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Reading 1CFA L1 QuantFull chapter

Time value of money in finance

In this chapter: Discount factors, present and future values · Annuities, perpetuities, growing perpetuities · Continuous compounding and effective rates

~6 min readLayer 4 · Professional CertificationsFree

If you remember one idea from your entire CFA program — make it this one. Every valuation, every bond price, every option premium, every retirement-corpus projection that follows is a specialised application of Time Value of Money (TVM). Get this reading right and most exam questions reduce to one operation: identify the cash flows, identify the rate, identify the timing. Get it wrong and the rest of the curriculum will feel unnecessarily hard. We will spend extra time here so the foundation is solid.

Foundation

₹100 today is worth more than ₹100 a year from now. Three reasons: opportunity cost (you could deploy it), inflation (rupee buys less later), and uncertainty (future cash isn't guaranteed). TVM puts a number on this. Two equations: • Future value: FV = PV × (1 + r)^n • Present value: PV = FV ÷ (1 + r)^n Where r is the periodic interest rate and n is the number of compounding periods. That is the entire toolkit. Annuities (regular cash flows), perpetuities (forever-cash flows), and bond pricing are all special cases of these two equations.

Deep Dive

Vocabulary that decides exam questions: • Periodic rate — interest rate per compounding period. If question says "8% per year, compounded quarterly", periodic rate is 8/4 = 2% per quarter, and n is the number of quarters. • Stated annual rate (SAR / nominal) — the quoted, headline rate. 8% in our example. • Effective annual rate (EAR) — what you actually earn over a year accounting for intra-year compounding. EAR = (1 + periodic)^m − 1, where m = periods per year. For 8% quarterly: EAR = (1.02)^4 − 1 = 8.24%. The 24-bps gap is real money over decades. • Continuous compounding — limit as m → ∞. EAR = e^r − 1. Annuity formulas (the workhorses): • PV of ordinary annuity (end-of-period payments): PV = PMT × [1 − (1+r)^−n] / r • FV of ordinary annuity: FV = PMT × [(1+r)^n − 1] / r • Annuity due (start-of-period payments): multiply ordinary by (1+r) • Perpetuity (forever): PV = PMT / r • Growing perpetuity: PV = CF₁ / (r − g), provided r > g The growing perpetuity formula is the foundation of the Gordon Dividend Discount Model — equity valuation in disguise.

Advanced

Three exam-day habits that protect TVM answers: 1. Always write n and r before touching the calculator. Half a minute of clarity saves three minutes of recovery if you input wrongly. 2. Convert compounding frequency at step one. If rate is annual but compounding is quarterly, calculator inputs are the periodic rate and period count. Do this once. 3. Sanity-check magnitudes against the Rule of 72. At 12%, money doubles every 6 years — 5 doublings in 30 years = 32×. If your calculator gives 5× or 5,000×, you misread something. Most-tested L1 trap: stated annual rate vs effective annual rate. When the question gives an annual rate but a non-annual compounding frequency, do NOT use the annual rate directly with annual periods. Always convert first. This single error trips more candidates than any other Quant question.

Regulatory references
  • CFA Institute Curriculum — Level 1, Quantitative Methods, Reading 1
  • Indian context: RBI repo rate as anchor for risk-free discount rate
  • NISM 10A — Investment Adviser Level 1 — overlapping TVM treatment
Common mistakes & pitfalls
  • Using stated annual rate directly with non-annual compounding frequency without converting to periodic rate.
  • Confusing ordinary annuity (end of period) with annuity due (start of period) — multiplying by (1+r) is needed for due.
  • Forgetting that perpetuity formula (PV = PMT/r) requires r > 0.
  • Confusing nominal and real returns — must use one consistent set throughout the calculation.
  • In growing perpetuity, mistakenly using r ≤ g — formula breaks down (negative or infinite value).

