Equity Research puzzles, solved step by step

The expected price is Rs 512, 2.4% above today. Weight each outcome by its chance: 0.6 x 560 plus 0.4 x 440 gives 336 plus 176. Today's Rs 500 sits exactly halfway between Rs 440 and Rs 560, so the market is pricing roughly a 50% chance of good results. Your 60% view is what separates Rs 512 from Rs 500, and that probability is what you would have to defend.

Why is the expected price not the likely price?

A cricket fan who thinks her team wins 60% of the time does not expect the team to win exactly 0.6 of a match; she expects the win most often and a loss sometimes. Expected value is the probability-weighted average of every outcome, and it can be a price that never actually trades. Here the stock will be at Rs 560 or Rs 440 tomorrow, never at Rs 512, yet Rs 512 is the right number to compare with today's price.

What does today's price tell you about the market's view?

Run the calculation backwards. If Rs 500 is the market's expected price, the chance q of the good outcome solves 560q + 440(1 - q) = 500, so q = 60 / 120 = 50%. The interesting number is not Rs 512 but the gap between your 60% and the market's 50%: that gap is the whole of your view. An analyst would next ask what evidence justifies seeing more upside than the market does.

Say the limitations. Two outcomes are a simplification of a whole spread of possible moves, and a 2.4% expected gain on one event is small next to the Rs 60 swing either way. The expected value is a way to state a view precisely, not a reason on its own to act on it.

EPS rose to Rs 16.1 and the P/E rose from 20x to 27.9x, so earnings explain about 59% of the gain and re-rating the rest. The price multiplied by 2.25. EPS multiplied by 1.1⁵ = 1.611, and the multiple by 1.397; the two multiply to 2.25. A year, that is 17.6% total: 10% from earnings and 6.9% from the multiple.

Why split a return into earnings and multiple at all?

A house bought for Rs 50 lakh sells for Rs 1 crore. Part of the gain came from the rent rising, part from buyers being willing to pay more years of rent for the same flat. A share price is EPS times P/E, so any change in price is a change in earnings multiplied by a change in the multiple. The split matters because earnings growth can continue while the business grows, but a multiple cannot rise for ever.

Does the rupee split depend on which step you take first?

Yes, and that is worth saying before the interviewer does. Because the effects multiply, the cross term goes to whichever leg is taken second, so a rupee split is a convention, while the log split of 59% and 41% is not. The analyst's conclusion: 41% of this return came from investors paying more for each rupee of profit. If the multiple drifts back to 20x, a holder earns only the EPS growth, and the next five years look very different from the last.

About 53.64 per share. Equity value is 5,000 plus 500 of net cash, 5,500. If the price is P, the options add 10 shares and their 400 of exercise cash buys back 400 / P shares, so P = 5,500 / (110 - 400/P). Solving gives P = 5,900 / 110 = 53.636. Iterating from the naive 55 gets there in three rounds.

Why can you not just divide by a share count?

Think of splitting a restaurant bill where one late guest pays a fixed Rs 40 whatever the bill, and the rest is shared. How much the others pay depends on the bill, and the bill depends on who is sharing it. Under the treasury stock methodA way to count option dilution: assume in-the-money options are exercised and the exercise cash is used to buy back shares at the current price., how many net new shares the options create depends on the share price, and the share price depends on how many shares there are. The dilution and the answer have to be found together.

How do you solve it in one line instead of looping?

Write P x (110 - 400/P) = 5,500. The P cancels in the second term, leaving 110P - 400 = 5,500, so P = 5,900 / 110 = 53.636. When the options are in the money, the consistent price is simply equity value plus exercise cash, divided by all shares including the options. The check: at 53.636, the 400 of cash buys back 7.458 shares, so 2.542 net new shares take the count to 102.542, and 5,500 over that is 53.636.

Say the limitation. The method ignores the time value of options and assumes exercise today. A model that values the options properly would subtract their value from equity instead, and the answer would come out slightly lower.

EPS and net worth fall; cash and net debt to EBITDA do not move. Assuming no tax deduction for the charge, reported profit falls by Rs 200 crore, taking EPS from Rs 5.00 to Rs 3.00 on 100 crore shares, and net worth falls by Rs 200 crore. No cash leaves, and EBITDA sits above the charge, so a covenant at 2.5x net debt to EBITDA is untouched. A covenant set on debt to equity would move.

Why does a Rs 200 crore loss leave cash untouched?

