Equity Research puzzles, solved step by step
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074A stock trades at Rs 400 and its EPS is Rs 20. What are its P/E and its earnings yield, and where does the price go if EPS rises 25% and the P/E stays the same?Sell-side equity researchIndian brokerage research
Try it first
EPS rises 25% and the P/E holds at 20x. Where is the price?
Show the worked solution
A P/E of 20x, an earnings yield of 5%, and a price of Rs 500. P/E is price over EPS, 400 / 20 = 20. The earnings yield is the inverse, EPS over price, 20 / 400 = 5%. If EPS rises 25% to Rs 25 and the market still pays 20 times, the price is 20 x 25 = Rs 500, also 25% higher: at a constant multiple, price moves one for one with earnings.
What does a P/E of 20 actually say?
Buy a small shop for Rs 20 lakh that earns Rs 1 lakh a year and you have paid twenty years of today's profit, a 5% return on your price before any growth. A P/E is the number of years of today's earnings you pay for, and its inverse, the earnings yield, is those earnings as a return on the price. Neither says whether the stock is cheap; that depends on how fast the earnings grow and how risky they are.
Price is the area of EPS times P/E, so a 25% rise in EPS at a constant 20 times widens the rectangle by a quarter from Rs 400 to Rs 500, while a fall in the P/E to 16x would leave the price at Rs 400. The relationshipP share price EPS earnings per share P/E the multiple the market pays for each rupee of earnings EY earnings yield, EPS over price What it says in wordsThe price is earnings per share times the multiple, and the earnings yield is one over the multiple.When does the price not follow earnings?
When the multiple moves. If EPS rises 25% but the P/E falls from 20x to 16x, the price does not move at all: 25 x 16 = Rs 400. The price change is the earnings change and the multiple change compounded: 1.25 x 0.8 = 1.0. That is how a company can grow profit for years while its share price stands still, and why analysts separate earnings growth from rerating when they explain a move.
Why is the earnings yield worth quoting?
It puts the price on the same scale as other returns. A 5% earnings yield can be set beside a deposit rate or a bond yield, as long as you say what differs: earnings are not all paid out, they can grow, and they are not promised. When bond yields rise above a stock's earnings yield, the case for the stock rests more heavily on growth, which is one reason multiples tend to fall when rates rise.
Where candidates lose it
The slip is adding instead of multiplying: taking the extra Rs 5 of EPS, or 25 rupees, onto the Rs 400 price. The market pays twenty rupees for each rupee of earnings, so Rs 5 more of earnings is Rs 100 more of price.
The quieter loss is saying the earnings yield is a return you receive. It is earnings on price, most of which the company keeps; say that once and the interviewer knows you understand the number.
What the interviewer asks next
- EPS rises 25% and the price rises only 10%. What is the new P/E? (17.6x)
- What P/E corresponds to an earnings yield of 8%?
- Why might two companies with the same EPS growth trade on very different P/Es?
075A company writes Rs 20 crore of inventory down to zero. The tax rate is 25% and the write-down is deductible for tax this year. What happens on each of the three statements?Sell-side equity researchIndian brokerage research
Try it first
What happens to cash, counting the tax saving?
Show the worked solution
Net income falls Rs 15 crore and cash rises Rs 5 crore; inventory falls Rs 20 crore while equity falls Rs 15 crore. The write-down is a Rs 20 crore expense, cut to Rs 15 crore after the Rs 5 crore tax saving. It costs no cash today, so the cash flow statement adds the Rs 20 crore back, leaving cash up by the tax saved. Assets fall Rs 15 crore, and so does equity.
Why does a Rs 20 crore loss raise cash?
A shopkeeper who bought stock last year and finds this year that it has spoiled is not paying for it again; the money went out when the goods came in. Admitting the loss costs nothing new, and if the tax rules let her deduct it, she pays less tax this year. The cash for inventory leaves when it is bought; the write-down only records the loss, and the tax deduction is the one real cash effect today. So profit falls by the loss after tax, while cash rises by the tax saved.
