Time Value of Money — Chapter 5

Move value through time: FV = PV(1+r)^n (compounding) and PV = FV/(1+r)^n (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 (annuity/loan).
  • Compounding — forward growth; Discounting — present valuation.
  • APR — nominal annual %; EAR — effective annual %.
Units must match: if r is monthly, then n is in months and cash flows are monthly.

Formulas & Excel (FIN301 cheat sheet)

Math Formulas

  • FV = PV *(1+r)^n
  • PV = FV / ((1+r)^n)
  • N = ln(FV/PV) / ln(1+r)
  • Rate = (FV/PV)^(1/n) - 1

Annuity: solve for N

  • N = ln((FV/C)*r + 1) / ln(1+r)
  • N = ln(1/(1-(PV/C)*r)) / ln(1+r)

Excel Formulas

  • FV: =ABS(FV(rate, nper, pmt, pv))
  • PV: =ABS(PV(rate, nper, pmt, fv))
  • Rate: =RATE(nper, pmt, pv, -fv)
  • Years (NPER): =NPER(rate, pmt, pv, -fv)
  • Annuity payment: =PMT(rate, nper, pv, -fv)
  • EAR: =EFFECT(nominal_rate, npery)
  • APR: =NOMINAL(effective_rate, npery)
Tools: NPV/NFV calculator · TVM calculator

Excel sign rules: If results look negative, wrap the result in ABS(...). If both PV and FV appear, use opposite signs (cash out vs cash in).

Quick Guide: PMT / APR / EAR / NPV

Chapter Add-Ons

PMT (Payment): Excel PMT(rate, nper, pv, [fv], [type]). Ordinary annuity uses type=0 (end of period). Annuity due uses type=1 (beginning).

PMT = recurring cash flow Annuity (type=0) Annuity Due (type=1)

Ordinary Annuity (type = 0)

Blue marker jumps at the end of period → Excel type=0.

Annuity Due (type = 1)

Pink marker jumps at the beginning → Excel type=1.

APR — Annual Percentage Rate

Nominal yearly rate (no within-year compounding). Monthly rate = APR/12.

Excel (APR → EAR): =EFFECT(nominal_rate, npery)

EAR — Effective Annual Rate

True annual return including compounding.

EAR = (1 + APR/m)^m − 1

Excel: =EFFECT(APR, m) and =NOMINAL(EAR, m)

Quick Cheats: NPV, NFV, type

  • NPV: NPV(rate, CF1..CFn) discounts t=1..n. If there’s C0 at time 0, do C0 + NPV(...).
  • NFV: compute PV first, then compound: FV(rate, T, 0, -PV, 0).
  • Annuity timing: type=0 end-of-period; type=1 beginning.
NFV sandbox: jufinance.com/nfv

Videos

Practice Questions (Q1–Q10 with timelines)

Q1 — Find FV (compounding)

Invest $5,000 (PV) at 4% for 8 years. Find FV.

Excel: =ABS(FV(4%,8,0,5000)) • Math: 5000*(1+4%)^8
Compounding from t=0 to t=n

Q2 — Find FV

Invest $3,000 (PV) at 3% for 12 years. Find FV.

Excel: =ABS(FV(3%,12,0,3000)) • Math: 3000*(1+3%)^12
Compounding

Q3 — Find PV (discounting)

Need $20,000 in 10 years; earn 3%. Find PV.

Excel: =ABS(PV(3%,10,0,20000)) • Math: 20000/(1+3%)^10
Discounting back to t=0

Q4 — Find PV

Need $15,000 in 5 years; earn 2%. Find PV.

Excel: =ABS(PV(2%,5,0,15000)) • Math: 15000/(1+2%)^5
Discounting

Q5 — Find rate

PV=$5,000 grows to FV=$6,500 in 5 years. Find rate.

Excel: =RATE(5,0,5000,-6500) • Math: r=(6500/5000)^(1/5)-1
Solve r given PV, FV, n

Q6 — Find rate

PV=$8,000 grows to FV=$10,000 in 6 years. Find rate.

Excel: =RATE(6,0,8000,-10000) • Math: r=(10000/8000)^(1/6)-1
Solve r

Q7 — Find NPER

PV=$5,000 at 4% grows to $6,000. Find NPER.

Excel: =NPER(4%,0,5000,-6000) • Math: n=ln(6000/5000)/ln(1+0.04)
Solve n

Q8 — Find NPER

PV=$10,000 at 5% grows to $15,000. Find NPER.

Excel: =NPER(5%,0,10000,-15000) • Math: n=ln(15000/10000)/ln(1+0.05)
Solve n

Q9 — Monthly payment

Borrow $30,000 at 4% APR for 5 years. Find monthly payment.

Excel: =PMT(4%/12,5*12,30000,0) • Math: PMT=(r·PV)/(1-(1+r)^(-n)), r=0.04/12, n=60
Monthly payments over n months

Q10 — Monthly payment

Borrow $20,000 at 3% APR for 10 years. Find monthly payment.

Excel: =PMT(3%/12,10*12,20000,0) • Math: PMT=(r·PV)/(1-(1+r)^(-n)), r=0.03/12, n=120
Monthly payments

Quiz 1 & Quiz 2 (Chapter 5 Concepts)

Complete Quiz 1 first, then Quiz 2.

Quiz 1 (T/F) Quiz 2 (T/F)

Homework (due with the first midterm)

Answers are hidden — expand each item for the Excel setup and numeric answer.

Appendix: More Practice

Extra practice problems (optional). Filter and expand to reveal the Excel setup and answer.

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