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).