Equity Research puzzles, solved step by step

Rs 112. For every four shares worth Rs 120, a holder pays Rs 80 for one new share. The five shares then hold Rs 480 of old value plus Rs 80 of new cash, Rs 560 in all, so each is worth Rs 560 divided by 5, which is Rs 112. The right to buy one new share is worth Rs 112 minus Rs 80, or Rs 32.

Why does the share price fall when nobody has lost anything?

Picture four friends who each own a Rs 120 share in a shared tiffin business. A fifth friend joins by paying only Rs 80. The business is now worth Rs 560 and there are five equal owners, so each stake is worth Rs 112. After a rights issue the old and new shares are identical, so the price has to settle at the weighted average of the old value and the new cash per share. The fall from Rs 120 to Rs 112 is not a loss to existing holders, because they were offered the cheap share too.

How do you prove an existing holder is no worse off?

Take a holder of four shares. If she subscribes, she had Rs 480 of shares and Rs 80 of cash; now she has five shares at Rs 112, still Rs 560. If she sells her right instead, she keeps four shares worth Rs 448 and receives about Rs 32 for the right, still Rs 480. Her wealth is unchanged either way; only a holder who ignores the right loses its value.

The analyst's follow-through: because the issue is priced below market, it carries a bonus element. Historical EPS is restated by dividing by the factor 120 over 112, or 1.0714, so per-share figures before and after the issue compare like with like.

LTV is Rs 1,500, LTV to CAC is 1.67x, and payback is 15 months per retained user, about 22 months for the cohort. Lifetime value is Rs 60 divided by 4% churn. CAC over monthly contribution, Rs 900 over Rs 60, gives the usual 15 month payback. Counting users who leave along the way, the cohort's cumulative contribution only reaches Rs 900 after about 22.4 months.

Why does churn set the lifetime value?

Think of a leaking water tank. If 4% of the water drains out every hour, a litre poured in stays, on average, 25 hours. With a constant monthly churn rate, the average customer life is one divided by churn, so lifetime value is monthly contribution divided by churn. Rs 60 over 0.04 is Rs 1,500. Halve churn to 2% and LTV doubles to Rs 3,000; that is why subscription analysts watch churn more closely than price.

Which payback number do you give the interviewer?

Give both and say which is which. The common convention is CAC divided by monthly contribution, 15 months, which describes a user who stays. A cohort of acquired users earns less than that each month because some have left, so the money spent on the cohort comes back later, after about 22.4 months. By month 15 only 54% of the cohort is still paying.

Then give a view. A ratio of 1.67x is thin: the business keeps Rs 600 per user over the user's life before any overhead, and a common rule of thumb looks for about three times. The two levers are churn and acquisition cost, and the numbers show churn is the stronger one.

About 33%. Take a price of 100 and a cost of 60, so profit is 40 a unit. Cut the price by 10 and profit falls to 30 a unit, because the cost does not move. To keep gross profit the same you need 40 divided by 30 as many units, 1.33 times, so volume must rise by a third. The formula is cut divided by (margin minus cut).

Why is the volume needed so much bigger than the cut?

A tea stall sells a cup for Rs 10 that costs Rs 6 to make, keeping Rs 4. Knock Re 1 off the price and the stall keeps Rs 3, a quarter less on every cup. A price cut comes straight out of the margin, because costs do not fall with price, so the percentage fall in profit per unit is the cut divided by the margin. Here a 10% price cut removes a quarter of the 40% margin, and volume has to make up the whole quarter.

How does the answer change with the margin?

This is why analysts treat price cuts in thin-margin businesses with suspicion. A retailer on a 20% gross margin that cuts prices 10% needs twice the volume just to stand still, while a software company on 80% needs only 14% more. Say the limitation too: the puzzle holds unit cost fixed. If volume brings purchasing discounts or spreads fixed factory costs, the bar is lower.

