ECON1003 · Quantitative Methods In Economics
Financial Mathematics
Financial mathematics is the most procedural topic in ECON1003: every question is 'plug the right numbers into the right formula and grind', and nearly every formula is on the provided sheet. The skill being tested is the small decisions — compound versus continuous interest, the m (compounding periods per year), and a future-value annuity versus a present-value one. Get those right and the rest is calculator work. This chapter covers arithmetic and geometric series, simple / compound / continuous interest, present value and NPV, and annuities (future and present value), plus reducing-balance depreciation and debt repayment. The marks are routinely lost on choosing the wrong formula or misreading 'term' for 'sum', so the chapter drills the decision tree as hard as the arithmetic.
What this chapter covers
- 013.1 Sequences and series — arithmetic vs geometric
- 023.2 Simple, compound and continuous interest
- 033.3 Present value and net present value (NPV)
- 043.4 Annuities — future value and present value
- 05Reducing-balance depreciation
- 06Debt repayment
Worked example: compound interest and the periods-per-year trap
- +1(a) Identify the inputs. Principal P = 5000, nominal rate i = 0.08, periods per year m = 4, years t = 3. Compound formula: FV = P(1 + i/m)mt.
- +1Substitute. FV = 5000(1 + 0.08/4)4×3 = 5000(1.02)12.
- +1Compute. 1.0212 ≈ 1.2682, so FV ≈ $6,341.21.
- +1(b) Continuous compounding. Use FV = P·eit = 5000·e0.08×3 = 5000·e0.24.
- +1Compute and compare. e0.24 ≈ 1.2712, so FV ≈ $6,356.25 — slightly more than quarterly, because more frequent compounding earns more.
Key terms
- Geometric series
- A sum whose terms multiply by a constant ratio r each step: Sₙ = a(1 − rⁿ)/(1 − r). When |r| < 1 the infinite sum converges to S∞ = a/(1 − r). Compound growth and annuities are geometric series.
- Compound interest
- Interest earned on principal plus previously accrued interest: FV = P(1 + i/m)mt, where m is the number of compounding periods per year. More frequent compounding (larger m) earns more.
- Continuous compounding
- The limit of compounding as m → ∞: FV = P·eit. It uses the natural exponential e and gives the highest value for a given nominal rate.
- Present value (PV)
- The value today of a future cash flow, found by discounting it back at the interest rate: PV = FV/(1 + i)ᵗ. Net present value (NPV) sums the discounted inflows minus the cost; a positive NPV means the project adds value.
- Annuity
- A stream of equal payments over time. A future-value annuity accumulates the payments to the end; a present-value annuity discounts them to today. Telling the two apart is the most common annuity exam decision.
Financial Mathematics FAQ
How do I avoid the most common financial-maths mistakes?
Three decisions cause most lost marks. First, compound vs continuous interest — quarterly uses (1 + i/m)^(mt) with m = 4; continuous uses e^(it). Second, getting m (compounding periods per year) right and matching it to t. Third, telling a future-value annuity from a present-value one. Get those three right and the rest is calculator arithmetic.
What's the difference between a 'term' and a 'sum' question?
Read the wording carefully. 'Output/value in week 20' wants the 20th term of the sequence; 'total over 20 weeks' wants the sum S₂₀ of the series. Mixing them up is a classic slip. And the infinite-sum formula S∞ = a/(1 − r) is only valid when |r| < 1.
Do I have to memorise the interest and annuity formulas?
No — nearly every financial-maths formula (compound and continuous interest, NPV, annuity FV and PV) is on the provided formula sheet. The exam tests whether you pick the right one and substitute correctly, not whether you remember it. Always write your substitution so method marks bank even if the final arithmetic rounds off.
Exam move
Treat financial maths as free procedural marks once the decision is right. Build a one-line decision tree: simple vs compound vs continuous interest; how many periods per year m; FV vs PV; term vs sum. Then it is pure substitution from the sheet plus a calculator. Drill the periods-per-year conversion (a 'nominal 8% compounded quarterly' means i/m = 0.02 per quarter over mt periods) because that is where most errors hide. Always write the substitution line before the number, so the method mark is locked in even if the final figure rounds off.
