Equity Research puzzles, solved step by step

About 11.7%. Scale the dollar rate by relative inflation: (1 + 9%) x (1 + 5%) / (1 + 2.5%) = 1.1166, so the rupee cost of equity is 11.66%. Adding the 2.5 point gap gives 11.5%, close but slightly low. The conversion keeps the value the same in both currencies, because rupee cash flows grow faster by exactly the same factor.

Why must the rate change when the business has not?

Think of a salary quoted two ways. A Rs 10 lakh salary rising with 5% inflation and the same salary quoted in dollars rising with 2.5% inflation are one job. If you discount the faster-rising rupee salary at the slower dollar rate, you make the same job look more valuable. A discount rate carries the inflation of its currency, so moving between currencies means moving the rate by the inflation gap, in the same way the cash flows move.

How do you prove the value is the same in both currencies?

Take a cash flow worth 100 in today's money, received in year five. In dollars it grows at 2.5% inflation and is discounted at 9%: 1.025 to the fifth over 1.09 to the fifth gives a discount factor of 0.7353. In rupees it grows at 5% and is discounted at 11.66%: the factor is 0.7353. The two factors match to the fourth decimal in every year, which is the proof that the conversion is right.

Say the limitation. This converts the currency, nothing else. If the dollar rate was built for a US listed peer, it may not carry any premium for country risk, and whether to add one is a separate judgement you should state and defend, not a number to fold silently into the conversion.

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.

About 0.5% of the starting stake. One head and one tail together multiply wealth by 1.5 x 0.6 = 0.90, so the typical growth factor per flip is the square root of 0.90, or 0.949, a loss of about 5.1% a flip. The median path has 50 heads and 50 tails and ends at 0.9 to the power 50, about 0.005. The average ends near 131.5x, carried by rare lucky paths.

Why does a plus 5% game shrink the typical player?

Think of a shop that raises a price 50% one month and cuts it 40% the next. The two changes average plus 5%, yet an item tagged Rs 100 ends at Rs 90. Wealth compounds by multiplying, not by adding. So the rate that decides where one player ends up is the geometric mean of the multipliers, not their arithmetic average. Here the geometric mean is the square root of 1.5 x 0.6, which is 0.949 a flip, and 0.949 applied a hundred times is a very small number.

If the typical player loses, where does the plus 5% average come from?

From a very small number of paths with far more heads than tails. You need at least 56 heads in 100 flips just to finish ahead, and only about 13.6% of players get there. The average is pulled up by the few players who land 70 or more heads and finish hundreds of thousands of times richer, while most players finish near zero. A player with exactly 70 heads ends at about 468,733 times the stake. The mean is a true number, but almost no individual player experiences it.

What does this have to do with running money?

A portfolio compounds exactly like the game. Volatility pulls the growth rate below the average return by roughly half the variance, so a strategy with a positive expected return can still shrink a typical account if it is run at too much size. Here the average return is 5% with a swing of 45% either way; half of 45% squared is about 10%, which is why the typical path loses about 5% a flip. The fix is sizing, not the odds: staking a quarter of wealth each flip, the Kelly fractionThe share of wealth to stake on each bet that maximises the long-run growth rate of wealth. here, lifts the typical player to about 1.86x after 100 flips.

About Rs 1,560 crore a year, on stated assumptions. Take 95 crore adults and 10% prevalence for 9.5 crore people with diabetes. Half are diagnosed, 4.75 crore; half of those take regular tablets, 2.38 crore; 60% of them are on this molecule, 1.43 crore patients. At Rs 3 a day for 365 days each patient spends Rs 1,095 a year, which gives about Rs 1,560 crore.

How do you structure a market size before you pick any number?

Sizing a drug market is like working out how many raincoats a town buys: not everyone gets caught in the rain, not everyone who gets wet buys a coat, and those who buy choose among brands. Write the chain first and say it out loud: people, times the share with the disease, times the share who know they have it, times the share treated, times the share on this molecule, times the annual cost. Stating the chain before any number shows the interviewer the logic, and lets them correct one input without the estimate collapsing.

