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)
Quick Excel Rule:
• FV / PV: you may use positive values inside the function and wrap with ABS(...) to display a positive result.
• RATE / NPER: do not use ABS. Instead, make PV and FV opposite signs.
Example: =RATE(4,0,-5000,6650) and =NPER(6%,0,-2000,3500).
If signs are entered correctly, the answer should already come out positive.

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%

📽️ Chapter 5 Slides (PPT)

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Video: TVM Basics

Practice Questions (Q1–Q9 with timelines)

How to use this section: You can change the inputs (PV, FV, rate, and years) and then click Calculate. The timeline below each question will update automatically, so you can clearly see where the money starts, where it ends, and whether we are compounding forward or discounting backward.

Excel rule for this chapter: For FV and PV questions, you may enter the cash flow as a positive number inside the Excel function and then put ABS(...) in front to make the result positive. Example: =ABS(FV(5%,5,0,1000)) or =ABS(PV(8%,17,0,150000)).

For RATE and NPER, Excel needs opposite signs for PV and FV because they represent two cash flows. Do not use ABS for RATE or NPER. Use formulas like =RATE(4,0,-5000,6650) and =NPER(6%,0,-2000,3500). If the signs are entered correctly, the result should already be positive.

Q1 — Find FV (compounding)

Invest $1,000 today (t=0) for 5 years at 5%. How much will you have at t=5?

Math: 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
Click Calculate to update the formula bar.

Q2 — Long-run FV intuition

$10 today (t=0) at 5.5% for 200 years. What is the future value?

Math: PV=10, r=5.5%, n=200 → FV ≈ $447,189.80
Excel: =ABS(FV(5.5%,200,0,10))
Big n shows the power of compounding
Click Calculate to update the formula bar.

Q3 — Find FV

Invest $500 today at 8% for 15 years. What is the future value?

Math: PV=500, r=8%, n=15 → FV ≈ $1,586.09
Excel: =ABS(FV(8%,15,0,500))
Compounding forward
Click Calculate to update the formula bar.

Q4 — Find PV (discounting)

You need $150,000 at t=17 and the rate is 8%. How much should you invest today at t=0?

Math: 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
Click Calculate to update the formula bar.

Q5 — Find PV from known FV

A trust fund is worth $19,671.51 at t=10 and earned 7%. What was the original deposit at t=0?

Math: FV=19,671.51, r=7%, n=10 → PV = $10,000.00
Excel: =ABS(PV(7%,10,0,19671.51))
Discounting backward
Click Calculate to update the formula bar.

Q6 — Solve for Rate r

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

Math: 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)
Important: Do not use ABS here. RATE needs opposite signs for PV and FV, and if entered correctly, the answer should already be positive.
We know PV and FV over n periods, so we solve for r
Click Calculate to update the formula bar.

Q7 — Solve for Rate r (another set)

$1,200 today becomes $1,800 over 6 years. What annual rate is implied?

Math: PV=1200, FV=1800, n=6 → r = (1800/1200)^(1/6) − 1 = 6.99%
Excel: =RATE(6,0,-1200,1800)
Important: Do not use ABS here. RATE needs opposite signs for PV and FV, and if entered correctly, the answer should already be positive.
Solve for the annual growth rate
Click Calculate to update the formula bar.

Q8 — Solve for Periods n

You deposit $2,000 today at 6%. How many years until it grows to $3,500?

Math: 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)
Important: Do not use ABS here. NPER needs opposite signs for PV and FV, and if entered correctly, the answer should already be positive.
We know PV, r, and FV, so we solve for n
Click Calculate to update the formula bar.

Q9 — Solve for Periods n (another set)

How long will it take for $750 today at 9% to become $1,500?

Math: PV=750, FV=1500, r=9% → n = ln(2)/ln(1.09) = 8.04 years
Excel: =NPER(9%,0,-750,1500)
Important: Do not use ABS here. NPER needs opposite signs for PV and FV, and if entered correctly, the answer should already be positive.
Solve for time n
Click Calculate to update the formula bar.

Chapter 6 Preview

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