Rate Function Calculator

Determine the periodic rate of return or growth required to reach a specific value over a set number of cycles.

e.g., Number of months or years.

Value added per cycle. Use negative for outflows.

Initial amount. Use negative for initial investment.

The desired end state value.

End of Period Beginning of Period

Standard starting guess is 0.1 (10%).

Calculated Rate Per Period: 0.00%

Understanding the Rate Function Logic

The Rate Function Calculator is used to solve for the interest rate or growth rate per period in an annuity. This is a complex calculation because the rate ($r$) cannot be isolated in the standard formula; instead, the calculator uses the Newton-Raphson iteration method to converge on the correct value.

The Mathematical Formula

The relationship between the variables follows this fundamental equation:

PV(1+r)^n + PMT(1+rt)((1+r)^n – 1)/r + FV = 0

• n: Number of periods (total cycles).
• PMT: The constant value added or subtracted each cycle.
• PV: The value at the beginning of the sequence.
• FV: The value at the end of the sequence.
• t: Timing (0 for end, 1 for beginning).

Example Calculation

Imagine you are analyzing a growth project with the following parameters:

• ✅ Starting Value (PV): -5,000 (representing an initial cost/investment)
• ✅ Target Value (FV): 15,000
• ✅ Cycles: 60 months
• ✅ Monthly Addition (PMT): -100 per month

By inputting these values, the calculator will iterate through possible rates until it finds that the monthly growth rate required is approximately 1.58%.

Common Use Cases

1. Yield Analysis: Finding the yield to maturity on a fixed-income instrument.
2. Growth Benchmarking: Determining the required periodic growth needed to reach a corporate milestone.
3. Equipment Leasing: Calculating the implied rate within a lease structure where the periodic payments and residual values are known.

function calculateRateResult() { var nper = parseFloat(document.getElementById('nper').value); var pmt = parseFloat(document.getElementById('pmt').value); var pv = parseFloat(document.getElementById('pv').value); var fv = parseFloat(document.getElementById('fv').value); var type = parseInt(document.getElementById('type').value); var guess = parseFloat(document.getElementById('guess').value); if (isNaN(nper) || isNaN(pmt) || isNaN(pv) || isNaN(fv)) { alert("Please fill in all primary fields with valid numbers."); return; } var rate = guess; var epsMax = 1e-10; var iterMax = 100; var i = 0; var f = 0; var df = 0; // Iterative solving using Newton-Raphson while (i < iterMax) { if (Math.abs(rate) < 1e-10) { // Special case for r = 0 f = pv + pmt * nper + fv; df = 0.5 * nper * (nper – 1) * (pmt * (type === 1 ? 2 : 1) + pv); // approximation of derivative at r=0 } else { var y = Math.pow(1 + rate, nper); var y_minus_one = y – 1; var one_plus_rate_t = 1 + rate * type; f = pv * y + pmt * one_plus_rate_t * (y_minus_one / rate) + fv; // Derivative df/dr var df_pv = nper * pv * Math.pow(1 + rate, nper – 1); var df_pmt = pmt * ( type * (y_minus_one / rate) + one_plus_rate_t * ( (nper * Math.pow(1 + rate, nper – 1) * rate – y_minus_one) / (rate * rate) ) ); df = df_pv + df_pmt; } var nextRate = rate – f / df; if (Math.abs(nextRate – rate) < epsMax) { rate = nextRate; break; } rate = nextRate; i++; } var resultBox = document.getElementById('rate-result-box'); var output = document.getElementById('rate-output'); var annualOutput = document.getElementById('annual-output'); if (isNaN(rate) || !isFinite(rate)) { output.innerText = "Error"; annualOutput.innerText = "Check your inputs; the calculation did not converge."; } else { var ratePercent = (rate * 100).toFixed(4); output.innerText = ratePercent + "%"; annualOutput.innerText = "Calculated across " + nper + " cycles."; } resultBox.style.display = 'block'; }

Professional Rate Analysis Tool – v2.1