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Finance & banking · The method — the desk, the system, and the two models

Accretive is not the same as good

The P/E rule of thumb everyone quotes is a special case of a comparison between two yields, and it names the wrong price. This lesson builds the merger model, derives the general condition, finds the exact premium at which each funding source flips to dilutive, and shows why all-cash deals look best for a reason that has nothing to do with the deal.

21 min read Free to read Patterns: merger-model

What you'll be able to do

Before this: the-three-statements-as-one-system

There is a rule of thumb every candidate brings to a merger-model question, and interviewers ask for it by name — sometimes with the hint attached: "tell me the rule for whether an acquisition will be accretive or dilutive (hint: P/E ratio)."

The rule: in an all-stock deal with no synergies, the deal is accretive if the target's P/E is lower than the acquirer's. A company at 20x buying one at 15x is accretive. At 20x buying 25x, dilutive.

The rule is true. It is also a special case of something more general, and as usually stated it names the wrong price. Both of those matter, because the follow-up question is always "and what if we paid in cash?" or "and what if we paid a 60% premium?" — at which point the rule of thumb has run out and you need the thing underneath it.

WatchFinanceKid · 23:40

A full worked accretion/dilution walkthrough, including the cash-versus-stock-versus-debt comparison this lesson generalises. Watch it before the model below if you have never built one; watch it after if you have, because the sequence of the arithmetic is what you will be graded on.

The general condition

Buying a company is buying an earnings stream, and paying for it costs something. EPS goes up when

the earnings you bought, per dollar spent, exceed the after-tax cost of the dollars you spent.

That is the whole of it. Both sides are yields, so it is a comparison of two percentages, and each funding source has its own cost:

FundingAfter-tax annual cost per dollar spent
New stockThe acquirer's own earnings yield, 1 / P/E — issuing shares dilutes existing earnings across more of them
New debtinterest rate × (1 − tax rate) — interest is deductible
Balance-sheet cashforegone interest × (1 − tax rate) — the cash was earning something, and now isn't

The stock row is where the rule of thumb comes from. If funding is all stock, the cost is the acquirer's earnings yield, so the condition becomes earnings yield bought > acquirer's earnings yield — and inverting both sides, P/E paid < acquirer's P/E.

Running it

JavaScript
// A merger model, and then the thing the merger model is secretly computing.
//
// The famous rule of thumb -- "accretive if the acquirer's P/E is higher than
// the target's" -- is not a rule. It is one special case of a comparison
// between two yields, and this cell derives the general form, then finds the
// exact price at which each funding source flips from accretion to dilution.
// Where the closed form and the full model disagree, one of them is wrong; they
// are checked against each other below.

const TAX = 0.25;

const ACQ = { netIncome: 500, shares: 250, price: 40 };   // EPS 2.00, P/E 20.0x
const TGT = { netIncome: 120, shares: 75,  price: 20 };   // EPS 1.60, P/E 12.5x on market

const PREMIUM   = 0.20;    // offer at a 20% premium to the target's market price
const DEBT_RATE = 0.060;   // pre-tax coupon on acquisition debt
const CASH_RATE = 0.030;   // what the acquirer's balance-sheet cash earns today
const SYNERGY   = 0;       // pre-tax cost synergies, run-rate

const acqEps = ACQ.netIncome / ACQ.shares;
const acqPe  = ACQ.price / acqEps;
const tgtEps = TGT.netIncome / TGT.shares;
const tgtPe  = TGT.price / tgtEps;

const offerPrice = TGT.price * (1 + PREMIUM);
const purchase   = offerPrice * TGT.shares;      // equity purchase price
const offerPe    = purchase / TGT.netIncome;     // P/E paid, not P/E quoted

