The accumulation function describes how an initial investment grows over time with compound interest. This function is crucial in finance, helping to determine the future value of an investment or loan.
1. Accumulation Function Formula
The accumulation function for compound interest is given by:
A(t) = P (1 + r/n)^(nt)
- A(t) = Accumulated amount at time t
- P = Principal (initial investment)
- r = Annual interest rate (decimal)
- n = Number of compounding periods per year
- t = Number of years
2. Example of Annual Compounding
Scenario: You invest $10,000 at an annual interest rate of 5%, compounded once per year for 10 years.
A(10) = 10,000 × (1 + 0.05/1)^(1×10)
A(10) = 10,000 × (1.05)^10
A(10) = $16,288.95
3. Example of Monthly Compounding
Scenario: You invest $10,000 at a 5% annual interest rate, compounded monthly for 10 years.
A(10) = 10,000 × (1 + 0.05/12)^(12×10)
A(10) = 10,000 × (1.004167)^120
A(10) = $16,470.09
4. Example of Continuous Compounding
When interest is compounded continuously, the formula becomes:
A(t) = P × e^(rt)
Example: Investing $10,000 at 5% interest for 10 years:
A(10) = 10,000 × e^(0.05×10)
A(10) = 10,000 × e^0.5
A(10) = $16,487.21
5. Comparison of Different Compounding Methods
For the same investment ($10,000 at 5% for 10 years), the accumulated amounts are:
- Annual Compounding: $16,288.95
- Monthly Compounding: $16,470.09
- Continuous Compounding: $16,487.21