Equity Research puzzles, solved step by step

With three cars the last car reaches about 1,833 miles; with n cars it reaches 1,000 x (1 + 1/2 + ... + 1/n). The cars drive together until they have burned one tankful between them, then one car refills the rest and stops. That happens after 1,000/n miles, then 1,000/(n - 1), down to the last car's 1,000. The sum has no ceiling, but it grows only like the logarithm of n.

Why does a car drop out after exactly 1,000/n miles?

Picture a group of hikers sharing water on a long desert walk. Once the group has drunk one person's worth between them, that person can pour what is left into the others' bottles, fill every one back up, and wait by the path. Carrying that hiker any further only costs water. The cars work the same way. After d miles each car has burned d miles of fuel and has room for d more. One car can refill the other n - 1 cars exactly when (n - 1) x d equals what it has left, 1,000 - d, which solves to d = 1,000/n.

Why do three cars buy 1,833 miles and not 3,000?

Follow the road left to right. Three cars run 333 miles together and burn 1,000 miles of fuel between them, one full tank. Car 3 has 667 miles of fuel left and hands 333 to each of the other two, which fills them. Two cars then run 500 miles, burning another tankful, and car 2 refills car 1. Car 1 runs its own 1,000. Every stretch burns exactly one tank, but each extra car buys a shorter stretch, because its tank has to move more cars.

What happens as n goes to infinity?

The harmonic series never converges, so enough cars can reach any distance at all. But the growth is logarithmic: ten cars reach about 2,929 miles and a hundred cars about 5,187, so every tenfold increase in cars adds only about 2,300 miles. When asked for the asymptotic answer, say both halves: unbounded, and slow. State the assumptions too: fuel moves between cars without loss, and a car that drops out is simply left behind. If the helper cars had to drive home, the answer would shrink sharply.

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 6.9% a month, and a 95% one-month value at risk of about Rs 1.13 crore. Variance adds across independent days, so volatility scales with the square root of time: 1.5% x the square root of 21, which is 4.58, gives 6.87%. The one-tailed 95% point of a normal distribution is 1.645 standard deviations, so a bad month costs 1.645 x 6.87% = 11.3% of Rs 10 crore, about Rs 1.13 crore.

Why does volatility grow with the square root of time?

Think of someone taking random steps left or right along a lane. After 100 steps they are not 100 steps from where they began; many steps cancelled, and the typical distance is about 10 steps, the square root of 100. Daily returns behave the same way when one day does not predict the next. Variances add across days, so the standard deviation, which is the square root of the variance, grows with the square root of the number of days.

How do you turn a volatility into a rupee value at risk?

Value at riskA loss threshold that a position should exceed only with a stated small probability over a stated period, such as 5% over one month. answers one question: how much would a bad month cost, where bad means the worst month in twenty. Under a normal distribution 5% of outcomes fall below the mean minus 1.645 standard deviations, so the 95% one-month VaR is 1.645 x 6.87% = 11.3% of the position. On Rs 10 crore that is about Rs 1.13 crore. Assume a zero expected return for the month; over one month the drift is small against the volatility.

What does the number leave out?

Two things, and saying them separates an analyst from a calculator. The square root rule needs independent days; when bad days cluster, as they do in a sell-off, real monthly risk is higher than 6.9%. And VaR marks the edge of the tail, not what sits inside it: it says nothing about how bad the worst 5% of months can get. Real returns have fatter tails than the normal curve, so a one-in-twenty month can cost well over Rs 1.13 crore.

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.

EPS rises 13.75%, from Rs 5.00 to about Rs 5.69. The cash was earning 6% before tax, Rs 60 crore, or Rs 45 crore after tax, so net income falls to Rs 455 crore. Rs 1,000 crore at Rs 50 retires 20 crore shares, leaving 80 crore. Rs 455 crore over 80 crore shares is Rs 5.69. The buyback helps because the earnings yield on the shares, 10%, beats the 4.5% after-tax yield on the cash.

Why does a buyback change EPS at all?

Four friends own a small shop and split its profit. One sells out to the other three, paid from the shop's savings account. Profit dips a little, because the savings no longer earn interest, but it is now split three ways instead of four. A buyback lowers earnings by the lost return on the cash and lowers the share count by the shares retired; EPS rises when the second effect is larger. Here the share count falls 20% and net income falls only 9%.

When does a buyback lift EPS?

Compare two yields. The earnings yieldEPS divided by the share price: the earnings each rupee spent on the shares buys. It is one over the P/E. is Rs 5 over Rs 50, 10%: what each rupee spent on shares buys in earnings. The after-tax yield on cash is 6% x (1 - 25%), 4.5%: what each rupee was earning where it sat. A buyback lifts EPS whenever the earnings yield beats the after-tax yield on the cash spent, which is the same as a P/E below one over that yield. Here the break-even P/E is 22.2x, so the same buyback would leave EPS flat at a share price of about Rs 111.

Does higher EPS mean each share is worth more?

Not by itself. EPS rose 13.75%, but the company also handed out Rs 1,000 crore of cash, and a buyback at a fair price leaves each remaining share worth the same. Before, the equity was worth Rs 5,000 crore over 100 crore shares; after, Rs 4,000 crore over 80 crore shares, still Rs 50 each. Holders own a bigger slice of a company that has less cash and so carries a little more risk, which is why the P/E often slips when EPS is lifted this way.

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 Rs 61 crore. A perpetuity of Rs 10 crore at 10% is worth Rs 100 crore. A perpetuity that starts only after year 10 is worth that Rs 100 crore discounted back ten years; 1.1 to the power 10 is about 2.59, giving Rs 38.6 crore. The ten-year stream is the difference, Rs 100 crore less Rs 38.6 crore, about Rs 61.4 crore, or roughly six years of undiscounted cash.

Why subtract two perpetuities?

Think of a shop lease that pays you rent for ten years. It is the same as a lease that pays you forever, minus a lease that starts in year eleven and pays you forever after. A ten-year annuity equals a perpetuity starting now less a perpetuity starting in year eleven, and a perpetuity is one division: payment over rate. That turns an awkward ten-term sum into two numbers you can hold in your head: Rs 100 crore, and the same Rs 100 crore pushed ten years into the future.

How do you get 1.1 to the power 10 in your head?

Square, don't multiply. 1.1 squared is 1.21; squared again is about 1.46 for four years; squared again is about 2.14 for eight years; one more 1.21 for years nine and ten gives about 2.59. Three squarings and one multiplication get you to 1.1^10 = 2.59, and 100 divided by 2.59 is about 38.6. A second route checks it: the rule of 72 says 10% doubles money in about 7.2 years, so ten years is a little more than one doubling, around 2.6.

What sanity checks can you say out loud?

Bracket it first. Every payment is worth between Rs 3.86 crore, the year ten payment, and Rs 9.09 crore, the year one payment, so the total sits between Rs 38.6 crore and Rs 90.9 crore. A tighter check treats all ten payments as arriving at the midpoint, year 5.5: Rs 100 crore over 1.1^5.5 is about Rs 59 crore, close to 61.4. The midpoint shortcut always comes out a little low, because discounting curves, which is itself a useful thing to say.