Finding a Rate You Were Not Told
Lenders sometimes present a deal as an amount and a payment without naming a rate — "$20,000, $400 a month for five years" — which makes it hard to compare with anything else. Working the rate out is the only way to know whether the offer is good.
Unlike the payment formula, there is no closed-form solution for the interest rate. The equation
PMT = P × i ÷ (1 − (1 + i)−n)
cannot be rearranged to isolate i algebraically. Financial calculators and spreadsheets solve it iteratively, and so does this one: it repeatedly narrows a range until the computed payment matches the one you entered to within a fraction of a cent.
A Quick Sanity Check
Before calculating, multiply the payment by the number of payments. If that total is not larger than the amount borrowed, no positive interest rate can explain the deal — the arithmetic is wrong somewhere, or there are fees or a balloon payment not being mentioned.
$400 × 60 = $24,000 against $20,000 borrowed means $4,000 of interest, which is plausible. $340 × 60 = $20,400 on the same loan implies well under 1% — possible only as a promotional rate.
Nominal Rate, Effective Rate and APR
| Rate | What it is | Where it appears |
|---|---|---|
| Nominal (stated) rate | The headline annual rate, ignoring compounding within the year | Loan documents, savings adverts |
| Effective annual rate (APY / EAR) | The nominal rate adjusted for compounding frequency | Savings account comparisons |
| APR | The nominal rate plus lender fees, annualised | Legally required on U.S. consumer loans |
The conversion from nominal to effective is:
APY = (1 + r ÷ n)n − 1 (continuous: APY = er − 1)
| Nominal 6% compounded | Effective annual rate |
|---|---|
| Annually | 6.0000% |
| Semi-annually | 6.0900% |
| Quarterly | 6.1364% |
| Monthly | 6.1678% |
| Daily | 6.1831% |
| Continuously | 6.1837% |
The gap widens sharply at high rates: a 24% credit card compounded daily has an effective rate of about 27.1%.
The Real Interest Rate
A 7% return during 3% inflation is not a 7% gain in purchasing power. The Fisher equation gives the exact relationship:
Real rate = (1 + nominal) ÷ (1 + inflation) − 1
which gives 3.883%, not the 4% that simple subtraction suggests. The approximation is close at low rates and increasingly wrong at high ones: at 50% nominal and 40% inflation, subtraction says 10% while the exact answer is 7.14%.
Real rates can be negative. A savings account paying 2% during 5% inflation loses about 2.9% of purchasing power a year, even though the balance grows.
What Determines the Rate You Are Offered
- The policy rate. Central bank rates set the floor that everything else prices from.
- Credit risk. The lender's estimate of default probability, expressed through your credit score and debt-to-income ratio.
- Collateral. Secured loans price several points below unsecured ones because the lender's loss on default is limited.
- Term. Longer exposure usually costs more, though an inverted yield curve occasionally reverses this.
- Inflation expectations. Lenders require compensation for the expected erosion of the money they will be repaid in.
- Competition. Credit unions and online lenders frequently undercut large banks on the same borrower profile.
Typical Rate Ranges
| Product | Typical range | Secured? |
|---|---|---|
| Mortgage (30-year fixed) | 6–8% | Yes — the property |
| Home equity loan | 7–10% | Yes |
| Auto loan (new, good credit) | 5–8% | Yes — the vehicle |
| Auto loan (subprime) | 15–21% | Yes |
| Personal loan | 8–25% | No |
| Credit card | 18–29% | No |
| Federal student loan | 5–8% | No |
| Payday loan | 300–600% APR | No |
Frequently Asked Questions
Why does the calculated rate differ from the APR I was quoted?
APR includes origination fees, points and certain closing costs. This calculator sees only the cash flows you enter, so it returns the nominal rate implied by them. If fees were financed into the amount, the APR will be higher than the result here.
Can the rate be negative?
The implied rate on a loan can be, if total payments are less than the amount borrowed — which happens with deep subsidies or promotional deals. The calculator flags this rather than returning a misleading number.
Which rate should I compare between lenders?
APR for borrowing, because it includes fees. APY for saving, because it includes compounding. Comparing a nominal rate against an APR is the most common way to be misled.
How accurate is the iterative solution?
It converges to well under a thousandth of a percentage point, which is far finer than any lender quotes.