Financial Mathematics
The Financial Mathematics domain applies percentage reasoning to money: percent change, markups and discounts, simple and compound interest, effective rates, savings plans, and loan payments. The arithmetic is easy, so the marks are lost on timing conventions and on treating rates as if they add.
Percent change and successive changes
- Percent change = (new − old)/old × 100%.
- Successive changes multiply: a 20% increase then a 20% decrease gives 1.20 × 0.80 = 0.96, a net loss of 4%.
- A price after a p% discount is P(1 − p/100). A markup followed by a discount on the same base is not symmetric with the reverse order.
Interest
- Simple interest: A = P(1 + rt).
- Compound interest: A = P(1 + r/n)^(nt), where r is the nominal annual rate and n the number of compounding periods per year; the periodic rate is r/n.
- Continuous compounding: A = P·e^(rt).
- Effective annual rate = (1 + r/n)^n − 1. More frequent compounding raises the effective rate for the same nominal rate.
Annuities and loans
- Future value of an ordinary annuity: FV = PMT[((1 + i)^n − 1)/i], with i = r/n and n = total number of payments.
- Loan payment: PMT = P·i/[1 − (1 + i)^(−n)].
- Rule of 72: doubling time ≈ 72 divided by the annual percentage rate.
Worked example
$2,000 is invested at 6% compounded monthly for 3 years. i = 0.06/12 = 0.005 and n = 36, so A = 2000(1.005)^36 ≈ 2000(1.19668) = $2,393.36. The effective annual rate is (1.005)^12 − 1 ≈ 0.0617, or about 6.17%. Simple interest would give 2000(1 + 0.06·3) = $2,360; the $33 difference is the effect of compounding.
Traps
- Using the annual rate r directly as the periodic rate instead of r/n.
- Mixing years with months: n must count compounding periods, not years.
- Adding successive percentage changes instead of multiplying them.
- Quoting the nominal APR when the question asks for the effective yield.