Equity Research puzzles, solved step by step

The answer is 1/6. After a first roll of 1, 3 or 5 the game ends on 2 with chance 1/3; after a 2 it cannot; after a 4 or 6 it ends on 2 with chance 1/2. Overall the game ends on 2 with chance 1/3, so a first roll of 1 carries (1/6 x 1/3) / (1/3) = 1/6. The 4 and 6 absorb the share the 2 lost.

Why does 1/5 feel right, and where does it go wrong?

Think of a cricket team told that the match was won off the last ball. That news does not just rule out one scenario; it makes the close games far more likely than the easy ones. Conditioning works the same way here. Being told the game ended on 2 is evidence, and evidence reweights every starting point by how well it explains what you saw. The 1/5 answer treats the news as if it only deleted the 2 and left the other five faces equally likely.

So ask, for each first roll, how likely it is that 2 ends up the last even number. After a 1, 3 or 5, all three evens are still missing and each is equally likely to come last, so 1/3. After a 2, the 2 is already seen and cannot come last, so 0. After a 4, only 2 and 6 are missing, and 2 comes last exactly when 6 shows first, so 1/2. The same holds after a 6.

How do you turn those chances into the answer?

Apply Bayes ruleThe chance of a cause given what you saw equals its prior chance times how likely it made the observation, divided by the total chance of the observation.. Each face starts at 1/6. Multiply by its chance of ending on 2 and add them up: three faces at 1/18, one at 0, two at 1/12, which totals 1/3. The 1 contributes 1/18 of that 1/3, which is 1/6, exactly its starting weight. The 4 and the 6 rise from 1/6 to 1/4 each, and the 2 falls to zero.

The research version of this is reading a data point. A company that beats estimates is more likely to be one that guided low, not just one that is doing well. Before you update, ask which starting stories make the thing you observed more likely, and shift weight towards them.

Bid nothing: every positive bid loses money on average. If you bid b, the owner accepts only when the value is below b, so accepted deals average b/2. Worth 1.5 times that to you, they return 0.75b for a price of b, a loss of a quarter of the bid each time. At Rs 60 crore you would win 60% of the time and lose Rs 15 crore on every win.

Why does the average value of Rs 50 crore mislead?

Imagine buying a used car from a private seller who has driven it for five years. If the seller happily accepts your first offer, the most useful thing you have learnt is what the seller knows about the car. The same logic runs here. The owner says yes only when your bid is above what the company is worth to them, so acceptance itself is bad news about the value. Across all companies the average is Rs 50 crore, but you never get to buy the average company; you buy the ones worth less than your bid.

How do you show that no bid works?

Take a general bid b. The owner accepts with chance b/100, and given acceptance the value is spread evenly from 0 to b, averaging b/2. To you that is worth 1.5 x b/2 = 0.75b. You pay b and receive 0.75b, so each accepted deal loses 0.25b, whatever b is. Multiply the loss by the chance of acceptance and the expected result is minus b squared over 400, which is zero only at b = 0. At a bid of Rs 60 crore that is minus Rs 9 crore.

This is the winner's curseThe tendency of the winning bid in an auction or negotiation to be the one that most overestimated the value, because the other side or other bidders knew better., and it is why a buy-side analyst asks who is on the other side of a trade. When the seller knows more than you do, the times you get filled are skewed towards the times you were wrong. The answer changes only if your edge is large enough: with a multiple of 2 instead of 1.5, accepted deals exactly break even.

About 325 to 420 million tonnes, on these illustrative inputs. The per capita route, 1,400 million people at 0.30 tonnes each, gives 420. Adding up housing, repairs, infrastructure and commercial building gives 325. The 95 million tonne gap points at the two softest inputs: the per capita anchor and the number of homes built each year. Check either against published industry data.

How does the per capita route work?

It is the way a household guesses its monthly rice: people times how much each eats. Take a population of about 1,400 million and an assumed consumption of 0.30 tonnes, 300 kg, a head. That gives 420 million tonnes. The route is quick but hangs on a single number you cannot see, the per capita figure, so it is only as good as your anchor. Say that you would check the anchor against published data rather than quoting one from memory.

How do you build the end use route, and why does it disagree?

Now count what gets built. Assume 10 million new homes a year at 600 sq ft and a builder's thumb rule of about 20 kg of cement per sq ft: 120 million tonnes. Repairs and extensions: 250 million existing homes, 6% doing a job a year, about 2 tonnes each: 30. Infrastructure: assume Rs 15 lakh crore of spending a year, cement at 5% of project cost and Rs 6,000 a tonne: 125. Commercial and industrial: 2,000 million sq ft at 25 kg: 50. Total 325.

