Equity Research puzzles, solved step by step

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.

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.

Stake 20% of capital each time; at 40% your money slowly shrinks. The Kelly fraction for an even-money bet is the win chance minus the loss chance, 0.6 minus 0.4. At 20% the typical path grows about 2.0% a bet. At 40% each bet still has a positive expected gain, but growth is about -0.24% a bet, below zero, because big losses compound harder than big wins.

Why is the bet with the highest expected value not the one that grows fastest?

Imagine a shopkeeper who puts half the till into stock every morning. A good day lifts the till by half; a bad day cuts it in half. One of each leaves 1.5 x 0.5 = 0.75 of where she started, even though the good and bad days were the same size. Wealth compounds, so what matters over many bets is the average of the logarithm of each outcome, not the average outcome. Expected value per bet rises in a straight line with the stake; growth rises, peaks and then falls.

What do the numbers look like at each stake?

Read the two columns against each other. Every row has a positive expected gain, yet the 40% and 50% rows lose money on the path you will actually live through. Half Kelly at 10% keeps about 75% of the maximum growth with far smaller swings, which is why many desks size below full Kelly. Say the limitation too: the formula assumes you know the 60% exactly. Real edges are estimates, and overestimating one pushes you towards the overbetting side of the curve.

About Rs 19,000 crore a year on these assumptions. Take an assumed fleet of 25 crore two-wheelers, 1.8 crore in each of the last five years and 1.6 crore in each older year. All young vehicles are insured; after year five the insured share falls from 60% to 20%. That leaves 14.6 crore insured vehicles paying Rs 800 to Rs 1,500 a year, a pool near Rs 18,700 crore.

Where do you start, the vehicles or the policies?

Think of a gym. Counting everyone who ever signed up tells you little; the revenue comes from the members who still renew. Start from the fleet, but the number that decides the pool is how many vehicles are still insured, not how many are on the road. In India third-party cover is compulsory by law and new two-wheelers are sold with a multi-year third-party policy, so the young fleet is close to fully insured. Confirm the current rules before quoting them. Once that upfront cover runs out, many owners of older, cheaper bikes let it lapse.

How do you show the interviewer which assumption matters?

Run one sensitivity out loud. If ten more vehicles in every hundred older than five years renewed their cover, the pool would rise by about Rs 1,440 crore, roughly 8% of the total. That is the lever an insurer or a regulator can pull: enforcement of the third-party requirement at the roadside. The premium per vehicle matters less because it is set in narrow bands for third-party cover. The limitation to say: premiums, fleet and lapse rates here are assumptions for the method, and an analyst would replace each with a sourced number before writing it into a note.

B trades lower, at 13.3x against A's 16.7x, and they match when B's debt costs 8%. A earns 80 x 0.75 = 60 on equity of 1,000. B pays 30 of interest, earns 37.5 on equity of 500. The P/Es match when the after-tax cost of debt equals A's earnings yield of 6%: r x 0.75 = 6%, so r = 8%. Cheaper debt lowers B's P/E; dearer debt raises it.

Why would the same business carry two different P/Es?

Suppose a flat earns rent of Rs 6 for every Rs 100 of its price. Buy it with half cash and half a loan at 4.5% after tax, and your cash earns more than 6%, because the borrowed half costs less than it yields. P/E divides the equity's value by the equity's earnings, and debt changes both, by different amounts. Swapping equity for debt that costs less than the equity's earnings yield lowers the P/E; swapping for dearer debt raises it.

Why is 8% the crossing point?

Swapping 500 of equity for 500 of debt removes equity that was earning its share of A's 6% earnings yield, 30 of net income, and adds an after-tax interest cost of 500 x r x 0.75. If that cost is exactly 30, net income halves along with equity and the P/E does not move; 30 over 375 is 8% before tax. The general rule: leverage cuts P/E when the after-tax cost of debt is below the earnings yield, E/P. The limitation worth saying: B's lower P/E is not a sign it is cheaper. Its equity is riskier, so investors should demand a lower multiple for the same business.

Cash tax falls by Rs 3.75 crore, reported tax does not change, and the deferred tax liability rises by Rs 3.75 crore. Tax is paid on taxable profit, which uses the Rs 25 crore tax depreciation. The P&L charges tax on book profit, which uses Rs 10 crore. The Rs 15 crore gap at 25% is Rs 3.75 crore of tax postponed, not saved: it comes back when tax depreciation later falls below book.

Why can two depreciation numbers exist for the same machine?

Think of a salaried employee who is allowed to claim a deduction early in the year for an expense she will actually use over three years. Her tax bill falls now, but she cannot claim it again later. Tax rules often allow depreciation faster than the books show it, so the same asset produces a larger deduction early and a smaller one later. The total over the asset's life is the same; only the timing differs. Confirm the current depreciation rates in the tax rules before using real numbers.

Where does each number land in the three statements?

The income statement shows a tax expense of Rs 25.00 crore, split into current tax of Rs 21.25 crore and deferred tax of Rs 3.75 crore, so net income is Rs 75 crore either way. The cash flow statement adds back the Rs 3.75 crore of deferred tax as a non-cash charge, so operating cash flow is Rs 3.75 crore higher than it would be if both depreciation numbers matched. On the balance sheet the deferred tax liability rises by Rs 3.75 crore, matched by the extra cash.

The analyst's point: a growing company that keeps buying assets keeps adding new early-year gaps, so its deferred tax liability can grow for years and behave almost like permanent free funding. When capex slows, the reversals arrive and cash tax rises above reported tax. That is worth one sentence in the room.