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Interest compounding is a powerful concept in finance that refers to the process where earned interest is added to the principal amount, and then, that interest too earns interest in the subsequent periods. This means, instead of earning a fixed amount of interest each period, the interest amount grows because it is calculated on an increasing principal.

For instance, if you invested \(1,000 at an interest rate of 5% per year, compounding annually, after one year, you'd earn \)50 in interest, giving you a total of \(1,050. If the interest were compounded, that new amount would be the basis for calculating interest in the second year, resulting in slightly more than \)50 of interest earned.

When the compounding frequency increases (weekly, daily, or even continuously), the effective interest earned over time increases. This is because interest calculations occur more frequently, each time adding a small amount of interest to the principal.

The nominal interest rate, also known as the stated or advertised rate, is the rate of interest before taking into account the effects of compounding. This rate does not reflect the actual financial benefit of an investment or loan because it does not consider how often the interest is compounded.

For example, a savings account with a nominal interest rate of 5% compounded monthly may have a different effective return than a different account with the same nominal rate compounded daily. The nominal rate provides a baseline from which we can calculate the real rate of return, known as the effective annual yield, by incorporating the compounding periods.

Annual Percentage Yield (APY), also known as Effective Annual Rate (EAR) or Effective Annual Yield (EAY), is the real rate of return on an investment after the effects of compounding interest are included. Unlike the nominal interest rate, APY provides a more accurate measure of how much you will earn or owe within a year.

In our case, when we calculated the APY, we compounded the daily nominal rate of 5.4% over the course of the year. The formula we used, \( AER = (1 + i/n)^{n*t} - 1 \), showed how the compounding effect increased the actual yield to about 5.62%, which is higher than the nominal rate. It's crucial to understand the APY to compare different financial products effectively because it standardizes the effects of compounding.

Financial mathematics is a field of applied mathematics concerned with financial markets and the economic theories that govern them. It involves a variety of mathematical tools and formulas that are used to calculate values related to money, such as interest rates, returns on investments, and the future value of an investment.

Understanding the elements of financial mathematics, like the concept of time value of money, allows us to compare different financial instruments on a like-for-like basis. The calculations behind the effective annual yield are an example of financial math in action; they allow investors to assess the true return on investments considering different compounding frequencies. Mastery of these concepts is key for making informed financial decisions.