Chapter 02 — Finance
Time Value of Money & DCF
A Dollar Today vs. A Dollar Tomorrow — What is this cash flow worth today?
🎯 The Core Idea — Start Here
One sentence: Money today is worth more than the same amount of money in the future — because today's money can be invested and grow.
Everything in this chapter — PV, FV, NPV, IRR, WACC, DCF — is built on this single idea. If you understand it intuitively, the rest is just math filling in the details.
Everything in this chapter — PV, FV, NPV, IRR, WACC, DCF — is built on this single idea. If you understand it intuitively, the rest is just math filling in the details.
📈 Watch $100 Grow Over 10 Years
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Start: $100After 10 years at 10%: $259Total growth: +159%
Hover over any bar to see the exact value. Each bar is the Future Value (FV) at that year — the Present Value (PV) is always the leftmost bar ($100). Switch the rate dropdown to see how small differences in return rate create enormous differences over time (this is compounding).
🧩 Quick Warm-Up — Test Your Intuition First
Before diving into formulas, answer these. They require no math — just common sense. Click "Show Answer" when ready.
1
If you put $100 in a savings account earning 10% per year, how much do you have after 1 year? After 2 years?
2
Your grandma promises to give you $121 in exactly 2 years. The bank pays 10%/year. What is that gift worth in today's dollars?
3
Why would you prefer to receive $500 today rather than $500 in one year, even if both amounts are guaranteed?
Topic & Why It Matters
Every investment decision boils down to one question: is what you are getting worth more than what you are giving up? The time value of money is the mathematical framework for answering that question. It underlies DCF valuation, bond pricing, capital budgeting, and virtually every quantitative decision in corporate finance.
The curse of discounting. Forget the "magic of compounding" — use the reverse. You are owed $1,000,000, but you won't receive it for 30 years. At a 10% discount rate, that $1M is worth only $57,000 today. This single fact explains why long-duration assets — growth stocks, 30-year bonds — are so violently sensitive to interest rate changes. A small rise in the discount rate destroys enormous present value. This is not a technicality; it is why Federal Reserve rate decisions move equity markets by trillions.
This chapter gives you the tools to discount any stream of cash flows, compute NPV and IRR for any project, understand WACC as the cost of capital, and see why DCF is both the most powerful and most dangerous tool in finance.
Knowledge Points
Each card shows the technical definition first, then a real-life analogy to make it click. Click "Key Terms" to expand definitions of every term used.
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1. Present Value (PV) and Future Value (FV)
FV = PV × (1+r)^t → PV = FV / (1+r)^t. r is the discount rate (opportunity cost of capital); t is the number of time periods. Intuition: money has a time price, just like goods have a market price. A dollar promised in the future is worth less than a dollar in hand today — because today's dollar can be invested elsewhere.
🧃 Everyday Example: The Friend Who Owes You $110
Your friend owes you $110 and will pay you back exactly one year from now. Meanwhile, a bank offers a savings account earning 10% per year.
If you deposited $100 today, it would grow to exactly $110 in one year. So your friend's IOU is worth $100 to you right now — not $110. The extra $10 is the "time cost" of waiting.
$100 today = $110 in one year (at 10%)
Flip it: $110 in one year = $100 today
That's all PV and FV are doing — converting money across time at a given rate.
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2. Net Present Value (NPV)
NPV = Σ [CFₜ / (1+r)^t] − Initial Investment. Decision rule: invest if NPV > 0; reject if NPV < 0. NPV > 0 means the project earns more than the cost of capital and therefore creates value. Common misconception: 'A positive NPV guarantees profit.' It does not — it is the expected value given the assumptions you feed into the model. The model is only as good as the business judgment behind the inputs.
🍋 Everyday Example: Should You Buy This Lemonade Stand?
You can buy a lemonade stand for $1,000. It earns $400 profit each year for 3 years. Naively: $400 × 3 = $1,200. That's $200 profit — great deal, right?
Not so fast. If you could instead invest that $1,000 in an index fund at 10%/year, those future $400 payments are worth less than face value today.
Year 1: $400 ÷ 1.10¹ = $364 today
Year 2: $400 ÷ 1.10² = $331 today
Year 3: $400 ÷ 1.10³ = $301 today
Total PV of future earnings = $996
NPV = $996 − $1,000 = −$4 ❌
Barely negative! The stand barely covers the opportunity cost. If it earned $450/year instead: PV ≈ $1,118 → NPV = +$118 ✓ — now it's worth it.
