Summary statistic for the average probability of default?

Question

I have the following scenario:

Let $X_i$ denote the event where some institution $i$ 'defaults' (don't worry about the exact definition of a default here, it is not relevant to the question at hand). Now, I have 10 institutions in my sample and I have calculate the following probabilities:

$P(X_1 \cap X_2 \cap \cdots \cap X_9 | X_{10})$

$P(X_1 \cap X_2 \cap \cdots \cap X_{8} \cap X_{10} | X_{9})$

$\vdots$

$P(X_2 \cap \cdots \cap X_{9} \cap X_{10} | X_{1})$

In words, the above expressions represent the (joint) probability of default of the 'remaining' institutions given that a particular institution has defaulted.

To calculate these probabilities, I know the underlying probability distribution that describes the entire system, i.e., $p(x_1, x_2, \cdots. x_{10})$, so for example, calculating the probability $P(X_1 \cap X_2 \cap \cdots \cap X_9 | X_{10})$ just simply requires me to find $\frac{P(X_1 \cap \cdots \cap X_{10})}{P(X_{10})}$ where both the denominator and numerator can be calculated by integrating over certain regions of the probability density function $p(x_1, x_2, \cdots. x_{10})$.

My question is, I wish to find one summary value that describes the average probability of default of this system, e.g. say $P(X_1 \cap X_2 \cap \cdots \cap X_9 | X_{10})= 0.4$, $P(X_1 \cap X_2 \cap \cdots \cap X_{8} \cap X_{10} | X_{9})= 0.3$ etc, how can I "combine" this $0.4$, $0.3$, etc into one value that describes the "average" probability of default of this system? My initial method is just to take the arithmetic average of each conditional probability, but that is mathematically incorrect conditional probabilities aren't summable (except when they are conditioned on the same event). So are there any other measures/techniques I can use to somehow "combine" these single probabilities into "one" value?

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Let me put this into context to make things more concrete.

Pretend each event is the event where a bank defaults. By default, I mean that the bank's assets drop below some pre-determined threshold. Then, say we have a sample of 10 banks. The entire "system" is the universe of these 10 banks. I want to find what is the "contribution" of each bank's default on the rest of the system, that is, given that one bank defaults, how does that affect the probability of default of the rest of the system (i.e., the remaining 9 banks). To do this, I have modeled the underlying (joint) asset distribution of this system of banks. Then, computing the probability $P(X_1 \cap X_2 \cap \cdots \cap X_9 | X_{10})$ represents the "contribution" of the default of the 10th bank on the rest of the system. Similarly, $P(X_1 \cap X_2 \cap \cdots \cap X_{8} \cap X_{10} | X_{9})$ represents the impact of the default of the 9th bank on the rest of the system. Now given I have computed the "contribution" of each bank's default, I want to find a single value that describes the average "contribution" of each bank's default on the rest of the system.

Answer 1

I have an Idea perhaps it helps you a bit (even though it deviates somewhat from your original setup). Let's assume you know the "anaffected" default probabilities for each bank $P(X_1<=C_1), \dots, P(X_n<=C_n)$. (Here I assumed that bank $i$ defaults when it's value falls below a certain value $C_i$)

Now e.g. for bank $n$ you can calulate $P(X_1<=C_1|X_n<=C_n), \dots, P(X_{n-1}<=C_{n-1}|X_{n}<=C_{n})$. Thus the default probabilites conditioned on the default of bank $n$.

Now you could calculate $d_i(n)=P(X_1<=C_1|X_n<=C_n) - P(X_1<=C_1)$ for $i\in\{1,\dots , n-1\}$ to determine whether the default probability of $X_1$ increased or decreased after the default of $X_n$.

Now add some weights $(w_1, \dots , w_{n-1})$ bcause the default of one bank might be seen as having more impact than the default of another.

Finally by taking $\sum_{i=1}^{n-1} w_i d_i(n)$ you would get a number telling you how the default of $X_n$ "impacts" the system.

Here I still haven't used all the "correlation-information" into account (some of it is already contained in the probablities $P(X_1<=C_1|X_n<=C_n)$) This information (or at least some of it) could also be included into the weights $w_i$.

One way could be to calculate the average number of banks that would default if e.g. $X_i$ defaulted. To do that you will have to calculate the conditional expectation $\mathbb{E}[ \sum_{j=1,j\neq i}^{n} 1_{X_j<=C_j}|X_i<=C_i]$.

Setting

$w_i=\mathbb{E}[ \sum_{j=1,j\neq i}^{n} 1_{X_j<=C_j}|X_i<=C_i]$

will give you weights that incoporate more information on the correlation among the banks.