When two routes disagree, the gap tells you which assumption to test, not which answer to average. Close the 95 million tonne gap from each side in turn. A per capita figure of 0.23 tonnes instead of 0.30 would close it alone. So would roughly 18 million new homes instead of 10, which is a big move, so the home count is less likely to be the whole story. Self-built rural homes are the category most often missed, which is where you would dig.

The index ends at 0.951, down 4.9%, and the 2x product ends at 0.815, down 18.5%. Each up-down pair multiplies the index by 1.1 x 0.9 = 0.99 and the product by 1.2 x 0.8 = 0.96. Five pairs give 0.99 to the fifth and 0.96 to the fifth. The product loses nearly four times as much, not twice.

Why does a flat-looking path lose money at all?

A shop marks a shirt up 10% and then runs a 10% sale: the tag ends at 99% of where it began, because the discount is taken on the higher price. A gain and an equal percentage loss never cancel; the pair always leaves you with one minus the square of the move. For 10% that is 1 minus 0.01, so each pair costs 1%. Five pairs cost a little under 5%.

Why is the 2x product four times worse and not twice?

Doubling the daily move doubles the swing, and the pair loss is the square of the swing. Twice the move means four times the loss per pair: 0.2 squared is 0.04 against 0.1 squared at 0.01. Compounded over five pairs the product lands at 0.815, a 18.5% loss, about 3.8 times the index's 4.9%. Someone who expected twice the index would have looked for 0.902.

The general name is volatility dragThe gap between the average of a set of returns and the compound return they produce, roughly half the variance of the returns.. It is why a daily leveraged product can fall over a month in which its index ended flat, and why it is built for short holding periods. The limitation is honest: in a steady trend with little back and forth, daily compounding can leave the product ahead of twice the index.

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.

Today: cash up Rs 50 crore, deferred revenue up Rs 50 crore, and no change to the income statement. Operating cash flow rises by 50 through the change in deferred revenue. On delivery, revenue of 50 and tax of 12.5 give net income of 37.5; deferred revenue falls by 50, so operating cash flow that year is minus 12.5, the tax paid, and equity rises 37.5.

Why is there no revenue on the day the cash arrives?

Think of a gym that sells a year's membership in January. The money is in the bank on day one, but the gym has not yet provided a single workout; if it shut down in February, it would owe most of that money back. Revenue is recorded when the service is delivered, so cash received in advance is a debt of service owed to the customer, and it sits on the balance sheet as deferred revenue.

On payment day the balance sheet grows on both sides: cash up 50 and deferred revenueCash received for goods or services not yet delivered. It is a liability until the company delivers, when it moves to revenue. up 50. The cash flow statement starts from net income of zero and adds the 50 rise in the liability, so operating cash flow is plus 50. The income statement is untouched.

What happens on the day the service is delivered?

Now the company has earned it. Revenue rises 50, tax at 25% is 12.5, net income rises 37.5. The cash flow statement starts at 37.5 and subtracts the 50 fall in deferred revenue, which leaves operating cash flow of minus 12.5: the profit was paid for in cash a year earlier, so the only cash that moves on delivery is the tax. The balance sheet balances: cash down 12.5 on one side, deferred revenue down 50 and retained earnings up 37.5 on the other.

For a research analyst this is the pattern behind subscription and advance-booking businesses: cash flow runs ahead of profit while bookings grow, and falls behind when they shrink. A growing deferred revenue balance flatters operating cash flow, so read the two together. The tax assumption matters too: some tax systems tax advances when received, so confirm the rule that applies before modelling it.

EBIT grows 50% and net income 100%. Revenue of 110 leaves 55 after variable costs; less fixed costs of 40, EBIT is 15 against 10. Less interest of 5, pre-tax profit is 10 against 5, and after 25% tax net income is 7.5 against 3.75. Fixed costs magnify the change five times and interest doubles it again.

Why does a 10% revenue change become a 50% EBIT change?

Think of a tea stall with a fixed monthly rent. Once the rent is covered, every extra cup sold is almost pure profit, so a busy month feels far better than the extra sales alone suggest. Fixed costs do not grow with revenue, so the whole extra contribution lands in EBIT, and a thin EBIT makes that addition a large percentage. Here revenue up 10 adds 5 of contribution, and 5 on an EBIT of 10 is 50%.

Why does net income grow twice as fast as EBIT?

Interest is a second fixed charge, sitting below EBIT. With interest of 5 taking half of an EBIT of 10, the next 5 of EBIT doubles pre-tax profit, and tax at a flat rate keeps that doubling intact. The shortcut: operating leverageHow much operating profit moves for a given change in revenue, driven by the share of costs that are fixed. is contribution over EBIT, 50 over 10, which is 5; financial leverage is EBIT over pre-tax profit, 10 over 5, which is 2. Together 5 x 2 = 10, and 10 x 10% = 100%.

