Equity Research puzzles, solved step by step

About 6.9% a month, and a 95% one-month value at risk of about Rs 1.13 crore. Variance adds across independent days, so volatility scales with the square root of time: 1.5% x the square root of 21, which is 4.58, gives 6.87%. The one-tailed 95% point of a normal distribution is 1.645 standard deviations, so a bad month costs 1.645 x 6.87% = 11.3% of Rs 10 crore, about Rs 1.13 crore.

Why does volatility grow with the square root of time?

Think of someone taking random steps left or right along a lane. After 100 steps they are not 100 steps from where they began; many steps cancelled, and the typical distance is about 10 steps, the square root of 100. Daily returns behave the same way when one day does not predict the next. Variances add across days, so the standard deviation, which is the square root of the variance, grows with the square root of the number of days.

How do you turn a volatility into a rupee value at risk?

Value at riskA loss threshold that a position should exceed only with a stated small probability over a stated period, such as 5% over one month. answers one question: how much would a bad month cost, where bad means the worst month in twenty. Under a normal distribution 5% of outcomes fall below the mean minus 1.645 standard deviations, so the 95% one-month VaR is 1.645 x 6.87% = 11.3% of the position. On Rs 10 crore that is about Rs 1.13 crore. Assume a zero expected return for the month; over one month the drift is small against the volatility.

What does the number leave out?

Two things, and saying them separates an analyst from a calculator. The square root rule needs independent days; when bad days cluster, as they do in a sell-off, real monthly risk is higher than 6.9%. And VaR marks the edge of the tail, not what sits inside it: it says nothing about how bad the worst 5% of months can get. Real returns have fatter tails than the normal curve, so a one-in-twenty month can cost well over Rs 1.13 crore.

Pull again straight away: you survive with probability 3/4, against 2/3 after a fresh spin. The four empty chambers sit together in a row and the cylinder advances one chamber per pull. The click you heard came from one of those four, each equally likely. Only one of them, the last empty chamber before the bullets, is followed by a bullet, so the chance of dying without a spin is 1/4. A fresh spin puts you back at two bullets in six.

Why does the first click carry information?

Picture a street with four dark houses in a row, then two lit ones, laid out in a circle. You are told you are standing at a dark house and that you will walk to the next door. Three of the four dark houses have another dark house next door. The click tells you that you are somewhere in the run of four empty chambers, and because the bullets sit together, most of that run is followed by another empty chamber. A fresh spin throws that information away.

Would the answer change if the bullets were not next to each other?

Yes, and saying so shows you understand the mechanism rather than the trick. Put the bullets in chambers 1 and 4. The empty chambers are 2, 3, 5 and 6, and two of them, 3 and 6, sit in front of a bullet. With the bullets apart, pulling straight away survives only 50% of the time, so a fresh spin at 66.7% becomes the better choice. The adjacency is what makes the empty chambers cluster, and the clustering is what makes the click informative in your favour.

Why would an investing firm ask this?

Because it is a small version of the question every analyst faces after new information: does what just happened change the odds of what comes next? An outcome you have already seen is data about the state of the world, and ignoring it in favour of the unconditional average is a common and costly habit. Say that link in one line after the numbers.

About Rs 450 crore. Revenue is 5,000 students x Rs 3 lakh, Rs 150 crore a year. A 25% EBITDA margin gives Rs 37.5 crore, and at 12 times that is an enterprise value of Rs 450 crore, before subtracting any debt. The case for buying is fees paid in advance, a campus and approvals that are slow to copy, and demand that renews with a new batch every year.

How do you structure an open question like this?

The question has two halves and the interviewer wants both. It is like selling a flat: first you say what makes it worth having, the light, the location, the building, and then you give a price per square foot times the area. Give the reasons to own it in three short points, then build the price as revenue times margin times multiple, naming every input so the interviewer can push on any one of them. Leading with the structure also buys you time to do the arithmetic.

What makes the case for buying convincing?

Pick the features an investor would pay for. The strongest is that customers pay a term or a year in advance and come back every year, so the business funds its own working capital and its revenue is visible well before the year starts. Add that a campus, a faculty and regulatory approvals take years to replicate, which protects the fee level, and that demand refreshes with every new batch. Then name the risk that decides the price: how full the seats are. One caution belongs here too: in many countries, including India, degree-granting colleges are commonly run through not-for-profit trusts or societies, so confirm the legal structure first, because it decides whether an investor can own the profit at all.

Which number would you defend hardest?

The margin and the multiple, because each moves the answer one for one. At a 20% margin the value falls to Rs 360 crore, and at 10 times EBITDA it falls to Rs 375 crore; every input in the chain carries equal weight. Justify 12 times with the visibility of fee income and 25% with a campus that is nearly full. If asked for a range rather than a point, Rs 360 to 450 crore on those two sensitivities is honest.

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.

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.

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.

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.

Stickers: 6n squared. Cubes on the surface: n cubed minus (n minus 2) cubed. Each of the six faces is an n by n grid of stickers, so there are 6n squared, which is 54 for n of 3. For the cubes, count the hidden core: peeling one layer off every side leaves (n minus 2) cubed, so 27 less 1 is 26 on show. Both formulas need n to be at least 2.

Why are stickers and surface cubes different counts?

A corner cube carries three stickers, an edge cube two and a face-centre cube one. The sticker count measures surface area, while the cube count measures how many pieces touch the surface, and the corners and edges make the first larger than the second. The surface area the question asks for is the sticker count: six faces, each n by n, so 6n squared unit squares.

What is the fast way to count the cubes on show?

Picture a box of laddoos packed three by three by three: to find how many touch the box, count the ones that do not. Only a core with one layer peeled from every side is hidden, so the hidden cubes form an (n minus 2) by (n minus 2) by (n minus 2) block, and everything else shows. For n of 3 that core is a single cube, so 26 show. The subtraction is quicker and harder to get wrong than adding corners, edges and faces.

How do you check the formula before you commit?

Test the smallest case you can picture. A 2 by 2 by 2 cube has no hidden centre, so all 8 pieces show, and the formula gives 8 minus 0, which is 8. Then look at the expanded form: 6n squared counts every face-cube once per face, minus 12n plus 8 removes the double and triple counts along the 12 edges and at the 8 corners. The formula breaks for n of 1, where it gives 2 for a single cube, so state that it holds from n of 2 upwards.