Financial Analysis puzzles, solved step by step

Weigh four against four with four left off, then use known good coins to mix the suspects. If the first weighing balances, the fake is among the four left off, and two more weighings find it. If it tips, the fake is one of four possibly heavy or four possibly light coins; weighing 1, 2 and 5 against 3, 6 and a good coin splits those eight into groups of three or fewer, and the third weighing names the coin and its direction.

How do you know three weighings can be enough?

Count the possible answers before touching a coin. Any of 12 coins could be the fake, and it could be heavy or light, so there are 24 answers. A balance has three outcomes, left down, right down or level, so three weighings give 3 x 3 x 3 = 27 outcomes. Twenty four answers fit inside twenty seven outcomes, but only just, so every weighing must split the remaining possibilities into three nearly equal groups. Think of the game of twenty questions: a question with three possible answers is only worth asking if all three answers are about equally likely.

Why four against four, and what happens next?

Six against six can never balance, which throws away one of the three outcomes. Four against four with four left off splits the 24 answers into 8, 8 and 8. If it balances, the eight coins on the scale are proven good, and you can use them as reference weights in the next weighing. Weigh three of the suspects, 9, 10 and 11, against three good coins. Level means coin 12 is fake and one last weighing against a good coin says heavy or light. A tip tells you both that the fake is among 9 to 11 and which way it is off, so weighing 9 against 10 finishes it.

If the first weighing tips with the left side heavy, the fake is one of 1 to 4 and heavy, or one of 5 to 8 and light. Now move coins between pans. Weigh 1, 2 and 5 against 3, 6 and good coin 9: each outcome leaves at most three suspects, each with a known direction. Level leaves 4 heavy, 7 light or 8 light, and weighing 7 against 8 settles it. Left heavy leaves 1 heavy, 2 heavy or 6 light; weigh 1 against 2. Left light leaves 3 heavy or 5 light; weigh 3 against a good coin.

The schemes cost the same at 2,000 units, Rs 1,90,000 each. The all-units deal costs 95 times the quantity; the incremental deal costs Rs 1,00,000 for the first 1,000 and Rs 90 after that, which equals 95Q only at 2,000. Below that, all-units is cheaper. It also creates a cliff: 990 units cost Rs 99,000 but 1,000 cost Rs 95,000, so anyone ordering more than 950 should round up to 1,000.

How do you set the two schemes side by side?

Think of two mobile data plans, one with a flat lower price on everything once you cross a usage level and one that only discounts the extra data. They feel similar, but they reward different amounts of use. Write each scheme as total cost against quantity, then find where the two formulas are equal. All-units: 95Q once Q reaches 1,000. Incremental: 1,00,000 for the first 1,000 units, plus 90 for each unit beyond.

A quicker way to see it: at 1,000 units the all-units scheme is Rs 5,000 cheaper. Each extra unit costs Rs 95 under all-units and Rs 90 under incremental, so the incremental deal claws back Rs 5 a unit. It needs another 1,000 units to recover Rs 5,000, so the schemes meet at 2,000. Above that the incremental scheme wins: at 3,000 units it costs Rs 2,80,000 against Rs 2,85,000.

Why would anyone order more than they need?

Because the all-units scheme makes buying more cost less. 990 units at Rs 100 cost Rs 99,000, while 1,000 units at Rs 95 cost Rs 95,000, so ten extra parts arrive with Rs 4,000 back in your pocket. The same logic holds for any order above 950 units, because 1,000 units cost Rs 95,000 and anything from 951 to 999 costs more. A buyer with somewhere to store the spares should always round up.

Say the limit from both sides of the deal. Extra units carry storage, handling and the risk they are never used, so the saving is only real if the spares have a use. For the supplier, the cliff means giving away Rs 4,000 on a sale that would have happened anyway, which is why incremental schemes are common where buyers order close to a threshold.

