Estimated study time: 45 minutes
Content:
The time value of money (TVM) is the foundational concept underpinning nearly all of finance: a dollar available today is worth more than a dollar available in the future, because today's dollar can be invested to earn a return. The core TVM variables are: present value (PV), future value (FV), interest rate per period (I/Y or r), number of periods (N), and periodic payment (PMT). Mastery of TVM calculations is essential for CFA Level 1 — it underpins bond valuation, equity valuation, capital budgeting, and retirement planning problems throughout the curriculum. The basic future value formula is: FV = PV × (1 + r)^N. The present value formula reverses this: PV = FV / (1 + r)^N. These formulas assume a single lump sum; annuity problems add a PMT component.
An annuity is a series of equal payments made at regular intervals. An ordinary annuity (annuity-immediate) makes payments at the end of each period, while an annuity due makes payments at the beginning. The present value of an ordinary annuity is: PV = PMT × [1 – (1 + r)^(–N)] / r. The future value is: FV = PMT × [(1 + r)^N – 1] / r. An annuity due is worth more than an ordinary annuity by a factor of (1 + r) because each payment compounds for one additional period. A perpetuity is an annuity that pays forever; its present value is simply: PV = PMT / r. For example, a preferred stock paying $2 per year with a required return of 8% is worth $2 / 0.08 = $25. Recognizing whether a problem involves a lump sum, ordinary annuity, annuity due, or perpetuity is the first step in every TVM problem.
Compounding frequency matters significantly. When interest compounds more frequently than annually, the effective annual rate (EAR) exceeds the stated annual percentage rate (APR). The EAR formula is: EAR = (1 + stated rate / m)^m – 1, where m is the number of compounding periods per year. Continuous compounding represents the limit as m approaches infinity: EAR = e^r – 1. For example, a 12% APR compounded monthly gives an EAR of (1 + 0.12/12)^12 – 1 = 12.68%. When comparing investment alternatives with different compounding frequencies, always convert to EAR before comparing. On the CFA exam, problems frequently test whether candidates adjust the interest rate and number of periods to match the payment frequency.
TVM concepts are directly applied in loan amortization and retirement planning scenarios. In a mortgage, each monthly payment covers interest on the outstanding balance plus a portion of principal. As the loan amortizes, the interest portion decreases and the principal portion increases. For retirement planning, a common problem type involves computing how much must be saved per period (PMT) to achieve a target nest egg (FV), or how much can be withdrawn per period (PMT) given a starting balance (PV) and expected return (r) over N years. Multi-stage TVM problems — where cash flows change at different points in time — are solved by working through each stage sequentially, using the ending PV/FV of one stage as the starting point of the next.
Key Terms:
Quiz Questions:
Q1. An investor plans to deposit $5,000 per year at the end of each year for 10 years into an account earning 6% per year. What is the future value of this investment at the end of 10 years?
A) $50,000 B) $65,904 C) $59,874 D) $44,161
Answer: B — This is an ordinary annuity problem. FV = 5,000 × [(1.06)^10 – 1] / 0.06 = 5,000 × 13.181 = $65,904. Option A ignores compounding. Option C is the result of incorrectly using PV of annuity instead of FV. Option D applies a wrong rate or period.
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Q2. A loan of $200,000 is taken at an annual interest rate of 6%, compounded monthly, for 30 years. What is the monthly payment?
A) $1,199 B) $1,432 C) $1,667 D) $955
Answer: A — The monthly rate is 6%/12 = 0.5%, and N = 30 × 12 = 360 periods. PV = 200,000; solve for PMT: PMT = 200,000 × [0.005 / (1 – (1.005)^(–360))] ≈ $1,199. This is a standard mortgage amortization problem — the key is converting the annual rate and term to monthly equivalents before solving.
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Q3. An investment offers a stated annual interest rate of 12% compounded quarterly. What is the effective annual rate?
A) 12.00% B) 12.55% C) 12.36% D) 12.68%
Answer: C — EAR = (1 + 0.12/4)^4 – 1 = (1.03)^4 – 1 = 1.1255 – 1 = 12.55%. Wait — let me recalculate: (1.03)^4 = 1.12551, so EAR = 12.55%. The correct answer is B: 12.55%. Option C (12.36%) corresponds to monthly compounding (1 + 0.12/12)^12 = 12.68%, which is Option D. Option B (12.55%) is correct for quarterly compounding.
Answer: B — EAR = (1 + 0.12/4)^4 – 1 = (1.03)^4 – 1 ≈ 12.55%. The more frequent the compounding, the higher the EAR relative to the stated rate. Monthly compounding (Option D at 12.68%) would be even higher. Option A assumes no intra-year compounding.
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Q4. A wealthy donor establishes an endowment that will pay a university $500,000 per year in perpetuity. If the endowment earns a 5% annual return, what lump sum must be donated today?
A) $2,500,000 B) $5,000,000 C) $10,000,000 D) $7,500,000
Answer: C — PV of a perpetuity = PMT / r = $500,000 / 0.05 = $10,000,000. This formula works because at a 5% return on $10 million, the endowment generates exactly $500,000 per year in perpetuity while preserving principal. Options A and B use wrong divisors.
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Q5. An investor will receive $10,000 in 5 years and $15,000 in 8 years. At a discount rate of 7%, what is the combined present value of these cash flows today?
A) $25,000 B) $16,362 C) $15,868 D) $14,773
Answer: C — PV of $10,000 in 5 years = 10,000 / (1.07)^5 = 10,000 / 1.4026 = $7,130. PV of $15,000 in 8 years = 15,000 / (1.07)^8 = 15,000 / 1.7182 = $8,730. Combined PV ≈ $7,130 + $8,730 = $15,860 ≈ $15,868. Option A ignores discounting entirely. Multi-cash-flow problems require discounting each flow separately and summing.
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