Financial Analysis puzzles, solved step by step

Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. The 4 already holds 3 litres, so it takes only 1 more. That is four moves. A longer route fills the 4 first and ends with 2 litres in the 4-litre bottle after six moves.

How do you avoid getting lost halfway through?

Think of following a recipe that says pour half the milk into the other pan: if you do not write down how much is in each pan, two steps later you have no idea. Track the pair of levels after every single move, written as (3-litre, 4-litre), and the puzzle becomes a short walk through a few states instead of a juggling act. There are only three kinds of move: fill a bottle from the tap, empty a bottle, or pour one into the other until the source is empty or the target is full.

Say the states out loud as you go: (0, 0), (3, 0), (0, 3), (3, 3), (2, 4). The interviewer can check each one in a second, and if you slip, you can see where.

Which amounts can these two bottles measure at all?

Every pour moves water in steps of 3 and 4, so the amounts you can isolate are the combinations of 3 and 4 with whole numbers in front: 4 minus 3 is 1, 3 times 2 minus 4 is 2, and so on. Because 3 and 4 share no common factor bigger than 1, these bottles can measure every whole number of litres from 1 to 4, and up to 7 if you count both bottles together. With a 4-litre and a 6-litre bottle you could never get an odd number, since every amount would be a multiple of 2. Saying that rule turns a party trick into reasoning.

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.

EPS falls from Rs 10.00 to Rs 9.125; the buyback breaks even at a P/E of 14.8x. After-tax interest is 400 x 9% x 75% = Rs 27 crore, so net income drops to Rs 73 crore over 8 crore shares. The shares cost 20x earnings, a 5% earnings yield, but the debt costs 6.75% after tax. EPS rises only if the P/E is below 1 / 0.0675 = 14.8x.

What decides whether a buyback raises or lowers EPS?

Think of borrowing at 9% to buy a shop that pays you 5% of its price in profit each year. Your income falls, even though you now own more. Buying your own shares is buying their earnings: it raises EPS only if the earnings yield on the shares, one over the P/E, beats the after-tax cost of the money used. At Rs 200 a share and EPS of Rs 10, the shares yield 5%. The debt costs 9% x (1 - 25%), which is 6.75%. You pay 6.75% to buy a 5% stream, so EPS falls.

Does a lower EPS mean the buyback destroys value?

Not by itself. EPS is an accounting test, and it ignores what happens to risk. Replacing equity with debt raises the risk borne by every remaining share, so a fall in EPS can sit alongside an unchanged or higher share price if the debt tax shield is worth more than the extra risk. Equally, a rise in EPS at a low P/E is not proof of value created. Say both sides: the EPS answer is Rs 9.125, the breakeven is 14.8x, and the value question turns on whether the shares were bought below what they are worth and on what the extra leverage does to the cost of equity.

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.

Stop on a 5 or 6 after the first roll, on 4 or more after the second, and take whatever the third gives. The game is worth 14/3, about 4.67. Solve it from the end: a last roll is worth 3.5, so with two rolls left you keep anything above 3.5, which makes two rolls worth 4.25. With three rolls left you keep only what beats 4.25.

Why do you start from the last roll?

Think of house hunting with three viewings booked and a rule that you must take the last flat if you get that far. You cannot judge the first flat until you know what walking away from it is worth, and that depends on the viewings still to come. A stop or continue decision is only as good as your value for continuing, so you price the last stage first and carry that value backwards. The method is called backward inductionSolving a sequence of decisions from the final step back to the first, so each earlier choice is made knowing what the later ones are worth., and it is how an option to wait is valued in finance too.

On the third roll there is no choice: you get the face, and a fair die averages (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. That 3.5 is the price of walking away from the second roll. So on the second roll you keep a 4, 5 or 6, each of which beats 3.5, and re-roll a 1, 2 or 3. Half the time you keep an average of 5; half the time you collect 3.5. Two rolls are worth 0.5 x 5 + 0.5 x 3.5 = 4.25.

What changes when you hold three rolls?

The bar goes up. With three rolls in hand, walking away from the first roll is worth 4.25, so a 4 is no longer good enough: only a 5 or a 6 beats it. Two faces in six you keep, averaging 5.5; four faces in six you roll on and collect 4.25. That is (2/6) x 5.5 + (4/6) x 4.25 = 1.83 + 2.83 = 4.67.

