# 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 $$Pr(portfolio.default) = 1 - (q_1 \times ... \times q_{13})$$