Week 21: Equity Valuation — DCF, Multiples, and the Fed Model

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1. Why This Is Important

By Week 21 you can read three financial statements (Week 8), you understand capital structure (Week 19), and you can tell accounting earnings from real cash (Week 20). The natural next question — the one every beginner thinks valuation answers — is: *given all that, what is the stock actually worth?*

The honest answer is: nobody knows, and the people who claim to know to the second decimal place are usually selling something. Valuation is not a measurement; it is a structured argument about the future in present-value form. You build the argument anyway, because without it you have nothing to compare price against, and price without a value reference is just a moving number on a screen.

Four reasons to do this work, even knowing the output is fuzzy:

It anchors decisions to cash, not narrative. A discounted

cash-flow model forces you to write down the actual cash you expect the business to throw off. If you cannot defend the numbers, you cannot defend the position. That alone filters out most "story stocks" before they hit your portfolio.

It surfaces the assumptions you would otherwise hide. A

stock at 35× earnings is implicitly forecasting 12% earnings growth for a decade. The market does not announce that forecast on a banner. A reverse DCF makes the implicit forecast explicit, and you can ask: do I believe that?

It separates "valuation alpha" from regime alpha. Alpha

is rare. Most apparent "value" outperformance over the long run is multiple compression risk in disguise — buying things that are statistically cheap and waiting for the multiple to mean- revert. That works in some regimes, fails in others (1998, 2017, 2020). Knowing what kind of bet you are actually making is half the battle.

It calibrates expectations. When the S&P 500 trades at a

Shiller CAPE of 36 — where it is in April 2026 — long-run real returns implied by the historical relationship are around 3% per year, not 7%. You can still own the index. You should just stop pencilling in 7% real to your retirement spreadsheet.

This week covers absolute valuation (DCF, terminal value, sensitivity, reverse DCF), relative valuation (P/E, P/B, EV/EBITDA, P/FCF, the Shiller CAPE), and one cross-asset signal — the Fed model — that compares the earnings yield of stocks against the 10-year Treasury yield. The hands-on tool at the end is a DCF lab where you move the sliders and watch the per-share intrinsic value swing.

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2. What You Need to Know

2.1 The Two Schools — Absolute and Relative

Every valuation method falls into one of two camps.

Absolute valuation asks: what cash will this business produce over its life, and what is that cash worth today? The canonical tool is the discounted cash-flow (DCF) model. You forecast free cash flow for some explicit period (usually 5–10 years), assume a steady state after that (the "terminal value"), discount everything back at a rate that reflects the riskiness of the cash, and divide by share count. The output is a per-share number you can compare against the market price.

Relative valuation asks: how is this company priced compared to similar companies, or compared to its own history? The tools are multiples — P/E, P/B, EV/EBITDA, P/FCF, dividend yield, earnings yield. You don't need a forecast; you just need a peer group or a historical range.

Neither school is right or wrong. They answer different questions. DCF answers "what should I pay?" Multiples answer "what is the market currently paying for businesses like this?" In practice you do both and you look at where they disagree, because the disagreement is where the interesting question lives. If your DCF says $80 and the market is paying $200 at 40× earnings, the burden is on you to explain either why the market sees something you don't, or why the market is wrong. Both happen. Knowing which one you are betting on is the discipline.

A useful warning before we start: as Horace puts it, most "valuation alpha" is actually multiple-compression risk dressed up as insight — alpha is rare. Buying something at 12× earnings instead of 22× can be a free lunch — or it can mean you are buying a structurally deteriorating business at the moment its multiple stops being able to defend itself. The valuation toolkit doesn't tell you which one. Cash-flow stability does.

