Financial Analysis puzzles, solved step by step

Market capitalisation is Rs 2,000 crore. With no other liabilities, assets equal debt plus equity. A D/E of 0.5 means debt is half of equity, so assets are 1.5 times equity and book equity is Rs 1,500 crore over 1.5, which is Rs 1,000 crore, with Rs 500 crore of debt. At 2.0x book, the market values that equity at Rs 2,000 crore.

How does a debt-to-equity ratio split the balance sheet?

Think of a Rs 15 lakh car bought with a loan that is half the size of your down payment. If the loan is half the down payment, the car cost one and a half down payments, so the down payment was Rs 10 lakh and the loan Rs 5 lakh. A D/E of 0.5 does not mean half the assets are debt; it means debt is half of equity, so assets are 1.5 times equity. Rs 1,500 crore over 1.5 gives book equity of Rs 1,000 crore and debt of Rs 500 crore.

What assumptions sit under the answer, and what could you add?

The question rules out other liabilities, such as payables and provisions, and that matters: a real balance sheet has plenty, and then assets are not just debt plus equity. Say the identity you are using, assets equal liabilities plus equity, before you lean on it. You can also go one step further than asked. With Rs 500 crore of debt and no cash given, enterprise value is about Rs 2,500 crore. And a P/B of 2.0 says the market thinks the assets earn more than their cost of capital; read it next to return on equity rather than on its own.

About 15.4%. Picture 10,000 invoices. 100 are fraudulent and the screen flags 90 of them. 9,900 are clean and the screen flags 5% of them, 495. The flagged pile holds 585 invoices, of which 90 are fraud, so a flag means fraud only 90 times in 585. The rare base rate swamps the screen's accuracy.

Why is a 90% accurate screen right only 15% of the time?

Think of a smoke alarm that never misses a fire but also goes off whenever someone makes toast. In a house where fires are rare and toast is daily, almost every alarm is toast. When the thing you are hunting is rare, even a small false positive rate on the large innocent pile produces more false alarms than true hits. Here 5% of 9,900 clean invoices is 495 false flags, against only 90 true ones.

What would make the screen useful, and how would an auditor use it?

Cutting the false positive rate does far more than raising the catch rate. At a 0.5% false positive rate the flagged pile would be 90 frauds and about 49.5 clean invoices, and a flag would mean fraud 65% of the time. Raising the catch rate from 90% to 100% would only move the answer from 15.4% to about 16.8%. In practice a screen like this is a triage tool: it shrinks 10,000 invoices to 585 for a human to review, and that review is where the 495 false alarms are cleared. Say the limit too: the 1% base rate is itself an estimate, and the answer is only as good as it is.

Unlevered free cash flow is Rs 1,00,000 a year and the NPV is about Rs 1,43,308. Sales are Rs 4,50,000, goods Rs 2,70,000 and rent Rs 60,000, so EBITDA is Rs 1,20,000. Less Rs 40,000 of depreciation is EBIT of Rs 80,000; after 25% tax, NOPAT is Rs 60,000. Adding back depreciation gives Rs 1,00,000. Five years of that at 14% is worth Rs 3,43,308, less the Rs 2 lakh machine.

How do you get from sales to unlevered free cash flow?

Think of a tea stall owner counting what she can take home each year: takings, less tea and milk, less rent, less the tax man's share. The stall's old kettle wearing out is a cost on paper, but no cash leaves her purse for it each year. Unlevered free cash flow is the cash the asset throws off before any financing: operating profit after tax, plus non-cash charges, less the capital spending and working capital it needs. Here: sales of Rs 4,50,000 less goods of Rs 2,70,000 and rent of Rs 60,000 is EBITDA of Rs 1,20,000. Depreciation of Rs 40,000 gives EBIT of Rs 80,000, tax takes Rs 20,000, and NOPAT is Rs 60,000.

Why does depreciation matter only through tax?

Depreciation is subtracted to reach EBIT and added back to reach cash flow, so on its own it washes out. Its only cash effect is the tax it saves: Rs 40,000 of depreciation at 25% cuts tax by Rs 10,000 a year. That is why taxing EBITDA directly is wrong: it would give Rs 90,000 a year and lose the shield. The answer leans on assumptions you should say out loud: no working capital for stock in the machine, no repairs, no salvage value, a steady 50 sales a day, and a 14% rate that reflects the risk of the location. The NPV is positive by a wide margin, so the decision is most sensitive to daily volume, the one number nobody can check in advance.

