Why 1.5% a month is not 18% a year

Multiplying a monthly rate by 12 gives the nominal annual rate, but if unpaid interest is added to the balance each month, you pay interest on interest. Then the true yearly cost is higher: 1.5% a month compounds to about 19.56% a year, not 18%. To convert, use (1 + monthly rate)¹² − 1.

Two different “annual” rates

• Nominal rate: monthly rate × 12. Simple, and the convention lenders use when they divide an annual rate by 12 to calculate monthly interest. 1.5% × 12 = 18%.
• Effective rate: what a balance actually grows by over a year if interest is added monthly and nothing is paid off. (1.015)¹² − 1 = 19.56%.

The gap grows with the rate. At 2% a month, nominal is 24% but effective is about 26.8%. At 0.5% a month, nominal 6% becomes about 6.17%.

Worked through month by month

Borrow 1,000 at 1.5% a month and pay nothing:

• Month 1: interest 15.00, balance 1,015.00
• Month 2: interest 15.23, balance 1,030.23
• Month 3: interest 15.45, balance 1,045.68

Each month’s interest is a little larger than the last because it is charged on the previous interest too. After 12 months the balance is about 1,195.62 — 195.62 of interest, not 180.

Converting either way

• Monthly to effective annual: (1 + m)¹² − 1. For m = 1.5%: 1.015¹² − 1 = 19.56%.
• Effective annual to monthly: (1 + a)^(1/12) − 1. For a = 12%: 1.12^(1/12) − 1 ≈ 0.949% a month, not 1%.
• Nominal annual to monthly: divide by 12. 12% nominal = 1% a month.

The Compound Interest Calculator lets you pick the compounding frequency, so you can see the difference between monthly, quarterly and yearly compounding on the same nominal rate.

Where this matters

• Credit cards and store finance often quote a monthly rate. A card at 3% a month costs about 42.6% a year if the balance is left to grow.
• Informal loans and moneylenders commonly quote “2 per cent a month” — about 26.8% a year effective, more than many people expect.
• Savings accounts work in your favour: interest credited monthly earns slightly more than the same nominal rate credited yearly. That is why some banks quote an AER or APY, which is the effective rate.

Daily rates

Some cards and overdrafts calculate interest daily. A daily rate of 0.05% sounds tiny, but × 365 is 18.25% nominal, and compounded daily it is about 20% a year. The same conversion applies with 365 in place of 12: (1 + daily rate)³⁶⁵ − 1. Short-term loans quoted “per week” work the same way with 52, and the effective annual figures can be startling. Whenever a rate is quoted over a short period, convert it to an effective annual rate before comparing it with anything else.

What about EMIs?

Standard loan EMIs use the nominal convention: an annual rate of 12% means 1% on the outstanding balance each month. Because each EMI pays that month’s interest in full, interest does not pile up on interest, and the 12% figure describes the cost fairly. Enter the annual rate in the EMI Calculator or the Loan Calculator; do not convert it to an effective rate first.

Reading the small print

Look for the words that tell you which rate you are seeing: “per month”, “p.m.” or “monthly” for a monthly rate; “p.a.” or “per annum” for yearly; “APR” for a yearly figure that, in many countries, must also include certain fees; “AER” or “APY” for an effective savings rate.

Mistakes to avoid

• Typing a monthly rate into a calculator that expects an annual one, or the reverse.
• Comparing one lender’s monthly rate with another’s annual rate.
• Assuming a savings rate and a loan rate with the same number cost the same.

Checklist

• Is the quoted rate per month or per year?
• Nominal or effective (APR vs AER/APY)?
• Does interest compound if unpaid?
• Comparing like with like across offers?