/**
 * Pro forma EPS for an arbitrary funding mix. Weights are fractions of the
 * equity purchase price; they must sum to 1.
 */
function proForma({ stock = 0, debt = 0, cash = 0 }, price = offerPrice, synergy = SYNERGY) {
  const consideration = price * TGT.shares;
  const newShares  = (consideration * stock) / ACQ.price;
  const newDebt    = consideration * debt;
  const cashUsed   = consideration * cash;

  const netIncome =
    ACQ.netIncome +
    TGT.netIncome +
    synergy * (1 - TAX) -                 // synergies are pre-tax, so tax them
    newDebt * DEBT_RATE * (1 - TAX) -     // interest on new debt
    cashUsed * CASH_RATE * (1 - TAX);     // interest the spent cash no longer earns
  const shares = ACQ.shares + newShares;
  const eps = netIncome / shares;
  return { eps, netIncome, shares, newShares, accretion: eps / acqEps - 1 };
}

/**
 * The general condition, in one line. A deal is accretive when the earnings you
 * buy, per dollar spent, exceed the after-tax cost of the dollars you spent.
 */
function costOfFunding({ stock = 0, debt = 0, cash = 0 }) {
  return (
    stock * (1 / acqPe) +                  // issuing equity costs its earnings yield
    debt * DEBT_RATE * (1 - TAX) +
    cash * CASH_RATE * (1 - TAX)
  );
}

const MIXES = [
  ["all stock",            { stock: 1 }],
  ["all new debt",         { debt: 1 }],
  ["all balance-sheet cash", { cash: 1 }],
  ["50 cash / 50 stock",   { cash: 0.5, stock: 0.5 }],
  ["60 debt / 40 stock",   { debt: 0.6, stock: 0.4 }],
];

console.log("standalone");
console.log("  acquirer  EPS " + acqEps.toFixed(2) + "   price " + ACQ.price.toFixed(2) +
            "   P/E " + acqPe.toFixed(1) + "x   earnings yield " +
            ((1 / acqPe) * 100).toFixed(2) + "%");
console.log("  target    EPS " + tgtEps.toFixed(2) + "   price " + TGT.price.toFixed(2) +
            "   P/E " + tgtPe.toFixed(1) + "x");
console.log("  offer     " + offerPrice.toFixed(2) + " per share (" +
            (PREMIUM * 100).toFixed(0) + "% premium)  ->  purchase price " +
            purchase.toFixed(0) + ",  P/E PAID " + offerPe.toFixed(1) + "x");
console.log("  earnings bought per dollar spent: " + TGT.netIncome + " / " + purchase.toFixed(0) +
            " = " + ((TGT.netIncome / purchase) * 100).toFixed(2) + "%");
console.log("");

console.log("funding mix                  new shares   pro forma EPS   accretion   cost of funding   predicted");
for (const [name, mix] of MIXES) {
  const r = proForma(mix);
  const cost = costOfFunding(mix);
  const yieldBought = (TGT.netIncome + SYNERGY * (1 - TAX)) / purchase;
  console.log(
    "  " + name.padEnd(24),
    r.newShares.toFixed(1).padStart(9),
    r.eps.toFixed(4).padStart(15),
    ((r.accretion >= 0 ? "+" : "") + (r.accretion * 100).toFixed(2) + "%").padStart(11),
    ((cost * 100).toFixed(2) + "%").padStart(17),
    (yieldBought > cost ? "accretive" : "dilutive").padStart(11)
  );
}
console.log("");
console.log("The last two columns never disagree with the third, and that is the");
console.log("whole point: EPS accretion is a comparison between the earnings yield");
console.log("you bought (" + ((TGT.netIncome / purchase) * 100).toFixed(2) +
            "%) and the after-tax cost of what you spent.");
console.log("");

// ── Where the rule of thumb comes from, and the word it gets wrong ─────────
console.log("the P/E rule of thumb, and the price it is measured at");
console.log("  All-stock funding costs the acquirer's earnings yield, 1/" +
            acqPe.toFixed(1) + " = " + ((1 / acqPe) * 100).toFixed(2) + "%.");
console.log("  So all-stock is accretive exactly when P/E PAID < acquirer P/E.");
console.log("  Here: paid " + offerPe.toFixed(1) + "x < " + acqPe.toFixed(1) +
            "x, so accretive -- with " + ((acqPe / offerPe - 1) * 100).toFixed(0) +
            "% of headroom left in the premium.");
console.log("");

