Equity Research puzzles, solved step by step

27 reports. Find the unit rate first: 3 analysts writing 3 reports in 3 days means one analyst writes one report in 3 days, a third of a report per analyst-day. Nine analysts working nine days put in 81 analyst-days, and 81 x 1/3 = 27. Tripling the team triples the output and tripling the time triples it again: 3 x 3 x 3 = 27.

Why does the fast answer of 9 feel right?

The sentence has a rhythm, three, three, three, then nine and nine, and the ear wants to finish the pattern with another nine. It is the same pull as a kitchen puzzle where three cooks bake three cakes in three hours. Matching the pattern of the words is the trap; the fix is to reduce the problem to one worker and one day before scaling anything. Once you have the rate per analyst-day, the answer is multiplication.

How do you do it in ten seconds?

Scale each dimension on its own. Three times the analysts gives three times the reports, and three times the days gives three times again, so the answer is 3 reports x 3 x 3 = 27. The general rule is output equals the rate per worker per day, times the workers, times the days, and it handles any version of the item, including the ones where the numbers do not match so neatly.

What is a timed test like this really checking?

Wonderlic-style tests put many short items against a tight clock, so the difficulty is the time, not any single item. The format, number of items and time allowed vary by version, so confirm what the firm uses. The habit that scores on the test is the same one that keeps a model right: write the unit rate, then scale it. On a timed test, skipping an item you cannot set up in a few seconds and coming back to it is usually worth more than grinding.

200 miles. Ignore the zigzag and ask how long the bird is in the air. The car takes 100 / 50 = 2 hours to reach B, and the bird flies the whole time at 100 miles an hour, so it covers 2 x 100 = 200 miles. Summing the legs gives the same answer: 133.3 miles in the first round trip, then a third as much each round, and 133.3 / (1 - 1/3) = 200.

What is the shortcut, and why does it work?

Picture walking home while your dog runs back and forth between you and the front door. Ask how far the dog ran and the back and forth looks like the hard part, but the dog simply ran at its own speed for exactly as long as your walk took. When something moves at a constant speed for a known time, the distance is speed times time, however many times it turns around. The car's trip fixes the time at 2 hours, and the bird's speed does the rest.

How do you check it the long way?

The first meeting comes when the gap of 100 miles closes at 150 miles an hour, after 40 minutes, 33.3 miles from A. The bird has flown 66.7 miles and flies 66.7 back to B, a first round trip of 133.3 miles. By then the car is two thirds of the way, and every later round trip covers a third of the remaining gap, so each is a third of the last. A geometric series with first term 133.3 and ratio one third sums to 133.3 / (2/3) = 200, the same answer the shortcut gives in one line.

Why would an investing firm ask a puzzle like this?

Because choosing the quantity that stays simple is most of the skill. Here time is simple and distance is messy, so solving in time turns an infinite sum into one multiplication. The same habit finds total cash generated over a period from an average rate instead of adding every quarter, or total interest paid on a loan from the average balance. Say the frame you chose before you give the number.

About Rs 4.4 crore a year, on stated assumptions. Five shows a day on 200 seats at 30% occupancy is 300 visitors a day, about 1,09,500 a year. At Rs 250 a ticket that is Rs 2.74 crore; at Rs 120 of food per head, Rs 1.31 crore; screen advertising adds about Rs 0.30 crore. Food is under a third of revenue, but because a distributor takes about half of each ticket, it is well over a third of what the screen keeps.

How do you build the visitor count?

Count it the way you would count lunches at an office canteen: seats, times sittings, times how full each sitting is, times days. Visitors are shows x seats x occupancy x days, and occupancy is the number to defend, because most screens are far from full outside weekend evenings. Five shows on 200 seats at 30% gives 300 visitors a day and 1,09,500 a year. Every figure here is an assumption for the exercise, and prices are taken net of GST, whose rates should be confirmed.

Why does food matter more than its price suggests?

Because the two rupees are not shared the same way. A ticket rupee is split with the film's distributor, while a food rupee mostly stays with the screen, so Rs 120 of popcorn and a drink keeps about Rs 90, close to the Rs 125 the screen keeps from a Rs 250 ticket. That is why chains price food high and push it hard, and why an analyst tracks spend per head as closely as ticket prices. The distributor's share and the food cost here are assumptions to check against a company's disclosures.

Which assumption would the interviewer push on?

Occupancy first. It moves both big lines at once: at 35% instead of 30%, revenue rises to about Rs 5.03 crore. A screen's economics turn on filling empty seats, because rent, staff and power are the same whether the hall is a third full or packed. Then the average ticket price, which blends weekday discounts with weekend premiums, and the number of shows, which depends on film lengths. Name all three and you have shown how the business works, not just its size.