2.2 Discounted Cash Flow — The Mechanics

The DCF formula is one of the few places in finance where the maths is genuinely simple. Free cash flow each year, discounted at a rate $r$, summed:

$$ V_0 = \sum_{t=1}^{N} \frac{\text{FCF}_t}{(1+r)^t} + \frac{\text{TV}_N}{(1+r)^N} $$

Three pieces:

Explicit-period FCF. You forecast free cash flow for 5–10 years. "Free cash flow" here means cash from operations minus capex — the number from Week 20. For a growing business, you start from the trailing FCF, apply growth rates by year (often a fade — 12% in Y1 sliding down to 4% by Y5), and write down the projection.

Terminal value (TV). After the explicit period, the business keeps existing. You collapse the rest of the future into one number at year N. Two standard methods:

• Gordon growth (perpetuity): $\text{TV}_N = \dfrac{\text{FCF}_{N+1}}{r - g}$,

where $g$ is a perpetual growth rate. The $g$ has to be lower than long-run nominal GDP growth (4–5%) — you cannot grow faster than the economy forever — or the formula explodes.

• Exit multiple: $\text{TV}_N = \text{FCF}_{N} \times M$, where $M$ is

some multiple drawn from peers or history (e.g. 18× FCF). Useful cross-check; weakness is the multiple itself is a relative-value number, so the "absolute" calculation imports a relative assumption.

Discount rate $r$. This is the weighted average cost of capital (WACC) for the firm — Week 19's number. For a typical large-cap US equity, WACC sits in the 7–10% range. For a speculative growth name, 12–15%. The choice drives the answer more than people realise.

The terminal value typically accounts for **60–80% of the total DCF output** for a normal-growth company. That is uncomfortable, and honest. Most of what you are valuing is the assumption about steady state.

2.3 Sensitivity — The WACC and g Knobs

Two numbers — the discount rate $r$ and the terminal growth $g$ — do most of the work in any DCF. Move either by 100 bps and the intrinsic value can swing 30%. The example below uses a $10/share starting FCF, growing 6% for five years then 3% terminal, with sensitivity around $r$ and $g$:

$g = 2\%$	$g = 3\%$	$g = 4\%$
$r = 7\%$	$238	$300	$440
$r = 8\%$	$182	$221	$290
$r = 9\%$	$148	$174	$215
$r = 10\%$	$124	$143	$169

Two takeaways: the formula is most violent in the upper-left corner where $r - g$ is small (the perpetuity denominator goes to zero), and a "cheap" stock at one $(r, g)$ pair can be expensive at the neighbouring pair. The honest output of any DCF is not a number; it is a range.

This is also why analyst price targets cluster. Twenty analysts running the same DCF will output twenty answers within ±10% of each other — not because the model is precise, but because they all gravitate to the same defensible $(r, g)$ region. The herd in price targets is a herd in WACC assumptions.

The hands-on lab at §2.7 lets you turn the knobs and watch.

2.4 Comparable Multiples — The Field Guide

When you don't have the patience for a DCF (or the business is too complicated for a clean cash-flow forecast), multiples do the job. Five matter.

P/E (price to earnings). Most cited, often misleading. Use diluted EPS, not basic. Use forward P/E (next 12 months) when you have analyst estimates you trust, trailing P/E when you don't. A "normal" S&P 500 P/E sits around 16–17×; the long-run Shiller CAPE mean (10-year smoothed real earnings) is also about 17. Sectors diverge: utilities 14–18×, banks 8–12×, consumer staples 18–25×, software 25–40×. Compare within sector, not across.

P/B (price to book). Market cap divided by shareholders' equity. Useful for banks and insurers because their assets are already mark-to-market-ish (loans, securities), so book value is meaningful. Useless for software, brand-led consumer, biotech, where intangibles are the asset and accounting doesn't capture them.

EV/EBITDA. Enterprise value (market cap + debt − cash) over earnings before interest, tax, depreciation, amortisation. Strips out capital-structure differences, which is why it is the standard in M&A. Weakness: EBITDA ignores capital intensity. A company with EBITDA of $1B and capex of $1.2B has an EV/EBITDA in line with peers and zero free cash flow. Use EV/EBITDA with a separate look at capex intensity, never alone.

