Financial Mathematics and Interest Theory
Financial Mathematics and Interest Theory
This domain tests accumulating and discounting cash flows, nominal and effective rates, annuities, and interest-rate sensitivity. Items frequently combine a rate conversion with a present-value calculation.
Core rules and concepts
- Compound accumulation: A(t)=P(1+i)^t; discount factor v=1/(1+i).
- Nominal rate i^(m) convertible m times per year: effective i=(1+i^(m)/m)^m−1.
- Force of interest δ=ln(1+i); accumulation under varying δ is exp(∫δ dt).
- Annuity-immediate a-angle-n=(1−v^n)/i; annuity-due ä=(1+i)a.
- NPV discounts each cash flow at its own time; IRR solves NPV=0.
- Macaulay duration D=Σt·PV(CF_t)/PV; modified duration D*=D/(1+i) measures price sensitivity.
- Spot rates discount single payments; forward rates link spot rates across maturities.
Worked example
A nominal rate of 8% convertible quarterly has quarterly rate 0.02. Effective annual rate=(1.02)^4−1=1.08243216−1=0.082432, or 8.2432%. The present value of 1,000 due in one year is 1,000/1.082432≈923.85. Using 8% as if it were effective would give 925.93 and lose marks.
Traps that cost marks
- Treating a nominal rate as effective.
- Mixing annuity-due and annuity-immediate timing.
- Discounting a cash flow by the wrong number of periods.
- Assuming simple interest for long horizons; it is not time-consistent.
- Forgetting that IRR can have multiple sign changes and is not a ranking rule when projects differ in scale.