⏳ Time Value of Money (TVM)

Core Principle: A dollar today is worth more than a dollar in the future because money today can be invested to earn a return.

Covered at: All MBA programs — typically Week 1 of Finance

🔑 The Intuition

Money has a time cost. If someone owes you $1, 000, g e tt in g i tt o d a y i ss t r i c tl y b e tt er t han g e tt in g i t ina ye a r — b ec a u seyo u c anin v es t$ 1,000 today and have more than $1,000 by next year.

This is why we discount future cash flows to their present value when making decisions.

📐 Core Formulas

Future Value (FV)

$F V = P V \times (1 + r)^{n}$

PV = Present Value
r = interest rate per period
n = number of periods

Example: $1, 000 in v es t e d a t 8$ 1,469`

Present Value (PV)

$P V = \frac{F V}{( 1 + r ) ^{n}}$

Example: What’s $1, 469 in 5 ye a rs w or t h t o d a y a t 8$ 1,000`

Net Present Value (NPV)

$NP V = \sum_{t = 0}^{n} \frac{C F _{t}}{( 1 + r ) ^{t}}$

Accept a project if NPV > 0
Reject if NPV < 0
NPV = value created for shareholders

Perpetuity

$P V = \frac{C}{r}$

Example: $100/ ye a r f ore v er a t 5$ 2,000`

Growing Perpetuity (Gordon Growth Model)

$P V = \frac{C}{r - g}$

💡 Key Concepts

Why the Discount Rate Matters

The choice of discount rate is everything in valuation:

Low rate → high PV (future cash flows are valuable)
High rate → low PV (future is heavily discounted)
Companies use WACC as their discount rate for projects

Rule 72

A quick mental math shortcut: $Years to double \approx \frac{72}{r}$

At 8%: money doubles in ~9 years. At 6%: ~12 years.

Annuity

A stream of equal cash flows for n periods: $P V_{ann u i t y} = C \times \frac{1 - ( 1 + r ) ^{- n}}{r}$

📊 Decision Rules

Investment Rule	Criterion	Notes
NPV	NPV > 0	Theoretically best rule
IRR	IRR > hurdle rate	Can mislead with non-normal CFs
Payback Period	PB < threshold	Ignores time value (bad!)
Discounted Payback	DPB < threshold	Better than simple payback
Profitability Index	PI > 1	Useful for capital rationing

🚫 Common Mistakes

Using nominal CFs with real rates (or vice versa) — be consistent
Forgetting the terminal value in DCF models
Using wrong n — make sure periods match the discount rate frequency
Adding incremental cash flows incorrectly — only incremental matters

🔗 Connected Concepts

DCF Valuation — applies TVM to entire companies
WACC — the discount rate for most corporate decisions
CAPM — how to derive the cost of equity
Capital Structure — how financing choices affect value
Options and Derivatives — derivatives pricing uses risk-neutral discounting

🏫 School Context

HBS Finance I: Taught via case studies; emphasis on which discount rate to choose, not the mechanics
Wharton FNCE 611: Heavily mathematical; derives PV rigorously from first principles
Booth: Context of Chicago School — markets are efficient, so TVM is purely mathematical
Columbia: Emphasis on Benjamin Graham’s margin of safety — conservative discounting

← 📊 Finance MOC | Related: DCF Valuation · WACC · NPV vs IRR

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Time Value of Money