Equity Research puzzles, solved step by step

EBITDA rises by the full Rs 12 crore, EBIT by about Rs 4.3 crore, net debt by a lease liability of about Rs 77 crore, and EV/EBITDA falls from 10.0x to 9.40x. Rent leaves operating costs and returns as depreciation of Rs 7.7 crore and interest of Rs 6.9 crore. The cash paid is identical, so the only safe comparison is between companies measured on the same basis.

Why does the rent move out of EBITDA at all?

Think of the difference between renting a flat and buying it with a home loan. The owner pays an EMI that is partly interest and partly repayment, and the flat sits on the family balance sheet as an asset with a loan against it. The lease standard treats a long lease the same way: the right to use the store becomes an asset, the promise to pay rent becomes a debt, and the rent itself disappears from operating costs. Discount ten payments of Rs 12 crore at an assumed borrowing rate of 9% and the lease liabilityThe present value of the lease payments still to be made, carried on the balance sheet like a loan. is Rs 77.0 crore.

What replaces the rent in the income statement?

Two lines, both below EBITDA. The right-of-use assetThe asset recognised for the right to use a leased item, depreciated over the lease term. of Rs 77.0 crore is depreciated over ten years, Rs 7.7 crore a year, and the liability accrues interest at 9%, Rs 6.9 crore in year one. So EBITDA gains the whole Rs 12 crore, EBIT gains only Rs 4.3 crore, and year-one pre-tax profit actually falls by Rs 2.6 crore, because interest is heaviest while the liability is largest. Over the ten years the charges add up to Rs 120 crore, the same as the rent; only the timing moves.

Why does this matter when you compare two retailers?

Because the multiple now depends on whether a company rents or owns its stores. A retailer that leases every store will look cheaper on EV/EBITDA and more levered on net debt to EBITDA than an identical one that owns its stores, unless you put both on the same basis. Here leverage rises from 3.33x to 3.85x while the multiple falls. Say which basis you are using, and either include leases for everyone or strip them out for everyone. The standard's exemptions for short-term and low-value leases also vary in use, so check the notes before comparing.

Diluted EPS is about Rs 1.90, against basic EPS of Rs 2.00. Assume the bond converts. Earnings rise by the after-tax interest no longer paid: Rs 300 crore at 8% is Rs 24 crore, Rs 18 crore after tax. Shares rise by 15 crore. Rs 218 crore over 115 crore shares is Rs 1.90, a dilution of 5.2%. Check first that the bond dilutes at all.

Why do earnings go up when the bond converts?

Imagine a friend who lent you money and agrees to take a share of your business instead of repayment. From that day you stop paying them interest, so your profit rises, but you now split it with one more owner. A convertible works the same way: conversion removes the coupon and adds the shares, and the if-converted methodThe standard way to include a convertible in diluted EPS: assume it converts at the start of the year, add back its after-tax interest and add the new shares. captures both. The add-back is after tax, because the interest was saving tax: Rs 24 crore of interest cost only Rs 18 crore of profit.

When would you leave the bond out altogether?

When including it would raise EPS. Work out what the bond costs per new share: Rs 18 crore of after-tax interest over 15 crore shares is Rs 1.20 a share. If that figure is below basic EPS, conversion dilutes and the bond goes in; if it is above, the bond is antidilutiveA security whose assumed conversion would increase EPS. Accounting standards exclude it from diluted EPS. and is left out. Here Rs 1.20 is below Rs 2.00, so it goes in. If the same bond converted into only 5 crore shares, its cost per share would be Rs 3.60 and it would be excluded.

An analyst uses diluted EPS for valuation because a convertible that is in the money will convert, and the market prices the stock on the larger share count. The limit is that diluted EPS is a snapshot: it counts only securities that dilute at today's numbers, and a rise in the share price can bring more of them in.

13.6%. Start from the 7% risk-free rate and add each factor's loading times its premium: market 1.1 times 6% is 6.6 points, size 0.3 times 2% is 0.6, and value minus 0.2 times 3% is minus 0.6. So 7 plus 6.6 plus 0.6 minus 0.6 gives 13.6%. Here the size and value terms cancel, so the answer matches a plain CAPM with the same beta, which is a coincidence of these numbers.

How does a factor model build a required return?

