The simple-interest accumulation function
- a(t)
- Accumulation factor after time t
- r
- Simple interest rate per year, as a decimal
- t
- Time in years
- P
- Original principal
If an amount P is invested or loaned, the accumulated value after time t is A(t) = P × a(t) = P(1 + rt). The interest earned is I = Prt.
Example: $10,000 at 6% simple interest for 3 years
Use r = 0.06 and t = 3:
The accumulation factor is 1.18. Multiply it by the original $10,000 principal:
The account or loan balance has increased by $1,800 of simple interest.
Example with part of a year
For $5,000 at 8% simple interest for 9 months, convert 9 months to 0.75 year:
The accumulated value is $5,000 × 1.06 = $5,300. The simple interest is $300.
Why the function is linear
Under simple interest, interest is calculated only on the original principal. The accumulation factor therefore increases by the same amount each period. At a 6% annual simple rate, the factor is 1.06 after one year, 1.12 after two years, and 1.18 after three years.
Simple interest vs. compound accumulation
| Method | Accumulation function | What earns interest? |
|---|---|---|
| Simple interest | a(t) = 1 + rt | Original principal only |
| Compound interest | a(t) = (1 + i)t | Principal plus previously credited interest |
The two functions can give very different answers over long periods. Use the method required by the loan, investment, contract, or financial-math problem.
Using a rate per period instead of per year
The same structure works when the rate is stated per period. If i is the simple rate per month and n is the number of months, the accumulation factor is 1 + in. Keep the rate and time units consistent.
Check the result with a calculator
Use the Simple Interest Calculator to enter principal, annual rate, and time in days, months, or years. For daily-accrual questions, use the Daily Interest & Per Diem Calculator.