Which assumption deserves the most care?

The one you are least sure of, because in a multiplicative chain every input moves the answer in the same proportion. Raising the diagnosis rate from 50% to 60% lifts the market by 20%, exactly as much as raising prevalence from 10% to 12%. Candidates spend their effort on prevalence because it is the headline statistic and wave the diagnosis rate through, when it is often the less certain number. Every figure here is an assumption for the exercise; check each against a published national survey before using it for anything.

How do you sanity check the answer?

Cross-check from the other end. Taking India's population as roughly 140 crore, also an assumption, Rs 1,560 crore works out to about Rs 11 per person per year, which is plausible for one cheap, widely used tablet. Then name what the estimate leaves out: combination tablets that contain the molecule, patients who take it irregularly, and the price gap between branded and generic packs. Each moves the number, and saying so is worth more than an extra decimal.

About 6.7x, on an implied stub value of Rs 2,000 crore. The 50% stake is worth half of Rs 16,000 crore, Rs 8,000 crore, at the subsidiary's own market price. Take that out of the holding company's Rs 10,000 crore and the market is paying Rs 2,000 crore for everything else. Against Rs 300 crore of earnings that is 6.7 times, assuming the holding company carries no debt or cash of its own.

How do you find the price of a business that has no price of its own?

A thali costs Rs 300 and includes a sweet the same restaurant sells alone for Rs 80. The rest of the meal is costing you Rs 220. When a company's value contains something with its own visible price, subtract that price to see what the market is paying for the rest. Analysts call what is left the stubThe value the market implicitly assigns to a holding company's own business after subtracting the market value of its listed stakes.. Here the visible item is the listed stake, worth Rs 8,000 crore at the subsidiary's share price.

Why might 6.7x not be the whole story?

Holding companies usually trade below the value of what they own. If the market applies a 20% holding-company discount to the stake, it is valuing the stake at Rs 6,400 crore, so the stub rises to Rs 3,600 crore and the implied multiple to 12x. The discount reflects tax on any eventual sale of the stake, dividends that may never reach the holding company's own shareholders, and the cost of running the holding company. Give both numbers and say which assumption produces each.

What would you check before calling the stub cheap?

Three things. Whether the holding company carries debt, which the stub has to absorb; whether the Rs 300 crore of earnings is recurring or flattered by one-off items; and whether the stake is ever likely to be sold or distributed. A discount that never closes is not a mispricing, so a low stub multiple is a question to investigate, not a conclusion.

EBITDA rises Rs 40 crore, EBIT rises Rs 30 crore, operating cash flow rises Rs 40 crore and investing cash flow falls Rs 40 crore; total cash does not change. Capitalising takes the spend off the income statement and puts Rs 10 crore of amortisation below EBITDA instead. On the cash flow statement the same Rs 40 crore moves from the operating section to the investing section. The company is no richer; it only looks richer on the two most quoted lines.

What does capitalising actually move?

A family that buys a Rs 40,000 laptop to last four years of college can tell itself it spent Rs 40,000 this month, or Rs 10,000 a year for four years. The bank balance is the same either way. Capitalising a cost is the second story: the money leaves in year one, but the income statement recognises it a quarter at a time. The spend becomes an asset on the balance sheet, and the cash paid for it is reported under investing activities instead of operating ones.

Which lines improve, and by how much?

Take a company with EBITDA of Rs 150 crore before this spend and nothing else going on. Expensed, EBITDA, EBIT and operating cash flow are all Rs 110 crore. Capitalised, EBITDA is Rs 150 crore, amortisation takes Rs 10 crore, EBIT is Rs 140 crore, operating cash flow is Rs 150 crore and investing cash flow is minus Rs 40 crore. EBITDA and operating cash flow each gain the full Rs 40 crore, EBIT gains only Rs 30 crore, and net cash is Rs 110 crore both ways.