Think of a taxi fare: a flag-down charge, then so much per kilometre, then so much per minute of waiting. Each meter runs at its own rate, and the fare is the sum. A factor model prices a stock the same way: the risk-free rate is the flag-down charge, and each factor adds how much the stock is exposed to it, the loading, times what that exposure pays, the premium. The {term('Fama-French three-factor model', 'An asset pricing model from Eugene Fama and Kenneth French that explains stock returns with three factors: the market, company size, and value against growth.')} uses three meters: the market, small against large companies, and cheap against expensive stocks.

Why does a negative loading subtract?

A negative value loading means the stock tends to do well when cheap stocks do badly: it behaves like a growth stock. The model says investors are paid a premium for holding value exposure, so a stock with the opposite exposure is priced to earn less, and its cost of equity falls. Push the value loading to minus 0.5 and the answer drops to 12.7%. Say the sign out loud; it is exactly what the interviewer is listening for.

Why does the answer equal a plain CAPM here, and should you trust it?

With the same market beta, CAPM gives 7 plus 1.1 times 6, which is 13.6%. The size and value terms happen to cancel in this question, so the two models agree only by coincidence. In practice the market loading from a three-factor regression is usually different from the CAPM beta, because the other factors absorb some of the movement. The bigger limit is the inputs: factor premiums are estimated from history, vary by market and period, and should be treated as assumptions to be stated, not facts to be quoted.

Revenue is Rs 219 in year three, and Rs 100 of it, 46%, comes from new customers. The three layers are the new batch at 100, the year-two batch at 70 and the year-one batch at 49. Growth falls from 70% to 28.8% even though the company signs just as many customers, because revenue creeps towards a ceiling of Rs 333, which is new spend divided by the share lost each year.

Why do you build revenue layer by layer?

Picture a gym that signs 100 members every January, where 30% of any class drops out each year. Its member count is not 100 a year times the years open; it is this year's class plus what is left of every earlier class. Revenue built from customers is a stack of cohortsA group of customers who joined in the same period, tracked together over time., each decaying at the retention rate, and the total is the sum of the layers. Writing the layers down stops you from projecting last year's growth rate forward, which is the mistake the question is set up to catch.

Why does growth slow when the company is signing just as many customers?

Because the loss grows with the base. In year three the company loses 30% of 170, which is 51, and adds 100, so net growth is 49. Revenue stops growing when the spend lost each year equals the spend added, which here is at Rs 333: 100 divided by 0.3. Retention sets that ceiling. At 80% retention, year-three revenue would be Rs 244 and the ceiling Rs 500; at 60% the ceiling is Rs 250.

What does an analyst do with the 46%?

It tells you how much of the business must be won again every year. When nearly half of revenue comes from customers signed in the last twelve months, sales hiring and marketing spend are carrying the top line, and any slowdown in new wins shows up in revenue almost at once. By year five the share from new customers falls to 36% as the older layers pile up. The limitation of the model is equal batches: a company that grows its sign-ups each year will show faster growth, and one that spends more per customer in year two, through upsell, can see a batch grow rather than decay.

10.5%. Weight each segment's growth by its share of revenue: 60% of 5% is 3 points, 30% of 15% is 4.5 points and 10% of 30% is 3 points, which add to 10.5%. The simple average of 16.7% is wrong because it gives the small, fast segment as much say as the large, slow one.

Why does the big slow segment decide most of the answer?

Think of a family where one parent earns most of the income and gets a 5% raise while a teenager's pocket money rises 30%. The household is not 17% richer. Group growth is a weighted average, and the weights are each segment's share of revenue, so the largest segment pulls the answer towards its own growth rate. The fast segment only adds 3 points, because it is only a tenth of the business.

How do you do it in your head without slipping?

Turn each segment into points of group growth, one at a time, and say them out loud. Six tenths of five is three; three tenths of fifteen is four and a half; one tenth of thirty is three; three plus four and a half plus three is ten and a half. Multiplying by tenths is just moving a decimal, so every step is a small whole-number product. Then sanity check: the answer must sit between the slowest and fastest rates and closer to the slow one, and 10.5% does.

What happens next year if each segment keeps its growth rate?

