Every DCF, every bond price, every mortgage payment, every pension liability is built on a single mathematical principle: a dollar received today is worth more than a dollar received tomorrow, because today's dollar can be invested and grow. The time value of money is the foundation of all valuation. Damodaran calls it 'the most important concept in finance — everything else follows from it.'
Time Value of Money — Core Framework
Every valuation model is a structured application of these four TVM building blocks
Present Value
PV = FV / (1+r)ⁿToday's worth of a future cash flow
$100 in 3 years @ 10% = $75.13 today
Future Value
FV = PV × (1+r)ⁿTomorrow's worth of money invested today
$100 today @ 10% for 5 years = $161.05
Annuity PV
PV = C × [1−(1+r)⁻ⁿ]/rPV of a stream of equal cash flows
$20/yr for 5 years @ 8% = $79.85
Perpetuity PV
PV = C / rPV of a stream that lasts forever
$10/yr forever @ 8% = $125
Present Value Factor Table — How Much Is $1 Worth Today?
| Discount Rate | 1 Year | 3 Years | 5 Years | 10 Years |
|---|---|---|---|---|
| 5% | 0.952 | 0.864 | 0.784 | 0.614 |
| 8% | 0.926 | 0.794 | 0.681 | 0.463 |
| 10% | 0.909 | 0.751 | 0.621 | 0.386 |
| 12% | 0.893 | 0.712 | 0.567 | 0.322 |
| 15% | 0.870 | 0.658 | 0.497 | 0.247 |
Reading: At 10% discount rate, $1 received 5 years from now is worth only $0.621 today. Higher rates = lower present values.
Gordon Growth Model — The Perpetuity Formula Behind Every DCF Terminal Value
P = D₁ / (Ke − g)
Where D₁ = next dividend, Ke = cost of equity, g = growth rate
D₁ = $3.00 (next dividend)
Cash returned to equity holders
Ke = 9% (cost of equity)
CAPM-derived required return
g = 3% (perpetual growth)
Must be < Ke or formula breaks
P = $3.00 / (0.09 − 0.03) = $50
Intrinsic value per share
Figure 4.1 — All four TVM building blocks and the PV discount table. The Gordon Growth Model connects perpetuity math to equity valuation.
The time value of money rests on three facts: (1) Inflation erodes purchasing power over time — $100 today buys more than $100 in 10 years. (2) Risk: a future payment may not materialize — a certain dollar today is worth more than an uncertain dollar tomorrow. (3) Opportunity cost: today's dollar, invested at the risk-free rate, grows — you must be compensated for the delayed receipt. Together, these create the discount rate — the rate at which future cash flows are reduced to their equivalent present value.
Future Value Formula
FV = PV × (1 + r)ⁿ
PV = present value, r = discount rate per period, n = number of periods
Present Value Formula
PV = FV ÷ (1 + r)ⁿ
Rearrangement of FV formula — discounting future cash flows back to today
| Years in Future | At 5% Discount Rate | At 10% Discount Rate | At 15% Discount Rate |
|---|---|---|---|
| 1 year | $952.38 | $909.09 | $869.57 |
| 5 years | $783.53 | $620.92 | $497.18 |
| 10 years | $613.91 | $385.54 | $247.18 |
| 20 years | $376.89 | $148.64 | $61.10 |
| 30 years | $231.38 | $57.31 | $15.10 |
This table reveals two critical insights for every valuation practitioner. First, at higher discount rates, future cash flows are worth very little in present value terms — a company whose value depends on cash flows 20+ years out (high-growth tech, pharma pipelines) is extremely sensitive to the assumed discount rate. Second, a small change in the discount rate (say, from 8% to 10%) has a massive impact on the present value of long-duration cash flows. This is why interest rate changes cause growth stocks to reprice dramatically — their terminal value is especially sensitive to small discount rate movements.
