Separating Principal From Interest Earned

Principal, Balance, and Interest

A deposit’s future account balance contains two conceptually different amounts: the money originally deposited and the growth accumulated since then. When a question asks for interest earned, it asks for the growth—not the entire balance. Keeping these quantities separate is a useful check on single-sum calculations.

The discussion assumes one hypothetical deposit, no later deposits or withdrawals, no fees or taxes, and a fixed positive effective annual interest rate. The deposit earns compound interest annually for a whole number of years. These assumptions make the accumulated balance and the interest portion straightforward to identify.

The principal is the starting deposit, \(P\). The balance is the total amount in the account at the stated future date. Cumulative interest is the difference between that balance and the original deposit. Thus, the balance equals principal plus cumulative interest: $$ P(1+i)^n=P+I_n. $$

These quantities are related but should not be used interchangeably. If a question asks how much interest has been earned, reporting the full account balance overstates the answer by exactly the original principal.

Why Subtracting the Principal Works

Under annual compounding, the balance at the end of each year becomes the amount on which interest is calculated in the next year. Starting with \(P\), the balance after one year is \(P(1+i)\). After two years it is \(P(1+i)(1+i)=P(1+i)^2\). Repeating this step gives the balance after \(n\) years.

The original principal remains part of that final balance: it has not vanished or been replaced. The part of the final balance in excess of that original amount is the interest earned. Subtracting \(P\) therefore gives the interest. Factoring \(P\) from the difference gives the alternate expression:

This derivation also explains why, with compounding, total interest is generally not \(Pin\). That expression would add the same amount \(Pi\) in every year, as if each year’s interest were calculated only on the original principal. Under annual compounding, later years also earn interest on interest accumulated earlier.

Calculating Cumulative Interest

Interest Earned in Each Year

Cumulative interest through a given year is different from the interest earned during that particular year. In year \(k\), the interest is the year-end balance less the balance at the start of that year. With a fixed annual rate and annual compounding, the balance at the start of year \(k\) is \(P(1+i)^{k-1}\). Multiplying that amount by \(i\) gives the year’s interest:

This expression follows directly from the annual interest rule: interest for a year is the opening balance for that year multiplied by the annual rate. The cumulative interest through year \(n\), by contrast, includes the interest from every year through \(n\).

Using the Future-Value Formula Carefully

The subtraction method works whether the future value is calculated directly from the formula or built up year by year. The key is to use the same principal and the same valuation date in both parts of the subtraction. The balance must be the balance at the end of the stated accumulation period, and the amount subtracted must be the original deposit—not an intermediate year-end balance.

For a positive interest rate and a positive principal, the accumulated balance after a positive number of years exceeds the original deposit, so cumulative interest is positive. At time zero, the balance is the principal and cumulative interest is zero. These are useful checks on whether the answer is being interpreted correctly.

The formula as written applies to a single deposit earning a fixed effective annual rate over whole annual periods, with annual compounding. If the rate changes, compounding occurs on a different schedule, or the account has deposits or withdrawals during the period, this setup does not directly describe the entire account. The cash flows and accumulation assumptions would need to be represented appropriately rather than treating the final balance as if it came from one unchanged deposit.

Common Mistake: Reporting the Balance as Interest

Suppose a deposit grows from $5,000 to $5,600. A frequent error is to state that $5,600 of interest was earned. The $5,600 is the total account balance. The correct interest calculation is:

A second error is to use simple interest, \(Pin\), without checking the accumulation method. For example, on $5,000 at 6% for two years, that calculation gives \(\$5{,}000\times0.06\times2=\$600\). Under annual compounding, however, the first year’s $300 of interest is included in the second year’s opening balance. The second-year interest is therefore \(\$5{,}300\times0.06=\$318\), making cumulative interest $618, not $600. The ending balance is $5,618.

The correct calculation depends on the stated model. If a problem specifies simple interest, the simple-interest method is appropriate; if it specifies annual compounding, use the compound accumulation formula. Do not substitute one method for the other merely because both involve the same quoted annual rate.