-19.8322%
-237.9865%
-92.9531%
$120,000.00
$20,000.00
$5,042.30
-19.8322%
-237.9865%
-92.9531%
$120,000.00
$20,000.00
$5,042.30
The Interest Rate Calculator (RATE) determines the interest rate implied by a set of loan or investment parameters. Given the loan amount, payment amount, and number of periods, this calculator reverse-engineers the interest rate — answering the question: 'What rate am I actually paying (or earning)?'
Unlike the other TVM functions (PV, FV, PMT, NPER), the RATE function cannot be solved algebraically in closed form. The equation PV × (1+r)n + PMT × [((1+r)n - 1) / r] = FV is transcendental in r and must be solved using numerical methods. This calculator uses the Newton-Raphson method, an iterative algorithm that converges rapidly to the solution through successive approximation.
Knowing the actual interest rate is critical for comparing financial products. A car dealer might advertise '$35,000 with $650/month for 60 months' without stating the interest rate. Using RATE, you can determine the implied rate: approximately 5.15% annually. Similarly, investors can calculate the rate of return implied by an investment's cash flows.
The RATE function also helps uncover the true cost of financing arrangements that disguise their rates behind fees, points, or creative payment structures. By entering the actual cash flows (what you borrow and what you pay), you can determine the effective interest rate and compare it to alternatives on a level playing field.
For investment analysis, RATE calculates the internal rate of return (IRR) for simple annuity cash flows. If you invest $100,000 and receive $1,000 per month for 120 months, RATE reveals your monthly return rate, which can be annualized for comparison with other investments.
The calculator solves: PV × (1+r)n + PMT × [((1+r)n - 1) / r] - FV = 0 for r using the Newton-Raphson method. Starting with an initial guess (1%), the algorithm iteratively refines the estimate using the function and its derivative. Each iteration approximately doubles the number of correct digits, so 8-10 iterations typically achieve precision to 10+ decimal places.
The Rate per Period is the interest rate for each payment period (monthly, quarterly, etc.). The Annual Interest Rate multiplies this by the number of periods per year for easy comparison. Total Amount Paid and Total Interest help you understand the full cost. If the result seems unreasonable, check that your inputs are consistent (PMT must be sufficient to cover interest on PV).
Inputs
Results
A $300K loan with $1,896/month for 30 years implies a 6% annual rate
Inputs
Results
$25K auto loan at $483/month for 5 years implies approximately 4.5% annual rate
The equation PV×(1+r)^n + PMT×((1+r)^n-1)/r = FV is transcendental — r appears both as a base of an exponent and in polynomial terms. No algebraic rearrangement can isolate r, so numerical methods (like Newton-Raphson) must be used.
It is an iterative numerical algorithm for finding roots of equations. Starting with a guess, each iteration improves the estimate using the formula: r_new = r_old - f(r_old)/f'(r_old), where f is the function and f' is its derivative. It typically converges within 8-10 iterations.
The algorithm can find negative rates if they exist mathematically, but most practical loan and investment scenarios involve positive rates. Negative rates would imply the lender pays you to borrow.
No solution exists if the payment amount is too small to ever cover the interest (PMT ≤ PV × r for any positive r). In this case, the algorithm may not converge or may return an unrealistic result.
RATE calculates the periodic interest rate from cash flows. APR (Annual Percentage Rate) includes additional fees and costs beyond the interest rate. APR is typically higher than the nominal rate because it accounts for origination fees, points, and other charges.
Yes. Enter your initial investment as PV, periodic income as PMT, final value as FV, and the number of periods. RATE will give you the per-period return rate, which you can annualize.
With 8 iterations starting from a 1% guess, the result is accurate to approximately 10-12 significant digits for typical financial calculations. This exceeds the precision needed for any practical application.
FV represents the remaining balance at the end of the period. For a fully amortized loan, FV = 0. For investments, FV is the final value of the account. For balloon loans, FV is the balloon payment amount.
Because of compounding. A monthly rate of 0.5% compounded 12 times gives an effective annual rate of 6.17%, not 6.0%. The nominal annual rate (0.5% × 12 = 6%) ignores this compounding effect.
Banks set rates based on the base rate (fed funds rate), their cost of funds, credit risk assessment, loan term, collateral, competition, and profit margin. The RATE calculator helps you verify the rate implied by the actual payment terms offered.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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