Equity Research puzzles, solved step by step

About 7.8% a year, forever. In a growing perpetuity the price equals next year's cash flow, Rs 30 x (1 + g), divided by 11% minus g. Setting that equal to Rs 1,000 and solving gives g = (110 - 30) / 1,030 = 7.77%. The quick version: an 11% required return minus a 3% cash yield leaves about 8% for growth.

Why solve for growth instead of assuming it?

Think of a shop listed for sale at a price that looks high for its current takings. Rather than argue about the price, ask what sales growth would justify it; then the argument becomes whether that growth is believable. A price is a bundle of assumptions, and solving for the growth it needs turns a vague sense that a stock is expensive into one number you can test. This is the idea behind a reverse DCFA valuation run backwards: start from the market price and solve for the growth or margins the price implies..

How do you solve it cleanly?

Write the growing perpetuity with next year's cash flow on top: 1,000 = 30 x (1 + g) / (0.11 - g). Multiply out: 110 - 1,000g = 30 + 30g, so 80 = 1,030g and g = 7.77%. The shortcut, cost of equity minus cash yield, gives 8% and is close because the yield is small; the exact answer is a little lower because the Rs 30 grows before it is paid.

Then judge it. Growing at 7.8% forever means growing faster than most economies can for ever, which is a demanding assumption. Look at how steep the curve is near the answer: at 7% the stock would be worth Rs 803, at 8.5% Rs 1,302. Small changes in the growth belief move the value a lot, which is why a single-stage model is a sense check here, not a valuation.

A implies a 20% margin and B 16.7%; B is cheaper for the margin you get. EV/sales divided by EV/EBITDA is EBITDA over sales, so A is 3 / 15 and B is 2 / 12. A pays 1.5 times as much per rupee of sales, but its margin is only 1.2 times B's. The rest of A's premium is a higher price per rupee of EBITDA, 15x against 12x, which has to be earned by faster growth or better quality.

How do two multiples reveal a margin?

If a flat costs Rs 60 lakh, which is 20 years of rent, and 3 times the owner's salary, you can work out that the rent is a fifteenth of the salary without seeing either. Two multiples on the same numerator divide to give the ratio of their denominators, so EV/sales over EV/EBITDA is EBITDA over sales, the margin. The enterprise value cancels, and the market has told you what margin it is capitalising.

So which is cheaper?

Split A's premium. A trades at 1.5 times B's sales multiple, and that decomposes exactly into 1.20 times the margin and 1.25 times the price per rupee of EBITDA. The margin explains part of the premium; the rest is the market paying 15x rather than 12x for each rupee of profit. On EBITDA, B is cheaper by a fifth. That does not make B the better stock: if A grows faster, converts more EBITDA to cash or carries less risk, the higher multiple may be deserved. The next question to ask is which of those it is.

Say the limitation: the implied margin is only as good as the EBITDA in the multiple. If one company's EBITDA is a forecast and the other's is last year's, or one capitalises costs the other expenses, the comparison is off before you start.

About 15.2x. Convert the numerator and the denominator separately. Enterprise value is 10 x 100 = 1,000; take off net debt of 200 and equity value is 800. EBITDA of 100 less D&A of 20 and interest of 10 is pre-tax profit of 70; after 25% tax, net income is 52.5. The P/E is 800 / 52.5 = 15.2x.

Why can't you read the P/E straight off the EV multiple?

A flat worth Rs 1 crore with a Rs 20 lakh loan on it is worth Rs 80 lakh to its owner, and the rent the owner keeps is what is left after the loan interest and the tax. The flat's value and the owner's stake are different numbers, and so are the rent and the owner's income. EV and EBITDA belong to lenders and shareholders together; equity value and net income belong to shareholders alone, so a multiple must pair one with the other of the same kind.

Why is the P/E higher than the EV multiple here?