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.

Net income falls Rs 15 crore and cash rises Rs 5 crore; inventory falls Rs 20 crore while equity falls Rs 15 crore. The write-down is a Rs 20 crore expense, cut to Rs 15 crore after the Rs 5 crore tax saving. It costs no cash today, so the cash flow statement adds the Rs 20 crore back, leaving cash up by the tax saved. Assets fall Rs 15 crore, and so does equity.

Why does a Rs 20 crore loss raise cash?

A shopkeeper who bought stock last year and finds this year that it has spoiled is not paying for it again; the money went out when the goods came in. Admitting the loss costs nothing new, and if the tax rules let her deduct it, she pays less tax this year. The cash for inventory leaves when it is bought; the write-down only records the loss, and the tax deduction is the one real cash effect today. So profit falls by the loss after tax, while cash rises by the tax saved.

How do you walk it through the statements in order?

Income statement first: a Rs 20 crore expense, usually inside cost of goods sold, lowers pre-tax profit by 20; tax falls by 25% of that, Rs 5 crore; net income falls Rs 15 crore. Cash flow statement next: start from net income of minus 15 and add back the Rs 20 crore, because no cash left; cash is up Rs 5 crore. Balance sheet last: cash up 5 and inventory down 20 take assets down 15, and retained earnings fall by the same 15, so the two sides move together.

What if the tax rules are different?

Tax treatment of write-downs varies by jurisdiction, and some rules allow the deduction only when goods are actually scrapped or sold, so confirm it for the case at hand. If the deduction comes later, book net income still falls Rs 15 crore, but cash tax does not change this year, so cash is flat and a Rs 5 crore deferred tax asset appears on the balance sheet instead. Saying this without being asked shows you know where the Rs 5 crore of cash came from.

What does an analyst read into a write-down?

Two things. The charge is usually treated as one-off, so analysts look at profit before it, but repeated write-downs say something about how the company buys and forecasts demand. A write-down also moves cost out of future periods: goods written off now cannot be expensed again when sold, so next year's gross margin can look better than the business really is.

About 31%: roughly two flags in three are false alarms. Take 10,000 companies. 200 manipulate and the screen catches 180 of them. 9,800 are clean and 4% of them, 392, are flagged anyway. Of 572 flags, 180 are real, which is 31.5%. The false alarms come from the huge clean group, so a rare problem plus a small error rate swamps the true hits.

Why is the answer not 90%?

Think of the smoke alarm in a kitchen. It goes off for every real fire, and it also goes off for burnt toast. Fires are rare and toast burns every week, so most of the times the alarm sounds, nothing is on fire. The 90% tells you how the screen treats a manipulator; the question asks how often a flag is right, and that depends on how many clean companies are standing there to be wrongly flagged. Those are two different conditional probabilities, and swapping them is the whole trap.

How do you set it up so the arithmetic is easy out loud?

Pick a round population and count people, not probabilities. With 10,000 companies the 2% who manipulate are 200, and 90% of them, 180, get flagged. The clean 9,800 produce 4% false flags, 392. Once the tree is drawn, the answer is just flagged guilty over all flagged: 180 over 572. This counting method is called natural frequenciesStating a probability problem as counts out of a round population, such as 180 out of 10,000, instead of as percentages. It makes conditional reasoning much easier to follow., and it is far harder to get wrong than juggling Bayes' formula in your head.

What does this mean for how an analyst uses a screen?

A flag is evidence, not a verdict. It moves the odds from 2 in 98 to 180 in 392, a jump of 22.5 times, which is exactly 90% divided by 4%. That is why a flag should start a deeper read of the notes to the accounts, never end one. If a second, independent check with the same accuracy also flags the company, the odds multiply again and the chance rises to about 91%. The limitation is the word independent: two ratio screens built on the same receivables line will fail together.

Randomise: pay Rs 3.15 with probability about 0.159 and Rs 3.14 otherwise. Pi is 3.14159, which is 15.9% of the way from 3.14 to 3.15. Weighting the two payable amounts by those odds gives an expected payment of exactly pi. Neither side is favoured on average, which is the only sense in which an amount you cannot pay can be paid fairly.

What does fair mean when the exact amount cannot be paid?

Five children and four mangoes: nobody can get four fifths of a mango each without a knife, so a fair parent draws lots and one child misses out, with every child facing the same chance. When the exact amount is impossible, fair means fair on average: the expected payment equals what you owe. Rounding to 3.14 short-changes the restaurant every time; rounding to 3.15 overpays every time. Only a random choice between the two neighbours can hit pi exactly.

