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Default probability is a term used to express the likelihood that a borrower, such as a corporation, will fail to make required payments on a financial instrument. In our case, it refers to the chance that a company could fail to make payments on a bond during specific periods.
The initial default probabilities for each year are given as:

• Year 1: 0.05
• Year 2: 0.07
• Year 3: 0.07
• Year 4: 0.07
• Year 5: 0.09

These numbers mean that in the first year, there's a 5% chance the company won't meet its financial obligation, increasing to a 9% risk by the fifth year. Thus, understanding default probability helps assess the risk inherent in an investment.

In probability theory, independent events are scenarios where the occurrence of one event does not affect the occurrence of another. In the context of our exercise, each year is an independent event when it comes to the probability of the company defaulting.
This means that the chance of default or no default in any given year has no effect on any other year's probability. Independence makes it easier to calculate the overall probability over several years, as we can analyze each year's probability separately without considering interdependencies.
Recognizing that defaults are independent allows us to use specific probability rules, such as the multiplication rule, to compute the overall likelihood of no defaults over multiple years.

The multiplication rule for independent events states that to determine the probability of multiple independent events occurring together, you must multiply their individual probabilities. Our task was to find the probability that no defaults occur over a five-year period.
For each year, we calculated the probability of not defaulting, which is simply 1 minus the probability of default. We then multiplied these probabilities:

• Year 1: 0.95
• Year 2: 0.93
• Year 3: 0.93
• Year 4: 0.93
• Year 5: 0.91

The multiplication of these probabilities gives \[0.95 \times 0.93 \times 0.93 \times 0.93 \times 0.91 \approx 0.704\]Therefore, there's about a 70.4% probability that the company will not default at any point over the five years.

Financial mathematics involves using mathematical methods to tackle problems in finance, providing tools to assess and manage risk. The concept applies to our exercise as we deal with default probability calculations for a corporate bond.
Understanding financial mathematics helps investors gauge potential risks and returns. Here, it helps in setting expectations about the probability of recovering their investment in full. Such calculations inform decisions, such as whether to invest in a company’s bonds or consider different investments based on risk appetites.
Furthermore, financial mathematics encompasses models beyond just simple probability – like options pricing, stock market analysis, and fixed income securities, offering a comprehensive toolkit for financial decision-making.