About 9.6% a year, not 14%. The 14% average only covers funds that survived, and funds usually close because they did badly. Weight both groups by their count: 60 funds at 14% and 40 at 3% gives 0.6 x 14% plus 0.4 x 3%, or 9.6%. A database that drops closed funds overstates the average investor's experience by 4.4 points a year.

Why does the survivor average mislead?

Ask the toppers of a coaching class how hard the entrance exam was, and they will say it was manageable. The students who failed have gone home and nobody asked them. Survivorship bias is judging a group by the members still standing, when the ones that dropped out left because their results were poor. A fund database that lists only live funds is the coaching class that only asks its toppers.

How big is the gap once it compounds?

Over ten years, 14% a year turns Rs 100 into Rs 371, while 9.6% turns it into Rs 250. A gap of 4.4 points a year looks modest, but it is the difference between multiplying money 3.7 times and 2.5 times. That is why performance studies that ignore closed and merged funds tend to make active management look better than it was.

Say the limitation too. Averaging annual returns across funds of different lives is a simplification; a careful study would weight by assets and by years in existence. The direction of the bias does not change, though: leaving out the losers always flatters the average.

EPS rose to Rs 16.1 and the P/E rose from 20x to 27.9x, so earnings explain about 59% of the gain and re-rating the rest. The price multiplied by 2.25. EPS multiplied by 1.1⁵ = 1.611, and the multiple by 1.397; the two multiply to 2.25. A year, that is 17.6% total: 10% from earnings and 6.9% from the multiple.

Why split a return into earnings and multiple at all?

A house bought for Rs 50 lakh sells for Rs 1 crore. Part of the gain came from the rent rising, part from buyers being willing to pay more years of rent for the same flat. A share price is EPS times P/E, so any change in price is a change in earnings multiplied by a change in the multiple. The split matters because earnings growth can continue while the business grows, but a multiple cannot rise for ever.

Does the rupee split depend on which step you take first?

Yes, and that is worth saying before the interviewer does. Because the effects multiply, the cross term goes to whichever leg is taken second, so a rupee split is a convention, while the log split of 59% and 41% is not. The analyst's conclusion: 41% of this return came from investors paying more for each rupee of profit. If the multiple drifts back to 20x, a holder earns only the EPS growth, and the next five years look very different from the last.

EPS and net worth fall; cash and net debt to EBITDA do not move. Assuming no tax deduction for the charge, reported profit falls by Rs 200 crore, taking EPS from Rs 5.00 to Rs 3.00 on 100 crore shares, and net worth falls by Rs 200 crore. No cash leaves, and EBITDA sits above the charge, so a covenant at 2.5x net debt to EBITDA is untouched. A covenant set on debt to equity would move.

Why does a Rs 200 crore loss leave cash untouched?

Imagine you paid Rs 20 lakh for a car three years ago and a valuer now says it is worth Rs 12 lakh. You are poorer on paper, but your bank balance did not change today; the money left when you bought it. A goodwill impairment admits that an acquisition was worth less than was paid, and the cash for that acquisition left the business when the deal closed. The goodwillThe part of an acquisition price above the fair value of the net assets bought, carried as an asset on the balance sheet. is written down, reported profit takes the charge and equity falls with it.

Which covenant does move, and why does the difference matter?

A covenant built on EBITDA is blind to the charge. A covenant built on net worth or on debt to equity is not: gross debt of Rs 1,300 crore against equity of Rs 2,000 crore is 0.65x, and after the write-down it is 0.72x. So the same accounting entry can be harmless under one loan and a breach under another. An analyst reads the covenant definitions before saying which.

Two more things to say. Most analysts strip the charge out of adjusted EPS, which stays at Rs 5.00, because it says nothing about next year's earnings. And whether the charge saves tax depends on local rules, which usually do not allow a deduction for goodwill impairment; confirm before assuming either way. The real signal is about management: the acquisition is now expected to earn less than was paid for it.