What happens once tax is added?

It depends on the tax rules, which vary by country and should be confirmed for the case at hand. If the tax authority allows the deduction on the spend whatever the accounts say, cash tax is the same both ways, and book profit after a 25% tax is Rs 22.5 crore higher, with a Rs 7.5 crore deferred tax liability. If tax instead follows the books, capitalising raises year-one cash tax by Rs 7.5 crore, so the choice that flatters profit actually costs cash.

Why does an analyst care?

Because the two lines that gain are the two most used in valuation and quality screens: EV/EBITDA and cash conversion. A company that capitalises heavily looks cheaper on EV/EBITDA and converts more of its profit into operating cash than a peer that expenses the same spend. The fair comparison is free cash flow after all capital spending, including capitalised development, which is Rs 110 crore in both cases. If the spend recurs every year, amortisation builds to Rs 40 crore by year four and EBIT converges, but EBITDA stays Rs 40 crore higher for good.

About 14%. In a growing perpetuity the price is next year's dividend divided by (r - g). The gap goes from 12% - 6% = 6 points to 13% - 6% = 7 points, so the price is multiplied by 6/7, a fall of 14.3%. With a Rs 6 dividend the price goes from Rs 100 to Rs 85.7. A stock growing only 2%, priced at the same Rs 100, falls just 9.1%.

Why does one percentage point matter so much?

Think of a shop you could buy whose rent rises every year. What you would pay depends on the return you want minus the rate at which the rent grows: the faster the rent grows, the less of your required return has to come from today's income. Price depends on the gap r minus g, not on r itself, so when growth is high and the gap is narrow, a one point move in r is a large change in the gap. Here the gap goes from 6 points to 7, a rise of a sixth, so the price falls to six sevenths.

Which stocks are most exposed to a rise in rates?

Compare a slow grower priced the same way. A stock with a Rs 10 dividend growing at 2% is also worth Rs 100 at 12%; at 13% it is worth Rs 90.9, a fall of 9.1%. The more growth is built into a price, the further in the future its cash arrives, and the more the price moves when the discount rate moves. The durationThe sensitivity of a price to its discount rate, roughly the percentage price change for a one point change in the rate. of a growing perpetuity is about 1/(r - g): 16.7 for the fast grower and 10 for the slow one. The duration shortcut predicts a 16.7% fall for the fast grower; the actual 14.3% is smaller because the curve bends.

What are the limits of this answer?

The perpetuity assumes 6% growth forever and a discount rate that shifts cleanly by one point with nothing else changing. Real companies do not grow steadily forever, and a rise in rates often arrives with a change in growth expectations too. Treat 14.3% as the sensitivity of the valuation to its discount rate, not as a forecast of the share price. The ranking survives the simplification: high-growth, long-dated cash flows are the most rate sensitive.

It must reinvest about 133% of its profit, so free cash flow is negative. Growth equals the reinvestment rate times the return on invested capital, so the reinvestment rate is 20% divided by 15%, or 1.33. On Rs 100 crore of after-tax operating profit the company must invest Rs 133 crore, leaving free cash flow of minus Rs 33 crore, which has to come from lenders or shareholders.

Where does the rule that growth equals reinvestment times return come from?

A tailor earns Rs 15 a year for every Rs 100 of sewing machines she owns. To earn 20% more next year she needs 20% more machines, and buying them takes cash. Profit grows only as fast as the capital that produces it, so growth equals the share of profit reinvested times the return that capital earns. Turn it round and the reinvestment rate is growth divided by the return on invested capitalAfter-tax operating profit divided by the capital tied up in the business, fixed assets plus working capital.: 20% over 15% is 133%.

Does negative free cash flow mean the business is bad?