What do you add to show you see the pattern?

Two observations. First, the bar rises with the number of chances left. A candidate who applies one rule, keep 4 or more, on every roll gets 4.625 instead of 4.667: a small loss that shows the continuation value was never priced. Second, each extra roll is worth less than the one before: the second roll adds 0.75, the third only 0.42, and a fourth would add 0.28. An extra option is worth less when the options you already hold are good. The limit to say out loud: this strategy maximises the average, which is right for a player who plays many times; someone playing once who needs at least 4 would play differently.

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.

The reported yield is 5.0%; with stock pay treated as a cost it is 3.1%. Market value is 10 crore shares at Rs 800, Rs 8,000 crore. Rs 400 crore over that is 5.0%; Rs 250 crore is 3.1%. You do not need a future share price: deduct stock pay as if it were cash salary, because whatever the price, the shares handed to staff are worth Rs 150 crore.

Why is stock pay a real cost when no cash leaves?

Think of a family shop that pays its manager with a slice of the business instead of a salary. The till looks fuller, but the family now owns less of the shop. Stock-based compensation is wages paid in ownership rather than cash, and adding it back to free cash flow counts the saving without counting the slice given away. For this company, Rs 150 crore of the Rs 400 crore reported, 37.5%, exists only because staff were paid in shares. Put the other way, the reported yield is 60% higher than the yield an owner actually earns.

How does stock pay reach equity value if you do not know the future share price?

There are two consistent routes. Treat stock pay as a cash expense in the DCF, leave it out of the add-backs, and divide the equity value by today's diluted share count. Or keep the add-back and forecast every share that will be issued in future, which needs the future share price you do not know. The first route avoids the circle, because the rupee value handed to staff is fixed by the grant, Rs 150 crore a year, whatever the price turns out to be. Only the number of shares depends on the price: at Rs 400 the grant is 37.5 lakh shares, at Rs 800 it is 18.75 lakh, at Rs 1,600 about 9.4 lakh. Owners give up Rs 150 crore of value each time.

What is the one line that shows you understand it?

The gap between the two yields, 5.0% less 3.1%, is 1.9%, exactly the share of the company given to employees each year. An investor who takes the 5.0% at face value is being paid partly in their own dilution. The limit to say: stock pay retains staff and may cost less than the cash salary it replaces, so it is a real expense rather than a waste. The only point is that it must be counted once, as a cost.

About Rs 156 crore on cash flow, but a buyer will not pay much above Rs 150 crore, the cost of building the same plant. Revenue is Rs 100 crore and cash costs Rs 70 crore. Taking depreciation equal to the Rs 5 crore of capex, tax is 25% of Rs 25 crore, Rs 6.25 crore, leaving Rs 18.75 crore a year: over 12% that is Rs 156.25 crore. Replacement cost caps it.

How do you get from units to a value?

Valuing a factory is the same sum as valuing a flat you rent out: the rent left after upkeep and tax, divided by the return you need. A business with no growth is worth its steady yearly free cash flow divided by the discount rate. All the work is in getting the yearly cash right, and the two lines people forget are tax and the capex needed just to stand still.

Why does the rebuild cost set the ceiling?

Suppose someone asks Rs 25 lakh for a used car when the same model costs Rs 20 lakh new at the showroom. You walk to the showroom. No sensible buyer pays much more for an asset than it would cost to build an identical one, so replacement cost caps the price even when the cash flows say more. Here the gap is small, Rs 156.25 crore against Rs 150 crore, so a defensible answer sits close to Rs 150 crore.

The cap is loose in three ways, and naming them is what separates a good answer. Building takes time, perhaps two years with no cash coming in, so a working factory earns a premium for the lost years. Land, permits and trained staff may be hard to copy. And if cash flow value stays well above rebuild cost, rivals build plants, supply rises and prices fall, which pulls the cash flows back down. That last force is why the two numbers tend to converge over time.

What if the numbers pointed the other way?

If the cash flow value were Rs 100 crore against a Rs 150 crore rebuild cost, the factory would be worth about Rs 100 crore. Replacement cost is a ceiling, not a floor: nobody pays Rs 150 crore for a plant whose cash flows justify only Rs 100 crore. The floor is what the land and machinery would fetch if sold off, which is a third number worth asking for before you commit.