How do you find the right odds?

Call the chance of paying 3.15 by the letter p. The expected payment is 3.14 plus p times one paisa, and you want that to equal 3.14159. So p is the fraction of the gap you still owe: 0.00159 over 0.01, which is 0.1593. Say it as a balance point: the further pi sits towards 3.15, the more often you pay 3.15.

How do you actually produce odds of 0.159 with a coin?

This is the follow-up that separates people. Write p in binary: 0.0010100011 and so on. Flip a fair coin to generate your own random binary digits, one at a time, and compare each with p's digit in the same place. At the first flip that differs, you know whether your random number is below p or above it, and you pay 3.15 or 3.14 accordingly. Each flip has a one in two chance of settling it, so on average two flips decide, even though p itself is irrational. The same idea, called randomised roundingRounding a fractional quantity up or down at random, with odds set so that the expected result equals the exact fraction., is used to split odd lots of shares fairly across client accounts.

About Rs 50,000 crore a year on these assumptions, with the two routes landing within 5% of each other. Bottom up, urban repaints give Rs 35,000 crore, rural repaints Rs 9,000 crore and new homes Rs 7,200 crore, Rs 51,200 crore in all. Top down, 140 crore people spending Rs 350 a head gives Rs 49,000 crore. Every input is an assumption to be checked against published data.

Where do you start the bottom-up build?

Start from what gets painted, not from who sells paint. A home is repainted every few years, and each repaint uses a number of litres set by the wall area. So the market is homes, divided by the repaint cycle, times litres per job, times the price of a litre, plus the new homes painted for the first time. Split urban and rural before you multiply anything, because they differ on every input: bigger walls, shorter cycles and costlier emulsions in cities; smaller homes, longer cycles and cheaper finishes in villages.

State each number as an assumption and move on. Assume 140 crore people in about 30 crore households, a third urban. Urban homes are repainted every five years, with 3,000 square feet of wall and ceiling; rural homes every eight years, with 1,200 square feet. A litre covers about 60 square feet for a two-coat finish, and a litre costs a blended Rs 350 in cities and Rs 180 in villages. About 60 lakh new homes a year take 40 litres each at Rs 300. None of these is a published figure; each is a round number you can defend in one sentence and replace later. Confirm household counts against the latest census or survey data.

How does a top-down check work, and why does it help?

Go the other way with a number you can feel. A family of four that repaints a flat for about Rs 7,000 of paint every five years spends Rs 1,400 a year, which is Rs 350 a head. Across 140 crore people that is Rs 49,000 crore. Two routes built from different inputs that land within a few percent of each other are worth more than one route worked to the last rupee. If they had landed a factor of two apart, you would know one assumption was wrong and go looking for it.

Then name your weakest input. Urban repaints are about 68% of the answer, so the urban cycle carries the most risk: at six years instead of five, the urban figure falls to Rs 29,167 crore. An analyst would close by saying which number to check first, and where: the decorative revenue that listed paint makers publish in their annual reports is the natural cross-check.

Rs 2.16 lakh on price alone and Rs 2.84 lakh on total return. Price compounds at 8% a year: 1.08 to the tenth is 2.16. Reinvested dividends lift the yearly return to 11%, and 1.11 to the tenth is 2.84. The gap of about Rs 68,050 is more than the Rs 30,000 that ten years of 3% seems to promise, because the dividends compound too.

Why is the answer not simply 8% growth plus 3% times ten?

Think of a bank fixed deposit with two options: interest paid out every quarter, or interest added to the deposit. The cumulative option ends with more money because each quarter's interest starts earning interest. A reinvested dividend is the cumulative option: it buys more shares, and those shares rise in price and pay dividends of their own. Adding 3% a year for ten years treats the dividend like the payout option on a fixed base, which undercounts twice: the base keeps rising and the reinvested money keeps compounding.

Where does the gap come from, piece by piece?

Split it into two layers. Dividends taken in cash are 3% of a price that grows 8% a year, so they start at Rs 3,000 and end near Rs 6,000, adding to about Rs 43,460 over ten years. Reinvesting them adds a further Rs 24,590, the return earned on dividends already received. So the Rs 68,050 gap is roughly two thirds the dividends themselves and one third compounding on them, and the second layer grows fastest in the later years.

What does this change about how you compare stocks?

Compare on total return, always. A high-yield stock with slow price growth can beat a faster grower on the number an investor keeps, and a price chart alone hides that. Index providers publish total return versions of their indices for exactly this reason. The limits are worth one sentence: tax on dividends leaks some of the gap, and reinvesting assumes you can buy at a fair price each year.

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.