BFC2140 · Corporate Finance
Financial Mathematics: Time Value of Money
Time value of money (TVM) is the foundation of every other topic in BFC2140 — a dollar today is worth more than a dollar later because it can earn a return, so all valuation is just discounting future cash flows to a present value. This chapter covers single sums, mixed streams, perpetuities and annuities (ordinary and due), loan amortisation, compounding frequency, APR vs EAR and continuous compounding. It is examined as numerical-answer questions in the MST and Section B of the final, where the marks come from picking the right cash-flow shape and applying the matching formula to two decimal places.
What this chapter covers
- 01Future and present value of a single sum: FV = C(1 + r)ⁿ and PV = C/(1 + r)ⁿ
- 02Solving for the unknown rate r or term n (logs, the Rule of 72 as a sanity check)
- 03Simple vs compound interest, and the value-additivity of mixed cash-flow streams
- 04Perpetuities (PV = C/r) and growing perpetuities (PV = C₁/(r − g))
- 05Ordinary annuities vs annuities due, and the present-value/future-value annuity factors
- 06Loan amortisation: solving for the level payment and splitting it into interest and principal
- 07Compounding frequency m: FV = C(1 + r/m)^(m·n)
- 08Nominal/APR vs EAR = (1 + r/m)^m − 1, and continuous compounding FV = C·e^(r·n)
EAR and continuous compounding compared
- 1 markFor monthly compounding use EAR = (1 + r/m)^m − 1 with r = 0.072 and m = 12: EAR = (1 + 0.072/12)¹² − 1 = (1.006)¹² − 1.
- 1 markCompute the monthly-compounding EAR: (1.006)¹² − 1 = 0.0744 = 7.44%.
- 1 markFor continuous compounding use EAR = e^r − 1 with r = 0.072: EAR = e^0.072 − 1 = 1.07466 − 1.
- 1 markCompute the continuous EAR: 1.07466 − 1 = 0.0747 = 7.47%.
- 1 markCompare on the common (effective annual) basis: 7.47% > 7.44%, so choose the continuous-compounding rival.
Key terms
- Future value (FV) / present value (PV)
- FV = C(1 + r)ⁿ compounds a cash flow forward n periods; PV = C/(1 + r)ⁿ discounts it back. PV is the workhorse of valuation — every bond, share and project value is a sum of discounted future cash flows.
- Compound vs simple interest
- Simple interest is earned only on the original principal (Interest = Principal × rate × periods); compound interest earns interest on accumulated interest too, which is why FV uses (1 + r)ⁿ. Over long horizons compounding dominates.
- Perpetuity
- A level cash flow that continues forever: PV = C/r. A growing perpetuity (cash flows rising at g) is PV = C₁/(r − g), valid only when r > g. Preference shares and the Gordon share model are perpetuity applications.
- Annuity (ordinary vs due)
- A finite stream of equal cash flows. An ordinary annuity pays at the end of each period: PV = C·[1 − (1 + r)^(−n)]/r. An annuity due pays at the start, so its PV (and FV) equals the ordinary value × (1 + r).
- Effective annual rate (EAR)
- The true annual rate after intra-year compounding: EAR = (1 + r/m)^m − 1, where r is the nominal/APR rate and m the periods per year. EAR is what you compare across products; APR alone is misleading.
- Continuous compounding
- The limit of ever-more-frequent compounding: FV = C·e^(r·n) and PV = C·e^(−r·n). Its effective annual rate e^r − 1 is the maximum achievable from a given nominal rate r.
Financial Mathematics: Time Value of Money FAQ
How do I know whether to use the annuity, perpetuity or single-sum formula?
Classify the cash-flow pattern first. One lump sum at a single date → use FV/PV of a single sum. A finite series of equal payments → annuity (ordinary if end-of-period, due if start-of-period). An equal payment forever → perpetuity (C/r), or a growing perpetuity (C₁/(r − g)) if it rises at a constant rate. An irregular mixed stream → discount each cash flow separately and add (value additivity). Drawing a quick timeline before you compute is the fastest way to pick the right tool, which is exactly the model-choice skill the exam rewards.
What is the difference between APR (nominal rate) and EAR?
The APR or nominal rate is the quoted annual rate that ignores intra-year compounding; the EAR is the rate you actually earn or pay once compounding is included. Convert with EAR = (1 + r/m)^m − 1. For example, 12% compounded monthly is an EAR of (1.01)¹² − 1 = 12.68%. Always compare loans or deposits on the EAR, and only discount cash flows with a rate whose compounding frequency matches the cash-flow frequency.
When do I use continuous compounding?
Use FV = C·e^(r·n) (and PV = C·e^(−r·n)) when the problem states the rate is compounded continuously, or asks for the limiting case of ever-more-frequent compounding. Its effective annual rate is e^r − 1, which is the highest EAR a given nominal rate r can produce — useful as the ceiling when comparing different compounding frequencies.
How do I solve for the rate or the number of periods?
Rearrange the single-sum FV formula. For the rate: r = (FV/PV)^(1/n) − 1. For the term: n = ln(FV/PV) / ln(1 + r). Use the Rule of 72 (doubling time ≈ 72/r) as a quick sanity check on your answer — if a result is wildly different from the rule, you have likely mis-keyed the logs or the exponent.
How is time value of money examined in BFC2140?
Heavily and early. It is the core of the Week 6 mid-semester test and appears throughout Section B (numerical) of the final, both directly and embedded inside bond, share and project questions. Expect to compute PV/FV of single sums and annuities, find loan payments, convert between APR and EAR, and handle continuous compounding — all to two decimal places with a financial calculator.
Exam move
Make TVM automatic, because it underpins every later topic. Start every problem by sketching a timeline and labelling each cash flow with its timing and sign, then name the pattern (single sum, annuity, perpetuity, or mixed) before reaching for a formula. Drill the financial-calculator keystrokes (HP 10bII+ or Casio FX) until PV/FV/PMT/N/I-YR are second nature, and always reconcile your calculator answer against the algebra so a mis-keyed sign or compounding frequency cannot cost you marks. Keep the rate's compounding frequency matched to the cash-flow frequency — semi-annual cash flows need a semi-annual rate — and convert to EAR whenever you must compare across frequencies. Finish each answer at two decimal places, the Section B convention, and use the Rule of 72 as a fast plausibility check on any doubling-time or rate calculation.
