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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. (a) What is the probability a randomly selected mortgage will not default (that is, pay off)? (b) What is the probability a bundle of five randomly selected mortgages will not default assuming the likelihood any one mortgage being paid off is independent of the others? Note: A derivative might be an investment in which all five mortgages do not default. (c) What is the probability the derivative becomes worthless? That is, at least one of the mortgages defaults? (d) In part (b), we made the assumption that the likelihood of default is independent. Do you believe this is a reasonable assumption? Explain.

Step by step solution

01

Calculate Probability of No Default for One Mortgage

Given the probability of default for one mortgage is 0.01, calculate the probability it does not default. This is simply 1 minus the probability of default. \( P(\text{no default}) = 1 - P(\text{default}) = 1 - 0.01 = 0.99 \)

02

Probability All Five Mortgages Will Not Default

Assuming the independence of each mortgage's outcome, calculate the probability that none of the five mortgages will default. This is the probability of no default for one mortgage raised to the power of 5. \( P(\text{no default for five}) = P(\text{no default})^5 = 0.99^5 \)

03

Probability the Derivative Becomes Worthless

The derivative becomes worthless if at least one of the five mortgages defaults. Calculate this by finding the complement of the event that no mortgage defaults out of the five. \( P(\text{at least one defaults}) = 1 - P(\text{no default for five}) = 1 - 0.99^5 \)

04

Analyze the Assumption of Independence

Evaluate if the assumption that the likelihood of default is independent is a reasonable one. In practice, defaults might be correlated due to economic conditions, interest rates, or other factors. Therefore, this independence assumption could be an oversimplification. Acknowledging dependencies might give a more accurate probability of default.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability of Default

In the context of financial derivatives, the probability of default is a crucial concept. This probability indicates the likelihood that a borrower, such as the holder of a mortgage, will be unable to meet their financial obligations.

For example, if a mortgage has a probability of default of 0.01, it means there is a 1% chance that the mortgage will not be paid off. This small probability is essential for risk assessment and pricing of financial assets.
Understanding the likelihood of default is key for investors and financial institutions to make informed decisions.

Independence Assumption

The independence assumption is a critical factor in calculating probabilities involving multiple events. For financial derivatives, this means assuming that the performance or default of each mortgage in a bundle is independent of others.

This assumption simplifies calculations. For instance, if defaults of five mortgages are independent, the probability calculation involves raising the probability of no default for a single mortgage to the power of the number of mortgages. But in reality, mortgages might be influenced by similar economic conditions or interest rates, making them not entirely independent.
This potential dependency should be considered for more accurate risk assessment.

Complement Rule

The complement rule is a fundamental principle in probability theory. It is used to determine the probability of an event not occurring by subtracting the probability of the event occurring from 1.

For instance, if the probability that a mortgage will default is 0.01, then the probability it will not default is 1 - 0.01, which equals 0.99. This is a straightforward application of the complement rule.
In the given exercise, the complement rule helps find the probability of no mortgages defaulting by subtracting the default probability from 1.

Financial Asset Valuation

Valuing financial assets, such as derivatives, involves understanding the underlying risks and probabilities. A derivative’s value depends significantly on the likelihood of events like defaults.

For example, if a derivative consists of a bundle of five mortgages, its value can be impacted if any mortgage defaults. Calculating the probability of all mortgages not defaulting is essential. This probability provides insight into the derivative’s potential return or whether it could become worthless.
Accurate valuation involves comprehensively assessing possible outcomes and their probabilities.

Random Selection in Finance

Random selection is commonly used in finance to assess risk and diversify investments. It involves selecting assets randomly from a broader set to mitigate biases.

For instance, when evaluating the risk of a mortgage derivative, assuming random selection of mortgages can help generalize the results. This process assumes each mortgage's risk is consistent with the overall mortgage pool.
Random selection helps in creating diversified portfolios, aiding in reducing the overall risk by spreading exposure across many independent assets.