Credit Portfolio Diversification
Overview
As with any portfolio of assets, a portfolio of instruments subject to credit risk benefits from deiversification of risk.
To show a simple example, we take a Portfolio to be the sum of a set of loans.
{% P = \sum _{i=1}^n \frac{1}{n} L_i %}
Here {% P %} is the value of the portfolio, and {% L_i %} is the value of the ith loan.
Then the expected value of the portfolio is given by
{% \mathbb{E}[P] = \sum _{i=1}^n \frac{1}{n} \mathbb{E}[L_i] %}
(see
expectation of a random variable)
Portfolio Variance
The portfolio
variance
is then given by
{% Var(P) = \sum_{i,j} (\frac{1}{n})^2 \times Cov(L_i, L_j) %}
If the loan defaults are uncorrlated, then {% Cov(L_i, L_j) = 0 %}
for {% i \neq j %}.
{% Var(P) = \sum_i (\frac{1}{n})^2 \times Var(L_i) %}
which goes to zero as {% n \rightarrow \infty %}
Identical Loans
When the loans are identical (that is drawn from identical distributions, but not independent)
{% Var(P) = \sum_{i,j} (\frac{1}{n})^2 \times Cov(L_i, L_j) %}
{% Var(P) = (\frac{1}{n})^2 \sum_i Var(L) + (\frac{1}{n})^2 \sum_{i \neq j} Cov(L_i, L_j) %}
{% = (\frac{1}{n}) Var(L) + (\frac{n-1}{n}) Cov(L_i, L_j) %}
When the loan defaults are uncorrelated
{% Var(P) = (\frac{1}{n})^2 \sum_i \times Var(L) = \frac{1}{n} Var(L) %}
which goes to zero as {% n \rightarrow \infty %}
otherwise
{% Var(P) \rightarrow Cov(L_i, L_j) %}
Loan Pricing and CAPM
The
Capital Asset Pricing Theory
asserts that the market only prices risk that cannot be diversified away.
In the case of a defaultable loan, when the loan default is uncorrelated with other assets, then the loan price should be
{% Price = discount \times \mathbb{E}(L) %}
that is, the price of the loan is just the discounted expected value of the loan. (here, expectation is with respect to the
real world measure, not the risk neutral measure).
When the loan default is correlated to the market of assets (in particular, other loan defaults), the market
will assign a value to undiverisifiable risk.
{% Price = discount \times \mathbb{E}(L) + risk \, premium %}
Typically, the undiverisifiable risk can be connected to a latent variable or set of latent variables.
(See
Latent Variables)
