Financial Analysis puzzles, solved step by step

Root 50 is about 7.07, and 1.2% a day is about 19% a year. Root 49 is 7, and near a known square the curve is nearly straight, so add the step divided by twice the root: 7 + 1 / 14 = 7.0714 against a true 7.0711. Volatility scales with the square root of time, so multiply by root 250, about 15.81: 1.2 x 15.81 = 19.0%.

How do you get root 50 without a calculator?

Think of walking up a gentle hill: one more step forward lifts you by the slope at the point you are standing, and for a single step the hill may as well be straight. Near a square you know, the root curve is nearly straight, so the root of 49 plus 1 is 7 plus 1 divided by the slope's denominator, twice 7. That gives 7 + 1/14 = 7.0714; the true value is 7.0711, an error of 0.0004. The slope of the root curve at a point is one over twice the root, which is why the denominator is 14 and not 7.

Why does volatility scale with the square root of time?

Toss a coin 100 times and count heads minus tails. The lead is almost never 100; it is typically about 10, the root of 100, because the tosses partly cancel. Independent moves add in variance, not in standard deviation, so the spread after n days is the daily spread times root n. Daily variance is 1.2 squared, 1.44; over 250 days it is 360; the root of 360 is 19.0. Root 250 by the tangent at 256 is 16 - 6/32 = 15.812, close enough to the true 15.811.

When does the root-of-time rule mislead?

The rule assumes each day's move is independent of the last and drawn from the same distribution. Trending markets, where moves follow moves, make the true spread wider than root n suggests; mean-reverting ones make it narrower. Volatility also clusters, so a 1.2% daily figure measured in a calm month understates a stormy one. The day count is a convention: 250, 252 and 365 are all in use, and annualised volatilityThe standard deviation of returns restated to a one-year horizon, usually by multiplying a daily figure by the square root of the number of trading days. quoted on each will differ, so state the one you used. The limit to say: the rule converts a horizon, it does not forecast the year.

Net income rises Rs 7.5 crore, cash falls Rs 2.5 crore, and net fixed assets are Rs 10 crore higher. Income statement: depreciation down 10, pre-tax profit up 10, tax up 2.5, net income up 7.5. Cash flow: net income up 7.5 but the add-back down 10, so cash from operations down 2.5. Balance sheet: cash down 2.5, fixed assets up 10, assets up 7.5, matched by retained earnings up 7.5.

Why does profit rise while cash falls?

A shopkeeper who decides his delivery van will last ten years instead of five writes off half as much each year. The van is the same van, the fuel bill is the same, and no customer paid him more. Changing a useful life changes when the cost of an asset is recognised, not whether it is paid, so the extra profit is an accounting rearrangement with one real consequence: a higher tax bill. Depreciation falls 10, pre-tax profit rises 10, tax at 25% rises 2.5, and net income rises 7.5.

On the cash flow statement, start from net income, up 7.5, and add back depreciation, which is now 10 lower than before. Cash from operations is 7.5 - 10 = -2.5. Nothing changes in investing or financing, so cash falls 2.5. On the balance sheet, cash is down 2.5 and net fixed assets are up 10, because less accumulated depreciation has been charged against them: assets up 7.5. Retained earnings carry the 7.5 of extra net income, so liabilities and equity are also up 7.5.

Why is the higher profit not good news?

Because nothing about the business improved. The same machines wear out on the same schedule; the company will replace them on the same date for the same money; and it has handed Rs 2.5 crore to the tax authority earlier than it needed to. A profit increase that comes with a cash decrease and no operating change is cosmetic, and the choice to make it is itself a signal: management may be reaching for a target. It also breaks comparability with peers who kept the shorter lives, and the total charge over the asset's life is unchanged, so the profit that appears now is borrowed from later years.

What if tax depreciation did not follow book?

In practice the tax authority sets its own depreciation rates, and in India book depreciation under the Companies Act and tax depreciation under the Income Tax Act are computed separately; confirm the current rules before relying on this. Then cash tax does not change: net income still rises 7.5, but a deferred tax liabilityTax that the accounts recognise as owed on profit already reported, but that the tax return has not yet charged, so it will be paid in a later period. of 2.5 is booked instead of extra cash tax. On the cash flow statement, net income up 7.5, add-back down 10, deferred tax up 2.5: cash from operations is 0.0. The limit: in that version the change is purely cosmetic, cash untouched, which is why analysts read the depreciation policy note before they read the profit line.

Play: one roll pays 30 x 3.5 = Rs 105 on average against a Rs 100 fee, an edge of Rs 5. With the re-roll, keep a 3 or better and re-roll a 1 or 2; the game is worth Rs 118.33, an edge of Rs 18.33. A fresh roll is worth 105 less the Rs 20 fee, 85, so re-roll only a result paying less than 85: 30 and 60 qualify, 90 does not.

