Time Value of Money — Chapter 5

Move value through time: FV = PV(1+r)^n (forward, compounding) and PV = FV/(1+r)^n (backward, discounting).

Theme: Single-page workbook

Glossary & Notation

  • PV — Present Value: value at t=0 (today).
  • FV — Future Value: value at t=n (future date).
  • r — interest rate per period (year if annual, month if monthly).
  • n — number of periods.
  • PMT — constant payment per period (for annuities/loans; Ch.6).
  • Compounding — forward growth; Discounting — present valuation.
  • APR — nominal annual % (no intra-year compounding); EAR — effective annual % (with compounding).
Units must match: if r is monthly, then n is in months and cash flows are monthly.

Core Formulas & Excel Twins

  • Future Value: FV = PV(1 + r)^n • Excel: =ABS(FV(rate, nper, 0, PV))
  • Present Value: PV = FV / (1 + r)^n • Excel: =ABS(PV(rate, nper, 0, FV))
  • Rate: r = (FV/PV)^(1/n) − 1 • Excel: =RATE(nper, 0, -PV, FV) (PV & FV must have opposite signs)
  • Periods: n = ln(FV/PV) / ln(1+r) • Excel: =NPER(rate, 0, -PV, FV) (PV & FV opposite signs)

Excel sign rules: Single cash flow (only PV or only FV): keep the amount positive and wrap the function with ABS(...) (e.g., =ABS(FV(...)), =ABS(PV(...))). Two cash flows (PV & FV both present): use opposite signs — one positive, one negative (e.g., =RATE(nper,0,-PV,FV), =NPER(rate,0,-PV,FV)). The on‑page toggle only affects the interactive calculators.

APR vs EAR

APR is the quoted nominal rate. EAR is the true annual growth including compounding.

  • APR→EAR (m times/year): EAR = (1 + APR/m)^m − 1
  • Excel: =EFFECT(APR, m)
  • Example: =EFFECT(12%, 12) ⇒ 12.68%

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Practice Questions (Q1–Q9 with timelines)

Q1 — Find FV (compounding)

Invest $1,000 today for 5 years at 5%. How much at time 5?

PV=1000, r=5%, n=5 → FV = 1000×(1.05)^5 = $1,276.28 • Excel: =ABS(FV(5%,5,0,1000))
Compounding from t=0 to t=n

Q2 — Long-run FV intuition

$10 at 5.5% for 200 years. What is FV?

PV=10, r=5.5%, n=200 → FV ≈ $447,189.80 • Excel: =ABS(FV(5.5%,200,0,10))
Big n shows power of compounding

Q3 — Find FV

$500 at 8% for 15 years. What is FV?

PV=500, r=8%, n=15 → FV ≈ $1,586.09 • Excel: =ABS(FV(8%,15,0,500))
Compounding

Q4 — Find PV (discounting)

Need $150,000 in 17 years at 8%. How much to invest today?

FV=150,000, r=8%, n=17 → PV ≈ $40,540.34 • Excel: =ABS(PV(8%,17,0,150000))
Discounting from t=n back to t=0

Q5 — Find PV from known FV

Trust fund now is $19,671.51 (after 10 years at 7%). What was the initial deposit?

FV=19,671.51, r=7%, n=10 → PV = $10,000.00 • Excel: =ABS(PV(7%,10,0,19671.51))
Discounting

Q6 — Solve for Rate r

You invest $5,000 and it grows to $6,650 in 4 years. What annual rate did you earn?

PV=5000, FV=6650, n=4 → r = (FV/PV)^(1/n) − 1 = (6650/5000)^(1/4) − 1 = 7.31%
Excel: =RATE(4,0,-5000,6650)
We know PV and FV over n periods → solve r

Q7 — Solve for Rate r (another set)

$1,200 becomes $1,800 over 6 years. What annual r?

PV=1200, FV=1800, n=6 → r = (1800/1200)^(1/6) − 1 = 6.92%
Excel: =RATE(6,0,-1200,1800)
Solve r

Q8 — Solve for Periods n

You deposit $2,000 at 6% annual. How many years to reach $3,500?

PV=2000, FV=3500, r=6% → n = ln(FV/PV)/ln(1+r) = ln(1.75)/ln(1.06) = 9.74 years
Excel: =NPER(6%,0,-2000,3500)
We know PV, r, FV → solve n

Q9 — Solve for Periods n (another set)

How long for $750 at 9% to become $1,500?

PV=750, FV=1500, r=9% → n = ln(2)/ln(1.09) = 8.04 years
Excel: =NPER(9%,0,-750,1500)
Solve n

Chapter 6 Preview

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