Segment C, up 2.5 percentage points, from 35.0% to 37.5%. Total revenue rises from Rs 1,000 crore to Rs 1,120 crore, 12% growth. C grew 20%, and 420 of 1,120 is 37.5%. A grew 15% but its 460 of 1,120 is 41.1%, only 1.1 points above 40.0%. B shrank 4% and lost 3.6 points, falling to 21.4%.

Why is growth not the same as gaining share?

Think of a household where every earner got a raise this year. Whose share of the family income went up most? Not the one with the biggest raise in rupees, and not necessarily the one with a big raise at all, but the one whose raise beat the household's overall rise by the widest margin. A segment gains share only by growing faster than the total, so the first number to compute is total growth, here 12%. A grew 15%, three points faster than the total; C grew 20%, eight points faster; B shrank.

How do you answer this fast in a timed test?

Sum each column first: 1,000 and 1,120. Compare every segment's growth with 12%, and only A and C can have gained. Then use the formula on those two: C is 0.35 x 8/112, which is 2.5 points; A is 0.40 x 3/112, about 1.07. Shares always add to 100%, so the gains and losses must net to zero, which gives a free check: 2.50 + 1.07 - 3.57 = 0. Watch the units in the answer options: C's share rose 2.5 percentage points, which is a 7.1% rise in its share. Timed numerical tests often offer both, and only one matches the question's wording.

About 2,520 ATMs, on stated assumptions. Of one crore people, 70 lakh are adults and 60% of them, 42 lakh, use ATMs about three times a month: 4.2 lakh withdrawals a day. If 12% fall in the busiest hour, that is 50,400 an hour. A machine handles 30 an hour at two minutes each; plan for two-thirds busy, 20 an hour, and you need about 2,520.

Where do you start, demand or supply?

Think of a canteen deciding how many counters to open. It does not divide the day's meals by 24 hours; it counts the lunch rush and opens enough counters for that. Size a service network to its peak load, so the estimate runs from people to withdrawals to the busiest hour, and only then to machines. Start with demand: 70% of one crore are adults, 70 lakh; assume 60% of them use ATMs for cash, 42 lakh, and that each withdraws three times a month. That is 1.26 crore withdrawals a month, or 4.2 lakh a day.

How much does the peak assumption change the answer, and how do you check it?

Machines stand idle at 3 a.m. and have queues at 6 p.m. If you spread 4.2 lakh withdrawals evenly over 24 hours, you would size for 17,500 an hour and buy 875 machines. The peak-hour share is the assumption that moves the answer most: at 12% of the day in one hour, the city needs nearly three times what the average implies. The same logic explains the planning margin: a machine busy 100% of the time has a queue that never clears, so plan for two-thirds use. Then sanity check the result: 2,520 machines for one crore people is about 25 per lakh, a ratio you can compare with the published ATMs-per-lakh figure for the city, which you should look up rather than quote from memory.

Say what you left out. Salary days bring a bigger rush at the start of the month; digital payments are shrinking cash use, so the user share and the withdrawals per month are both falling; and machines sit where people work and shop, so a city with a dense business district needs more than population alone suggests.

200 km. The car takes 100 / 50 = 2 hours to reach B, and the bird flies without stopping for exactly those 2 hours at 100 km/h. Distance is speed times time, so the bird covers 200 km. How many trips it makes and where each turn happens do not matter; only the time in the air does.

Why is the zigzag the wrong thing to count?

Picture a dog running back and forth between you and your front gate while you walk home. You could chart every lap, or you could notice that the dog runs for exactly as long as you walk. When something moves at a constant speed for a known time, its distance is speed times time, and the route it takes is irrelevant. The puzzle is built to pull you into the laps.

Here the clock belongs to the car. It has 100 km to cover at 50 km/h, so the whole episode lasts 2 hours. The bird flies at 100 km/h throughout, so it flies 200 km. That is the whole answer, and it takes one sentence.

How do you check it the long way if the interviewer asks?