Frequently asked

Why does the Rule of 72 work?
It's a mathematical approximation derived from continuous compounding. The exact formula is years to double = ln(2) / ln(1+r) ≈ 0.693 / r for small r. The "72" is a convenience constant that's reasonably accurate for r between 6% and 12% — exactly the range most retail investors think about. For very high or low rates, it gets less precise; for everyday financial discussions, it's perfect.
When should I use simple interest vs compound interest?
In modern Indian finance, almost everything compounds — bank FDs (quarterly compounding), mutual fund NAVs (continuous), home loans (monthly amortisation). Simple interest persists in only a few corners: short-term commercial paper, some inter-bank lending, and many bond coupons (which are simple within the period but the total return compounds via reinvestment). When in doubt, assume compound — and verify the compounding frequency.
How does TVM apply to bond pricing?
A bond is just a series of cash flows: periodic coupons + principal at maturity. Bond price = PV of all those cash flows discounted at the appropriate yield. Higher yield → lower price (inverse relationship). All of bond mathematics in CFA Fixed Income is direct application of TVM. Master Reading 1 here and you're halfway through Reading 1 of Fixed Income too.

Practice questions

Click each question to reveal the answer and explanation.

Q 1
A bank FD pays 8% per year compounded quarterly. The effective annual rate (EAR) is closest to:
  1. (a)8.00%
  2. (b)8.16%
  3. (c)8.24%
  4. (d)8.30%
Correct: (c) 8.24%
EAR = (1 + 0.08/4)^4 − 1 = (1.02)^4 − 1 = 0.0824 = 8.24%. The 24 bps gap between SAR (8%) and EAR (8.24%) compounds materially over decades. This is the most-tested CFA L1 Quant question type.
Q 2
₹50,000 invested today at 9% annual compounding for 10 years grows to (closest):
  1. (a)₹95,000
  2. (b)₹1,18,368
  3. (c)₹1,35,000
  4. (d)₹2,00,000
Correct: (b) ₹1,18,368
FV = 50,000 × (1.09)^10 = 50,000 × 2.3674 = ₹1,18,368.
Q 3
A perpetuity pays ₹500 per year. At a 6% discount rate, its present value is closest to:
  1. (a)₹500
  2. (b)₹3,000
  3. (c)₹5,000
  4. (d)₹8,333
Correct: (d) ₹8,333
PV = PMT/r = 500/0.06 = ₹8,333. Perpetuity formula is the most elegant in finance.
Q 4
A growing perpetuity has CF₁ = ₹100, growth 4%, discount rate 10%. PV is:
  1. (a)₹1,000
  2. (b)₹1,667
  3. (c)₹2,500
  4. (d)₹4,000
Correct: (b) ₹1,667
PV = CF₁ / (r − g) = 100 / (0.10 − 0.04) = 100 / 0.06 = ₹1,667. This is the Gordon Growth model.
Q 5
A SIP of ₹10,000 per month for 20 years at 12% per year (1% per month). The future value is closest to:
  1. (a)₹24 lakh
  2. (b)₹50 lakh
  3. (c)₹98 lakh
  4. (d)₹1.5 cr
Correct: (c) ₹98 lakh
FV = 10,000 × [(1.01)^240 − 1] / 0.01 = 10,000 × 989.255 = ₹98,92,554 ≈ ₹98 lakh. The total invested is ₹24 lakh — the rest is compounding magic.
Q 6
Using the Rule of 72, money doubles in approximately how many years at 9%?
  1. (a)6 years
  2. (b)8 years
  3. (c)12 years
  4. (d)15 years
Correct: (b) 8 years
72/9 = 8 years. The exact answer is ln(2)/ln(1.09) = 8.04 years. Rule of 72 is highly accurate in this range.
Educational purposes only. The numbers, returns, and examples used in this lesson are illustrative. Past performance does not guarantee future results. Mutual fund and securities investments are subject to market risks. This lesson is not investment advice; for advice tailored to your circumstances, consult a SEBI-registered Investment Adviser. Read our full disclaimer.