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# Euler’s Theorem

## What is Euler’s Theorem?

Euler’s theorem allows homogeneous risk functions in a portfolio to be decomposed into component contributions. This is valuable when we want to know how much each underlying loan contributes to the overall loan portfolio risk.A set of risk variables for functions F and a constant can be expressed as the homogeneous function.

## An example of Euler’s Theorem:

$Q_{i}\, = \, X_{i}\, \frac{\Delta F_{i}}{\Delta X_{i} }\, = \, \frac{\Delta F_{i}}{\frac{\Delta X_{i}}{X_{i}}}$

where: ΔXi = small change in variable i ΔFi = resultant small change in F Euler showed that as Xi gets smaller, the risk function F simplifies to the sum of the individual Qi components:

$F\, = \, \sum_{i=1}^{n}\, Q_{i}$

Consider a portfolio with three loans. Suppose that the standard deviation of losses for loan(1) = 1.2, loan(2) = 0.8 and loan(3) = 0.8. The correlation matrix for the three loans is given in the following table.
 Loan(1) Loan(2) Loan(3) Loan(1) 1 0 0 Loan(2) 0 1 0.6 Loan(3) 0 0.6 1
We can determine the portfolio’s standard deviation of total losses as:

$\sqrt{(1.2^{2} + 0.8^{2} + 0.8^{2} + 2 \times 0.6 \times 0.8 \times 0.8)}\, = \, 1.87$

We can then decompose this total risk into the individual contributions of the three loans. In order to do that, suppose that the size of loan(L) increased by 1%. The Standard deviation of the loss from loan(L) would then increase to 1.2 x 1.01 = 1.212 We now calculate the increase in the standard deviation of the loan portfolio:$\sqrt{(1.212^{2} + 0.8^{2} + 0.8^{2} + 2 \times 0.6 \times 0.8 \times 0.8)}$ – $\sqrt{(1.2^{2} + 0.8^{2} + 0.8^{2} + 2 \times 0.6 \times 0.8 \times 0.8)}\, = \, 0.007733$From our earlier description of Qi, QL is then 0.007733 / 0.01 = 0.7733 Similarly, Q2 = 0.5492 & Q3 = 0.5492. Thus Q1 + Q2 + Q3 = 0.7733 + 0.5492 + 0.5492 = 1.87
Owais Siddiqui