On the Application of Sample Coefficient of Variation for Managing Loan Portfolio Risks

Abstract

Banks and financial institutions are exposed with credit risk, liquidity risk, market risk, operational risk, and others. Credit risk often comes from undue concentration of loan portfolios. Among the diversity of tools available in literature for risk measurement, in our study the Coefficient of Variation (CV) was chosen taking into account that it reveals a very useful characteristic when loan portfolios comparison is desired: CV is unitless—it is independent of the unit of measure associated with the data. We obtain the lower and upper bounds for sample CV and the possibility of using it for measuring the risk concentration in a loan portfolio is investigated. The capital adequacy and the single borrower limit are considered and some theoretical results are obtained. Finally, we implement and illustrate this approach using a real data set.

Appendix

Let us first recall the Lyapanov Theorem.

Theorem 4

Let X ₁, …, X _n be independent random variables with different distribution. If

• (a) $E\left [\left |X_{i} -E(X_{i} )\right |{ }^{3} \right ]<\infty$

• (b) $\mathop {\lim } \limits _{n\to \infty } \frac {\sum _{i=1}^{n}E\left [\left |X_{i} -E(X_{i} )\right |{ }^{3} \right ] }{\left (\sum _{i=1}^{n}var(X_{i} ) \right )^{3/2} } =0$ ,

then, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\sum _{i=1}^{n}X_{i} -\sum _{i=1}^{n}E(X_{i} ) }{\sqrt{\sum _{i=1}^{n}var(X_{i} ) } } \to N(0,1). \end{array} \end{aligned} $$

Proof

See, e.g., Feller [9].

In order to apply this Theorem for L, we must check conditions (a) and (b). For the first condition, we have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} E\left[\left|L_{i} -E(L_{i} )\right|{}^{3} \right]=E\left[\left|l_{i} B_{i} -l_{i} p\right|{}^{3} \right]=l_{i}^{3} p(1-p)\left(p^{2} +(1-p)^{2} \right)<\infty. \end{array} \end{aligned} $$

For condition (b), we have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \mathop{\lim }\limits_{n\to \infty } \frac{\sum _{i=1}^{n}E\left[\left|L_{i} -E(L_{i} )\right|{}^{3} \right] }{\left(\sum _{i=1}^{n}var(L_{i} ) \right)^{3/2} } \\ &\displaystyle =&\displaystyle \mathop{\lim }\limits_{n\to \infty } \frac{\sum _{i=1}^{n}l_{i}^{3} p(1-p)\left(p^{2} +(1-p)^{2} \right) }{\left(\sum _{i=1}^{n}l_{i}^{2} p(1-p) \right)^{3/2} } \\ &\displaystyle =&\displaystyle \frac{\left(p^{2} +(1-p)^{2} \right)}{\sqrt{p(1-p)} } \mathop{\lim }\limits_{n\to \infty } \frac{\sum _{i=1}^{n}l_{i}^{3} }{\left(\sum _{i=1}^{n}l_{i}^{2} \right)^{3/2} } \\ &\displaystyle \le&\displaystyle {\frac{\left(p^{2} +(1-p)^{2} \right)}{\sqrt{p(1-p)} } \mathop{\lim }\limits_{n\to \infty } \frac{n\max l_{i} }{\left(n\min l_{i} \right)^{3/2} } =0}. \end{array} \end{aligned} $$

Therefore, using Lyapanov theorem, we can conclude that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{L-E\left(L\right)}{\sqrt{var(L)} } \to N(0,1). \end{array} \end{aligned} $$