Imagine you paid Rs 20 lakh for a car three years ago and a valuer now says it is worth Rs 12 lakh. You are poorer on paper, but your bank balance did not change today; the money left when you bought it. A goodwill impairment admits that an acquisition was worth less than was paid, and the cash for that acquisition left the business when the deal closed. The goodwillThe part of an acquisition price above the fair value of the net assets bought, carried as an asset on the balance sheet. is written down, reported profit takes the charge and equity falls with it.

Which covenant does move, and why does the difference matter?

A covenant built on EBITDA is blind to the charge. A covenant built on net worth or on debt to equity is not: gross debt of Rs 1,300 crore against equity of Rs 2,000 crore is 0.65x, and after the write-down it is 0.72x. So the same accounting entry can be harmless under one loan and a breach under another. An analyst reads the covenant definitions before saying which.

Two more things to say. Most analysts strip the charge out of adjusted EPS, which stays at Rs 5.00, because it says nothing about next year's earnings. And whether the charge saves tax depends on local rules, which usually do not allow a deduction for goodwill impairment; confirm before assuming either way. The real signal is about management: the acquisition is now expected to earn less than was paid for it.

The price falls to about Rs 400, EPS halves, and the P/E is unchanged. A 1:1 bonus gives one free share for each share held, so the share count doubles while the business, its profit and its value do not change. With EPS of, say, Rs 40, EPS becomes Rs 20 and the price Rs 400, leaving the P/E at 20x and the market value at Rs 8,000 crore.

If shareholders get free shares, why are they not richer?

Cut a pizza into eight slices instead of four and nobody gets more pizza. A bonus issue changes how many slices the company is cut into, not the size of the company, so each slice is worth proportionally less. A holder of 10 shares at Rs 800 had Rs 8,000; after the bonus she has 20 shares at Rs 400, still Rs 8,000. Nothing was paid and nothing was received.

What happens in the accounts?

The company moves an amount equal to the face value of the new shares out of reserves and into share capital. Net worth is unchanged, because money only moves between two lines inside equity. No cash moves, profit is untouched, and the dividend per share usually halves unless the board chooses to keep it, which would then be a real increase in payout.

The analyst's housekeeping: restate past EPS, dividends per share and price charts for the bonus, otherwise the history shows a false halving. Say the limitation too. In practice the price can drift from the exact half, because a lower share price can widen the pool of buyers, but that is a market effect, not a change in value.

A levered beta of 1.10. Relever with levered beta = unlevered beta x (1 + (1 - tax rate) x debt/equity). That is 0.8 x (1 + 0.75 x 0.5) = 0.8 x 1.375 = 1.10. The 0.8 is the risk of the business itself; the extra 0.3 is the financial risk shareholders take on because lenders are paid first.

Why does debt raise the equity's beta?

Think of two families with the same salary. One has a home loan EMI to pay first each month; the other has none. A 10% pay cut hurts the family with the EMI far more, because the EMI does not shrink. Lenders are paid a fixed amount first, so the same swing in the business moves the shareholders' leftover by more, and beta measures exactly that swing. The business risk is the 0.8; debt stacks financial risk on top.

Where do candidates go wrong with the inputs?

Two places. Debt to equity uses market values where you can, and it is debt over equity, not debt over total capital: 0.5 debt to equity is one third debt in the capital structure. Using 0.33 by mistake gives 0.8 x (1 + 0.75 x 0.33) = 1.00. And the tax rate is the one that applies to this company's interest deduction; confirm the current rate rather than assuming it.

Say the limitation. The formula assumes the debt itself carries no market risk and that the debt level stays constant. For a heavily indebted company, debt starts to behave like equity and the simple formula overstates the levered beta.

About 20.7%. Set the market at 100. The company's sales go from 20 to 22.4 and the market from 100 to 108, so its share is 22.4 over 108, or 20.74%. The gain is only 0.74 of a point, because share moves with relative growth: 1.12 divided by 1.08, about 3.7% more share, on a 20% base.

Why does growing 4 points faster add less than 1 point of share?

A student who scored 20 out of 100 last term and improves her marks by 12% while the class average improves by 8% has done a little better relative to the class, not a lot. Market share is a ratio, so it changes by the ratio of the two growth rates, 1.12 over 1.08, applied to the share you started with. A 3.7% relative gain on a 20% share is about 0.74 of a point.

How do you use this the other way round in a model?