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3. Internal Rate of Return (IRR)
The discount rate that makes NPV = 0. Decision rule: invest if IRR > hurdle rate (WACC). Three critical pitfalls: (1) Multiple IRRs — when cash flows change sign more than once, multiple valid IRRs can exist. (2) Reinvestment assumption — IRR assumes all intermediate cash flows are reinvested at the IRR rate, which is often unrealistic. (3) Scale blindness — a 50% IRR on $100 creates less value than a 20% IRR on $10M. Always compute NPV first; use IRR as a secondary check only.
🏦 Everyday Example: What Interest Rate Is This Deal Paying Me?
A friend offers you a deal: 'Give me $1,000 today and I'll pay you $1,200 in 2 years.'
IRR is asking: 'What annual interest rate does this represent?'
If a savings account turned $1,000 into $1,200 in 2 years, that's:
$1,000 × (1 + r)² = $1,200 → r ≈ 9.5%
So the deal's IRR ≈ 9.5%.
Now compare it to your hurdle rate:
• If your savings account earns 5% → this deal (9.5%) beats it → Accept ✓
• If your savings account earns 12% → this deal (9.5%) loses → Reject ✗
IRR tells you the rate. Hurdle rate tells you the bar. If IRR clears the bar, invest.
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4. Perpetuities and Annuities
Annuity PV = (CF / r) × [1 − 1/(1+r)^t]. Perpetuity PV = CF / r. Growing perpetuity PV = CF / (r − g). A bond is an annuity (coupon payments) plus a lump sum (face value at maturity). A stock, in theory, is a growing perpetuity of dividends — the Gordon Growth Model: P = D₁ / (r − g). This formula reveals why stocks are so sensitive to interest rates: r appears in the denominator.
🏠 Everyday Example: Mortgage Payments vs. a University Endowment
An annuity = fixed payment for a set number of years.
Your mortgage: you pay $2,000/month for 30 years → that's an annuity. The bank used the annuity PV formula to figure out your monthly payment from the loan amount.
A perpetuity = fixed payment forever.
A university gets a $1M donation and invests it at 5%. They can spend $50,000/year forever, since 5% of $1M = $50K and the principal ($1M) stays intact.
PV = $50,000 ÷ 0.05 = $1,000,000 ✓
The Gordon Growth Model (P = D₁ / (r − g)) treats a stock like a growing perpetuity of dividends. A stock pays $2 dividend, growing at 3%/year, and you want 8% return:
P = $2 ÷ (0.08 − 0.03) = $2 ÷ 0.05 = $40
Why stocks fall when interest rates rise: r goes up → denominator grows → P falls. Automatic.
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5. Weighted Average Cost of Capital (WACC)
WACC = (E/V)·Rₑ + (D/V)·Rᵈ·(1−T). E = equity value, D = debt value, V = E + D, T = corporate tax rate. Rₑ = cost of equity, estimated via CAPM: Rₑ = Rf + β·(Rm − Rf). Rᵈ = pre-tax cost of debt (yield on the company's bonds or its loan rate). WACC is the minimum return a project must earn to create value. It is the discount rate for a DCF valuation. Debt is cheaper than equity because (a) lenders bear less risk and (b) interest is tax-deductible.
🍹 Everyday Example: Mixing Two Types of Borrowed Money
You're funding a $1,000 project with two sources:
• $600 from a bank loan at 5% interest (cheap Debt: Rᵈ = 5%)
• $400 from your own savings that could earn 12% elsewhere (your Equity: Rₑ = 12%)
Blended cost = (60% × 5%) + (40% × 12%) = 3% + 4.8% = 7.8%
That's your WACC.
The tax bonus: governments let companies deduct interest payments from taxable income. So if the corporate tax rate T = 30%, the effective cost of that 5% debt is only:
5% × (1 − 0.30) = 3.5%
Debt gets cheaper after tax — that's why large companies borrow heavily.
Big picture: WACC is the minimum return a project must earn to create value. If a project returns 10% and WACC is 7.8%, it creates value. If it returns 6%, it destroys value.
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6. Free Cash Flow (FCF)
FCF = EBIT × (1−T) + D&A − ΔWorking Capital − CapEx. This is the cash the business generates for all capital providers after maintaining and growing its asset base. Not net income. Not EBITDA. FCF is what flows to debt holders and equity holders combined. Buffett's 'owner earnings' ≈ FCF adjusted for maintenance vs. growth CapEx — a more conservative version of the same concept, introduced in his 1986 shareholder letter.