The Rs 20 crore write-down cuts net income by Rs 15 crore after tax, the cash flow statement adds the non-cash Rs 20 crore back to leave cash up Rs 5 crore, and the balance sheet balances with assets and equity both down Rs 15 crore. How do you walk it through the statements in order?
Income statement first: a Rs 20 crore expense, usually inside cost of goods sold, lowers pre-tax profit by 20; tax falls by 25% of that, Rs 5 crore; net income falls Rs 15 crore. Cash flow statement next: start from net income of minus 15 and add back the Rs 20 crore, because no cash left; cash is up Rs 5 crore. Balance sheet last: cash up 5 and inventory down 20 take assets down 15, and retained earnings fall by the same 15, so the two sides move together.
What if the tax rules are different?
Tax treatment of write-downs varies by jurisdiction, and some rules allow the deduction only when goods are actually scrapped or sold, so confirm it for the case at hand. If the deduction comes later, book net income still falls Rs 15 crore, but cash tax does not change this year, so cash is flat and a Rs 5 crore deferred tax asset appears on the balance sheet instead. Saying this without being asked shows you know where the Rs 5 crore of cash came from.
What does an analyst read into a write-down?
Two things. The charge is usually treated as one-off, so analysts look at profit before it, but repeated write-downs say something about how the company buys and forecasts demand. A write-down also moves cost out of future periods: goods written off now cannot be expensed again when sold, so next year's gross margin can look better than the business really is.
Where candidates lose it
The common slip is saying cash falls by Rs 20 crore, as if the write-down were a payment. The money left when the goods were bought; today's write-down only admits the loss.
The second slip is forgetting tax, which leaves net income down 20 and cash unchanged. With a deductible loss the tax saving is the only cash that moves, and it moves the other way: up Rs 5 crore.
What the interviewer asks next
- What changes if the write-down is not deductible for tax until the goods are scrapped?
- Next year the written-off goods are sold for Rs 4 crore. What happens on the statements?
- Why might a new management team write down inventory heavily in its first year?
088Two stock pitches. Pitch A returns 5x your money with a 10% probability and zero otherwise. Pitch B returns 1.5x with a 60% probability and 0.5x otherwise. Which has the higher expected multiple?Buy-side equity researchLong-only asset management
Try it first
Which pitch has the higher expected multiple of your money?
Show the worked solution
Pitch B, at 1.1x against 0.5x for Pitch A. Weight each outcome by its probability. A gives 10% of 5x plus 90% of nothing, which is 0.5x: on average it loses half the money. B gives 60% of 1.5x plus 40% of 0.5x, which is 0.9 plus 0.2, 1.1x. A would need better than a one in five chance of the 5x just to break even.
Why does the big number not win?
A lottery ticket that costs Rs 100 and pays Rs 500 one time in ten is a bad ticket, however good Rs 500 sounds. An expected value multiplies each outcome by its probability, so a large payoff is shrunk by a small chance before it counts. For A, 5x shrinks to 0.5x. For B, a modest win that happens more often than not, plus a partial loss that still returns half the money, adds up to 1.1x.
Pitch A's 5x outcome has only a 10% chance and the other 90% returns nothing, so it expects 0.5x, while Pitch B's 60% chance of 1.5x and 40% chance of 0.5x expect 1.1x, above the line where you get your money back. The relationshipE[A], E[B] the expected multiple of money for each pitch 0.1, 0.9, 0.6, 0.4 the probabilities of each outcome 5, 0, 1.5, 0.5 the multiples of money in each outcome What it says in wordsWeight every outcome by its probability and add: that is what you get on average per rupee put in.What would make Pitch A worth taking?
Work backwards from breakeven. A returns your money on average only if the chance of the 5x is one in five, 20%. So the real question about A is not how big the upside is but whether you can defend a probability above 20%, double what the pitch claims. That is how a buy-side analyst would push back on a story built around one dramatic outcome: ask for the odds, then check them.
Is the higher expected value the whole answer?