Because the two sides shrink by different amounts. The price side loses 20% on the way from EV to equity, but the earnings side loses nearly half on the way from EBITDA to net income, so the P/E ends above the EV multiple. Leverage changes the balance: with net debt of 400 and interest of 20, equity value is 600, net income is 45 and the P/E is 13.3x. Heavy D&A or a high tax rate pushes the P/E up; debt that costs less than the earnings yield pulls it down.

When would an analyst do this conversion for real?

Whenever two sources quote different multiples for the same company, or when a sector trades on EV/EBITDA but a client thinks in P/E. Saying out loud which claims each number belongs to, all funders or shareholders only, is the habit that stops the conversion going wrong. Assume here that D&A is tax deductible in full and that the company has no minority interests or associates, and say so.

The price implies 10% growth a year, which needs a return on equity of about 16.7%. A forward P/E equals the payout ratio over cost of equity minus growth, so 20 equals 0.4 over 12% minus g, and g is 10%. Growth is funded by retained profit, 60% here, so ROE must be 10% divided by 60%, about 16.7%, well above the 12% cost of equity.

How do you pull growth out of a P/E?

Start from the dividend model and divide both sides by earnings. Price is next year's dividend over cost of equity minus growth, and dividend over earnings is the payout ratio. So a forward P/E is the payout ratio divided by the gap between cost of equity and growth: 20 equals 0.4 over 12% minus g. The gap must be 2%, which puts growth at 10%. Think of it like working out a car's speed from the distance it covered and the time it took: the multiple and the payout are the readings, growth is what they imply.

Why does the growth need a particular ROE?

Growth is not free; it is paid for with retained profit. A family that saves 60% of its income and wants its wealth to grow 10% a year must earn 10% divided by 60% on its savings, about 16.7%. A company is the same: sustainable growthThe growth a company can fund from its own retained profit: return on equity times the share of profit retained. is ROE times the share retained, so 10% growth on 60% retention needs an ROE of 16.7%. That is the number to test against the company's history, not the P/E itself.

What does the curve tell you that the formula does not?

Two things. First, when ROE equals the cost of equity, growth adds nothing: the P/E is 8.3x, one over 12%, whatever the growth. Growth only earns a higher multiple when each retained rupee earns more than shareholders demand. Second, the curve is steep near 20x, so a small change in the growth assumption moves the justified multiple a lot. The limit is the single perpetual rate: a real company grows fast for some years and then slows, and a two-stage model would put less weight on the far future.

15x. Dividend yield is dividend over price, and payout is dividend over earnings. Dividing the first by the second cancels the dividend and leaves earnings over price, the earnings yield: 4% over 60% is 6.67%. The P/E is the inverse, 1 over 0.0667, which is 15x. On a Rs 100 share, the dividend is Rs 4 and earnings are Rs 6.67.

How do the two ratios combine?

Suppose a friend tells you her rent is 30% of her salary and her rent is Rs 15,000. You know her salary without asking: Rs 15,000 over 30%. The dividend works the same way: if the dividend is 4% of the price and 60% of earnings, earnings must be 4% over 60% of the price, an earnings yieldEarnings per share divided by the share price, the inverse of the P/E. of 6.67%. Once you have earnings as a share of price, the P/E is just that number turned upside down.

How do you check it on a single share?

Pick a price of Rs 100. A 4% yield means a dividend of Rs 4. A 60% payout means that Rs 4 is 60% of earnings, so EPS is Rs 6.67. Price over EPS, Rs 100 over Rs 6.67, is 15x, which agrees. Doing it on one share is the fastest way to catch an inverted ratio, and it is worth saying out loud because the interviewer is listening for whether you can move between the three numbers.

Where does this shortcut mislead?

It assumes the payout ratio is stable and measured on the same earnings as the P/E. A company paying a special dividend, or keeping its dividend flat while profits fall, will show a payout ratio that does not describe normal earnings, and the implied P/E will be wrong. Check whether the yield is trailing or forward, and whether the payout is on reported or adjusted earnings, before you trust the answer.