Methodology

1. Future Value with Contributions

The compound interest formula with regular contributions combines the growth of an initial lump sum and a series of periodic payments:

FV = PV × (1 + r/n)^(n×t) + PMT × [(1 + r/n)^(n×t) − 1] / (r/n)

Where: FV = future value, PV = present value (initial principal), PMT = periodic payment, r = annual interest rate (decimal), n = compounding periods per year, t = number of years.

2. Effective Annual Rate (EAR)

The Effective Annual Rate converts a nominal rate with compounding frequency into the equivalent annual rate:

EAR = (1 + r/n)^n − 1

This is used throughout the calculator to display the true annual return accounting for compounding frequency.

Bisection Method (for r and t)

The bisection method is a root-finding algorithm that repeatedly bisects an interval and selects the subinterval containing the root. It is used because the future value equation cannot be algebraically rearranged to isolate r or t in closed form.

1. Define f(x) = calculated FV − target FV, where x is the unknown variable (r or t).
2. Start with a bracket [a, b] known to contain the root (f(a) × f(b) < 0).
3. Compute midpoint m = (a + b) / 2 and evaluate f(m).
4. Replace either a or b with m based on the sign of f(m), halving the interval.
5. Repeat until the interval is smaller than the tolerance (1e-8 for r, 1e-4 for t).

Solving for Payment (PMT)

The periodic payment is solved by rearranging the future value formula algebraically:

PMT = (FV − PV × (1 + r/n)^(n×t)) × (r/n) / [(1 + r/n)^(n×t) − 1]

Solving for Present Value (PV)

Present value is solved analogously by rearrangement:

PV = [FV − PMT × ((1 + r/n)^(n×t) − 1) / (r/n)] / (1 + r/n)^(n×t)

1. Constant Interest Rate

The calculator assumes a fixed annual interest rate for the entire investment period. In reality, interest rates fluctuate with market conditions, central bank policy, and economic cycles.

2. Simplified Tax Model

A single flat capital gains tax rate is applied uniformly to all gains. Real-world tax systems feature progressive brackets, different rates for short-term vs. long-term gains, tax-advantaged accounts (IRA, 401(k), ISA), and jurisdiction-specific rules.

3. Constant Inflation Rate

A fixed annual inflation rate is used to compute real (inflation-adjusted) values. Actual inflation varies year to year and across different goods and services.

4. No Transaction Costs or Management Fees

The model assumes zero trading commissions, management fees, expense ratios, bid-ask spreads, or other costs. Real investment products carry fees that reduce net returns.

5. Continuous Compounding Approximation

For daily compounding (n=365), the calculator uses discrete compounding. The difference between daily discrete and continuous compounding is negligible for most practical purposes.

6. No Market Risk or Volatility

The model projects smooth, deterministic growth. Real investments experience volatility, drawdowns, and sequence-of-returns risk that can significantly impact outcomes.

7. Reinvestment Assumption

Interest and dividends are assumed to be reinvested at the same rate. In practice, reinvestment rates may differ from the original yield.