To understand what it means for 8% to be compounded semiannually, we need to break down the concept of compounding and the frequency at which it occurs. Compounding refers to the process of earning interest on both the initial amount of money and any previously earned interest. In other words, it is the process of reinvesting the interest earned.

When interest is compounded semiannually, it means that interest is calculated and added to the principal twice a year. This is different from annual compounding, where interest is added only once a year.

Let’s consider an example to illustrate this. Suppose you have $100 that earns 8% annual interest compounded semiannually. After the first six months, you would earn 4% interest on the initial $100, which amounts to $4. This $4 is then added to the principal, resulting in a new total of $104.

For the second six-month period, you would earn 4% interest on the new principal of $104, which amounts to $4.16. This $4.16 is added to the principal, resulting in a new total of $108.16.

So, after one year (two semiannual periods), your initial $100 would grow to $108.16. This means that 8% compounded semiannually results in an effective annual interest rate slightly higher than 8%. In this case, it is approximately 8.16%.

Now, let’s apply this concept to a larger amount of money. If you have $10,000 and the interest rate is 8% compounded semiannually, you can calculate the future value after two semiannual periods.

The formula for calculating the future value is:

Future Value = Present Value * (1 + i)^n

Where:

– Present Value is the initial amount of money ($10,000 in this case)

– i is the interest rate per compounding period (4% in this case)

– n is the number of compounding periods (2 in this case)

Using this formula, we can calculate:

Future Value = $10,000 * (1 + 0.04)^2

Future Value = $10,000 * (1.04)^2

Future Value = $10,000 * 1.0816

Future Value ≈ $10,816

Therefore, the present value of $10,000 will grow to a future value of approximately $10,816 at the end of two semiannual periods when the 8% annual interest rate is compounded semiannually.

It’s important to note that the frequency of compounding can have a significant impact on the growth of investments or loans. Compounding more frequently, such as daily or monthly, can result in higher overall returns compared to less frequent compounding. However, the difference between compounding annually and semiannually is relatively small.