Equity Research puzzles, solved step by step

The price implies 10% growth a year, which needs a return on equity of about 16.7%. A forward P/E equals the payout ratio over cost of equity minus growth, so 20 equals 0.4 over 12% minus g, and g is 10%. Growth is funded by retained profit, 60% here, so ROE must be 10% divided by 60%, about 16.7%, well above the 12% cost of equity.

How do you pull growth out of a P/E?

Start from the dividend model and divide both sides by earnings. Price is next year's dividend over cost of equity minus growth, and dividend over earnings is the payout ratio. So a forward P/E is the payout ratio divided by the gap between cost of equity and growth: 20 equals 0.4 over 12% minus g. The gap must be 2%, which puts growth at 10%. Think of it like working out a car's speed from the distance it covered and the time it took: the multiple and the payout are the readings, growth is what they imply.

Why does the growth need a particular ROE?

Growth is not free; it is paid for with retained profit. A family that saves 60% of its income and wants its wealth to grow 10% a year must earn 10% divided by 60% on its savings, about 16.7%. A company is the same: sustainable growthThe growth a company can fund from its own retained profit: return on equity times the share of profit retained. is ROE times the share retained, so 10% growth on 60% retention needs an ROE of 16.7%. That is the number to test against the company's history, not the P/E itself.

What does the curve tell you that the formula does not?

Two things. First, when ROE equals the cost of equity, growth adds nothing: the P/E is 8.3x, one over 12%, whatever the growth. Growth only earns a higher multiple when each retained rupee earns more than shareholders demand. Second, the curve is steep near 20x, so a small change in the growth assumption moves the justified multiple a lot. The limit is the single perpetual rate: a real company grows fast for some years and then slows, and a two-stage model would put less weight on the far future.

About Rs 82 crore. Receivables were Rs 1,000 crore times 60 over 365, about Rs 164 crore. Now they are Rs 1,200 crore times 75 over 365, about Rs 247 crore. Of the Rs 82 crore increase, about Rs 33 crore is growth at the old terms and about Rs 49 crore is customers paying 15 days later. That cash comes straight out of operating cash flow.

How do receivable days turn into rupees?

A kirana store that lets customers settle at month end is carrying about a month of its own sales as money owed. Receivable daysReceivables divided by revenue, times 365: roughly how many days of sales customers owe at any time. work the same way: receivables are sales per day times the days customers take to pay. Rs 1,000 crore a year is about Rs 2.74 crore a day, and 60 days of that is Rs 164.4 crore. At Rs 1,200 crore and 75 days it is Rs 246.6 crore.

Why split the increase into two pieces?

Because the two pieces tell different stories. Growth at unchanged terms is the ordinary cost of selling more: Rs 32.9 crore. The extra 15 days is a change in behaviour, Rs 49.3 crore, and it is the piece an analyst asks management about. Customers might be under stress, the company might be offering longer credit to win sales, or revenue might be booked earlier than the cash warrants. Any of those makes reported growth lower quality.

Where does it show up in the statements?

The income statement books the full Rs 1,200 crore of revenue. The cash flow statement subtracts the Rs 82.2 crore rise in receivables from operating cash flow, because that revenue has not been collected. So profit grows while cash lags, which is the pattern behind the question of why a company's revenue is growing but its cash is not. The limit of the day count is seasonality: a year-end balance can mislead if sales are lumpy, so compare the same quarter across years.

Rs 80 crore. Consolidation adds 100% of a controlled subsidiary's profit, even though the parent owns only 60%. The outside shareholders' 40% of Rs 50 crore, Rs 20 crore, is then deducted as the non-controlling or minority interest. Rs 100 crore less Rs 20 crore leaves Rs 80 crore, which is the figure to use for EPS and P/E.

Why does consolidation count profit the parent does not own?

Think of two brothers who run a shop together, one owning 60% and the other 40%. The elder brother runs it, signs for it and reports its full takings in his family budget, then sets aside his brother's share. Accounting does the same: a parent that controls a subsidiary shows all of its revenue, costs and profit, then carves out the part owned by others as the non-controlling interestThe share of a subsidiary owned by shareholders other than the parent. Also called minority interest.. Control decides consolidation; ownership decides who keeps the profit.

How do you check the answer another way?

Build it from the parent's side. The parent's own business earns Rs 100 crore less the subsidiary's Rs 50 crore, which is Rs 50 crore. Add its 60% share of the subsidiary's Rs 50 crore, Rs 30 crore, and you reach the same Rs 80 crore. Two routes that agree are what the interviewer wants to hear, and the second route also shows you which business drives the parent's profit.

What goes wrong if you use the Rs 100 crore?

Every per-share number is overstated. If the company is worth Rs 1,200 crore, the P/E on Rs 100 crore is 12x, but on the Rs 80 crore that shareholders actually own it is 15x. Using consolidated profit makes the stock look 20% cheaper than it is. The same logic runs through enterprise value: EBITDA includes all of the subsidiary, so EV must add the value of the minority stake to stay consistent. Indian groups with listed subsidiaries make this a frequent test.