The weights move. After a year the fast segment is a bigger slice, about 11.8% instead of 10%, and the core shrinks to about 57.0%. So group growth rises to about 11.1% next year with no segment accelerating at all: mix alone lifts it. That is worth saying in the room, because management commentary often credits acceleration to execution when it is arithmetic. The limit runs the other way too: a fast segment rarely keeps 30% growth as it gets bigger.

About 1.54 standard deviations too high. Each forecast is right on average, but choosing the highest selects for the biggest positive error. The expected maximum of ten independent draws from a normal distribution is 1.54 standard deviations above the mean. If each analyst's error has a spread of Rs 2 around a true EPS of Rs 50, the top forecast averages Rs 53.08.

Why does choosing the highest create bias when nobody is biased?

Weigh yourself on ten bathroom scales, each accurate on average but each a little off, and write down only the heaviest reading. You will always look heavier than you are. The bias is not in any one estimate; it comes from the choosing, because the maximum picks whichever estimate had the largest positive error. Auction theory calls the same effect the winner's curseThe tendency for the winning bid in an auction to overestimate the value of the prize, because the highest estimate of many noisy ones is usually too high.: the bidder with the highest estimate wins, and usually overpaid.

How do you get to 1.5 without tables?

Ask where the maximum has an even chance of landing. All ten estimates must fall below a level for the maximum to fall below it, so you need the level where the chance for one estimate, raised to the tenth power, is one half. That single-estimate chance is 0.5 to the power one tenth, about 0.933, which a normal table puts at 1.50 standard deviations. The mean of the maximum is a touch higher, 1.54, because the distribution of the maximum has a longer right tail. Either number is a fine answer in the room if you show the route.

Where does this bite an analyst?

Anywhere the top of a list is chosen after the fact. The most bullish forecast in the consensus, the best of ten back-tested strategies and the top-ranked fund of the year all carry a selection premium that will not repeat. The effect grows with the list: the highest of five is 1.16 standard deviations high, of twenty about 1.87. The fix is to shrink the chosen number back towards the average in proportion to how noisy the estimates are. The limit of this answer is the independence assumption: analysts who talk to the same management team share errors, and correlated errors shrink the gap.

Switching wins two times in three. Your first pick is right with probability one third, and nothing the host does changes that. In the other two thirds of cases the prize is behind one of the two doors you did not pick, and the host, who must open an empty door, is forced to leave the prize door shut. One half is wrong because it treats the host's reveal as random when it is not.

Why does one half feel right?

Two closed doors, one prize: the picture in front of you looks like a coin toss, and intuition prices the picture rather than the process that produced it. One half would be right only if the two remaining doors were equally likely before the reveal and the reveal told you nothing about either of them. Neither is true. Your door was chosen blind; the other door survived a filter run by someone who knew the answer.

What exactly does the host's knowledge change?

Think of a shopkeeper who knows which of three mangoes is ripe. You pick one; she then removes an unripe one from the two you did not pick, always careful never to remove the ripe one. She has concentrated the two thirds chance that the ripe mango was among the other two onto the single one she left. Your own pick is untouched at one third, because she would have been able to remove an unripe mango whatever you picked. The reveal carries information about the other door, not about yours.

When would one half actually be the right answer?

This is the part the interviewer is asking for when they ask why intuition fails. If the host did not know and opened one of the other doors at random, and it just happened to be empty, then switching would win only one half of the time. A random reveal that could have shown the prize rules out the branches where it would have, and what is left is symmetric. So the answer depends on the host's rule, not on the doors. Scale it up to make the point land: with 100 doors, you pick one, and a knowing host opens 98 empty ones; nobody thinks the last closed door is a coin toss.

The analyst's version is reading a signal from someone who knows more than you do. A management team that chooses which numbers to disclose is running a filter, and what survives the filter is not a random sample.

Pitch B, at 1.1x against 0.5x for Pitch A. Weight each outcome by its probability. A gives 10% of 5x plus 90% of nothing, which is 0.5x: on average it loses half the money. B gives 60% of 1.5x plus 40% of 0.5x, which is 0.9 plus 0.2, 1.1x. A would need better than a one in five chance of the 5x just to break even.

Why does the big number not win?

A lottery ticket that costs Rs 100 and pays Rs 500 one time in ten is a bad ticket, however good Rs 500 sounds. An expected value multiplies each outcome by its probability, so a large payoff is shrunk by a small chance before it counts. For A, 5x shrinks to 0.5x. For B, a modest win that happens more often than not, plus a partial loss that still returns half the money, adds up to 1.1x.

