"Derivatives
In finance, a derivative is a financial asset whose value is determined (derived) from a bundle of various assets, such as mortgages. Suppose a randomly selected mortgage has a probability of 0.01 of default.


d. In part (b), we made the assumption that the likelihood of default is independent. Do you believe this is a reasonable assumption? Explain."

Verified step by step guidance

1. Step 1: Understand the concept of independence in probability. Two events are independent if the occurrence of one does not affect the probability of the other occurring.
2. Step 2: In the context of mortgage defaults, independence means that whether one mortgage defaults does not influence the likelihood of another mortgage defaulting.
3. Step 3: Consider real-world factors that might affect mortgage defaults, such as economic conditions, housing market trends, or borrower characteristics, which could create dependencies between defaults.
4. Step 4: Reflect on whether these factors could cause defaults to be correlated, meaning the assumption of independence might not hold true in practice.
5. Step 5: Conclude by explaining that assuming independence simplifies modeling but may not be fully realistic because defaults can be influenced by common external factors, leading to potential dependence.

Key Concepts

Probability of Default

The probability of default is the chance that a borrower fails to meet their debt obligations. In this context, it quantifies the risk that a mortgage will default, given as 0.01 or 1%. Understanding this helps assess the risk embedded in financial derivatives based on these mortgages.

Independence of Events

Independence means the occurrence of one event does not affect the probability of another. Assuming mortgage defaults are independent implies one default does not influence another, which simplifies modeling but may overlook real-world factors like economic conditions causing correlated defaults.

Correlation and Systemic Risk

Correlation measures how events move together; in finance, defaults can be correlated due to shared economic factors. Systemic risk arises when many defaults happen simultaneously, violating independence assumptions and impacting the accuracy of risk assessments in derivatives pricing.