A.3.1 Equity Valuation from Zero

Basic Prerequisites: none - this page assumes no accounting or finance background.

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1. Intuition & Practical Objective

This page builds the why of equity valuation with no prior knowledge needed. The objective is one idea: a share is a claim on a stream of future cash, so its value today is that stream discounted back at a rate that reflects how risky the stream is.

Start with the dumbest question: why is a share worth anything at all? Not because someone else might pay more for it tomorrow - that is a bet on a greater fool. It is worth something because owning it entitles you to a share of the cash the business will generate: dividends, buybacks, or the proceeds of a sale. Strip the ticket-tape away and a stock is a claim on cash flows that have not happened yet.

Three steps:

1. Value is about the future, not the past. A firm with beautiful historical earnings and no prospect of future cash is worth little; a firm with losses today but a dominant, cash-generating future can be worth a fortune. Accounting measures what happened; valuation measures what is expected.

2. Discounting is the price of waiting and of risk. A dollar promised in ten years is worth less than a dollar today - partly because you could invest today’s dollar (the time value), and partly because the promise might not be kept (the risk premium). The discount rate bundles both.

3. Growth is only worth paying for when it earns above the cost of capital. Investing more cash to grow faster destroys value if the return on that investment is below the discount rate. This single insight separates value investing from “growth at any price”.

The one-sentence essence. “Intrinsic value $={\sum }_t\frac{{\text{expected cash}}_t}{(1+r{)}^t}$ - and growth is a good thing only when the return on reinvested cash exceeds $r$.“

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2. Mathematical Ground Truth & Derivations

Present value of an explicit stream. If the owner expects cash $C_1,C_2,\dots ,C_n$ and then a terminal amount $V_n$, then today’s value is

$$V_0=\sum _{t=1}^n\frac{C_t}{(1+r{)}^t}+\frac{V_n}{(1+r{)}^n}.$$

The growing perpetuity (Gordon) shortcut. If cash starts at $C_1$ and grows forever at constant $g<r$,

$$V_0=\sum _{t=1}^{\infty }\frac{C_1(1+g{)}^{t-1}}{(1+r{)}^t}=\boxed{\frac{C_1}{r-g}}.$$

This is the algebraic backbone of every terminal value (page 04). Two facts follow immediately and are worth memorising:

• The value is dominated by the tail. At $C_1=100$, $r=10\%$, $g=3\%$, the perpetuity is worth $1428.57$; the first ten years are only $48.2\%$ of it. More than half of the value sits in years 11 and beyond - which is exactly why the terminal value (page 04) and its assumptions dominate DCF output.
• The denominator is fragile. Both $r$ and $g$ move value inversely and non-linearly: raising $g$ from $3\%$ to $4\%$ adds $238$, a $16.7\%$ jump, on a single percentage point.

Where growth creates value. A firm growing at $g$ must reinvest to fund that growth. With a return on capital $\text{ROC}$, the reinvestment rate is $\text{RR}=g/\text{ROC}$, so the cash flow is

$${\text{FCF}}_1={\text{NOPAT}}_1(1-\text{RR})={\text{NOPAT}}_1(1-\frac{g}{\text{ROC}}),V_0=\frac{{\text{NOPAT}}_1(1-g/\text{ROC})}{r-g}.$$

When $\text{ROC}=r$ this collapses to $V_0={\text{NOPAT}}_1/r$ - growth adds nothing. When $\text{ROC}>r$ growth adds value; when $\text{ROC}<r$ growth destroys it. This is the fundamental growth equation that page 02 makes concrete.

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3. Computational Implementation - present value, from scratch

Stdlib only. Part A shows the mechanics of discounting and the tail-dominance fact; Part B shows the growth/ROC value logic.

Read the table carefully: the same earnings and the same growth rate produce values from $588.6$ to $1324.3$ purely as a function of the return on reinvested capital. A firm earning $5\%$ on reinvestment while its cost of capital is $10\%$ is worth less than a no-growth firm - it is destroying value by growing.

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4. Failure Modes & First-Principles Breakdowns

1. The “greater fool” confusion. Treating price as value substitutes market sentiment for cash-flow fundamentals. Intrinsic value is a claim on cash; market price is a negotiation. Confusing them is the root of momentum-bubble losses.
2. Growth ≠ value. The single most expensive beginner error: assuming faster growth is always worth more. Growth only creates value when $\text{ROC}>r$; below that it is a value incinerator (see Part B: value falls from $1030$ to $588.6$).
3. Ignoring the terminal tail. Because $>50\%$ of value sits beyond year 10, a “conservative” five-year spreadsheet can still hide an aggressive perpetuity assumption. Never read a DCF without asking what fraction of value is terminal (page 04).
4. Time-value/risk conflation. The discount rate is not one number: it is $r_f$ (waiting) + a risk premium (uncertainty). Using a single “hurdle rate” for every firm ignores that risk differs across businesses, and that cash flows and discount rates must be matched (page 03).

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5. References

• John Burr Williams, The Theory of Investment Value (1938)
• Damodaran, Investment Valuation
• Graham, The Intelligent Investor
• Pinto et al. (CFA Institute), Equity Asset Valuation

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6. Connected Graph Bridges

• Forward: 02 · Cash-Flow Forecasting · Index Hub
• Base: Statistics & Inference
• Downstream: Fundamental Multi-Factor Models (value/quality factors)