About 11.8%. Weight each source of money by its share and use debt after tax. Equity contributes 70% of 14%, 9.8 points. Debt costs 9% before tax but only 9% times 75%, 6.75%, after it, so it contributes 30% of 6.75%, about 2.0 points. The total is 11.825%. Using debt before tax would overstate WACC at 12.5%.

Why is WACC a weighted average rather than a simple one?

A household that pays for a flat with 70% of its own savings and a 30% home loan has a blended cost of money closer to what its savings could have earned than to the loan rate, because most of the money is savings. WACC weights each source by its share of the funding, measured at market value, because that is the mix the company's investments must pay for. A simple average of 14% and 9% would give debt far more say than 30% of the money deserves.

Why does the tax rate touch only the debt?

Interest is deducted before tax is worked out; dividends are not. So every Rs 100 of interest cuts the tax bill by Rs 25, and debt at 9% really costs the company 9% times 75%, which is 6.75%. The tax shieldThe reduction in tax a company gets because interest is deductible: interest times the tax rate. is why WACC uses the after-tax cost of debt, and why the equity cost stays as it is. Confirm the applicable tax rate for the company before using a headline figure.

What would make you distrust this number?

The weights and the inputs both move. Weights should be market values, not book values, and they should reflect the mix the company will hold over the forecast, not a single year-end snapshot. Adding debt does not lower WACC forever: as borrowing rises, lenders charge more and shareholders demand more for the extra risk. And the tax shield only exists if the company has profits to shield. A loss-making company's debt costs the full 9%, which would lift WACC to 12.5%.

EBITDA rises from Rs 140 crore to Rs 260 crore, up about 86%. At 70%, 7 million tonnes earn Rs 840 crore of contribution, less Rs 700 crore of fixed cost. At 80%, 8 million tonnes earn Rs 960 crore, less the same Rs 700 crore. Volume rises 14.3%, but because the plant runs just above its 58% breakeven, EBITDA nearly doubles.

Why does profit move so much more than volume?

An autorickshaw driver pays a fixed daily rent for the vehicle. On a slow day the fares barely cover the rent; one extra long ride can double what he takes home, because the rent was already paid. When most costs are fixed, every extra tonne adds its full contribution to profit, so profit grows far faster than volume. This is operating leverageHow strongly profit responds to a change in volume, because fixed costs do not move with sales.. A cement plant is almost all fixed cost: the kiln, the staff and the depreciation are there whether it runs at 60% or 90%.

How do you get the percentage quickly?

Use the degree of operating leverage: contribution over EBITDA. At 70% that is Rs 840 crore over Rs 140 crore, 6 times. So a 14.3% rise in volume becomes roughly 6 times 14.3%, about 86%. The multiplier falls as utilisation rises, because EBITDA grows while fixed cost stays the same, so the next 10 points would add less in percentage terms. The breakeven, Rs 700 crore over Rs 120 crore per million tonnes, is 58.3%, and the closer a plant runs to it, the bigger the multiplier.

What does an analyst take from this?

Utilisation is the swing variable for cement earnings. EBITDA per tonne moves from about Rs 200 to Rs 325 as utilisation rises from 70% to 80%, even though price and cost per tonne have not changed. That is why cement analysts track regional demand and capacity additions so closely. The limit runs both ways: the same leverage cuts EBITDA by the same Rs 120 crore if utilisation falls to 60%, and in practice contribution per tonne also moves with prices and fuel costs, which can swamp the volume effect.

The average of the three P/Es is 43.3x, but the sector trades at 18.75x. The P/Es are 10x, 100x and 20x, and their simple average is 43.3x. The sector is worth 300 in total and earns 16 in total, so its P/E is 300 over 16, 18.75x. The average is pulled up by the company earning only 1, which says more about that company's depressed profit than about the sector's valuation.

Why does one company dominate the average?

Think of three friends who each spend Rs 100 on lunch, one buying ten samosas, one buying one fancy sandwich and one buying five. Average the price per item and the sandwich makes lunch look absurdly expensive; divide total spend by total items and you get the real average price. A ratio with a small denominator explodes, and a simple average of ratios gives that explosion full weight, however little of the sector's earnings it represents. Company B earns 1 of the sector's 16 but contributes 100 of the 130 P/E points summed.

What is the right way to get a sector multiple?

Sum first, then divide. The sector P/E is total market value over total earnings, 300 over 16, 18.75x, which is what you would pay for a slice of the whole sector's profit. An equivalent route is to average the earnings yields, 10%, 1% and 5%, which gives 5.33%, and turn that upside down: 18.75x. Earnings yields do not explode as profit falls, so averaging them is safe when the companies are the same size.

When is the median or the average still useful?

When you want a typical company rather than the sector as a whole. The median, 20x, ignores the outlier and is a fair description of a normal company in the group. But when you compare a stock with its sector, or value a basket, you need the aggregate multiple, because that is the price of the sector's earnings. In practice, the fix for a comps table is to drop or flag companies with depressed or negative earnings, whose P/Es mean little, rather than let them skew the answer.