Why is the first game worth playing?

A fair die averages 3.5, so the average payout is 30 x 3.5 = Rs 105, more than the Rs 100 you pay. On expected value the game pays Rs 5 a play, so a player who can repeat it should play. Half the time you lose money, since a 1, 2 or 3 pays 30, 60 or 90, and the other half you win 120, 150 or 180; the wins are larger than the losses, which is where the edge sits.

How does the re-roll change the rule?

Think of returning a shirt. You keep it if what you have is worth more than what the shop will hand you after the restocking fee. Re-roll only when the roll in hand is worth less than a fresh roll net of its fee, which is 105 - 20 = Rs 85. A 1 pays 30 and a 2 pays 60, both below 85: re-roll. A 3 pays 90, above 85: keep it, even though 3 is below the average face. The fee is what makes a 3 worth keeping.

What does the option itself cost and earn?

The re-roll lifts the game from 105 to 118.33, so the option is worth Rs 13.33 on average, and it also cuts the chance of losing money from a half to a third, because a bad first roll gets a second chance. Re-rolling a 3 as well would give 117.50, worse by 0.83; a free re-roll would make re-rolling the 3 right and lift the game to 127.5. The threshold moves with the fee: below a fee of Rs 15 the 3 is worth re-rolling, above it the 3 is kept. The limit to say: the worst path, a 1 or 2 followed by a 1, costs Rs 90, and a player who cannot afford that should not value the game by its average.

From shrinking sales, not better collecting. With days sales outstanding fixed at 73, receivables are a fifth of revenue, so they fell from Rs 200 crore to Rs 150 crore as revenue fell Rs 250 crore, releasing Rs 50 crore. Unchanged days means customers paid no faster. The release is a one-off that reverses when sales recover, and it was bought with a quarter of the revenue.

Why does falling revenue release cash?

A tailor who gives customers two months to pay always has about two months of sales outstanding. If orders halve, the amount owed to him halves too, and for a while he collects old bills faster than he issues new ones: cash arrives. Receivables are revenue times the collection period, so when the period is fixed, receivables move with revenue, and a fall in sales releases cash once, mechanically. Here 73 days is 73 / 365, one fifth of a year, so receivables are a fifth of revenue: 200 on 1,000, 150 on 750. The Rs 50 crore is a fifth of the Rs 250 crore of sales that disappeared.

How do you separate the volume effect from the collections effect?

Split the change in receivables into two pieces. The volume effect is the change in revenue times the old days: (750 - 1,000) x 73 / 365 = -50. The collections effect is the new revenue times the change in days: 750 x (73 - 73) / 365 = 0. Every rupee of the release is volume; the collections effect is exactly zero, and that is the number management's claim rests on. Had days improved to 60, receivables would be Rs 123 crore, a release of 76.7: 50 of volume and 26.7 of genuine improvement, and only that 26.7 would deserve the word collections.

What does the analyst say about the quality of the Rs 50 crore?

Three things. It is one-off: receivables cannot keep falling unless sales keep falling. It reverses: if revenue climbs back to Rs 1,000 crore at 73 days, receivables return to 200 and the Rs 50 crore is reabsorbed. And it was expensive: at a 30% contribution margin, Rs 250 crore of lost revenue costs about Rs 75 crore of contribution every year, more than the cash released once. The limit: this uses year-end revenue and year-end receivables, and a company whose sales fell late in the year can show a days figure that flatters or punishes it; check the quarterly pattern before concluding.

Enterprise value stays at Rs 1,000 crore at every step; only the split between lenders and shareholders moves. Raising Rs 200 crore and holding it leaves net debt at 300 and equity at 700. Paying it out takes net debt to 500 and equity to 500, with shareholders holding the Rs 200 crore in cash. As PIK interest accrues at 10%, the notes grow to 242 after two years and equity falls to 458, value moving to the lenders.

Why does borrowing not change enterprise value?

Suppose your house is worth Rs 1 crore and you take a Rs 20 lakh loan against it, keeping the money in the bank. The house is still worth Rs 1 crore; you owe 20 lakh more and hold 20 lakh more, and your net position is unchanged. Enterprise value is the value of the operating business and does not depend on how it is funded; a financing step only changes who has a claim on that value. Here the PIK notes add 200 of debt and 200 of cash, so net debt stays 300, equity stays 700, and EV stays 1,000.

What does the dividend do?

The cash leaves the company and lands in shareholders' pockets. Net debt rises from 300 to 500 because the cash that offset the notes is gone, and equity value falls from 700 to 500. Shareholders are no richer and no poorer: they hold Rs 500 crore of shares plus Rs 200 crore of cash, 700 in all, exactly the 700 they started with. What changed is that the lenders now have a 500 claim ahead of them on the same Rs 1,000 crore of operations, so the equity is riskier and the dividend was financed by that risk.

