6.6 Multiples as compressed DCFs, and the reverse DCF

Motivation

Most market conversation is in multiples: "trading at 60× earnings". A multiple carries a growth, payout and risk assumption. Making it explicit tells you what you are betting on.

6.6.1Justified multiples#

Proposition 6.9Justified P/E and P/B

In the Gordon model with cost of equity $k$ , growth $g < k$ , payout ratio $p$ and return on equity ROE, with $g = (1-p)\text{ROE}$ :

\frac{P_0}{E_1} = \frac{p}{k-g} = \frac{1 - g/\text{ROE}}{k-g}, \qquad \frac{P_0}{E_0} = \frac{p(1+g)}{k-g}, \tag{6.17}

\frac{P_0}{B_0} = \frac{\text{ROE} - g}{k - g}. \tag{6.18}

Proof

$P_0 = D_1/(k-g)$ and $D_1 = pE_1$ give the first form; $p = 1 - b = 1 - g/\text{ROE}$ by Proposition 6.5 gives the second; $E_1 = E_0(1+g)$ gives the trailing form. For (6.18), $E_1 = \text{ROE}\cdot B_0$ , and book equity grows at $g$ , so retained earnings in year 1 are $gB_0$ and $D_1 = (\text{ROE} - g)B_0$ . Divide $P_0 = D_1/(k-g)$ by $B_0$ .∎

Reading the formulas

If $\text{ROE} = k$ , then $P_0/E_1 = 1/k$ and $P_0/B_0 = 1$ for every $g$ . Growth only earns a premium multiple when returns exceed the cost of equity: Proposition 6.7 again.
For banks, (6.18) is the working model: a bank with ROE 18%, growth 10% and $k$ = 14% is worth $(0.18 - 0.10)/(0.14 - 0.10) = 2.0\times$ book; one with ROE 12%, growth 8% is worth $0.04/0.06 = 0.67\times$ book. PSU banks traded at large discounts to book through 2015–2020 because markets priced ROE below cost of equity (6.10).
For a high-ROE compounder, the P/E is explosive in $k - g$ . With ROE 30% and $k$ = 13%: $g$ = 10% gives forward P/E 22.2; $g$ = 11% gives 31.7; $g$ = 12% gives 60.0. A one-point change in perpetual growth doubles the multiple. Multiples in the 60–90 range for Indian consumer franchises are statements that $k - g$ is very small for a very long time.
Worked example (lab V5). Payout 0.5, $g$ = 8%, $k$ = 13% gives trailing P/E $0.5 \times 1.08/0.05 = 10.8$ .

The PEG ratio (P/E divided by growth in percent) is a heuristic with no derivation behind it: P/E is not linear in $g$ in (6.17).

6.6.2Solving for implied growth#

Reverse DCF Instead of forecasting growth and computing value, take the market's enterprise value $V^{mkt}$ and solve for the growth that justifies it. Mauboussin & Rappaport call this "expectations investing".

Proposition 6.10Unique implied growth

In (6.9) fix $F_0 > 0$ , $W > g_T$ and $N \ge 1$ . Then $V_0(g)$ is continuous and strictly increasing on $(-1, \infty)$ with $V_0 \to 0$ as $g \to -1$ and $V_0 \to \infty$ as $g \to \infty$ . Hence for every $V^{mkt} > 0$ there is exactly one $g^*$ with $V_0(g^*) = V^{mkt}$ .

Proof

Each explicit term $F_0(1+g)^t/(1+W)^t$ and the terminal term (proportional to $(1+g)^N$ ) is continuous, positive and strictly increasing in $g$ for $g > -1$ , and tends to 0 as $g \to -1$ . The terminal term tends to $\infty$ . The intermediate value theorem gives existence; strict monotonicity gives uniqueness.∎

Root-finding Bracket the root on $[lo, hi]$ with $V_0(lo) < V^{mkt} < V_0(hi)$ . Bisection halves the bracket each step, so after $k$ steps the error is at most $(hi - lo)/2^k$ ; 40 steps on $[-0.5, 1]$ give $10^{-12}$ . Brent's method (scipy.optimize.brentq) combines bisection with secant and inverse quadratic steps and converges superlinearly while keeping the bracket guarantee. Newton's method needs $V_0'(g)$ and can overshoot outside $(-1, \infty)$ ; avoid it without safeguards.

6.6.3Implied growth for the mid-cap#

Worked example

Same company as 6.3 ( $F_0$ = ₹1,000 cr, WACC 12%, $g_T$ 5%, $N$ = 5). If the market EV is ₹30,000 cr, the implied explicit growth is $g^* = 22.43\%$ a year for five years (reverse_dcf(3000, 100, 0.12, 0.05, 5) = 0.224344). If the EV is ₹15,000 cr, $g^* = 5.0\%$ : the market prices no excess growth at all. The question is then a business question: has this company ever compounded free cash flow at 22% for five years? What does its 10-year revenue CAGR, margin trend and reinvestment history say?

For the value-driver model the reverse problem has a closed form: solving $V(W-g) = N(1 - g/R)$ for $g$ gives $g^* = (VW - N)/(V - N/R)$ . With $V$ = 9,166.7, $N$ = 500, $R$ = 30%, $W$ = 12% this returns $g^* = 8.0\%$ , recovering the example of 6.4.

Practitioner notes

A reverse DCF is more useful than a forward one because it avoids anchoring on your own optimism. Solve for implied growth under two or three WACCs and two or three RONICs; if every combination requires growth the company has never achieved, the price assumes a different company.