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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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  • $\begingroup$ What do you mean default of a portfolio? $\endgroup$ – Gordon Dec 16 '16 at 13:52
  • $\begingroup$ The portfolio is in default if the enterprise value of the sum of the companies is less than a certain threshold. $\endgroup$ – jacob Dec 19 '16 at 10:22
  • $\begingroup$ You should add this to your question to make it clear. $\endgroup$ – Gordon Dec 19 '16 at 13:36

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