What happens as the PIK interest accrues?

A PIK noteA loan whose interest is paid in kind: added to the principal each period instead of paid in cash, so the amount owed compounds. charges no cash interest; the coupon is added to the principal. At 10%, the Rs 200 crore becomes 220 after a year and 242 after two. With operations unchanged, EV is still 1,000, so equity is 1,000 - 542 = 458: the 42 of accrued interest is value transferred from shareholders to lenders without a rupee moving. In practice EV does move, but for operating reasons: earnings grow or shrink. Two second-order effects can touch EV through financing: interest that is deductible lowers tax and adds a shield, and heavy leverage raises the chance of distress, which costs value; confirm whether accrued but unpaid interest is deductible under the current tax rules before counting the shield.

The trap in the question is the word increase. A candidate who hears PIK and reaches for a bigger EV is adding debt to a number that already includes it. The market capitalisation does fall, from 700 to 500 and then 458, and adding net debt of 500 back to a market cap of 500 gives 1000, the same 1,000. The limit: this holds the operating business fixed to isolate the financing; a real company's EV changes every day for other reasons.

Lifetime value is about Rs 11,667, LTV to CAC is about 1.9x, and payback is about 17 months. Each customer brings Rs 350 of gross profit a month, 70% of Rs 500. At 3% churn the average customer stays 1 / 0.03 = 33.3 months, so LTV is Rs 350 x 33.3. The Rs 6,000 acquisition cost divided by Rs 350 a month is paid back in 17.1 months.

Why is the average customer life one over churn?

Think of a hostel where 3 residents in every 100 move out each month and are replaced. A resident faces a 3% chance of leaving every month, so on average the wait before leaving is 1 / 0.03 months. With a constant monthly churn rate, the expected customer life is one divided by that rate: at 3%, 33.3 months. Lifetime value is the gross profit the customer brings each month times that life.

Use gross profit, not revenue. The Rs 150 a month it costs to serve the customer is spent whether or not the customer was worth acquiring, so it cannot pay back the acquisition cost. Rs 350 x 33.33 months is Rs 11,667, and against a CACCustomer acquisition cost: marketing, sales and onboarding spend divided by the number of customers it won. of Rs 6,000 that is 1.94x.

Why are there two payback answers?

Payback is the month in which cumulative gross profit covers the Rs 6,000 spent on day zero. For a customer who stays, that is 6,000 / 350 = 17.1 months, the standard answer. Across a whole cohort payback is slower, because some customers leave before they have repaid their share of the acquisition cost. Weight each month by the chance the customer is still there and the average customer crosses Rs 6,000 in month 23.7. Give 17 months and name the cohort figure in one sentence.

Is 1.9x good enough?

Many subscription investors use a rule of thumb of 3x or better with payback inside about a year; treat those as conventions, not laws. At 1.9x this business recovers its acquisition cost with a thin cushion, and the ratio flatters it, because LTV here is undiscounted and ignores any cost of retaining customers. Churn is the strongest lever: cutting it from 3% to 2% lifts LTV by half to Rs 17,500, while a 10% price rise at the same churn only reaches Rs 12,833.

Not on this evidence: most of the drop is regression to the mean. A branch that tops the table usually had skill plus a good year. The good year does not repeat, so the same branches fall back toward the average whatever the manager does. Here only 2 points of a 20 point lead survived, so most of last year's lead was luck. Judge the head against similar branches that kept their managers, not against their own lucky year.

Why do the best performers fall back with no cause at all?

Think of a class test. The student who topped it probably knows the subject and also had a good day: the questions suited her and two guesses landed. In the next test the knowledge stays but the luck is drawn again, so her score is likely to be lower while staying above average. Any result that is part skill and part luck will, when it is extreme, usually be followed by a less extreme one. Francis Galton called this regression toward the mean when he compared the heights of parents and their children.

How much of the 30% was luck?

Measure the lead, not the level. The top five were 20 points above average last year and 2 points above this year. Keeping 2 points of a 20 point lead means about a tenth of last year's outperformance was persistent, and nine tenths was noise that happened to land on those five branches. Look at the bottom of the chart as well: the weakest five rose from -10.0% to 8.3% with no new manager, which is the same drift running the other way.

How would you test whether the new head made a difference?

Build a comparison. Take branches that were about as strong last year but kept their managers, and see how they grew this year. If they also fell to around 12%, the new head is doing exactly what chance predicts. If they held near 20%, the head has a real question to answer. The fair benchmark for a group picked for being extreme is a similar group picked the same way, never its own best year. The same trap sits in sales incentives, fund selection and bonus pools, where praising last year's winners and punishing last year's losers both appear to work.

