Equity Research puzzles, solved step by step

The terminal value is about 77% of the value, and a one point rise in WACC cuts the total by about 16%. At 10% the five years are worth 379 and the terminal value 1,304 in today's money, total 1,683. At 11% they are worth 370 and 1,039, total 1,408. The explicit years barely move; the terminal value drops by a fifth.

Where does most of the value sit?

Think of valuing a flat you plan to rent out for five years and then keep forever. The five years of rent are real, but the flat itself, the part you keep, is most of what you are paying for. A DCF is the same. The five explicit years are worth 379; everything after year five, the terminal value, is worth 1,304 in today's money, 77.5% of the total. The terminal value at year five is 100 x 1.05 / (0.10 - 0.05) = 2,100, discounted back five years.

Why does one point of WACC move the value so much?

The terminal value divides by the gap between WACC and growth. Moving WACC from 10% to 11% widens that gap from 5% to 6%, cutting the undiscounted terminal value from 2,100 to 1,750, a sixth, before the extra discounting takes more. The explicit years fall only from 379.1 to 369.6. Together the value drops from 1,683 to 1,408, down 16.3%.

This is why a research note shows a sensitivity table of WACC against terminal growth rather than a single number. It is also why you check the implied exit multiple: 2,100 is 21 times year-five cash flow, and if peers trade nowhere near that, the inputs need a second look. The limitation is that the table shows sensitivity; it does not tell you which rate is right.

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 is cheaper on PEG, 1.2 against 1.9, but PEG ignores how long the growth lasts and what it costs. PEG divides the P/E by the growth rate, so it rewards fast growth whatever its duration. If A's 25% lasts only three years, today's price is 15.4 times year-three earnings for A against 11.9 times for B, and A is the dearer stock.

What does PEG measure?

Think of two rented flats, one expensive in a fast-improving area and one cheap in a quiet one. Comparing rent per square foot is not enough; you would also ask how quickly each area is improving. PEG does that for stocks. It divides the P/E by the growth rate, so it expresses price per unit of growth: A pays 1.2 times per point of growth, B pays 1.88. A PEG near 1 is often used as a rough marker of fair value, which is a convention, not a law.

What does PEG leave out?

Two things. First, duration: a growth rate is a speed, and PEG ignores how long the speed lasts. If A grows 25% for three years and then slows to B's pace, you are paying 15.4 times its year-three earnings for a company then growing like B, which costs 11.9 times. Second, the cost of growth: a company that must reinvest most of its earnings to grow is worth less than one that grows with little capital, and PEG cannot see the difference.

How a research analyst would use it: as a quick screen across a sector, then replaced by something that sees duration and reinvestment, such as a DCF or a return on capital comparison. The limitation in one line: PEG compares price to the speed of growth, not to its length or its cost.

B trades lower, at 13.3x against A's 16.7x, and they match when B's debt costs 8%. A earns 80 x 0.75 = 60 on equity of 1,000. B pays 30 of interest, earns 37.5 on equity of 500. The P/Es match when the after-tax cost of debt equals A's earnings yield of 6%: r x 0.75 = 6%, so r = 8%. Cheaper debt lowers B's P/E; dearer debt raises it.

Why would the same business carry two different P/Es?

Suppose a flat earns rent of Rs 6 for every Rs 100 of its price. Buy it with half cash and half a loan at 4.5% after tax, and your cash earns more than 6%, because the borrowed half costs less than it yields. P/E divides the equity's value by the equity's earnings, and debt changes both, by different amounts. Swapping equity for debt that costs less than the equity's earnings yield lowers the P/E; swapping for dearer debt raises it.

Why is 8% the crossing point?

Swapping 500 of equity for 500 of debt removes equity that was earning its share of A's 6% earnings yield, 30 of net income, and adds an after-tax interest cost of 500 x r x 0.75. If that cost is exactly 30, net income halves along with equity and the P/E does not move; 30 over 375 is 8% before tax. The general rule: leverage cuts P/E when the after-tax cost of debt is below the earnings yield, E/P. The limitation worth saying: B's lower P/E is not a sign it is cheaper. Its equity is riskier, so investors should demand a lower multiple for the same business.

About 53.64 per share. Equity value is 5,000 plus 500 of net cash, 5,500. If the price is P, the options add 10 shares and their 400 of exercise cash buys back 400 / P shares, so P = 5,500 / (110 - 400/P). Solving gives P = 5,900 / 110 = 53.636. Iterating from the naive 55 gets there in three rounds.

Why can you not just divide by a share count?

Think of splitting a restaurant bill where one late guest pays a fixed Rs 40 whatever the bill, and the rest is shared. How much the others pay depends on the bill, and the bill depends on who is sharing it. Under the treasury stock methodA way to count option dilution: assume in-the-money options are exercised and the exercise cash is used to buy back shares at the current price., how many net new shares the options create depends on the share price, and the share price depends on how many shares there are. The dilution and the answer have to be found together.

How do you solve it in one line instead of looping?

Write P x (110 - 400/P) = 5,500. The P cancels in the second term, leaving 110P - 400 = 5,500, so P = 5,900 / 110 = 53.636. When the options are in the money, the consistent price is simply equity value plus exercise cash, divided by all shares including the options. The check: at 53.636, the 400 of cash buys back 7.458 shares, so 2.542 net new shares take the count to 102.542, and 5,500 over that is 53.636.