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.

About 1.5 lakh cups a day on these assumptions. Take 5 lakh passengers. The 2 lakh long-distance travellers wait long enough that one in two buys a cup: 1 lakh cups. The 3 lakh commuters rush through, one in ten buys: 30,000. Staff, porters and drivers, say 5,000 people at three cups, add 15,000. That gives 1,45,000, and 50 stalls selling about 3,000 cups each agrees.

Why start from people and not from stalls?

If you wanted to know how many samosas a school canteen sells, you would count students and ask how many buy one at break, not count the frying pans. Demand comes from the people passing through, so the estimate starts with footfall and a buying rate; the stall count is useful only as a check on capacity. Starting from stalls forces you to guess sales per stall, which is the very number you are trying to find.

State every number as an assumption. Assume a busy junction handles about 5 lakh passengers a day. Split them by how long they stay: long-distance travellers wait on the platform for trains that may be late, while suburban commuters walk straight through. The split matters because waiting time drives tea buying far more than the station's size does. Give long-distance travellers a rate of one cup for every two people and commuters one in ten.

How do you check it from the supply side?

Count the sellers. Suppose a big station has about 50 stalls and trolleys. A busy stall can pour a cup every 20 seconds for much of a 17-hour day, about 3,000 cups. 50 sellers at 3,000 cups is 1,50,000, close to the 1,45,000 from the demand side, so the two routes agree. If they had disagreed by a factor of three, you would know one buying rate or the footfall figure was off and could say which one you distrust.

What would you refine with more time?

The long-distance buying rate carries most of the answer, so refine it first. Waiting time varies with delays, time of day and weather: a winter morning sells far more tea than a summer afternoon. A sharper estimate would split the day into blocks and apply a rate to each, which is what the hourly chart does. The limitation is that footfall itself is an assumption here; railway data on passengers per day for the specific station would replace it.

15x. Dividend yield is dividend over price, and payout is dividend over earnings. Dividing the first by the second cancels the dividend and leaves earnings over price, the earnings yield: 4% over 60% is 6.67%. The P/E is the inverse, 1 over 0.0667, which is 15x. On a Rs 100 share, the dividend is Rs 4 and earnings are Rs 6.67.

How do the two ratios combine?

Suppose a friend tells you her rent is 30% of her salary and her rent is Rs 15,000. You know her salary without asking: Rs 15,000 over 30%. The dividend works the same way: if the dividend is 4% of the price and 60% of earnings, earnings must be 4% over 60% of the price, an earnings yieldEarnings per share divided by the share price, the inverse of the P/E. of 6.67%. Once you have earnings as a share of price, the P/E is just that number turned upside down.

How do you check it on a single share?

Pick a price of Rs 100. A 4% yield means a dividend of Rs 4. A 60% payout means that Rs 4 is 60% of earnings, so EPS is Rs 6.67. Price over EPS, Rs 100 over Rs 6.67, is 15x, which agrees. Doing it on one share is the fastest way to catch an inverted ratio, and it is worth saying out loud because the interviewer is listening for whether you can move between the three numbers.

Where does this shortcut mislead?

It assumes the payout ratio is stable and measured on the same earnings as the P/E. A company paying a special dividend, or keeping its dividend flat while profits fall, will show a payout ratio that does not describe normal earnings, and the implied P/E will be wrong. Check whether the yield is trailing or forward, and whether the payout is on reported or adjusted earnings, before you trust the answer.

Rs 3.00, on 110 crore weighted average shares. The 100 crore opening shares count for the full year. The 20 crore new shares existed only from 1 October, half the year, so they count as 10 crore. Rs 330 crore over 110 crore is Rs 3.00. Dividing by the year-end 120 crore would understate EPS at Rs 2.75.

Why weight the shares by time?

A flat shared by two people for six months and three people for the next six has housed two and a half people on average over the year, and splitting the annual electricity bill by three would overcharge the one who arrived late. EPS divides a full year's profit by the shares that were outstanding while it was earned, so shares issued mid-year count only for the part of the year they existed. The money raised on 1 October only helped earn profit in the second half.

When is the time weighting not used?

When no new money comes in. A bonus issue or a share split changes the number of shares without changing the company's resources, so it is applied to the whole year and to comparative years as if it had always happened. If the 20 crore shares had been a bonus issue, the count would be 120 crore for the full year and EPS Rs 2.75, with last year's EPS restated on the same basis. Confirm the exact treatment against the current accounting standard on earnings per share, Ind AS 33 in India.

Why does an analyst care about the difference?

Because next year's EPS starts from 120 crore shares for the full twelve months. If profit stays at Rs 330 crore, EPS falls to Rs 2.75 next year even though nothing got worse, simply because the new shares count for a full year. An analyst forecasting growth in EPS has to model the share count forward, not just the profit, or the forecast will look better than the business.