The real raise is 3.77%, so the 4% shortcut overstates it by about a quarter of a point. Real growth divides growth factors: 1.10 / 1.06 = 1.0377. The shortcut subtracts the rates, which ignores that the raise is itself paid in money worth 6% less. At low rates the gap is small; at a 30% raise with 20% inflation the shortcut says 10% against a true 8.33%.

Why divide rather than subtract?

Measure your pay in lunches instead of rupees. Last year a thali cost Rs 100 and you earned Rs 100 a day: one thali. This year the thali costs Rs 106 and you earn Rs 110, so you can buy 110 / 106 = 1.0377 thalis. Your real pay rose 3.77%, because real growth is the ratio of two growth factors, not the difference between two rates. The relation is usually credited to the economist Irving Fisher.

The second form shows exactly where the shortcut goes wrong. The 4% gap is right in the numerator; it just has to be shrunk by dividing by 1.06. The error is small when inflation is small and grows as inflation rises.

When does the shortcut become a real error?

Three places. When rates are high, as the chart shows. When the rate compounds: over ten years, 4% a year grows purchasing power by 48.0% but the true 3.77% grows it by 44.8%, a gap of about three points. And when a budget is being judged: an FP&A analyst splitting revenue growth into price and volume uses exactly this division, and subtracting instead misstates the volume growth the business really delivered. For quick talk in a meeting, 4% is fine; in a model, divide.

Roughly Rs 70,000 to 75,000 crore a year, on the assumptions below. Start from an assumed 1.8 crore two-wheelers sold a year, assume a little over half are bought on credit, and lend about three quarters of the price. Splitting the market into commuter motorcycles, scooters and premium bikes gives about 99 lakh loans averaging Rs 73,600, or Rs 72,832 crore. Every input is an assumption to state and then confirm.

What is the structure before any number?

To estimate what a college canteen sells on credit, you would count meals, the share put on a tab, and the average tab. Loans work the same way. A finance market size is units times financing penetration times ticket size, and saying that structure first lets the interviewer follow every number after it. Here the units are new two-wheelers sold in a year, penetration is the share bought on a loan, and the ticket is the price less the down payment.

Assume about 1.8 crore two-wheelers are sold in India a year. Treat that as an assumption to confirm against the industry body's current sales data, never a fact to quote. Prices range from a basic commuter bike to a premium motorcycle, so one average price is fragile; split the market three ways instead.

How do you check it a second way?

Run it top-down in one line: 1.8 crore units x 55% financed x 75% of an average Rs 1,00,000 price is Rs 74,250 crore. Two routes landing within about 2% of each other is the check, and the segment split earns its place by showing where the uncertainty lives. The softest input is the financed share: every 5 points on it moves the answer by about Rs 6,621 crore. Then a feel check: 99 lakh loans a year is about 27,123 loans a day across the country, which is plausible for a mass market sold through thousands of dealers.

Say what the number is not. It is a yearly flow of new loans. The loan book outstanding at any time is a stock: with loans running about two and a half years and repaid evenly, the average loan is half outstanding for that period, so the book is roughly 1.25 times a year's disbursement, about Rs 91,041 crore. Used-vehicle loans are excluded.

12% a year at a 40% payout, and 6% at 70%. With no new equity and fixed leverage, the only fuel for growth is retained profit. Equity grows by ROE times the share of profit kept: 20% x 60% = 12%. Debt grows in step to hold the ratio, so assets, and at a steady asset turnover sales, can grow 12%. Keep only 30% and the ceiling halves to 6%.

Why is retained profit the only fuel?

A family shop that refuses outside partners and refuses to borrow more than it already does relative to its size can grow only on the profit it leaves in the till. With new equity ruled out and leverage fixed, equity can grow only by retained profit, and debt can grow only as fast as equity. So the whole balance sheet is capped at the rate equity compounds.

Put numbers on it. Opening equity of Rs 100 crore earns Rs 20 crore at a 20% ROE. Rs 8 crore is paid out and Rs 12 crore kept, so equity closes at Rs 112 crore, 12% higher. Next year's profit on Rs 112 crore is Rs 22.4 crore, also 12% higher, and the pattern repeats.

What assumptions are hiding inside the formula?

The formula, often taught through Robert Higgins' sustainable growth rate, assumes the 20% ROE holds on every new rupee of capital. Split ROE the DuPont way and it could be a 5% net margin x 2.0 asset turnover x 2.0 assets to equity. Growth above 12% therefore needs one of five things: a better margin, faster asset turnover, more leverage, new equity, or a lower payout. That list is the useful part of the answer, because it is exactly the set of levers a finance team argues about when a plan grows faster than its funding.

The limitation is worth one sentence: ROE rarely stays flat as a company grows, because new projects are usually less profitable than the best existing ones. The 12% is a ceiling under today's economics, not a forecast.