P/FCF (price to free cash flow). The cleanest of the lot. Free cash flow is hard to fake (Week 20). For mature businesses, P/FCF in the 15–25× range is normal. Below 12×, you are either buying genuine cheapness or a value trap. Above 35×, you are paying explicitly for growth that may or may not arrive.

Earnings yield (E/P). The reciprocal of P/E. A stock at 20× earnings has a 5% earnings yield. The reason this matters separately from P/E: it puts equity earnings on the same axis as bond yields, which is exactly what the Fed model needs (§2.6).

2.5 Shiller CAPE — The Long History

Robert Shiller's cyclically-adjusted P/E (CAPE) takes the S&P 500 price and divides by the average of the last ten years of real earnings. The smoothing kills the cyclical noise that makes the trailing P/E spike to absurd levels in recessions (when the E collapses) and look reasonable at the top of a cycle (when E is at peak). The chart below plots CAPE from 1881 through April 2026.

Two patterns matter.

First, **CAPE mean-reverts on long horizons but is useless as a short-term timing signal**. CAPE was 25 in 1996, 30 in 1997, 38 in 1998, 44 in 1999, and only peaked in March 2000. Anyone who shorted the index at CAPE 25 lost their job before they got proven right. The signal is real on 10-year-forward returns; it is noise on 1-year- forward returns.

Second, the 80-year low (1981) and the modern range diverge. CAPE in 1981 was around 8 because Treasury yields were 14% — the cost of capital was crushing equity multiples. CAPE in 2009 was 13 because earnings had collapsed. CAPE in 2026 at 36 is because rates are 4% and tech-sector earnings dominate the index. The number is the same metric across the whole history, but the regime supporting it is wildly different.

Use CAPE as a temperature reading, not a buy/sell signal. When CAPE is in the top decile of history, expected real returns over the next decade are mid-single-digits at best. That is the calibration. It does not tell you to sell.

2.6 The Fed Model — Earnings Yield vs the 10-Year

The Fed model compares the earnings yield of equities (E/P) to the yield on 10-year Treasuries. The thesis: if E/P is much higher than the bond yield, equities are cheap relative to bonds, because their cash-flow yield exceeds the risk-free alternative. If E/P is below the bond yield, the bond market is offering a better risk-adjusted deal.

Two warnings about the Fed model.

It is not in any Fed publication. The name comes from a 1997 Humphrey-Hawkins report chart. Greenspan never endorsed it as a valuation tool. Practitioners adopted the framework anyway because it is intuitive.

It conflates real and nominal. The earnings yield is a real quantity (earnings grow with inflation in the long run). The Treasury yield is nominal. Subtracting one from the other is a type error. The signal works best when inflation is stable; it breaks down precisely when inflation is the dominant variable (the 1970s, where the Fed model said equities were cheap for a decade while real returns on stocks were near zero).

Treat the Fed model as one cross-asset reference, not a verdict. At the April 2026 reading near zero, both stocks and bonds price in mediocre real returns. There is no obvious arbitrage between the two — which is itself information.

2.7 Reverse DCF — The Sanity Check

The reverse DCF flips the question. Instead of forecasting cash flow and computing intrinsic value, you take the *current market price* as given and ask: what growth rate, at the prevailing discount rate, makes this price the answer?

You hold $r$, $g$, and starting FCF fixed, and solve for the growth rate $g_e$ in the explicit period that produces $V_0 = \text{Price}$. The output is the growth implied by the price.

Example: AAPL at $215 with 14.5B shares is a $3.1T market cap. Trailing FCF is about $109B. If WACC is 9% and terminal growth is 3%, the explicit-period growth implied by the current price is roughly 5–6% per year over the next decade. Apple has grown FCF about 9% annualised over the last decade. The implied 5–6% is reasonable: it bakes in some maturation but not collapse. The price passes a basic sanity check.

