Marginal Distributions and Joint Probability simply explained

When exploring the probability of events, such as bond defaults, it's critical to grasp that knowing the individual likelihoods, or marginal distributions, of each bond defaulting does not inherently inform us about the likelihood of multiple bonds defaulting at the same time. This concept is key because even if two sets of bonds have identical marginal probabilities, their joint probabilities can differ dramatically based on how their defaults are correlated.

Marginal distributions tell us about the individual behavior of each variable (in this case, each bond's default probability) without regard to any other variables. For example, if Bond 1 and Bond 2 both have a 50% chance of defaulting, these are their marginal probabilities. However, these probabilities don't reveal how Bond 1's default might be influenced by or correlate with Bond 2's default.

The joint distribution, however, tells us about the probability of various combinations of these events occurring simultaneously. If bonds are independent, as in Scenario 1, their joint probability distribution simply reflects the product of their marginal probabilities. Here, the joint probability table reflects this independence:

| B_1B_2 | 0 (No Default) | 1 (Default) | P(B_1) |
|---------|---------------|------------|--------
| 0 | 0.25 | 0.25 | 0.5
| 1 | 0.25 | 0.25 | 0.5
| P(_2) | 0.5 | 0.5

However, if we consider Scenario 2, where Bond 3 and Bond 4 are perfectly correlated, the joint probability distribution changes dramatically, despite the marginal probabilities remaining the same as in the first scenario.

In this case, the default of Bond 3 guarantees the default of Bond 4, and vice versa, leading to a joint distribution that focuses on the extremes, with no probability spread between them:

| B_3B_4 | 0 (No Default) | 1 (Default) | P(B_3)
|----------|---------------|------------|-----
| 0 | 0.5 | 0 | 0.5
| 1 | 0 | 0.5 | 0.5
| P(B_4) | 0.5 | 0.5

This contrast between Scenarios 1 and 2 illustrates a fundamental statistical truth: identical marginal distributions do not guarantee the same joint distribution. The underlying dependency structure, which can be modeled using copulas, plays a crucial role in shaping the joint distribution.

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