An annuity is a series of equal cash flows at regular intervals. Mortgages, lease payments, bond coupons, and pension distributions are all annuities. Rather than discounting each payment individually, the annuity formula provides an efficient shortcut:
Present Value of Ordinary Annuity
PV = C × [1 − 1/(1+r)ⁿ] / r
C = cash flow per period, r = discount rate per period, n = number of periods
| Discount Rate | PV of 10-Year Annuity | Intuition |
|---|---|---|
| 5% | $7,721.73 | $1,000 × [1 − 1/(1.05)¹⁰] / 0.05 |
| 8% | $6,710.08 | Higher rate → each payment worth less → lower total PV |
| 10% | $6,144.57 | Continues to decline as rate rises |
| 15% | $5,018.77 | At very high rates, even 10-year annuity worth only 5× annual payment |
| 20% | $4,192.47 | Discount rate above growth completely dominates duration effects |
This is an annuity in reverse — you know the PV (the loan amount) and need to find C (the monthly payment). Solving the annuity formula for C: C = PV × r / [1 − 1/(1+r)ⁿ]. Monthly: r = 7%/12 = 0.5833%, n = 360 months. Monthly payment = $300,000 × 0.005833 / [1 − 1/(1.005833)³⁶⁰] = $1,995.91/month. Total paid over 30 years = $718,525 — more than twice the original loan. This is the time value of money working against the borrower.
A perpetuity is an annuity that never ends — equal cash flows forever. Mathematically, the infinite sum converges to a simple formula because future payments become progressively less valuable as they recede further into the future. The perpetuity formula is arguably the most important formula in all of valuation because terminal value in a DCF is almost always calculated as a growing perpetuity:
Present Value of Perpetuity
PV = C / r
C = annual cash flow, r = discount rate. Example: $100/year forever at 8% = $100/0.08 = $1,250
Present Value of Growing Perpetuity (Gordon Growth Model)
PV = C₁ / (r − g)
C₁ = next year's cash flow, r = discount rate, g = perpetual growth rate. Requires r > g
| Growth Rate (g) | Value = C₁ / (r − g) | Change vs. g=0% | Implication |
|---|---|---|---|
| 0% | $1,000 | Baseline | No growth — $100 / 0.10 |
| 2% | $1,250 | +25% | Modest growth — GDP-level assumption |
| 3% | $1,429 | +43% | Common 'stable growth' assumption in terminal value |
| 5% | $2,000 | +100% | High growth — doubles value vs. no-growth case |
| 7% | $3,333 | +233% | Aggressive — approaching discount rate |
| 9% | $10,000 | +900% | Approaching r — mathematically explodes; unrealistic |
The growing perpetuity table above reveals why terminal value assumptions dominate DCF models. Moving the terminal growth rate from 2% to 3% increases value by 14%. Moving it from 3% to 5% increases value by 40%. Moving it from 5% to 7% increases value by 67%. And as growth approaches the discount rate, value mathematically approaches infinity. This is why Damodaran insists that the terminal growth rate in any DCF must be justified by a credible story about long-run competitive dynamics — not simply chosen to produce a convenient answer.
The Gordon Growth Model (also called the Dividend Discount Model in its simplest form) applies the growing perpetuity formula directly to stock valuation: a stock is worth the present value of all future dividends, where dividends grow at a constant rate forever. While the model has limitations (most companies don't pay stable dividends, and the assumption of constant growth is unrealistic for most businesses), it provides an important analytical baseline and directly links valuation to fundamentals:
Gordon Growth Model
P₀ = D₁ / (Ke − g)
P₀ = current stock price, D₁ = next year's expected dividend, Ke = cost of equity, g = sustainable growth rate in dividends
| Application | Use | Example |
|---|---|---|
| Direct valuation | Value of a mature, dividend-paying stock with stable growth | D₁ = $3.00, Ke = 9%, g = 3% → P = $3.00 / (0.09 − 0.03) = $50 |
| Implied discount rate | Back-calculate the cost of equity from market price | Stock at $50, D₁ = $3, g = 3% → Ke = $3/$50 + 0.03 = 9% |
| Terminal value calculation | Calculate the terminal value at the end of the explicit forecast period | Year 10 FCF = $200M, stable g = 2%, WACC = 9% → TV = $200M / (0.09 − 0.02) = $2,857M |
P₀ = D₁ / (Ke − g) is deceptively simple but contains the entire logic of fundamental equity valuation. It says: stock price is determined by three things — next year's cash flow (D₁), the discount rate (Ke), and the growth rate (g). When interest rates rise, Ke rises, and the denominator grows, reducing P₀. When growth expectations rise, g rises, and the denominator shrinks, increasing P₀. The entire discussion of why growth stocks are more interest-rate-sensitive than value stocks is embedded in this formula: growth stocks have small (Ke − g) denominators, so small changes in either variable produce large changes in value.
Key Takeaways
A company is expected to generate $50M in free cash flow next year, growing at 4% per year forever. The appropriate discount rate is 9%. What is the company's intrinsic value using the growing perpetuity formula?