A.3.6 Relative Valuation & Advanced Extensions

Basic Prerequisites: 05 · Failure Modes and 04 · Terminal Value & EV→Equity.

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1. Intuition & Practical Objective

DCF is absolute; relative valuation (“comps”) is market-based. Instead of forecasting cash flows, you ask: what are similar businesses trading for, and how does this one compare? Multiples are fast, communicable, and - crucially - they tell you what the market is paying, which is often a more practical question than what an asset is “worth”.

This page (a) derives each multiple from the DCF so you know what it embeds, (b) builds a comps table and applies it, and (c) shows how to make a DCF robust rather than precise: sensitivity grids and Monte Carlo. The closing lesson is the central tension of the whole folder: absolute and relative valuation answer different questions and can disagree wildly - know which one you are using and why.

The one-sentence essence. “A multiple is a compressed DCF: $\text{PE}=\frac{\text{payout}(1+g)}{k_e-g}$ - so a ‘cheap’ multiple is only cheap relative to the growth, risk, and returns it embeds.”

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2. Mathematical Ground Truth & Derivations

2.1 Every multiple is a rearranged DCF (Damodaran Ch 17–19)

Dividing the stable-growth DCF by the relevant denominator gives the fundamental multiple:

$$\text{PE}=\frac{\text{Payout}(1+g_n)}{k_e-g_n},\text{PBV}=\frac{\text{ROE}\cdot \text{Payout}(1+g_n)}{k_e-g_n},\text{PS}=\frac{\text{Margin}\cdot \text{Payout}(1+g_n)}{k_e-g_n},$$ $$\frac{V_0}{{\text{FCFF}}_1}=\frac{1}{K_c-g_n},\text{EV/EBITDA}\text{and}\text{EV/EBIT}\text{follow from the same FCFF identity}.$$

This is the why behind the “companion variable”: each multiple has a dominant fundamental driver - growth + risk for P/E, ROE for P/BV, margin for P/S, reinvestment/ROIC for EV multiples. Two firms with the same P/E are not equally cheap if their growth or ROE differs.

2.2 Multiples and the risk of circularity

• Equity multiples (P/E, P/BV, P/S) apply to equity value per share; enterprise multiples (EV/EBITDA, EV/EBIT, EV/Sales) apply to the whole firm and must be bridged EV → equity by subtracting net debt (exactly as in page 04).
• Comparable ≠ same industry. The textbook definition is firms with similar cash flows, growth and risk - industry is only a convenient proxy. The trade-off: a broad industry gives more peers but a noisier median; a narrow one gives fewer but cleaner comps.
• Controlling for fundamentals: regress the multiple on the companion variables across peers or the market, then compare each firm to its predicted multiple - the only rigorous way to use comps across heterogeneous firms.

2.3 Robustness: from a point estimate to a distribution

Because value is very sensitive to $g$ and $K_c$ (page 05), the professional output is a range:

$$V_{\text{lower}}\le V\le V_{\text{upper}}\text{from a }\pm \text{range on each driver (scenario / sensitivity grid)},$$

or, treating the drivers as random, a Monte Carlo distribution: draw $g_1,g_n,K_c,\text{reinvestment}$ from distributions, recompute the DCF, and read the percentiles. The point is not a better number - it is an honest band and a probability of being below price.

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3. Computational Implementation - comps, implied multiples, and Monte Carlo

Stdlib only. Part A derives the multiples from fundamentals; Part B builds a comps table, takes the median, and applies it; Part C runs the Monte Carlo on the page-04 DCF.

Read the results:

• The fundamentals-implied P/E is $8.40$, not some market-average $15$–$20$. A firm with $40\%$ payout, $5\%$ growth and a $10\%$ cost of equity should trade at $\approx 8.4\times$; a P/E of $14.5$ for that firm would embed either faster growth or lower risk.
• The two comps disagree by $70\%$ ($58.00 vs $34.25). This is not a bug - different multiples capture different fundamentals (earnings vs operating cash), and a small, mismatched peer set can’t resolve which is right. Relative valuation is only as good as the peer set.
• The Monte Carlo reframes the “value”: median $11.31 but a $5$–$95$ band of $6.48–$21.08. The range is three times wide around the median - a $11.29 point estimate is a fiction; the honest output is “roughly $6–$21, central $\approx$$11”.

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4. Failure Modes & First-Principles Breakdowns

1. Comparing unlike firms. Using industry average multiples across firms with different growth, margins, or leverage mistakes composition for cheapness. Control for the companion variable (regression or matched peers).
2. Ignoring the bridge. Applying EV/EBITDA to equity value (or forgetting to subtract net debt) misstates per-share value by the capital structure. Enterprise multiples apply to the firm, then bridge.
3. Linear-multiple fallacy. PEG and similar shortcuts assume P/E is linear in growth; the DCF shows it is not (it is convex and blows up as $g\to k_e$).
4. Multiples inside a DCF. Using an exit multiple for terminal value smuggles relative assumptions into an absolute model - the two philosophies silently conflict (page 04).
5. False precision via a beautiful model. More line items and a Monte Carlo around wrong distributions still produce a wrong range. The distribution inputs are as much a judgment as the point estimate; document them.
6. No cross-check. Absolute and relative valuation should be run together: a DCF that says $11 and comps that say $58 is a signal to find the disagreement - usually in growth or risk assumptions - not to average them blindly.

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5. References

• Damodaran, Investment Valuation
• Pinto et al. (CFA Institute), Equity Asset Valuation
• Graham & Dodd, Security Analysis
• Koller et al. (McKinsey), Valuation
• Green & Hand & Zhang (2017), “The Characteristics That Provide Independent Information About Average U.S. Monthly Stock Returns”, RFS

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Base: Statistics & Inference (regressions controlling for fundamentals) · Econometrics & Time Series
• Research: Fundamental Multi-Factor Models · Backtesting Hygiene
• Risk: Stress Testing & Scenario Analysis (scenario + Monte Carlo robustness)