Compound Interest: Calculate For Various Periods

Nov 14, 2025 by Alex Johnson 49 views

Understanding how compound interest works is a cornerstone of smart financial planning. It's not just about the initial amount you invest or borrow; it's about how frequently that interest starts earning its own interest, a phenomenon often described as "money making money." This article will delve into calculating the interest earned for each compounding period on an initial principal of $50,000 over 2.5 years at an annual interest rate of 4.3%. We'll break down the calculations for various compounding frequencies, from annually to continuously, to illustrate the powerful effect of compounding.

The Magic of Compounding: Understanding the Basics

Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. Essentially, it's interest on interest. This is different from simple interest, where interest is only calculated on the original principal amount. The more frequently interest is compounded, the faster your money grows (or the faster debt accrues). The formula for compound interest is $A = P(1 + r/n)^(nt)$ , where:

A is the future value of the investment/loan, including interest.
P is the principal investment amount (the initial deposit or loan amount).
r is the annual interest rate (as a decimal).
n is the number of times that interest is compounded per year.
t is the number of years the money is invested or borrowed for.

While this formula gives us the total amount, we are interested in the interest earned during each compounding period. To find this, we first calculate the total amount A and then subtract the initial principal P to get the total interest earned over the entire term. However, the question asks for the interest for each compounding period. This requires a slightly different approach. For discrete compounding (annually, semiannually, quarterly, monthly, daily), we can calculate the value at the end of each period and subtract the value at the beginning of that period. For continuous compounding, we use a different formula and concept.

Our starting principal (P) is $50,000. The annual interest rate (r) is 4.3%, or 0.043 in decimal form. The time period (t) is 2.5 years.

a. Annually Compounded Interest

When interest is compounded annually, it means the interest is calculated and added to the principal once a year. Over 2.5 years, there will be two full compounding periods and then a partial period. However, the question asks for the interest for each compounding period. This implies we should consider the interest earned within each full year and then the interest earned in the final partial year. For simplicity and clarity in demonstrating the compounding effect, let's calculate the interest earned in Year 1, Year 2, and the interest earned in the first half of Year 3 (since 2.5 years is 2 full years plus half a year).

Year 1:
- Interest = Principal * Rate = $50,000 * 0.043 = $2,150$ . This is the interest earned in the first year. The new principal at the end of Year 1 is $50,000 + $2,150 = $52,150$ .
Year 2:
- Interest = New Principal * Rate = $52,150 * 0.043 = $2,242.45$ . This is the interest earned in the second year. The new principal at the end of Year 2 is $52,150 + $2,242.45 = $54,392.45$ .
First Half of Year 3 (0.5 years):
- Since we are compounding annually, the interest for this partial period is calculated based on the principal at the start of the year. The interest for the full third year would be $54,392.45 * 0.043$ . However, for just half a year, we can prorate this: Interest = (Principal * Rate) * (Time in years) = ( $54,392.45 * 0.043$ ) * 0.5. A more precise way for partial periods when annual compounding is specified is to use the formula with t as the partial time: $A = P(1 + r)^t$ . So, for the interest earned during the first half of the third year, it's the difference between the amount at 2.5 years and the amount at 2 years. Amount at 2.5 years ( $A_{2.5}$ ) = $50,000 * (1 + 0.043)^2.5$ . Using a financial calculator or software for $(1.043)^2.5 \approx 1.11388$ . So, $A_{2.5} \approx 50,000 * 1.11388 = $55,694.00$ . The interest earned during this half-period is $A_{2.5} - A_2 = $55,694.00 - $54,392.45 = $1,301.55$ .

So, for annual compounding, the interest earned in the first compounding period (Year 1) is $2,150. The interest earned in the second compounding period (Year 2) is $2,242.45. The interest earned in the partial compounding period (first 0.5 year of Year 3) is $1,301.55.

b. Semiannually Compounded Interest

Compounding semiannually means interest is calculated and added to the principal twice a year. In 2.5 years, there are 2.5 * 2 = 5 compounding periods. The interest rate per period is the annual rate divided by the number of periods per year: $r/n = 0.043 / 2 = 0.0215$ . The number of periods is $n*t = 2 * 2.5 = 5$ .