No, and saying so earns the extra point. B still loses half the money 40% of the time, so the expected value tells you which pitch to prefer, not how much to put in. Position size depends on how bad the bad outcome is and how often it comes. The limitation of the question is that it hands you the probabilities. In practice those are the hardest number to estimate, and a pitch with a vivid upside tends to come with an optimistic probability attached.
Where candidates lose it
Candidates are pulled to A by the 5x and justify it with language about asymmetric upside. The interviewer is checking whether you multiply by the probability before you get excited.
The second loss is stopping at the expected value. Mention that B still halves your money 40% of the time, so the choice of pitch and the size of the position are separate questions.
What the interviewer asks next
- What probability of the 5x makes A as attractive as B?
- If you could hold both, each with half your money, what is the expected multiple and the chance of losing money?
- Why might a fund still take a small position in something like Pitch A?
094A company is funded 70% by equity costing 14% and 30% by debt costing 9% before tax. The tax rate is 25%. What is its weighted average cost of capital?Sell-side equity researchIndian brokerage research
Try it first
Pick the WACC.
Show the worked solution
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.
Equity at 70% of funding and a 14% cost contributes 9.8 points and debt at 30% and an after-tax cost of 6.75% contributes 2.025, so WACC is about 11.8%, while using debt before tax would wrongly give 12.5%. The relationshipw_E, w_D shares of equity and debt in the funding, at market value k_E, k_D cost of equity and pre-tax cost of debt t the tax rate, 25% What it says in wordsWeight each source's cost by its share of the money, and cut the cost of debt by the tax it saves.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%.
Where candidates lose it
The fast wrong answer is 12.5%, using debt before tax. Candidates remember the formula but drop the one term that makes debt cheaper.
The second loss is applying the tax rate to equity as well, or using book weights without saying so. State market weights and after-tax debt in the first sentence and the interviewer moves on.
What the interviewer asks next
- What happens to WACC if the company moves to 50% debt and the cost of equity rises to 16%?
- Why should you use market weights rather than book weights?
- How does WACC change for a company that pays no tax because of past losses?
098Estimate how many cups of tea are sold in a day at a busy railway station.Consulting style estimationResearch KPO and GCC
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Where should the estimate start?
Show the worked solution
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.
Passenger flow peaks in the morning and evening, and applying buying rates to 2 lakh long-distance travellers and 3 lakh commuters gives about 1,30,000 cups, to which station staff add 15,000, about 1,45,000 cups a day. 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.
Where candidates lose it
Candidates start from stalls, guess a sales figure per stall and multiply, which makes the answer a single unchecked guess. Others multiply every passenger by one cup, ignoring that a commuter running for a local train rarely stops.
The second loss is giving a number with no cross-check. The supply-side count takes twenty seconds and turns an estimate into a reasoned range.
What the interviewer asks next
- How would the answer change on a foggy winter day with long delays?
- What would the annual tea revenue of the station be?
- How would you estimate the number of stalls the station can support?
100A company starts its April to March financial year with 100 crore shares and issues 20 crore new shares on 1 October. Net income for the year is Rs 330 crore. What is EPS?Indian brokerage researchResearch KPO and GCC
Try it first
Which share count goes under the Rs 330 crore?
Show the worked solution
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.
The company had 100 crore shares from April to September and 120 crore from October to March, a time-weighted average of 110 crore, so Rs 330 crore of net income gives EPS of Rs 3.00. The relationshipN bar the weighted average number of shares, crore 6/12 the share of the year the new shares were outstanding 330 net income for the year, Rs crore What it says in wordsCount each share for the fraction of the year it existed, then divide the year's profit by that average.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.
Where candidates lose it
The common slips are dividing by the closing 120 crore, which gives Rs 2.75, or by the opening 100 crore, which gives Rs 3.30. Both ignore when the shares were issued.
The follow-up catches the rest: candidates who time-weight a bonus issue as if it were a fresh issue for cash have missed the one case where the rule changes.
What the interviewer asks next
- The 20 crore shares were a bonus issue instead. What is EPS?
- If the issue had been on 1 January, what would the weighted count be?
- What will EPS be next year if net income stays at Rs 330 crore?