For a richer comparison, run the reverse DCF on a meme stock at a $50B market cap with no FCF: the implied growth is "infinite from today's free cash flow." That is the moment you put the model down and admit you are not valuing this thing on cash.

The reverse DCF is the discipline that separates "buying because it's cheap on the model" from "buying because the model says miracles are required." The interactive lab in §2.7 includes a reverse-DCF readout for each preset stock — try it.

The hands-on tool: see interactive/week21_dcf_lab.html. Move the WACC, growth, and terminal-growth sliders, and watch the per-share intrinsic value swing for AAPL, MSFT, GOOGL, JPM, and KO.

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3. Common Misconceptions

"DCF gives you a precise number." It doesn't. It gives you a

range. The number is precise; the inputs are not. Treat any DCF output as ±25% at best.

"A low P/E means a stock is cheap." Not by itself. Banks,

tobacco, and oil majors trade at low P/Es structurally because of regulatory, terminal, or commodity-price risk. The market isn't being stupid; it is pricing in a real future hazard.

**"Terminal value is a small piece — most of the value is in

the explicit forecast." Wrong direction.** TV is typically 60- 80% of total DCF for a steady-state business. The explicit period mostly bridges you to the assumption you actually care about: the steady state.

**"The Fed model says when to be in stocks vs bonds." It

doesn't.** It is a rough cross-asset thermometer that ignores inflation regime entirely. It told you stocks were cheap in 1972 and you lost money in real terms for a decade.

**"Shiller CAPE is broken because it has been high since 2014

and stocks went up." It's not broken — the signal is on 10-year forward returns, not next-year.** From CAPE 26 in 2014, the subsequent decade gave roughly 9% nominal CAGR — well below the long-run 10% but consistent with the elevated CAPE.

**"P/B is always meaningful." For asset-light businesses (tech,

consumer brands, services), book value omits 80% of what makes the company valuable.** Use it for banks; ignore it elsewhere.

**"EV/EBITDA is a 'cleaner' P/E because it strips out capital

structure." Yes, and it also strips out capex, which is the line that separates a real business from a treadmill.** A cement producer at EV/EBITDA 8 may be more capital-hungry than a software company at EV/EBITDA 25.

**"You should buy whatever has the lowest P/E in a sector."

That's a deep-value heuristic, and it fails the moment one of the names is structurally impaired.** Cheap relative to peers is meaningful only when the underlying business is comparable.

"Higher discount rate = always more conservative." Almost.

A higher $r$ lowers the present value of all future cash, but it also lowers TV — which is most of the answer. The conservatism is real; the magnitude is usually larger than people expect.

**"Growth stocks should be valued differently because DCF

doesn't work for them." DCF works fine; the inputs are just harder.** What people mean is "I don't trust my own forecast," which is honest. The fix is wider sensitivity bands, not a different method.

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4. Q&A Section

Q1. What discount rate should I use for a typical S&P 500 stock? WACC is the right answer; for most large-cap US firms it lands in the 7–9% band. If you don't want to compute WACC from scratch, 8% is a defensible default. For high-leverage or commodity-cycle names, push toward 10%; for defensive utilities, 6–7%.

Q2. How many years should the explicit DCF period cover? Five to ten. Five if the business is mature and stable (KO, JNJ). Ten if the business is in a clear high-growth phase that should fade to maturity (NVDA, GOOGL). Going past ten is forecasting performance art; you are pretending to know about regime that doesn't exist yet.

**Q3. Why does the Gordon growth formula go crazy when $r$ is close to $g$?** Because $\dfrac{\text{FCF}}{r - g}$ has a denominator that goes to zero. Mathematically: a perpetual cash flow growing at rate $g$ discounted at rate $r$ has finite present value only if $g < r$. If $g \geq r$, the present value is infinite — the cash flows grow faster than the discount eats them. This is why $g$ in practice has to be capped at long-run nominal GDP (~4-5%); above that, you are claiming the firm grows forever faster than the economy it lives in, which is impossible.

