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.


c. What is the probability the derivative becomes worthless? That is, at least one of the mortgages defaults?

Verified step by step guidance

1. Identify the event of interest: the derivative becomes worthless if at least one mortgage in the bundle defaults.
2. Understand that the probability of at least one default is the complement of the event that no mortgages default.
3. Let the probability of a single mortgage defaulting be \(p = 0.01\). Then, the probability that a single mortgage does not default is \$1 - p = 0.99$.
4. Assuming the mortgages are independent, the probability that none of the mortgages default is the product of the probabilities that each individual mortgage does not default. If there are \(n\) mortgages, this is \((1 - p)^n\).
5. Therefore, the probability that at least one mortgage defaults (making the derivative worthless) is \$1 - (1 - p)^n$.

Key Concepts

Probability of an Event

Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1. For example, a 0.01 probability means there is a 1% chance the event (mortgage default) happens. Understanding this helps in calculating combined probabilities for multiple events.

Complement Rule in Probability

The complement rule states that the probability of an event occurring is 1 minus the probability that it does not occur. For instance, the probability that no mortgages default is 1 minus the probability that at least one defaults. This rule simplifies calculating 'at least one' type probabilities.

Independence of Events

Events are independent if the occurrence of one does not affect the probability of another. Assuming mortgage defaults are independent allows us to multiply individual probabilities to find the chance that none default, which is key to solving the problem about the derivative's value.