A levered beta of 1.10. Relever with levered beta = unlevered beta x (1 + (1 - tax rate) x debt/equity). That is 0.8 x (1 + 0.75 x 0.5) = 0.8 x 1.375 = 1.10. The 0.8 is the risk of the business itself; the extra 0.3 is the financial risk shareholders take on because lenders are paid first.

Why does debt raise the equity's beta?

Think of two families with the same salary. One has a home loan EMI to pay first each month; the other has none. A 10% pay cut hurts the family with the EMI far more, because the EMI does not shrink. Lenders are paid a fixed amount first, so the same swing in the business moves the shareholders' leftover by more, and beta measures exactly that swing. The business risk is the 0.8; debt stacks financial risk on top.

Where do candidates go wrong with the inputs?

Two places. Debt to equity uses market values where you can, and it is debt over equity, not debt over total capital: 0.5 debt to equity is one third debt in the capital structure. Using 0.33 by mistake gives 0.8 x (1 + 0.75 x 0.33) = 1.00. And the tax rate is the one that applies to this company's interest deduction; confirm the current rate rather than assuming it.

Say the limitation. The formula assumes the debt itself carries no market risk and that the debt level stays constant. For a heavily indebted company, debt starts to behave like equity and the simple formula overstates the levered beta.

0.648. At least two means exactly two or all three. Exactly two can happen three ways, each with chance 0.6 x 0.6 x 0.4 = 0.144, so 0.432 together. All three beat with 0.6 cubed, 0.216. Adding them gives 0.648. The complement checks it: none beat 0.064 plus exactly one 0.288 is 0.352, and 1 minus 0.352 is 0.648.

How do you avoid missing an outcome?

Think of three friends each deciding whether to come to dinner. At least two coming covers four different guest lists: each of the three possible pairs, and all three. List the outcomes by how many succeed, count the ways each count can happen, and multiply by the chance of one such way. With three stocks there are only eight results, so write them down; with more, the count of ways is a binomial coefficientThe number of ways to choose k items out of n, written n choose k; here 3 choose 2 is 3..

Is independence a fair assumption for three stocks you cover?

Usually not, and saying so is the part that sounds like an analyst. Stocks in one sector share demand, input costs and the same consensus-setting habits, so beats tend to cluster: when one beats, the others are more likely to. Clustering fattens both ends. The chance of all three beating and of none beating both rise above 0.216 and 0.064, and whether at least two beat can move either way, so the independent answer is a starting point, not a forecast.

About Rs 69 crore of revenue and Rs 2.2 crore of gross profit a year, on stated assumptions. Assume 12,000 vehicles a day pass on the pump's side, about 500 an hour. About 384 stop, filling 20,232 litres a day, most of it diesel for trucks. At assumed pump prices near Rs 90 to Rs 100 a litre that is Rs 18.8 lakh a day; at an assumed dealer margin of Rs 3 a litre the pump keeps about Rs 2.2 crore a year.

Where does the estimate start?

Think of a roadside dhaba: its takings depend less on how many vehicles pass than on which ones stop and how much each group eats. Start from the traffic on the pump's side of the road, split it by vehicle type, and apply a stop rate and litres per fill to each, because the three types differ by a factor of forty in what they buy. Every input below is an assumption for the method; fuel prices and dealer margins change and should be checked before use.

How do you turn litres into revenue and profit, and check the answer?

Revenue is litres times the pump price: petrol for two-wheelers and cars at an assumed Rs 100, diesel for trucks at Rs 90, which gives Rs 18.8 lakh a day and Rs 68.6 crore a year. The dealer does not keep the pump price; it earns a commission per litre, so gross profit is litres times margin, about Rs 2.2 crore a year at Rs 3 a litre. Sanity check the stops: 384 a day is about 16 an hour, one fill every four minutes, which a pump with a few nozzles handles easily. The weakest assumption is the truck stop rate, since fleet operators choose pumps by contract and credit terms, not by chance.

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.