
at risk of re-defaulting. Thus, the equation for the NPL ratios explored
in this study also has to include that variable. If the variable\(\varepsilon\) is set to represent a performing or reperforming loan
that is at risk of becoming non-performing, then the total amount of
these loans, represented by the variable \(\beta\) can be expressed by
the following equation below:
\begin{equation}
{\beta=\varepsilon}_{1}+\varepsilon_{2}+\varepsilon_{3}+\ldots+\varepsilon_{n}\nonumber \\
\end{equation}
All performing loans, as well as reperforming loans not at risk of
defaulting and non-performing loans that are expected to become
reperforming in the near future, can be represented by ρ and
the \(\gamma\) variable representing the total amount. Thus:
\begin{equation}
\gamma=\rho_{1}+\rho_{2}+\rho_{3}+\ldots+\rho_{n}\nonumber \\
\end{equation}
To make the mathematical calculations later in this paper easier to
understand, the variable ω is set to represent the total
amount (both non-performing and performing) of loans at a given bank (or
in a given economy) at any given time. Thus:
\begin{equation}
\omega=\alpha+\beta+\gamma\nonumber \\
\end{equation}
With these equations and variables above defined, it can be derived that\(\partial=\frac{\alpha+\beta}{\omega}\) where \(\partial\) is the
NPL ratio of the bank and where \(0\leq\partial\leq 1\) , which means
that \(\alpha+\beta\leq\omega\).
Using these equations, we can set \(f(\alpha+\beta)\) as the function
of total amount of effective non-performing loans at a bank and set\(f(\omega)\) as the function that defines the total amount of loans at
the bank (nonperforming, performing, and reperforming). National
treasuries often report NPL growth in percentage and relative growth
rate. This is represented by \(\mu\) in this paper. To calculate the
absolute growth within a given period of time based on the growth rate,
this formula must be used:
\begin{equation}
\text{aggregate\ growth}=\ {f\left(\alpha+\beta\right)}_{i}\nonumber \\
\end{equation}
Where \({f\left(\alpha+\beta\right)}_{i}\) is the initial amount of
total non-performing loans, \(f(\mu)\) is the growth rate of \(\mu\),n is the number of years in the time period, and i is the
number of additional doublings that the growth rate incurs, if any (due
to outside influence). With these definitions proved, the rest of this
following section will be dedicated to introducing (as well as proving)
other theorems and formulas used in paper’s results/findings section.
It should also be noted that the above equations can be summarized for
simple NPL cases by the following:
\begin{equation}
\text{Assuming\ }x=\alpha+\beta;\ y=\omega;\text{and}\ z=f(\mu)\nonumber \\
\end{equation}\begin{equation}
\downarrow\nonumber \\
\end{equation}\begin{equation}
x=\alpha+\beta{=\varphi}_{1}+\varphi_{2}+\varphi_{3}+\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\ldots+\varphi_{n}+\varepsilon_{n}\nonumber \\
\end{equation}\begin{equation}
\downarrow\nonumber \\
\end{equation}\begin{equation}
\partial=\frac{\alpha+\beta}{\omega}=\frac{x}{y}\nonumber \\
\end{equation}\begin{equation}
\mathbf{\downarrow}\nonumber \\
\end{equation}\begin{equation}
simple\ NPL\ aggregate\ growth=f\left(x\right)+\text{rate\ of\ growth\ of\ }f(x)\nonumber \\
\end{equation}
The simple rate of growth for the aggregate growth rate would be simply
the derivative
The above expression written out verbally would be:
\begin{equation}
\text{Simple\ NPL\ Ratio}=\frac{total\ defaulted\ loans+\left(\text{re}\right)\text{performing\ loans\ at\ risk\ of\ defaulting}}{\text{total\ amount\ of\ loans\ at\ bank}}\nonumber \\
\end{equation}
A simple calculation of the above statement is found commonly in this
paper. For example, a bank that has a portfolio of $600 million. The
bank has $80 million worth of non-performing loans and an additional
$25 million worth of performing loans that are at risk of becoming
non-performing. Its simple NPL ratio would be:
\begin{equation}
\text{Simple\ NPL\ Ratio}=\frac{\$80,000,000+\$25,000,000}{\$600,000,000}=\frac{\$105,000,000}{\$600,000,000}=0.175=17.5\%\nonumber \\
\end{equation}
The main calculation done in this paper will include or choose to not
include units per the situation revolving the problem at hand. If the
problems does call for it, it will do.
This method of calculation will be used to derive all NPL ratios in this
paper unless another method is otherwise stated. Mathematical reasoning
and work may be grouped into one step as repeating the same work
complicates the mathematics in the paper.
With all the examples and mathematical definitions and proofs given, the
next section will cover a review of literature in the topic of NPL
ratios and previous research done on the topic. Please note that this
method will not be repeated in subsequent sections, all mathematical
work done will be based on simple assumptions, logic, and the methods
presented in this section. If a new method arises but was not introduced
in this section, a short introduction will be made (but that method most
likely will not be relevant).