Interest and loans: study guide

Key ideas

Interest problems differ in whether interest is earned on the principal only or on the growing balance. Simple interest, compound interest, effective annual rate and loan payments all follow from a few formulas.

• Simple interest is I = P × r × t. The balance is P + I and grows by the same amount every year.
• Compound interest is A = P(1 + r ÷ n)^(nt), where n is the number of compounding periods per year.
• The effective annual rate EAR = (1 + r ÷ n)ⁿ − 1 puts rates with different compounding on one footing.
• A fixed loan payment is M = P × i ÷ (1 − (1 + i)^−N), where i is the rate per period and N the number of payments.

Key equations

• I = P × r × t
• A = P(1 + r ÷ n)^(nt)
• EAR = (1 + r ÷ n)ⁿ − 1
• M = P × i ÷ (1 − (1 + i)^−N)

Worked example

Simple interest on 1000 at 5 % for 3 years is 1000 × 0.05 × 3 = 150. Compounded yearly, the balance is 1000 × 1.05³ = 1157.63. A nominal 12 % compounded monthly has EAR = 1.01¹² − 1 = 12.68 %. A loan of 10 000 at 6 % per year over 60 monthly payments has i = 0.005 and M = 10 000 × 0.005 ÷ (1 − 1.005⁻⁶⁰) = 193.33.

Common mistakes

• Using the annual rate with the number of months, instead of dividing the rate by 12 for a monthly period.
• Comparing nominal rates with different compounding as if they were equal.
• Reporting the final balance when the question asks for the interest earned, which is the balance minus the principal.

Practice

Test yourself with practice questions on this topic: Interest and loans