**Q4. What's the difference between P/E and Shiller CAPE in practice?** Trailing P/E uses the last twelve months of earnings. CAPE uses the average of the last ten years of real earnings (inflation- adjusted). The smoothing matters enormously at cycle turning points: in 2009, trailing P/E spiked above 100 because earnings collapsed faster than price; CAPE was around 13. CAPE is the more honest measure across cycles. P/E is the more responsive measure within a cycle.

Q5. Should I use forward P/E or trailing P/E? Forward, when the analyst estimates are credible (large-cap, well- covered, low cyclicality). Trailing, when the business is volatile or estimates have a wide spread. The gap between trailing and forward P/E tells you the implied growth rate: if trailing is 18× and forward is 15×, the market is pricing in 20% earnings growth.

Q6. Is the Fed model a buy/sell signal in 2026? No — but it is informative. April 2026 reading is near zero (E/P ~2.8%, 10y ~4.0%). That means cash earnings yield on the index is below the risk-free rate. Historically, that condition has preceded sub-par 5–10 year equity returns. It does not mean sell; it means lower your forward expectations and consider tilting a modest weight toward bonds (consistent with the four-tranche framework from Week 14).

**Q7. Why is reverse DCF more useful than regular DCF for big benchmark stocks?** Because everyone has already done the regular DCF and the answer is "price ≈ intrinsic value, give or take 10%." That is not actionable. Reverse DCF tells you the implied growth rate the market has priced in. Now you can ask whether that growth rate is reasonable versus history, versus competitors, versus what you think management can deliver. You're betting on a difference of opinion about growth, not on an absolute valuation gap.

Q8. Banks don't fit the DCF mould. What do I use? Residual income or P/B-vs-ROE. A bank trading at $1.5\times$ book when it earns 15% ROE is fairly priced (because $\text{P/B} \approx \text{ROE} \times \text{P/E}$, and a 10× P/E with 15% ROE pencils to 1.5× P/B). The cleanest one-number summary for a bank is ROTCE — return on tangible common equity. JPM at 22% ROTCE in FY2024 is structurally the highest-quality bank in the index, which is why it trades at a premium to peers.

**Q9. Multiples vary so much across sectors. How do I compare a software company at 30× to a utility at 15×?** You don't compare them directly. You compare each within its sector, and you compare the cross-sector premium ratio across time. If software-vs-utility usually trades at 1.7× the multiple and currently trades at 2.5×, you have learned something. The cross-sector level itself is not the signal; the change in the ratio is.

**Q10. How does Horace's "alpha is rare" principle apply to valuation?** Tightly. Most "valuation alpha" is regime-specific multiple compression and expansion that flips to the loser of the previous cycle. The 2000-2010 decade rewarded value because tech multiples collapsed; 2010-2020 rewarded growth because rates compressed and tech earnings exploded. Both decades had their crowd of "valuation experts" who looked smart for a while and then looked foolish. Genuine, repeatable valuation alpha comes from one narrow place: businesses that compound faster than the market discounts, and whose multiples don't blow out at the entry. Find those rarely; the rest is multiple-compression risk in a suit.

Q11. What should the DCF lab teach me beyond the slider game? That the answer depends almost entirely on three numbers — WACC, terminal growth, and the year-1 starting FCF — and that the same business can be priced "cheap" or "expensive" by 50% with rounding- error changes to those inputs. The lesson is humility, not precision.

**Q12. Is there a case to ignore valuation entirely and just index?** Yes — most of the time, for most people. US-listed equities are the investable universe; per Week 2, the index is the default. Valuation discipline matters when you are choosing individual names or deciding whether to tilt the index allocation up or down at extremes. At CAPE 36, you don't have to sell — but you do have to lower the return you write into your retirement plan. That single discipline is worth more than most stock-picking.