Estimating Loan’s Contribution using Euler’s theorem
We will cover following topics
Introduction
In the realm of credit risk assessment, understanding the individual contribution of each loan within a portfolio to the overall risk is essential. Euler’s theorem provides a powerful tool to dissect and quantify this contribution, offering insights into the risk dynamics of a portfolio. By comprehending Euler’s theorem and its application, financial institutions can make informed decisions regarding risk management strategies. In this chapter, we delve into Euler’s theorem and its significance in determining the contribution of a loan to the overall risk of a portfolio.
Euler’s Theorem: A Brief Overview
Euler’s theorem, also known as the Euler equation, offers a way to assess the marginal impact of each asset within a portfolio on the portfolio’s overall risk. It is especially valuable in credit risk assessment, where understanding the risk contributions of individual loans is crucial.
Formula and Application
Euler’s theorem can be mathematically expressed as follows:
$$dV=\sum_{i=1}^n w_i dR_i+\frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n w_i w_j \rho_{ij} \sigma_i \sigma_j d\rho_{ij}$$
Where:
- $d V$ represents the change in portfolio value due to small changes in risk factors
- $w_i$ is the weight of the ith asset in the portfolio
- $d R_i$ is the change in the ith asset’s return
- $\rho_{i j}$ is the correlation coefficient between the returns of assets $i$ and $j$
- $\sigma_i$ and $\sigma_j$ represent the standard deviations of the returns of assets $i$ and $j$
- $d \rho_{i j}$ is the change in the correlation between the returns of assets $i$ and $j$
Example: Let’s consider a portfolio with two loans, Loan A and Loan B. Loan A has a weight of 60%, Loan $B$ has a weight of 40%, and the correlation coefficient between their returns is 0.5. The standard deviations of Loan A and Loan B returns are 0.12 and 0.08, respectively.
Suppose a small change in Loan A’s risk leads to a change in value $(dV)$ of USD 1,200, and a similar change in Loan B’s risk leads to a change in value of USD 800. Using Euler’s theorem, we can calculate the risk contributions as follows:
$$\begin{aligned} &dR_A=\frac{d V}{w_A}=\frac{1200}{0.60}=2000 \\ & dR_B=\frac{d V}{w_B}=\frac{800}{0.40}=2000 \end{aligned}$$
Conclusion
Euler’s theorem provides a profound understanding of how individual loans within a portfolio contribute to the overall risk. By considering both changes in asset returns and the interactions between asset returns, financial institutions can enhance their risk management strategies. Accurately assessing risk contributions empowers institutions to make informed decisions about asset allocation and risk mitigation, ultimately bolstering the stability and resilience of their portfolios. Through Euler’s theorem, the complex landscape of portfolio risk becomes navigable, offering valuable insights for effective credit risk management.