Not on its own. The check on the right of the figure proves the arithmetic: Rs 100 crore of profit on Rs 667 crore of capital is 15%; add Rs 133 crore and capital is Rs 800 crore, which earns Rs 120 crore next year, exactly 20% more. Negative free cash flow from growth creates value as long as the return on new capital beats the cost of that capital, and destroys value when it does not. The questions an analyst asks are how long the gap can be funded, and at what price.

What does this tell you about a growth story pitched in an interview?

That growth is never free, and its cost depends on returns. Two companies growing 20% can have opposite cash profiles: one earning 40% on new capital throws off half its profit, one earning 10% needs twice its profit again every year. When someone pitches a fast grower, ask for its return on new capital before you ask for its growth rate.

About 19%. Start from the sector's 10%, the base rate, and keep only the part of the outperformance that tends to persist. The excess was 40% - 10% = 30 points, and a coefficient of 0.3 keeps 9 of them, so the forecast is 10% + 0.3 x 30 = 19%. Extreme results are partly luck, and luck does not repeat, so the forecast sits much nearer the average than last year's number.

Why not forecast 40% again?

A student who scores 95 in one mock exam, when the class averages 60, is probably both able and lucky that day. On the next mock she is likely to beat the class again, but by less, because the luck part does not come back. An extreme result is part skill and part luck, and only the skill carries forward, so the best forecast pulls the result back towards the average. The persistence coefficient says how much pulling to do: 0.3 means 30% of the gap survives and 70% fades.

Where does the 0.3 come from, and what if you are not given it?

It is an empirical fact about a set of companies, not a law. You would estimate it by comparing one year's growth with the next across many companies in your universe over several years. Without data, the question to ask is how much of last year's jump came from something that recurs, such as a new plant or share won from rivals, and how much from something that does not, such as a one-off order, a weak base year or a tax credit. A jump built on recurring drivers deserves a higher coefficient.

How does this show up on a research desk?

Forecasts for last year's winners are often extrapolations with a haircut too small. A stock priced as if 40% growth continues is exposed when growth drifts back towards the sector, even if the company stays better than average. Say that link after the number: it shows you understand why the interviewer framed the question around a forecast rather than a fact.

About 31%: roughly two flags in three are false alarms. Take 10,000 companies. 200 manipulate and the screen catches 180 of them. 9,800 are clean and 4% of them, 392, are flagged anyway. Of 572 flags, 180 are real, which is 31.5%. The false alarms come from the huge clean group, so a rare problem plus a small error rate swamps the true hits.

Why is the answer not 90%?

Think of the smoke alarm in a kitchen. It goes off for every real fire, and it also goes off for burnt toast. Fires are rare and toast burns every week, so most of the times the alarm sounds, nothing is on fire. The 90% tells you how the screen treats a manipulator; the question asks how often a flag is right, and that depends on how many clean companies are standing there to be wrongly flagged. Those are two different conditional probabilities, and swapping them is the whole trap.

How do you set it up so the arithmetic is easy out loud?

Pick a round population and count people, not probabilities. With 10,000 companies the 2% who manipulate are 200, and 90% of them, 180, get flagged. The clean 9,800 produce 4% false flags, 392. Once the tree is drawn, the answer is just flagged guilty over all flagged: 180 over 572. This counting method is called natural frequenciesStating a probability problem as counts out of a round population, such as 180 out of 10,000, instead of as percentages. It makes conditional reasoning much easier to follow., and it is far harder to get wrong than juggling Bayes' formula in your head.

What does this mean for how an analyst uses a screen?

A flag is evidence, not a verdict. It moves the odds from 2 in 98 to 180 in 392, a jump of 22.5 times, which is exactly 90% divided by 4%. That is why a flag should start a deeper read of the notes to the accounts, never end one. If a second, independent check with the same accuracy also flags the company, the odds multiply again and the chance rises to about 91%. The limitation is the word independent: two ratio screens built on the same receivables line will fail together.