Period 1 (End of 6 months):
- Interest = $50,000 * 0.0215 = $1,075$ . New Principal = $50,000 + $1,075 = $51,075$ .
Period 2 (End of 12 months):
- Interest = $51,075 * 0.0215 = $1,097.91$ . New Principal = $51,075 + $1,097.91 = $52,172.91$ .
Period 3 (End of 18 months):
- Interest = $52,172.91 * 0.0215 = $1,121.72$ . New Principal = $52,172.91 + $1,121.72 = $53,294.63$ .
Period 4 (End of 24 months):
- Interest = $53,294.63 * 0.0215 = $1,145.84$ . New Principal = $53,294.63 + $1,145.84 = $54,440.47$ .
Period 5 (End of 30 months / 2.5 years):
- Interest = $54,440.47 * 0.0215 = $1,170.37$ . New Principal = $54,440.47 + $1,170.37 = $55,610.84$ .

The interest earned for each semiannual compounding period is approximately: $1,075.00, $1,097.91, $1,121.72, $1,145.84, and $1,170.37. Notice how the interest earned in each subsequent period is slightly higher than the previous one, a direct result of compounding.

c. Quarterly Compounded Interest

Compounding quarterly means interest is calculated and added to the principal four times a year. In 2.5 years, there are 2.5 * 4 = 10 compounding periods. The interest rate per period is $r/n = 0.043 / 4 = 0.01075$ . The number of periods is $n*t = 4 * 2.5 = 10$ .

We can calculate this iteratively, but for brevity, let's use the future value formula and then find the interest for the last period. The total amount after 10 periods can be calculated as $A = P(1 + r/n)^(nt) = 50,000 * (1 + 0.043/4)^(4*2.5) = 50,000 * (1.01075)^10$ . Using a calculator, $(1.01075)^{10} \approx 1.11498$ . So, $A \approx 50,000 * 1.11498 = $55,749.00$ . The total interest earned is $55,749.00 - $50,000 = $5,749.00$ .

To find the interest for each period, we would perform 10 separate calculations. Let's show the first two and the last one to illustrate.

Period 1 (End of 3 months):
- Interest = $50,000 * 0.01075 = $537.50$ . New Principal = $50,000 + $537.50 = $50,537.50$ .
Period 2 (End of 6 months):
- Interest = $50,537.50 * 0.01075 = $543.25$ . New Principal = $50,537.50 + $543.25 = $51,080.75$ .
... (Periods 3 through 9 would follow similarly)
Period 10 (End of 30 months / 2.5 years):
- To find the interest for the 10th period, we need the principal at the end of the 9th period. The amount at the end of 9 periods is $A_9 = 50,000 * (1.01075)^9 \approx 50,000 * 1.10291 = $55,145.50$ . Interest for Period 10 = $55,145.50 * 0.01075 = $592.81$ . The final amount is $55,145.50 + $592.81 = $55,738.31$ . (Slight discrepancy from the total calculated earlier due to rounding in intermediate steps). If we use the exact total amount of $55,749.00 (calculated from $A = 50,000 * (1.01075)^{10}$ ), then the interest for the last period is $55,749.00 - ( ext{Amount at end of period 9})$ . Amount at end of period 9 = $50,000 * (1.01075)^9 \approx 55,152.04$ . So, Interest for Period 10 = $55,749.00 - $55,152.04 = $596.96$ . It's crucial to maintain precision.

So, the interest for the first quarterly period is $537.50, the second is $543.25, and the interest for the final (10th) quarterly period is approximately $596.96.

d. Monthly Compounded Interest

Compounding monthly means interest is calculated and added to the principal twelve times a year. In 2.5 years, there are 2.5 * 12 = 30 compounding periods. The interest rate per period is $r/n = 0.043 / 12 \\approx 0.00358333$ . The number of periods is $n*t = 12 * 2.5 = 30$ .

Let's calculate the total amount first: $A = 50,000 * (1 + 0.043/12)^{30}$ . Using a calculator, $(1 + 0.043/12)^{30} \approx (1.00358333)^{30} \approx 1.11575$ . So, $A \approx 50,000 * 1.11575 = $55,787.50$ . The total interest earned is $55,787.50 - $50,000 = $5,787.50$ .

To find the interest for each period, we again perform iterative calculations. Let's show the first and the last.