/** Bisect for the offer price at which a mix goes exactly EPS-neutral. */
function breakEvenPrice(mix, synergy = SYNERGY) {
  let lo = TGT.price, hi = TGT.price * 20;
  for (let i = 0; i < 200; i++) {
    const mid = (lo + hi) / 2;
    if (proForma(mix, mid, synergy).accretion > 0) lo = mid; else hi = mid;
  }
  return (lo + hi) / 2;
}

console.log("break-even offer price, i.e. the premium at which accretion runs out");
console.log("  funding mix                price   premium   P/E paid   closed form");
for (const [name, mix] of MIXES) {
  const p = breakEvenPrice(mix);
  const paid = (p * TGT.shares) / TGT.netIncome;
  // Closed form: yield bought = cost of funding  ->  purchase = NI / cost.
  const closed = TGT.netIncome / costOfFunding(mix) / TGT.shares;
  console.log(
    "  " + name.padEnd(24),
    p.toFixed(2).padStart(8),
    ((p / TGT.price - 1) * 100).toFixed(0).padStart(7) + "%",
    paid.toFixed(1).padStart(10) + "x",
    closed.toFixed(2).padStart(11) + (Math.abs(closed - p) < 0.01 ? "  ok" : "  MISMATCH")
  );
}
console.log("");
console.log("Read the all-stock row: it goes neutral at a P/E paid of exactly " +
            acqPe.toFixed(1) + "x,");
console.log("the acquirer's own multiple. That is the rule of thumb falling out of");
console.log("the algebra -- and note it is the multiple PAID, not the target's");
console.log("quoted " + tgtPe.toFixed(1) + "x. A candidate who compares against the quoted");
console.log("multiple has no way to tell you where the line is at all.");
console.log("");

// ── Synergies move the line, and so does the funding you happen to own ─────
console.log("with pre-tax run-rate synergies, break-even premium moves:");
console.log("  synergies    all stock    all debt    all cash");
for (const syn of [0, 20, 40, 60]) {
  const cells = [{ stock: 1 }, { debt: 1 }, { cash: 1 }].map((m) => {
    const p = breakEvenPrice(m, syn);
    return (((p / TGT.price - 1) * 100).toFixed(0) + "%").padStart(11);
  });
  console.log("  " + (syn + " ").padStart(9) + "  " + cells.join(" "));
}
console.log("");
console.log("Two things to notice, and they point in opposite directions.");
console.log("");
const cashBe = breakEvenPrice({ cash: 1 });
const stockBe = breakEvenPrice({ stock: 1 });
console.log("1. Cash tolerates a " + ((cashBe / TGT.price - 1) * 100).toFixed(0) +
            "% premium versus stock's " + ((stockBe / TGT.price - 1) * 100).toFixed(0) + "%.");
console.log("   Cash looks best precisely BECAUSE it earns the least (" +
            (CASH_RATE * 100).toFixed(1) + "% pre-tax). You");
console.log("   are being scored for spending your cheapest money, which is not the");
console.log("   same as buying the asset at a sensible price.");
const synBe = breakEvenPrice({ stock: 1 }, 20);
console.log("2. Synergies raise the tolerable premium fast: 20 of pre-tax savings");
console.log("   moves the all-stock line from " + ((stockBe / TGT.price - 1) * 100).toFixed(0) +
            "% to " + ((synBe / TGT.price - 1) * 100).toFixed(0) + "%. And synergies are the");
console.log("   least verifiable number in the model, so any deal can be made");
console.log("   accretive on a slide by finding one more round of them.");
console.log("");
console.log("Which is why \"is it accretive?\" is a screening question, not a verdict.");
console.log("EPS is an accounting output: it ignores the cash spent, the risk added,");
console.log("and whether the price paid was below what the business is worth. A deal");
console.log("can be accretive and still destroy value, and the way to say so in a");
console.log("room is to give the accretion number first and then the reason it is");
console.log("not sufficient.");

Reading the output

The last two columns never disagree with the third. Every row's accretion sign matches the prediction from the yield comparison, and it does so for mixes as well as pure funding. That agreement is the evidence that the full pro-forma model and the one-line condition are the same statement. You can now answer "what if it were 60% debt and 40% stock?" in your head — blend the costs, 4.70%, compare to the 6.67% you bought, accretive — without building anything.