Analysts often forecast a company as market growth plus a share assumption. If you forecast 12% growth in an 8% market, you are assuming a share gain of about 0.74 points a year, and five years of that takes 20% to about 24.0%. Saying that out loud tests whether the forecast is believable: which competitor is giving up that share, and why? The limitation: this assumes the market figure and the company's sales are measured the same way, which is often not true when companies report by segment.

Both, if you say it properly: the margin rose 3 percentage points, which is a 25% relative rise. The difference between two percentages is measured in percentage points, here 15 minus 12, or 300 basis points. The relative change is 3 divided by 12, which is 25%. On flat sales of Rs 1,000 crore, that is EBITDA rising from Rs 120 crore to Rs 150 crore, 25% more.

Why are both numbers right and one of them misleading?

If a bank's loan rate moves from 8% to 9%, nobody says the rate rose 1%; they say it rose one percentage point, even though the interest bill rose 12.5%. A percentage point is the difference between two percentages; a per cent change is that difference measured against the starting value. Saying margin improved 3% leaves the reader guessing whether the margin went to 15% or to 12.36%, which is 3% more than 12%.

When does the 25% matter more than the 3 points?

When you are forecasting profit. At flat sales, a margin that rises by a quarter of itself lifts EBITDA by a quarter, so the 25% is what flows into earnings and valuation. On Rs 1,000 crore of sales, EBITDA moves from Rs 120 crore to Rs 150 crore. The same 3 points on a 30% margin would be only a 10% rise in EBITDA, which is why the same points of margin matter much more for a thin-margin business.

In the room, give the answer in one line: 3 percentage points, 300 basis points, a 25% relative improvement. Then add the one caution that shows judgement: check whether sales were flat, because a margin can rise while EBITDA falls if revenue shrinks.

0.648. At least two means exactly two or all three. Exactly two can happen three ways, each with chance 0.6 x 0.6 x 0.4 = 0.144, so 0.432 together. All three beat with 0.6 cubed, 0.216. Adding them gives 0.648. The complement checks it: none beat 0.064 plus exactly one 0.288 is 0.352, and 1 minus 0.352 is 0.648.

How do you avoid missing an outcome?

Think of three friends each deciding whether to come to dinner. At least two coming covers four different guest lists: each of the three possible pairs, and all three. List the outcomes by how many succeed, count the ways each count can happen, and multiply by the chance of one such way. With three stocks there are only eight results, so write them down; with more, the count of ways is a binomial coefficientThe number of ways to choose k items out of n, written n choose k; here 3 choose 2 is 3..

Is independence a fair assumption for three stocks you cover?

Usually not, and saying so is the part that sounds like an analyst. Stocks in one sector share demand, input costs and the same consensus-setting habits, so beats tend to cluster: when one beats, the others are more likely to. Clustering fattens both ends. The chance of all three beating and of none beating both rise above 0.216 and 0.064, and whether at least two beat can move either way, so the independent answer is a starting point, not a forecast.

About Rs 69 crore of revenue and Rs 2.2 crore of gross profit a year, on stated assumptions. Assume 12,000 vehicles a day pass on the pump's side, about 500 an hour. About 384 stop, filling 20,232 litres a day, most of it diesel for trucks. At assumed pump prices near Rs 90 to Rs 100 a litre that is Rs 18.8 lakh a day; at an assumed dealer margin of Rs 3 a litre the pump keeps about Rs 2.2 crore a year.

Where does the estimate start?

Think of a roadside dhaba: its takings depend less on how many vehicles pass than on which ones stop and how much each group eats. Start from the traffic on the pump's side of the road, split it by vehicle type, and apply a stop rate and litres per fill to each, because the three types differ by a factor of forty in what they buy. Every input below is an assumption for the method; fuel prices and dealer margins change and should be checked before use.

How do you turn litres into revenue and profit, and check the answer?

Revenue is litres times the pump price: petrol for two-wheelers and cars at an assumed Rs 100, diesel for trucks at Rs 90, which gives Rs 18.8 lakh a day and Rs 68.6 crore a year. The dealer does not keep the pump price; it earns a commission per litre, so gross profit is litres times margin, about Rs 2.2 crore a year at Rs 3 a litre. Sanity check the stops: 384 a day is about 16 an hour, one fill every four minutes, which a pump with a few nozzles handles easily. The weakest assumption is the truck stop rate, since fleet operators choose pumps by contract and credit terms, not by chance.