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

All-in-One TVM Calculator

PV, FV, PMT, NPER, RATE, annuities, NPV/NFV, APR, and EAR in one separate app.

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See the Time Value of Money

Time value of money means that money can change in value as time passes. If money earns a return, today’s dollar can grow into more than one dollar in the future. The pictures below make the roles of time (n) and the interest rate (r) visible before we calculate them.

Compounding: FV = PV(1+r)^n  →  each year’s interest becomes part of the balance and can earn interest in later years.
Time value of money chart showing one dollar compounded at 10 percent each year from Year 0 through Year 10, growing to 2.59 times the original value.
Watch compounding happen year by year. At 10% per year, $1.00 becomes $1.10 after one year, $1.21 after two years, and about $2.59 after ten years. Growth accelerates because interest earns interest.

Time + Rate: Why Both Matter

Start with the same $1.00. A higher return and a longer holding period create much larger future values.

Comparison of one dollar compounded at 10 percent for 10, 20, and 30 years, growing to 2.59, 6.73, and 17.45 dollars.
10% annual growth: $1.00 → $2.59 in 10 years → $6.73 in 20 years → $17.45 in 30 years.
Comparison of one dollar compounded at 3 percent for 10, 20, and 30 years, growing to 1.34, 1.81, and 2.43 dollars.
3% annual growth: $1.00 → $1.34 in 10 years → $1.81 in 20 years → $2.43 in 30 years.
Key idea: The future value depends on PV, r, and n. Time magnifies the effect of the interest rate. At 30 years, $1 growing at 10% becomes about $17.45, while the same $1 growing at 3% becomes about $2.43. Going backward from a future amount to today is discounting.

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

▶ Time Value of Money Made Simple — Excel & All-in-One TVM Calculator with Aya
Watch on YouTube Open All-in-One TVM Calculator

Practice Questions (Q1–Q15 with interactive 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.

Timeline: Receive $20,000 at month 0; make 120 equal payments at the end of months 1–120. This is an ordinary annuity.

Math: r=3%/12=0.25% per month, n=10×12=120
PMT=(r×PV)/(1-(1+r)^(-n))=(0.0025×20000)/(1-(1.0025)^(-120))=$193.12

Excel: =ABS(PMT(3%/12,10*12,20000,0,0))

Answer: $193.12 per month.

Monthly payments

Q11 — Annuity due versus ordinary annuity: Bridget and Jordan

Bridget saves monthly starting today (annuity due). Jordan saves monthly starting one month from today (ordinary annuity). Change any input and compare their ending balances.

Why Bridget’s first $150 is PMT—not PV

Watch on YouTube

Why Bridget's first deposit is PMT—not PV: it is the first payment in a repeating monthly stream. Because it occurs at month 0, use type=1. Jordan uses type=0.

Q12 — EAR from an 18% APR compounded monthly

Change the APR or the number of compounding periods per year. The periodic rate, EAR, and timeline update automatically.

Q13 — EAR from an 18% APR compounded quarterly

Change the APR or compounding frequency to explore how within-year compounding changes EAR.

Interactive NPV/NFV Calculator

Use the calculator below to practice discounting cash flows to today and compounding value to a future date.

Open jufinance.com/nfv/ →

Q14 — Project NPV and NFV

A project costs $100,000 today and generates $30,000, $40,000, $50,000, and $50,000 in Years 1–4. At a 10% required return, find NPV today and NFV at the end of Year 4.

Try the NPV/NFV website

Q15 — Compare another project using NPV and NFV

A second project costs $120,000 today and generates $25,000, $35,000, $45,000, and $70,000 in Years 1–4. At a 9% required return, find NPV today and NFV at the end of Year 4.

Try the NPV/NFV website
▶ Watch Aya’s Practice Q1–Q11 Walkthrough (7:49)

Aya explains each question while the blackboard shows the prompt, timeline, formula, solution, and Excel setup.

Watch on YouTube
▶ Watch Practice Q12–Q15 — APR/EAR, NPV & NFV with Mei

Follow the blackboard questions, math, and Excel solutions. Enlarge the player or use its full-screen control.

Watch on Synthesia

Chapter 5 Concept Quizzes

Complete Quizzes 1–5 in order. Each quiz provides instant feedback and a short explanation.

Quiz 1 (T/F) Quiz 2 (T/F) Quiz 3: Interest Rates & Value Quiz 4: Annuity Timing Quiz 5: APR, EAR, NPV & NFV

Homework (due with the first midterm)

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