Estimating Portfolio Default Probability Under Dependence Assumptions

Summary

This question asks how to estimate the chance that at least one borrower in a portfolio defaults when companies have different default probabilities. Under independence, the probability of no defaults is the product of the individual survival probabilities; subtracting that product from one gives the probability of at least one default. Because the individual probabilities differ, the number of defaults follows a Poisson-binomial distribution rather than an ordinary binomial distribution. That distribution can describe the full count of defaults, while the product formula directly answers the at-least-one question.

The question also raises scenarios with positive or negative dependence, but does not specify a joint probability model or provide a solution for them. Dependence can materially change portfolio default risk, and individual default probabilities alone do not determine the joint outcome. Its proposed scenario in which one default entails all others also needs a coherent joint structure, particularly if borrower probabilities differ. The note motivates careful assumptions, but gives no numerical estimates or bounds for correlated defaults.

Key ideas

• Under independence, the probability of at least one default is one minus the product of all survival probabilities.
• With unequal default probabilities, the count of defaults is modeled by a Poisson-binomial distribution.
• The probability of at least one default differs from the full distribution of the number of defaults.
• Individual default probabilities alone do not determine portfolio risk when defaults are dependent.
• A dependence assumption must be specified coherently, especially when borrower probabilities differ.

Tags

# Portfolio diversification on default risk


# Portfolio diversification on default risk












A portfolio of 13 different companies have loans. Company $i$ default on their loan with probability $p_i$ and survive with prob $q_i=1-p_i$. Let $Y_i=1$ denote default. Question: How could I get to a reasonable guess, preferrably using probability theory, on what is the probability of default for the portfolio?

I am pondering about the risk on a loan to the entire portfolio. I am free to make assumptions and be creative, for example not find an exact solution but bound the probability by some interval. I make three scenarios, each with different assumptions

- I) If one defaults, all other defaults.

- II) The firms are independent

- III) They are positively/negatively correlated in the sense that if one falls, it is a higher/lower probability of the other falling.

## I)

Pr(portfolio.defaults) = Pr(someone defaults) = 1 - Pr(no one defaults). Here my thinking stops, can I

## II)

Here it is like a binomial with $n=13$ although the probabilities $p$ are not the same but are changing. I could make an upper bound and say it is equal to $Bin(n=13, p=max[p_i])$ but surely there is a distribution for this? Maybe I could use the PDF of a Bin directly via ${\displaystyle \Pr(Y_i=k)={\binom {n}{k}}p_i^{k}(1-p_i)^{n-k}}$

## I) and II)

We have that Pr(portfolio.default) = by assummption I = Pr(someone defaults) = 1 - Pr(no one defaults). Since they are independent by assumption II we have $$Pr(portfolio.default) = 1 - (q_1 \times ... \times q_{13})$$

## III)

I have not started thinking about this scenario.

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