The rule of thumb falls out of the algebra. The all-stock break-even lands at a paid multiple of exactly 20.0x, the acquirer's own. That is not a coincidence to be memorised; it is what the condition reduces to when the funding cost is the acquirer's earnings yield.

And it happens at a 60% premium, not a 20% one. The target trades at 12.5x. The deal is accretive all the way up to a paid multiple of 20x — a 60% premium, three times the offer on the table. If you had applied the rule of thumb to the quoted 12.5x you would have had no way to know where that line was. This is the practical payoff of the correction: the rule tells you the sign, the general condition tells you the distance to the edge.

The two things that move the line, and why only one is honest

Funding cost. All-cash tolerates a 256% premium before it turns dilutive, versus 60% for all-stock. That is a spectacular-looking result and it is almost entirely an artefact: cash looks best because it earns the least. At 3% pre-tax, sitting cash is the cheapest funding on the balance sheet, so spending it barely dents net income. The model is rewarding you for using your cheapest money — a statement about your balance sheet, not about whether the price was right.

Synergies. Twenty of pre-tax cost savings moves the all-stock break-even premium from 60% to 80%. Forty moves it to 100%. Cost synergies in real deals commonly run 5%–15% of the target's cost base, and revenue synergies are conventionally haircut hard because they depend on customers behaving as predicted. Which means: any deal can be made accretive on a slide by finding one more round of synergies, and the synergy number is the least verifiable input in the model.

What accretion does not measure

The rule of thumb also breaks in ways worth naming explicitly. It holds only for all-stock deals with no synergies; introduce debt financing, cash, transaction costs or synergies and it stops being a reliable predictor — which is exactly why the general condition is the thing to carry.

More fundamentally, EPS accretion is an accounting output, and it is silent on:

  • Cash. An all-cash deal that empties the balance sheet is maximally accretive and may leave the company unable to fund its own capital plan.
  • Risk. Levering up to buy something is accretive at today's rates and is a different company afterwards. EPS does not carry a risk adjustment.
  • Price. Nothing in the calculation asks whether the business is worth what you paid. That is a DCF question, and the previous lesson is where it gets answered.
  • Time. Deals are frequently dilutive in year 1 — from transaction costs, integration spend and synergies that phase in — and accretive by year 3. A single-year answer misses the trajectory that the deal was actually justified on.

The complementary analyses to name if asked what you would look at instead: a DCF on the target standalone and with synergies, the deal IRR, and return on invested capital against the acquirer's cost of capital. Those measure value; accretion measures the accounting.

What to change and re-run

  1. Set PREMIUM = 0.60 and watch all-stock go exactly EPS-neutral while the other mixes stay accretive. This is the rule of thumb at its boundary, and seeing it land on zero is more convincing than being told it does.
  2. Set CASH_RATE = 0.05 — a higher-rate environment. Cash's advantage over debt narrows sharply, because the interest you give up rises. The ranking of funding sources is not a fact about funding sources; it is a fact about the rate environment.
  3. Set DEBT_RATE = 0.09 and re-check the debt row. Above the target's earnings yield on the price paid, debt-funded acquisition turns dilutive on its own, no premium required.

The rest of this track goes to the models these two lessons set up: the LBO, where the same funding arithmetic determines a return rather than an EPS; comparable companies and precedent transactions, where the multiples used as cross-checks here come from; and the stock pitch, where you have to produce a view rather than defend one.