🍕 Everyday Example: The Pizza Shop's Real Cash
A pizza shop reports $100,000 net profit this year. Sounds great!
But look what actually happened with cash:
• The oven broke → spent $20,000 replacing it (CapEx)
• Stocked extra ingredients for a big event → $5,000 more tied up in inventory (Working Capital ↑)
• Add back: $8,000 depreciation — accountants counted it as an expense, but zero cash left the door
FCF = $100,000 + $8,000 (D&A) − $5,000 (ΔWC) − $20,000 (CapEx) = $83,000
That $83,000 is the actual cash available to pay the bank (debt holders) or keep yourself (equity). The reported $100,000 'profit' was partly accounting — FCF is the real money.
Why DCF uses FCF, not profit: you can't pay dividends or debt with accounting numbers. Cash is king.
Formula Reference
Click the 💡 Example button on any row to see a plain-English explanation and a worked numerical example.
| Concept | Formula | Note | Guide |
|---|---|---|---|
| Future Value | FV = PV × (1 + r)^t | r = rate per period, t = periods | |
| Present Value | PV = FV / (1 + r)^t | Discount a future amount back to today | |
| Net Present Value | NPV = Σ CFₜ/(1+r)^t − I₀ | Accept if NPV > 0 | |
| Annuity PV | PV = (CF/r) × [1 − 1/(1+r)^t] | Fixed payment for t periods | |
| Perpetuity PV | PV = CF / r | Fixed payment forever | |
| Growing Perpetuity | PV = CF / (r − g) | Gordon Growth Model; r > g required | |
| WACC | (E/V)·Rₑ + (D/V)·Rᵈ·(1−T) | Minimum return to create value | |
| CAPM (Cost of Equity) | Rₑ = Rf + β·(Rm − Rf) | β > 1 = more volatile than market | |
| Free Cash Flow | EBIT·(1−T) + D&A − ΔWCA − CapEx | Cash to all capital providers |
Interactive Demo — NPV Calculator
Set the initial investment, five years of cash flows, and the discount rate. Watch NPV respond in real time, see how discounting shrinks distant cash flows, and track how NPV shifts across the full rate range from 1% to 30%.
$500M
9%
$120M
$150M
$160M
$140M
$130M
Net Present Value
+43.6M
▲ Accept — value created above cost of capital
Internal Rate of Return
How discounting shrinks future cash flows — nominal (purple) vs. present value (green)
Y1
$110M
Y2
$126M
Y3
$124M
Y4
$99M
Y5
$84M
Nominal CF
Present Value
NPV sensitivity — how NPV changes as discount rate rises from 1% to 30%
The dashed purple line is your current discount rate. Drag the WACC slider above to watch NPV shift. At high rates, years 4–5 contribute almost nothing — distant cash flows are nearly worthless when discounted aggressively.
Simplified model: five discrete annual cash flows, no terminal value. Try setting Year 4–5 CFs high and cranking WACC to 25% — watch how those distant flows collapse to near zero. That is the core intuition of DCF.
Step-by-Step Method — How to Calculate NPV
- Identify all cash flows: the initial outlay (negative, period 0) and all expected future operating cash flows.
- Determine the discount rate. For corporate projects, use WACC. For personal decisions, use your opportunity cost of capital — what you could earn elsewhere at equivalent risk.
- Discount each cash flow: PV of CFₜ = CFₜ / (1 + r)^t.
- Sum all discounted cash flows.
- Subtract the initial investment. The result is NPV.
- Decision rule: NPV > 0 → accept (project earns above cost of capital). NPV < 0 → reject. NPV = 0 → indifferent (project exactly earns its cost of capital).
Real-World Case — Should Amazon Build a New Warehouse?