Say the limitation. The method ignores the time value of options and assumes exercise today. A model that values the options properly would subtract their value from equity instead, and the answer would come out slightly lower.

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 6.7x, on an implied stub value of Rs 2,000 crore. The 50% stake is worth half of Rs 16,000 crore, Rs 8,000 crore, at the subsidiary's own market price. Take that out of the holding company's Rs 10,000 crore and the market is paying Rs 2,000 crore for everything else. Against Rs 300 crore of earnings that is 6.7 times, assuming the holding company carries no debt or cash of its own.

How do you find the price of a business that has no price of its own?

A thali costs Rs 300 and includes a sweet the same restaurant sells alone for Rs 80. The rest of the meal is costing you Rs 220. When a company's value contains something with its own visible price, subtract that price to see what the market is paying for the rest. Analysts call what is left the stubThe value the market implicitly assigns to a holding company's own business after subtracting the market value of its listed stakes.. Here the visible item is the listed stake, worth Rs 8,000 crore at the subsidiary's share price.

Why might 6.7x not be the whole story?

Holding companies usually trade below the value of what they own. If the market applies a 20% holding-company discount to the stake, it is valuing the stake at Rs 6,400 crore, so the stub rises to Rs 3,600 crore and the implied multiple to 12x. The discount reflects tax on any eventual sale of the stake, dividends that may never reach the holding company's own shareholders, and the cost of running the holding company. Give both numbers and say which assumption produces each.

What would you check before calling the stub cheap?

Three things. Whether the holding company carries debt, which the stub has to absorb; whether the Rs 300 crore of earnings is recurring or flattered by one-off items; and whether the stake is ever likely to be sold or distributed. A discount that never closes is not a mispricing, so a low stub multiple is a question to investigate, not a conclusion.

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.

A P/E of 20x, an earnings yield of 5%, and a price of Rs 500. P/E is price over EPS, 400 / 20 = 20. The earnings yield is the inverse, EPS over price, 20 / 400 = 5%. If EPS rises 25% to Rs 25 and the market still pays 20 times, the price is 20 x 25 = Rs 500, also 25% higher: at a constant multiple, price moves one for one with earnings.

What does a P/E of 20 actually say?

Buy a small shop for Rs 20 lakh that earns Rs 1 lakh a year and you have paid twenty years of today's profit, a 5% return on your price before any growth. A P/E is the number of years of today's earnings you pay for, and its inverse, the earnings yield, is those earnings as a return on the price. Neither says whether the stock is cheap; that depends on how fast the earnings grow and how risky they are.

When does the price not follow earnings?

When the multiple moves. If EPS rises 25% but the P/E falls from 20x to 16x, the price does not move at all: 25 x 16 = Rs 400. The price change is the earnings change and the multiple change compounded: 1.25 x 0.8 = 1.0. That is how a company can grow profit for years while its share price stands still, and why analysts separate earnings growth from rerating when they explain a move.

Why is the earnings yield worth quoting?

It puts the price on the same scale as other returns. A 5% earnings yield can be set beside a deposit rate or a bond yield, as long as you say what differs: earnings are not all paid out, they can grow, and they are not promised. When bond yields rise above a stock's earnings yield, the case for the stock rests more heavily on growth, which is one reason multiples tend to fall when rates rise.

About 9.7% a year, forever. For a bank, justified price to book is ROE minus growth over cost of equity minus growth. Setting 2.5 equal to (18% minus g) over (13% minus g) gives g of 9.67%. With no growth the bank would deserve 1.38x, so about 45% of today's price is paying for growth, which needs 54% of profits retained every year.

Why does price to book depend on ROE against cost of equity?

Picture a shop that earns Rs 18 a year on every Rs 100 its owner has put in, when the owner could earn Rs 13 elsewhere for the same risk. Nobody would sell that shop for Rs 100. A bank's book value is the owners' capital, so a bank earning more on it than shareholders demand is worth more than its book, and one earning less is worth less. With no growth, every Rs 100 of book produces Rs 18 a year forever, worth 18 over 13, or {pb0_80:.2f} times book.

How do you get the growth out of the price?

Growth needs capital. A bank growing its book at g must retain g over ROE of its profits, so the dividend on each Rs 100 of book is ROE minus g, not ROE. Put that dividend into the Gordon growth modelA valuation of a stream of dividends growing at a constant rate forever: next year dividend divided by cost of equity minus growth. and the price to book falls out. Set the formula equal to 2.5 and solve: 2.5 times (13% minus g) equals 18% minus g, so 1.5g equals 14.5%, and g is 9.67%.

Is 9.7% forever believable?

That is the question the interviewer really wants answered. A reverse valuation is only useful if you then judge the implied number. Growing book at 9.7% while paying out 46% of profits requires the 18% ROE to hold for decades, with asset quality intact. Notice how steep the curve is near the answer: a point more of growth, or a point less of cost of equity, moves the justified multiple a long way, which is why small changes in rate expectations swing bank valuations. The limitation is the model itself: one growth rate forever is a simplification, and a two-stage version is fairer to a bank growing fast today.