Sum the laps. On the first leg the bird and the car close at 150 km/h across 100 km, so they meet after 40 minutes, 33.3 km from A. The bird has flown 66.7 km and flies the same back to B, a round trip of 133.3 km. By then the car is 33.3 km from B, so the next round trip is 44.4 km. Each round trip is one third of the one before, so the laps form a geometric series that sums to 133.3 / (1 - 1/3) = 200 km.

Why would a finance interviewer ask this?

It tests whether you look for the quantity that controls a problem before you start calculating. Most analysis questions hide one controlling number under a lot of detail, and the skill is finding it first. In a model review that might be the single assumption the valuation hangs on; in a variance analysis it might be one volume line. Give the two hour answer first, then offer the series as a check. Speed plus a second method is exactly what the interviewer is listening for.

Revenue up 10% adds the most, Rs 4 crore, against Rs 3 crore for the COGS cut and Rs 0.75 crore for EBITDA up 5%. New revenue brings only its gross margin: 10% of Rs 40 crore of gross profit. The COGS cut saves 5% of Rs 60 crore. The two levers tie when 10% of gross profit equals 5% of COGS, at a gross margin of 33.3%; below that, cutting COGS wins.

Why is a revenue increase worth less than it sounds?

A tea stall that sells 10% more cups also buys 10% more milk and tea leaves. When costs move with sales, extra revenue adds only its gross margin to profit, not the whole rupee. Here each extra rupee of sales brings 40 paise of gross profit, so Rs 10 crore of new revenue adds Rs 4 crore. The Rs 25 crore of other costs is assumed fixed; if part of it rose with sales too, the revenue lever would shrink further.

The COGS cut is simpler: 5% of Rs 60 crore is Rs 3 crore straight to EBITDA. EBITDA up 5% is the decoy: 5% of a Rs 15 crore base is only Rs 0.75 crore, because a percentage of a small number is a small number.

At what gross margin does the answer flip?

Write both gains per rupee of revenue. The revenue lever adds 10% times the gross margin; the COGS lever adds 5% times the cost ratio, which is one minus the gross margin. They tie when 0.10 x GM = 0.05 x (1 - GM), which gives a gross margin of one third. Above 33.3%, growing sales does more; below it, as in grocery or commodity processing, the cost cut does more. At the tie each lever adds Rs 3.33 crore on Rs 100 crore of revenue.

What does the interviewer want to hear beyond the number?

Ask for the margin structure before answering, because the right lever depends on it. Then add the practical view: a 5% cost cut is often more within management's control than 10% more sales, which may need price cuts or marketing spend that eat into the gain. Finally, EBITDA up 5% can never beat the revenue lever, since EBITDA can never exceed gross profit; it beats the COGS cut only when EBITDA is larger than COGS, as in some software businesses. If half the other costs were variable, the revenue lever would fall to Rs 2.75 crore and the COGS cut would win.

The company's own cash repaid Rs 400 crore of debt, and every rupee repaid moved a rupee of the unchanged Rs 1,000 crore value from lenders to the fund. Equity went in at Rs 400 crore and came out at Rs 800 crore: a MOIC of 2.0x and, over five years, an IRR of about 14.9%. No growth and no change in multiple were needed.

Where did the gain come from if the price did not move?

Think of a flat bought for Rs 1 crore with a Rs 60 lakh home loan. The rent covers the loan payments and brings the loan down to Rs 20 lakh over five years. You sell the flat for exactly Rs 1 crore, yet your stake has grown from Rs 40 lakh to Rs 80 lakh. Enterprise value is a pie shared between lenders and owners, and each rupee of debt repaid from the business's own cash moves a rupee of that pie to the owners. The price never had to rise.

The return arithmetic follows. MOIC, the multiple on invested capital, is Rs 800 crore out over Rs 400 crore in, 2.0x. IRR is the yearly rate that turns 400 into 800 over five years: 2 to the power one fifth, less 1, which is 14.87%. The rule of 72 gives a quick check: doubling in five years is roughly 72 / 5, about 14.4%.

Is that skill or just leverage?

