When you take out a loan, the lender gives you the interest rate upfront. But sometimes you want to work backwards — from the payment amount and term, find the implied interest rate. This is useful for comparing loan offers, understanding credit card APRs, or checking if a car dealer's financing is competitive.
Simple Interest: Finding the Rate
For simple interest (used in short-term loans and some personal loans):
Interest rate = (Interest / Principal) / Time (years) × 100
Or rearranged:
r = I / (P × t)
Example: You borrowed £5,000 and repaid £5,600 after 1 year.
Interest = £5,600 − £5,000 = £600
Rate = £600 / (£5,000 × 1) = 0.12 = 12%
Example: You borrowed £3,000 and repaid £3,450 after 18 months (1.5 years).
Interest = £450
Rate = £450 / (£3,000 × 1.5) = 0.10 = 10%
Compound Interest: Finding the Rate
For compound interest (used in mortgages, savings, investments):
A = P(1 + r)ⁿ
Solving for r:
r = (A/P)^(1/n) − 1
Example: You invested £10,000 and it grew to £14,693 over 8 years.
r = (14,693 / 10,000)^(1/8) − 1
= (1.4693)^(0.125) − 1
= 1.0495 − 1
= 0.0495 ≈ 5%
The annual compound growth rate was approximately 5%.
Monthly Loan Payments: Finding the Rate
For standard amortising loans (mortgage, car loan, personal loan), the monthly payment formula is:
M = P × [r(1+r)ⁿ] / [(1+r)ⁿ−1]
Where:
- M = monthly payment
- P = loan principal
- r = monthly interest rate (annual rate ÷ 12)
- n = number of payments
Finding the rate from a known payment requires iteration (trial and error) or a financial calculator.
Practical approach: Use the APR calculator and work backwards.
Example: You're offered a car loan. Principal: £15,000, monthly payment: £285, term: 60 months (5 years).
Try 5% annual rate → monthly rate = 0.4167%:
M = 15,000 × [0.004167(1.004167)⁶⁰] / [(1.004167)⁶⁰ − 1]
= 15,000 × [0.004167 × 1.2834] / [0.2834]
= 15,000 × 0.005347 / 0.2834
= 15,000 × 0.01887
= £283/month
That's slightly under £285. Try 5.1%... this iterative process converges on the true rate.
Using the Rule of thumb: For rough estimates, the rate in percentage terms ≈ 24 × [(M × n − P) / (P × n)].
= 24 × [(285 × 60 − 15,000) / (15,000 × 60)]
= 24 × [(17,100 − 15,000) / 900,000]
= 24 × [2,100 / 900,000]
= 24 × 0.00233
= 5.6%
APR vs Nominal Rate
Nominal rate: The stated annual rate without accounting for compounding frequency.
APR (Annual Percentage Rate): Includes the effect of compounding — more comparable across products.
APR = (1 + nominal rate/n)ⁿ − 1
Where n = compounding periods per year.
Example: A savings account pays 4.8% nominal, compounded monthly.
APR = (1 + 0.048/12)¹² − 1
= (1.004)¹² − 1
= 1.04906 − 1
= 4.91%
The APR is 4.91%, slightly higher than the nominal 4.8%.
Credit Cards: How the Rate Works
Credit cards quote an APR, but interest is actually calculated daily:
Daily rate = APR / 365
Daily interest = Balance × daily rate
Monthly interest = Sum of daily interest charges
Example: £2,000 balance on a 22.9% APR card:
Daily rate = 22.9% / 365 = 0.0627%
Daily interest = £2,000 × 0.000627 = £1.25/day
Monthly interest ≈ £1.25 × 30 = £37.60
If you only make the minimum payment and don't pay it off, you'd pay roughly £451 in interest over a year on that £2,000 balance.
Quick Reference Formulas
| Situation | Formula |
|---|---|
| Simple interest rate | r = I / (P × t) |
| Compound annual rate | r = (A/P)^(1/n) − 1 |
| Daily to APR | APR = daily rate × 365 |
| Monthly to APR | APR = (1 + monthly rate)¹² − 1 |
| APR to monthly | monthly = (1 + APR)^(1/12) − 1 |