Maths⏱ 5 min read
How to Calculate Interest Rate from a Loan or Investment
When you know the payments but not the rate, you need to work backwards to find the interest rate. Here's how to calculate the implied rate on any loan or investment — and why this matters.
Most financial calculators ask you to input the interest rate. But sometimes you need to work backwards — you know the payments and the term, and want to know what rate you're actually paying. Here's how.
Why Finding the Rate Matters
Car dealers and retailers often quote monthly payments without prominently disclosing the interest rate. "Only £249/month for 36 months" on a £7,500 car sounds reasonable — until you calculate what rate that implies.
Car: £7,500
Payments: £249/month for 36 months
Total repaid: £249 × 36 = £8,964
Total interest: £8,964 − £7,500 = £1,464
Is that a good rate? You need to calculate the APR to know.
Finding Rate from Known Payments (Iteration Method)
There's no simple algebraic formula to isolate the interest rate — it requires iteration (trial and improvement), which is exactly what financial calculators and spreadsheets do automatically.
Loan formula: P = PMT × [1 − (1+r)^-n] / r
P = 7,500, PMT = 249, n = 36
Solve for r (monthly rate):
Try r = 1% monthly: PMT = 7,500 × 0.01 / [1−(1.01)^-36] = £249.0
→ Monthly rate = 1.0%
→ APR = (1 + 0.01)^12 − 1 = 12.68% APR
That's a fairly high rate for a car purchase —
a bank loan at 7–9% would save hundreds of pounds.
Simple Interest Rate (for Short Loans)
For simple interest loans:
r = (Total Interest ÷ Principal) ÷ Time in years
Example: Borrow £1,000, pay back £1,180 after 18 months
Interest = £180
r = (180 ÷ 1,000) ÷ 1.5 = 0.18 ÷ 1.5 = 12% per year (simple)
Note: APR would be slightly different as it compounds quarterly/monthly.
Investment: Implied Annual Return
CAGR (Compound Annual Growth Rate):
r = (Final Value ÷ Initial Value)^(1/years) − 1
Property bought for £200,000, sold 8 years later for £310,000
r = (310,000 ÷ 200,000)^(1/8) − 1
r = (1.55)^0.125 − 1
r = 1.0562 − 1 = 5.62% per year
Or for a savings product:
£5,000 grew to £6,800 over 5 years
r = (6,800 ÷ 5,000)^(1/5) − 1 = (1.36)^0.2 − 1 = 6.33%/year
The Rule of 72 (Quick Estimate)
Years to double = 72 ÷ Annual interest rate %
OR
Implied rate = 72 ÷ Years to double
Example: Money doubled in 9 years
Implied rate ≈ 72 ÷ 9 = 8% per year
Verify: (1.08)^9 = 1.999 ≈ 2 ✓
Example: At 6%, when does £10,000 become £20,000?
Years ≈ 72 ÷ 6 = 12 years
APR vs Flat Rate vs Monthly Rate
Rate TypeWhat It MeansRelationship
Monthly rateInterest per month× 12 = nominal annual rate
Nominal annualMonthly × 12Understates true cost
APR (UK legal standard)True annual cost including fees= (1+monthly)^12 − 1
Flat rateInterest on original principal onlyAPR ≈ flat rate × 1.8
Car finance often quotes "flat rates" which are misleadingly low. A "3% flat rate" on a declining balance loan is approximately 5.4% APR. Always convert to APR for fair comparison.