Compound Interest Formulas
Mathematical formulas for compound interest calculations and exponential growth models
Basic Compound Interest Formula
Future Value Formula
Where:
- A = Final amount (Future Value)
- P = Principal amount (Present Value)
- r = Annual interest rate (as a decimal)
- n = Number of times interest is compounded per year
- t = Time in years
Percentage Rate Formula
Where R is the annual interest rate as a percentage (e.g., 5% = 5).
Compound Interest Amount
Where CI is the compound interest earned.
Common Compounding Frequencies
Standard Frequencies
Annual Compounding (n = 1)
Semi-Annual (n = 2)
Quarterly (n = 4)
High-Frequency Compounding
Monthly (n = 12)
Daily (n = 365)
Continuous (n → ∞)
Derived Formulas
Calculate Principal (P)
Present Value formula - find initial investment needed.
Calculate Rate (r)
Find the interest rate required for growth.
Calculate Time (t)
Find time needed to reach target amount.
Calculate Compounding (n)
Find optimal compounding frequency (approximate).
Continuous Compounding Formulas
Continuous Compounding Formula
Where e ≈ 2.71828 is Euler's number (natural exponential base).
Continuous Interest Amount
Derived Continuous Formulas
Find Principal
Find Rate
Find Time
Doubling Time
Growth and Doubling Formulas
Rule of 72 (Approximation)
Where R is the annual percentage rate. This gives approximate doubling time in years.
Exact Doubling Time
Tripling Time
General Growth Factor
Time for investment to grow by factor k (e.g., k=2 for doubling, k=3 for tripling).
Effective Interest Rate
Effective Annual Rate (EAR)
The effective annual rate shows the actual annual return considering compounding frequency.
Continuous Effective Rate
Nominal to Effective Conversion
Example: 12% nominal rate compounded monthly
Related Annuity Formulas
Future Value of Ordinary Annuity
Where PMT is the periodic payment amount.
Present Value of Ordinary Annuity
Annuity Due Adjustment
Comparison with Simple Interest
Advantage of Compound Interest
Additional earnings from compounding vs simple interest.
Break-even Point
For very short periods, compound and simple interest are approximately equal:
Ratio of Compound to Simple Interest
Formula Applications
Example 1: Quarterly Compounding
Given: P = $10,000, r = 8% annually, n = 4 (quarterly), t = 5 years
Example 2: Finding Required Rate
Goal: $5,000 to grow to $8,000 in 6 years with monthly compounding
Example 3: Continuous vs Annual Compounding
Compare: $1,000 at 6% for 10 years