A simplified version of the kind of capital allocation decision Amazon faces every year. The lesson is not the numbers — it is how violently sensitive the outcome is to assumptions.
| Input / Output | Base Case | Bear Case | Bull Case |
|---|---|---|---|
| Upfront Cost | $500M | $500M | $500M |
| Annual FCF Boost (yrs 1–10) | $80M / yr | $55M / yr | $110M / yr |
| Terminal Growth Rate | 3% | 1% | 4.5% |
| WACC | 9% | 11% | 8% |
| PV of 10-yr CFs (approx) | ~$513M | ~$324M | ~$738M |
| PV of Terminal Value (approx) | ~$580M | ~$196M | ~$1,520M |
| Total NPV | +$593M ✓ | +$20M ✓ (barely) | +$1,758M ✓✓ |
The fragility of DCF. In the bear case, annual FCF is $55M instead of $80M (a 31% reduction), WACC rises 2 points to 11%, and terminal growth falls to 1%. The result: NPV collapses from +$593M to +$20M — a $573M swing that nearly erases the project's case entirely. Changing only the terminal growth rate from 3% to 1% at base-case WACC moves NPV by over $150M on its own. This is why DCF is called a telescope: powerful, but tiny adjustments in aim produce enormous changes in where it points. The model is not the answer. The business judgment behind the assumptions is the answer.
Note: in the base case, 60% of total value (+$580M out of +$1,093M before subtracting the investment) comes from the terminal value — cash flows beyond year 10. This is why “ignoring terminal value” is listed as a top-five DCF pitfall.
Common Pitfalls
| Mistake | Corrective Rule |
|---|---|
| Using net income instead of FCF | Net income includes non-cash items (D&A) and ignores CapEx. A DCF built on net income double-counts depreciation and ignores the cash cost of maintaining assets. Always use Free Cash Flow. |
| Ignoring terminal value | For most growing companies, 60–80% of DCF value sits in the terminal value — the infinite stream of cash flows beyond the explicit forecast period. Ignoring it or using an arbitrary exit multiple distorts the entire valuation. |
| Using one discount rate for all projects | High-risk projects need higher discount rates. Using firm-wide WACC to discount a speculative venture overstates its NPV. Adjust the discount rate for the specific risk profile of each project. |
| Circular reference in WACC | Equity value depends on WACC, which depends on the equity-to-total-value weight, which depends on equity value. Resolution: use current market-value weights, or iterate until convergence. Never use book-value weights for WACC. |
| Treating IRR as the primary criterion | Always compute NPV first. IRR fails when cash flows change sign multiple times, ignores the scale of investment, and assumes unrealistic reinvestment rates. NPV measures actual dollars of value created. |
Self-Check
Answer these from memory. If you cannot answer all three, re-read the relevant section.
- A project requires $1M upfront and returns $300K per year for 5 years. At a 10% discount rate, what is the approximate NPV? Should you accept or reject?
- IRR for Project A is 25%; for Project B is 18%. Project A costs $100; Project B costs $10M. Which project should you choose and why?
- Name the three components WACC depends on and explain why debt is cheaper than equity in the formula.
Answers:
- PV of $300K annuity at 10% for 5 years = 300 × [1 − 1/(1.10)^5] / 0.10 = 300 × 3.791 = $1,137K. NPV = $1,137K − $1,000K = +$137K. Accept the project — NPV > 0 means it earns more than the 10% cost of capital.
- Choose Project B. IRR ignores scale. Project A returns 25% on $100 — maybe $10 of value created. Project B returns 18% on $10M — millions in NPV. Always compare absolute NPV, not rates. A higher IRR on a tiny investment is worthless compared to a lower IRR on a large one.
- WACC has three drivers: (1) Cost of equity (Rₑ, via CAPM) — the return shareholders demand for bearing risk. (2) Cost of debt (Rᵈ) — the yield on bonds or loan rate. (3) The capital-mix weights (E/V and D/V). Debt is cheaper than equity for two reasons: lenders bear less risk than equity holders (they are paid first in bankruptcy), and interest is tax-deductible — the (1−T) term captures this 'tax shield.'
References & Further Learning
Click any card to jump directly to the course or resource and continue learning.
Damodaran: DCF Sessions 4–7
NYU Stern · Free
Sessions on TVM, risk-free rate, equity risk premium, and building a full DCF model from scratch. Slides and Excel models downloadable.
Introduction to Corporate Finance (Wharton)
Coursera · Audit Free
Wharton's flagship 4-week corporate finance course covering NPV, DCF, and WACC with applied problem sets.
Khan Academy: Time Value of Money
Khan Academy · Free
Interactive lessons on present value, future value, annuities, and perpetuities with worked examples and exercises.
MIT OpenCourseWare: Finance Theory I
MIT OCW · Free
Module 2 covers NPV, capital budgeting decision rules, and the full DCF framework at full MBA rigor.