What would make Pitch A worth taking?

Work backwards from breakeven. A returns your money on average only if the chance of the 5x is one in five, 20%. So the real question about A is not how big the upside is but whether you can defend a probability above 20%, double what the pitch claims. That is how a buy-side analyst would push back on a story built around one dramatic outcome: ask for the odds, then check them.

Is the higher expected value the whole answer?

No, and saying so earns the extra point. B still loses half the money 40% of the time, so the expected value tells you which pitch to prefer, not how much to put in. Position size depends on how bad the bad outcome is and how often it comes. The limitation of the question is that it hands you the probabilities. In practice those are the hardest number to estimate, and a pitch with a vivid upside tends to come with an optimistic probability attached.

About Rs 28,800 crore a year on these assumptions, most of it from private schools. Government schools: 13 crore children, 2 sets a year at Rs 300, Rs 7,800 crore. Private schools: 12 crore children, 2.5 sets at Rs 700, Rs 21,000 crore. A single blended average would have given Rs 20,000 crore, so the split matters more than any one input. Confirm enrolment against the latest official school data.

Why split by school type before anything else?

Think of two families on the same street. One sends a child to the government school, where the uniform is supplied through the state at a fixed allowance. The other pays a private school's appointed tailor for a blazer, a house t-shirt and two sets of the regular uniform. Who pays decides both how many sets are bought and what each costs, so the segments must be sized separately and then added. A national average price describes neither family.

State the assumptions plainly and treat each as a placeholder to be replaced with data. Assume about 25 crore children in school, 13 crore in government schools and 12 crore in private ones; confirm against the latest UDISE+ report. Assume 2 sets a year at about Rs 300 a set in government schools, where many states fund uniforms through schemes, and 2.5 sets at about Rs 700 in private schools, allowing for sports and house uniforms. Each number is round on purpose; the point is a structure you can defend.

How do you sanity check it?

Turn it into a number a parent can feel. The market works out to about Rs 1,152 a child a year, roughly Rs 1,750 for a private school child and Rs 600 for a government school child. If a private school parent you know spends about that on uniforms each year, the build is in the right range. A second check comes from the supply side: the number of uniform makers and school tailors in one town, times their annual sales, scaled by the number of similar towns.

What would you refine with more time?

The private segment carries almost three quarters of the value, so refine it first. Private schools range from low-fee neighbourhood schools, where uniforms cost little more than in government schools, to premium schools with branded kits. Splitting private schools into two or three fee bands is the next step, because price varies most there. Replacement is a second lever: younger children outgrow uniforms every year, while older ones may make one set last two.

The average return is 13.3% a year, but the compound annual return is 8.0%. The average adds 50, minus 30 and 20 and divides by three. The money multiplies: 1.5 times 0.7 times 1.2 is 1.26, so Rs 100 becomes Rs 126, and the rate that compounds to 1.26 in three years is 8.0%. The compound rate is what an investor actually earned.

Why is the simple average the wrong measure of what you earned?

A shop marks a Rs 100 shirt up 50% to Rs 150, then puts it on a 30% sale. The sale takes Rs 45 off, not Rs 30, because it is taken from the higher price. Each year's return is applied to whatever the previous year left you, so returns multiply, and a simple average of them overstates the growth of the money whenever returns vary. The loss in year two comes out of a larger base than the one the gain was earned on.

How do you estimate the gap without a calculator?

Use the rule that the compound rate is roughly the average minus half the variance. The returns sit 36.7, minus 43.3 and 6.7 points from their average; squared and averaged, they give a variance of about 0.109 in decimal terms. Half of that, about 5.4 points, is the drag volatility puts on compounding, and 13.3 minus 5.4 gives about 7.9%, close to the exact 8.0%. The approximation is rougher when swings are this large, but it shows the mechanism: the more a return bounces, the further the compound rate falls below the average.

When is the average the right number to use?

When you want the best guess of a single future year's return, the arithmetic average is the unbiased estimate, which is why cost of capital work often uses it. When you want to describe what happened to money held over several years, only the compound rate is honest. Fund factsheets report compound annual growth rates for that reason, and a pitch that quotes an average return for a volatile stock is flattering it.