Run the same business with no debt. The fund pays Rs 1,000 crore of equity, the company's cash flow piles up as at least Rs 400 crore of cash, and at exit the fund receives Rs 1,000 crore plus that cash, about Rs 1,400 crore. That is 1.4x and about 7.0% a year, before adding the interest it never paid. Debt did not create the Rs 400 crore; the business earned it. Debt let a smaller cheque claim all of it, which lifts the return from 1.4x to 2.0x.

Leverage cuts both ways, and saying so is the limit worth stating. Had the company sold for only Rs 500 crore, the levered fund would get back Rs 300 crore on Rs 400 crore, 0.75x, a loss, while the unlevered owner would hold 0.9x. The same debt that doubled the good outcome deepened the bad one.

What does the interviewer listen for?

Name the three ways a buyout makes money: EBITDA growth, a higher exit multiple, and debt paydown. This deal used only the third, which is the point of the question. Then add time: the same 2.0x earned over three years instead of five is about 26% a year, which is why funds care about IRR and not only the multiple. A finishing touch is noting that the Rs 400 crore repaid is cash after interest, so the business had to earn more than Rs 400 crore to do it.

About 67%. If the price is the probability-weighted average of the two outcomes, p x 500 + (1 - p) x 350 = 450, so p = (450 - 350) / (500 - 350) = 100 / 150 = 2/3. The price sits two thirds of the way from the failure value to the offer. The answer is only as good as the Rs 350 failure estimate, which nobody can observe directly.

Why does a price between two outcomes reveal a probability?

Picture a resale ticket for a cricket match that may be rained off. If the match is played the ticket is worth Rs 1,000; if it is washed out you get a Rs 400 refund. If tickets change hands at Rs 800, buyers are betting on play two times in three. When a price can end at one of two known values, where it sits between them is the market's probability, read off by distance from the bad outcome. A merger target is the same ticket: it ends at the offer price or falls back to where it would trade alone.

What changes once you allow for time?

Deals take months to close, and an arbitrageur who ties up Rs 450 wants paying for the wait. Say closing is six months away and the required return is 8% a year, 4% for the half year. Then the expected payoff must be 450 x 1.04 = Rs 468, and p = (468 - 350) / 150 = 78.7%. Ignoring time value understates the implied probability, because part of the gap to the offer is simply the return for waiting.

What would you check before trusting the number?

The failure value first, because it is an estimate and the answer swings on it. If the shares would fall only to Rs 380, say because the market has risen since the bid, the implied probability drops to 58.3%. Next the shape of the bet: Rs 50 to gain against Rs 100 to lose, so an arbitrage desk needs real confidence in the regulatory approvals, the buyer's financing and the shareholder vote. The limit to say aloud is that a probability read from prices also carries a premium for bearing deal risk, so it is not a pure forecast of completion.

On earnings yield alone equities look dear, 4% against 7%, but that ignores growth; on expected return they offer about 12% against 7%, a premium of about 5 points. The dividend yield is half of 1/25, 2%, and growing it at 10% gives roughly 12%. Whether 5 points is enough to pay for equity risk is the judgement the question is really after.

Why is 4% against 7% the wrong comparison?

A flat that rents for 3% of its price looks poor next to a 7% fixed deposit, yet people still buy flats, because rents rise over time and the deposit's interest never does. A bond's yield is close to its whole return if held to maturity, while an earnings yield is only the first year of a stream that grows, so setting one against the other treats a growing payment as a fixed one. The 4% earnings yield is 1 / 25; it says nothing yet about the 10% growth in the question.

How do you turn a multiple into an expected return?

Use the dividend growth relation: the return on a share held for the long run is the dividend yield plus the growth rate of the dividend. The payout is 50%, so the dividend yield is 0.5 / 25 = 2%. Growth is 10%, so the expected return is about 12%. Using next year's dividend, 2% grown by 10%, gives 12.2%; the difference does not change the verdict.

What does the growth assumption have to survive?

Growth carries the whole verdict, so test it. Retaining half the earnings and growing 10% forever requires a 20% return on every rupee reinvested, which is a demanding assumption for an entire market. If growth is 7% instead, the expected return is 9% and the premium shrinks to 2 points; at 5% growth equities offer no more than the bond. The answer to 'which is cheaper' is a statement about growth: at 25 times earnings, equities beat bonds by a healthy margin only if 10% growth is believable. Say that, and resist a one-word verdict.