Period 1 (End of 1 month):
- Interest = $50,000 * (0.043/12) \approx $179.17$ . New Principal = $50,000 + $179.17 = $50,179.17$ .
... (Periods 2 through 29)
Period 30 (End of 30 months / 2.5 years):
- The amount at the end of period 29 is $A_{29} = 50,000 * (1 + 0.043/12)^{29} \approx 50,000 * 1.11375 = $55,687.50$ . Interest for Period 30 = $55,687.50 * (0.043/12) \approx $200.00$ . The final amount is $55,687.50 + $200.00 = $55,887.50$ . (Again, slight discrepancies due to rounding). If we use the precise total amount $55,787.50$ , then Interest for Period 30 = $55,787.50 - ( ext{Amount at end of period 29})$ . Amount at end of period 29 = $50,000 * (1 + 0.043/12)^{29} \approx 55,588.30$ . Interest for Period 30 = $55,787.50 - $55,588.30 = $199.20$ . The interest earned in the first month is approximately $179.17, and the interest earned in the final (30th) month is approximately $199.20.

e. Daily Compounded Interest

Compounding daily means interest is calculated and added to the principal 365 times a year. In 2.5 years, there are 2.5 * 365 = 912.5 periods (approximately, we usually use 365 days per year for calculation). Let's use 365 days for a standard year. So, $n = 365$ . The number of periods is $n*t = 365 * 2.5 = 912.5$ . We'll calculate for 912 full periods and then a half period, or more practically, the total number of days. For simplicity, let's assume 2.5 years have $2.5 imes 365 = 912.5$ days. We can approximate this by using 913 periods, or calculate the exact fraction. A common approach for daily compounding over a non-integer number of years is to use the number of days. Let's use $N = 2.5 imes 365 = 912.5$ periods. The daily interest rate is $r/n = 0.043 / 365 \\approx 0.0001178$ . The total amount is $A = 50,000 * (1 + 0.043/365)^(365*2.5)$ . $(1 + 0.043/365)^{912.5} \approx (1.0001178)^{912.5} \approx 1.11645$ . So, $A \approx 50,000 * 1.11645 = $55,822.50$ . The total interest earned is $55,822.50 - $50,000 = $5,822.50$ .

Calculating the interest for each individual day is tedious. The interest for the first day is $50,000 * (0.043/365) \approx $5.89$ . The interest earned on the last day of the 912.5 periods would be based on the principal accumulated up to that point. The interest earned on the first day is about $5.89. The interest earned on the final day (or half-day period) will be slightly higher due to compounding.

f. Hourly Compounded Interest

Compounding hourly means interest is calculated and added to the principal 24 times a day, 365 days a year. So, $n = 24 * 365 = 8,760$ . In 2.5 years, there are $8,760 * 2.5 = 21,900$ compounding periods. The hourly interest rate is $r/n = 0.043 / 8,760 \\approx 0.00000490867$ . The total amount is $A = 50,000 * (1 + 0.043/8760)^{8760*2.5}$ . $(1 + 0.043/8760)^{21900} \approx (1.00000490867)^{21900} \approx 1.11666$ . So, $A \approx 50,000 * 1.11666 = $55,833.00$ . The total interest earned is $55,833.00 - $50,000 = $5,833.00$ .

The interest earned in the first hour is $50,000 * (0.043 / 8760) \approx $0.245$ . The interest earned in each subsequent hour will be slightly higher. The interest for the last hour will be based on the accumulated principal up to that point.

g. Continuously Compounded Interest

Continuous compounding is the theoretical limit as the number of compounding periods per year approaches infinity. The formula for continuous compounding is $A = P * e^(rt)$ , where e is the base of the natural logarithm (approximately 2.71828).

P = $50,000
r = 0.043
t = 2.5 years

$A = 50,000 * e^(0.043 * 2.5)$ $A = 50,000 * e^(0.1075)$

Using a calculator, $e^(0.1075) \approx 1.11350$ . So, $A \approx 50,000 * 1.11350 = $55,675.00$ .

The total interest earned with continuous compounding is $55,675.00 - $50,000 = $5,675.00$ . Unlike discrete compounding periods, continuous compounding doesn't have individual periods where you can easily isolate the interest earned