From a Deposit to an Accumulated Value
A deposit earns interest as time passes. With compound interest, interest credited in one period remains in the account and can earn interest in later periods. The accumulated value therefore depends on both the amount deposited and how many compounding periods elapse. A reliable calculation starts by matching the stated rate to the period count.
For a single deposit, the key inputs are the initial amount, the interest rate for each compounding period, and the number of periods. If the rate is stated per year, the period count must also be in years when compounding is annual. If the rate is stated per quarter, count quarters. The rate and period count must describe the same unit of time.
Here, \(r\) means the rate that applies over one stated compounding period. When periods are years and the rate is an effective annual rate, \(r=i\), using the course convention for \(i\). When a problem supplies a rate per quarter or per half-year, use that rate as \(r\) and count those periods. Do not treat a per-period rate as an annual rate unless the stated period is one year.
Why the Compound Factor Has This Form
The formula follows from applying the same accumulation rule once in each period. At the end of the first period, the deposit has grown by the fraction \(r\) of its starting amount. In the next period, interest is calculated on the amount then in the account, which already includes the first period’s interest.
Repeating the same step gives \(C_3=C_2(1+r)=C_0(1+r)^3\). Each additional period contributes one more factor of \(1+r\). After \(n\) periods there are \(n\) such factors, so \(C_n=C_0(1+r)^n\). This reasoning assumes the rate is unchanged from period to period and the balance stays invested; it does not describe a changing rate, withdrawals, or a different compounding arrangement.
Record the amount invested at time zero as \(C_0\).
Determine the time unit to which the stated interest rate applies, such as one year or one quarter.
Express the investment duration as a number \(n\) of those same periods.
Calculate \(C_0(1+r)^n\), then subtract \(C_0\) if the question also asks for interest earned.
Worked Applications
The examples assume a fixed rate, compounding at the end of each period, no deposits or withdrawals after time zero, and no fees or taxes. Amounts are rounded to the nearest cent only at the final step.
Worked Example: Annual Compounding Over Four Years
A small business places $4,800 in a reserve account earning an effective annual rate of 5%. The balance remains invested for four years, with interest compounded annually. Find the future value and the interest accumulated.
Method. Annual compounding means one period is one year, so \(r=0.05\) and \(n=4\). The fixed-rate compound-interest model applies.
Substitute the deposit, rate, and period count:
Calculate the factor: \(1.05^2=1.1025\), so \(1.05^4=1.1025^2=1.21550625\). Then
The future value is $5,834.43. Interest accumulated is \(5{,}834.43-4{,}800.00=\) $1,034.43. As a check, the future value exceeds the original deposit because the rate is positive; the difference between them is the accumulated interest.
Common mistake avoided: Simple interest would give \(4{,}800+4{,}800(0.05)(4)=\$5{,}760.00\). That calculation does not credit interest on earlier interest, so it is not the stated compound-interest model.
Worked Example: A Rate Stated Per Quarter
A hypothetical insurer deposits $12,500 in an account earning 3.2% per quarter. It leaves the money in the account for six quarters. Calculate the future value and the interest earned. Treat the supplied rate as the effective rate for each quarter; do not convert it to an annual rate.
Method. The rate period is one quarter, and the investment lasts six quarters. Thus \(r=0.032\) and \(n=6\). The fact that six quarters equal a year and a half does not change the rate period used in the formula.
For the factor, \(1.032^2=1.065024\), and \(1.032^3=1.065024(1.032)=1.099104768\). Therefore,
Multiplying by the deposit gives \(12{,}500(1.208031291\ldots)=\$15{,}100.39\) to the nearest cent. The interest earned is \(\$15{,}100.39-\$12{,}500.00=\$2{,}600.39\). The check is consistent: the accumulated amount is greater than the deposit, and their difference is positive.
Common mistake avoided: Using six as the number of years would give the wrong time scale. There are six compounding periods here, and each is a quarter. Both \(r\) and \(n\) must use that quarterly basis.
Worked Example: Half-Year Periods and an Interest Check
A community trust invests $9,200 for three years at 6% per six-month period, compounded at the end of each half-year. Find the accumulated amount and the interest earned. Then verify the accumulated amount by listing the balance at each period-end.
Method. Each period is six months. Three years contain \(3(2)=6\) half-year periods, so \(r=0.06\) and \(n=6\).
Build the factor step by step: \(1.06^2=1.1236\); \(1.06^3=1.1236(1.06)=1.191016\); and \(1.06^6=(1.191016)^2=1.418519112256\). Thus
Rounded to the nearest cent, the future value is $13,050.38. Interest earned is \(13{,}050.38-9{,}200.00=\) $3,850.38.
For a check using period-end balances, multiply by \(1.06\) each time: after period 1, the balance is $9,752.00; after period 2, $10,337.12; after period 3, $10,957.35; after period 4, $11,614.79; after period 5, $12,311.68; and after period 6, $13,050.38. These values are rounded for display; the formula calculation retains full precision until the final rounding. The final balance agrees.
Common mistake avoided: Three years is not three periods when interest is credited every six months. Using \(n=3\) would omit three actual compounding dates.
Reading the Result Correctly
A future value answers a specific question: what amount is in the account at the stated future date under the stated accumulation assumptions? It is not the interest amount alone. To distinguish the two, compare the ending balance with the original deposit:
This subtraction is an interpretation of the calculated balance, not a second accumulation rule. In the half-year example, the account ends with $13,050.38, while $3,850.38 is the amount above the original $9,200 deposit. Keeping these quantities separate prevents a common reporting error: calling the full ending balance “interest.”
Always label the rate’s period in your working. A statement such as “6% for three years” is incomplete if it is unclear whether 6% applies annually, every six months, or over some other interval. When a problem states the rate per period explicitly, use that period directly and count how many such periods occur. If the stated assumptions do not specify a fixed rate and regular compounding, the formula may not apply as written; the accumulation pattern must first be identified.
Check Your Understanding
For each problem, use the stated rate per compounding period and round money to the nearest cent. Assume the balance remains invested, the rate is fixed, and there are no fees, taxes, deposits, or withdrawals.
- A laboratory deposits $3,200 at 4% per year, compounded annually, for three years. Find the future value and interest earned. (Answer: future value $3,599.56; interest $399.56)
- A cooperative invests $8,500 at 2.5% per quarter for eight quarters. Calculate the future value and interest. (Answer: future value $10,356.42; interest $1,856.42)
- A fund places $7,500 in an account earning 6% per quarter for four quarters. What is the future value? (Answer: $9,468.58)
- A deposit of $6,400 earns 3.5% per six-month period for five periods. Calculate the accumulated value and the interest portion. (Answer: accumulated value $7,601.19; interest $1,201.19)
- A reserve of $11,000 earns 2% per quarter for twelve quarters. A colleague uses three years as \(n\), arguing that the investment lasts three years. Calculate the correct future value and identify the period-count error. (Answer: future value $13,950.66; there are twelve quarterly periods, not three annual periods)