The same arithmetic runs in reverse, which is the point a research analyst should add. A 10% revenue fall would halve EBIT and wipe out net income entirely. Leverage magnifies both directions, so a highly geared, high fixed cost company is the one whose earnings estimates move most on a small change in the top line. The limit of the shortcut: it holds only while costs behave as fixed, and over a few years most costs flex.

The cost of equity rises from 12% to 16% and WACC stays at 12%. With debt equal to equity, shareholders carry the same business risk on half the capital, so their required return rises by the spread between the unlevered rate and the debt rate, 4 points, times debt over equity. Half at 16% and half at 8% is still 12%. Without taxes, cheap debt only moves risk around.

Why can cheap debt not lower the cost of capital on its own?

Think of two friends buying a food truck. If one lends at a fixed rate and the other takes whatever is left after paying the loan, the truck's takings are no less risky; the owner just now carries all of the ups and downs on a smaller stake. Borrowing does not change the business, so it cannot change the total return the business must earn for all its funders; it only shifts risk from lenders to shareholders.

How do you get the 16%?

The Modigliani and MillerThe 1958 result, from Franco Modigliani and Merton Miller, that in a world without taxes or distress costs the value of a firm does not depend on how it is financed. relation gives it directly. The cost of equity is the unlevered rate plus the gap between the unlevered rate and the debt rate, scaled by debt over equity. Here that is 12% plus (12% minus 8%) times 1, which is 16%. Check with WACC: half at 16% plus half at 8% is 12%, exactly the unlevered rate.

Now add back what the puzzle removed. With tax, interest is deductible, so debt does lower WACC a little; at high debt, the cost of debt itself rises and distress costs appear. That is why an analyst who sees WACC fall sharply as a model adds debt should check whether the cost of equity was left unchanged. In practice this shows up as re-levering beta: the equity beta must rise when leverage rises.

Total revenue grows 12%, from Rs 1,000 crore to Rs 1,120 crore. Existing stores add 4%, Rs 40 crore. The 20 new stores sell at Rs 8 crore a year but trade for only half the year, adding Rs 80 crore, 8 points. Split the two when you present it, because they tell different stories about the business.

Why split growth into old stores and new stores?

Think of a restaurant owner who opens a second branch. Total takings rise, but that says nothing about whether the first branch is doing better. Total growth mixes two different things: how the existing stores are trading, which is like-for-like growthSales growth from stores open for the whole of both periods, so new openings and closures do not distort it. Also called same store sales growth., and how many stores were added. A retailer can grow 12% on openings while every existing store shrinks.

How do you work the new stores' contribution?

Three adjustments, one at a time. Twenty stores at a mature rate of Rs 10 crore would be Rs 200 crore. They run at 80%, so Rs 160 crore a year. They trade for half the year, so Rs 80 crore this year. Part-year openings and ramp-up each shrink a new store's first-year contribution, which is why store count growth overstates revenue growth in the year of opening.

The follow-up an analyst should volunteer: next year, even with zero like-for-like growth and no openings, the 20 stores trade a full year and add another Rs 80 crore. That annualisation is growth already in the bag, and it is worth saying separately from the growth the business still has to earn.

20% for four years, narrowly: 2.07x against 2.01x. The rule of 72 says 20% doubles in about 3.6 years and 15% in about 4.8, so both paths just pass double, the 20% path by more. Check exactly: 1.2 squared is 1.44 and 1.44 squared is 2.0736; 1.15 squared is 1.3225, squared again is about 1.749, times 1.15 is 2.0114.

How do you get close without multiplying anything?

Think of two savers racing to double their money. The rule of 72A shortcut: dividing 72 by a growth rate in per cent gives roughly the number of years it takes to double. tells you when each one doubles, and whoever has more time left after doubling is ahead. At 20%, 72 over 20 is 3.6 years, leaving 0.4 years of 20% growth. At 15%, 72 over 15 is 4.8 years, leaving 0.2 years of 15%. The 20% saver has twice as long left at a higher rate, so 20% for four years should win.

Why check the edge exactly, and how?

The rule of 72 is rough at high rates: 20% actually doubles in 3.8 years, not 3.6, and 15% in 4.96, not 4.8. When a shortcut says the answer is close, the gap it shows can be smaller than the shortcut's own error, so you check the edge with exact arithmetic. Here it is easy: square 1.2 twice to get 2.0736, and for 1.15 square twice and multiply once more to get 2.0114. The margin is about 3%, real but narrow.

Why a research interviewer asks it: growth comparisons of this kind turn up every day, a company growing earnings faster for fewer years against one growing slower for longer. Saying the approximation first and then checking it shows the habit they want, and the limitation is worth one line: the answer flips if the 15% path runs for six years.