Company Insolvency Risk

Solution & Explanation

To solve the problem of finding the probability that company X will become insolvent before reaching time T, we need to delve into the concept of conditional probability and exponential distribution.

Definitions and Assumptions

• A: Event where company X goes bankrupt between time T and T+dT.
• B: Event where company X does not go bankrupt until time T.
• B': Event where company X goes bankrupt before time T.
• K: The constant representing the rate of insolvency per unit time.

The problem states that the likelihood of company X becoming insolvent between T and T+dT, assuming it remains solvent until T, is given by:

$P(A|B) = K \cdot dT$

Probability Theory

1. Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

   Given: $P(A|B) = K \cdot dT$

2. Complementary Probability: The probability of the complement of B, denoted as B', is given by: $P(B') = 1 - P(B)$

Solution Steps

1. Rearrange the Conditional Probability Formula: Given: $K \cdot dT = \frac{P(A \cap B)}{P(B)}$

   Rearrange to express $P(B)$: $P(B) = \frac{P(A \cap B)}{K \cdot dT}$

2. Find P(B'): Since $P(B') = 1 - P(B)$, substitute the expression for $P(B)$:

   $P(B') = 1 - \frac{P(A \cap B)}{K \cdot dT}$

   This equation gives the probability that company X will become insolvent before reaching time T.

Exponential Distribution Insight

The problem can also be viewed through the lens of an exponential distribution, which models the time until an event occurs (such as insolvency) and is characterized by:

• Probability Density Function (PDF): $f(t) = K \cdot e^{-Kt}$

• Cumulative Distribution Function (CDF): $P(T \leq t) = 1 - e^{-Kt}$

• Survival Function (Complement): $P(T > t) = e^{-Kt}$

For our problem, the probability that company X becomes insolvent before time T is equivalent to:

$P(B') = 1 - e^{-KT}$

This formulation aligns with the exponential distribution's memoryless property, where the probability of insolvency within a future interval is independent of the past, given the current state.

Conclusion

The probability that company X will become insolvent before reaching time T can be derived using both conditional probability concepts and exponential distribution insights. The key takeaway is understanding the relationship between the rate of insolvency, time intervals, and the nature of the distribution governing the event's occurrence.