Financial Mathematics
This domain tests time value of money: compounding and discounting, effective rates, annuities, perpetuities, loan instalments and project appraisal by NPV and IRR. Cash-flow timing is the main source of error.
Core rules
- Future value = PV(1 + i)^n; present value = FV/(1 + i)^n.
- Effective annual rate = (1 + i/m)^m − 1 for nominal rate i compounded m times a year.
- Ordinary annuity PV = PMT × [1 − (1 + i)^−n]/i; annuity due multiplies this by (1 + i).
- Perpetuity PV = PMT/i; growing perpetuity PV = PMT/(i − g), with i > g.
- NPV = PV of inflows − initial outlay. Accept an independent project if NPV > 0; IRR is the rate at which NPV = 0.
- Loan instalment = principal/annuity factor.
Worked example
A project costs 10,000 now and returns 4,000 at the end of each of years 1–3. The discount rate is 10%. Annuity factor = [1 − 1.1^−3]/0.1 = 2.4869. PV of inflows = 4,000 × 2.4869 = 9,947.6. NPV = 9,947.6 − 10,000 = −52.4. Reject the project because NPV is negative; the IRR is slightly below 10%. The timing assumption matters: if the first 4,000 arrived immediately, the annuity-due factor would be higher and the decision could change.
Traps
- Discounting year 1 cash flow twice.
- Using a nominal rate for annual cash flows without adjusting for compounding.
- Ranking mutually exclusive projects by IRR alone when scale differs.
- Confusing an ordinary annuity with an annuity due.