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Semiannual compounding is a method used to calculate the growth of an investment when interest is applied to the principal amount twice a year. This concept is quite essential in the realm of finance as it determines how much one could earn from an investment over a period of time.

To get a clearer picture, let's delve into its mathematical representation. The formula for calculating the final amount with semiannual compounding is represented as \( A = P(1 + \frac{r}{n})^{nt} \), where \( A \) is the future value of the investment, \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year (which is 2 for semiannual), and \( t \) is the time in years.

Using this formula, an investment that compounds interest semiannually can grow more compared to simple interest because it takes advantage of interest on interest, effectively earning more over time. In our exercise example, the interest for Investment A compounds twice annually, thus allowing the investment to grow at regular intervals within the year.

Continuous compounding is the theoretical limit of compounding frequency. Unlike semiannual or quarterly compounding where the compound events are discrete and countable, continuous compounding assumes that the compounding effect occurs continuously, every moment, all year round.

The formula for continuous compounding is given by \( A = Pe^{rt} \), with \( e \) being the base of the natural logarithm, approximately equal to 2.71828. Here, \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time the money is invested for.

For instance, Investment B offers a continuous compounding interest, which means that the investment is growing at every instant, harnessing the power of exponential growth. Continuous compounding can yield higher returns over long periods compared to other compounding frequencies due to its perpetual compounding nature.

The future value of an investment refers to the amount of money that an initial deposit will grow to over time when interest is applied. Understanding this concept is critical as it allows investors to predict how much their current investments will be worth in the future.

When it comes to calculating the future value, it's crucial to understand that the frequency of compounding plays a significant role. The more frequently the interest is compounded, the greater the future value of the investment will be. This correlation is due to the concept of compound interest, where the interest earned itself earns interest.

In the exercise, we calculated the future values of two differing investments to determine their worth at the end of a 4-year period. As seen in the solution, Investment A, with semiannual compounding, resulted in a slightly higher future value than Investment B, which had continuous compounding, despite the fact that continuous compounding typically yields more frequent growth events.

The rate of return is a measure of the profitability of an investment over a set period. It is usually expressed as a percentage and is used to compare the efficiency of different investments. In essence, it tells you what percentage of the original investment amount has been gained or lost.

Two critical factors to understand when it comes to determining the rate of return are the time period and the nature of compounding. An investment with a higher rate of return over the same period is typically preferable. However, the compounding frequency can greatly influence the effective rate of return.

In our problem, while both investments have compounding interest, Investment A with semiannual compounding and a slightly higher nominal interest rate ended up with a greater rate of return over the four-year period compared to Investment B with continuous compounding. This illustrates how a higher nominal rate with less frequent compounding can sometimes outperform a lower rate with more frequent compounding.