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:
ΔX_i = small change in variable i
ΔF_i = resultant small change in F
Euler showed that as X_i gets smaller, the risk function F simplifies to the sum of the individual Q_i 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.

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