Roughly 21,800 to 26,200. Scale the 16% annual volatility by the square root of the time, not the time itself: four months is a third of a year, the square root of a third is about 0.58, and 16% times 0.58 is about 9.2%. That is about 2,217 points either side of 24,000, a band that holds the close about two times in three.

Why can you not just say where the index will close?

You cannot, and saying so is part of a good answer. The honest reply to a where-will-it-close question is a central point and a range, with the range doing most of the work. Today's level is the natural centre over a few months, since the expected drift is small next to the spread. The interviewer reported here asked a sales and trading candidate exactly this, and what they are testing is whether you can turn a volatility number into a range in your head.

Why does volatility scale with the square root of time?

Picture someone walking along a lane, taking each step forward or back at random. After four steps they are not usually four steps away; good and bad steps partly cancel, and the typical distance is about two steps, the square root of four. Price moves behave the same way: variance adds up with time, so the standard deviation grows with the square root of time. A third of a year is not a third of the annual volatility; it is the square root of a third, about 58% of it.

How do you check it, and what are you leaving out?

Check by another route. Monthly volatility is 16% over the square root of 12, about 4.6%, and four months is the square root of 4, which is 2, times that: about 9.2%. Same answer. Two standard deviations, about 19,600 to 28,400, covers roughly 95% of outcomes, and naming that wider band shows you know a one standard deviation range will be wrong a third of the time. Say the limits too: prices compound, so the upper side is slightly wider, about 21,900 to 26,300 on a log basis; the drift is ignored; and real index returns have fatter tails than a bell curve.

Net income falls Rs 60 crore, cash falls Rs 10 crore and debt rises Rs 50 crore. All Rs 100 crore of interest is an expense, so pre-tax profit drops 100 and tax drops 40. On the cash flow statement, add back the Rs 50 crore paid in kind because no cash left, so cash from operations falls 10. On the balance sheet, cash is down 10, debt up 50 and retained earnings down 60: both sides fall 10.

What does paid in kind actually mean?

Think of a credit card bill where you pay half the interest and the bank adds the other half to what you owe. You have still been charged the full interest; you just have not paid all of it in cash. PIK interestPaid-in-kind interest: interest that is settled by adding it to the loan principal instead of paying cash, so the debt grows each period. is a real expense that is settled by growing the loan instead of draining the bank account. That single sentence tells you where each half goes: the expense hits profit in full, and the PIK half turns up as more debt.

How does it move through each statement?

Income statement: interest of Rs 100 crore cuts pre-tax profit by 100. Assuming all of it is deductible, tax falls by Rs 40 crore, so net income falls Rs 60 crore. Cash flow statement: start from net income, down 60, and add back the Rs 50 crore of PIK as a non-cash charge. Cash from operations falls by only Rs 10 crore, because the tax saving on the whole Rs 100 crore nearly covers the Rs 50 crore of cash interest. Balance sheet: cash down 10 on the assets side; debt up 50 and retained earnings down 60 on the other side. Both sides fall by 10.

What should you add after the walk-through?

Two points earn credit. First, the tax assumption: some tax systems limit how much interest a company can deduct, and the rules on PIK can differ, so say you have assumed full deductibility and would confirm the current rule. If the PIK half were not deductible, net income would fall Rs 80 crore and cash Rs 30 crore. Second, PIK compounds: next year's interest is charged on a loan Rs 50 crore larger, which is why lenders price it higher and why it shows up in leveraged buyouts where cash is tight early on.

About Rs 157.5 crore. Start from potential rent of Rs 20 crore and take off 10% vacancy to reach effective gross income of Rs 18 crore. Operating expenses at 30% of that are Rs 5.4 crore, leaving net operating income of Rs 12.6 crore. Divide NOI by the 8% cap rate: Rs 12.6 crore over 0.08 is Rs 157.5 crore, which is 12.5 times NOI.

Why does a cap rate apply to net income and not to rent?

Think of a flat you rent out for Rs 30,000 a month. Some months it sits empty, and the society charges, repairs and property tax come out of your pocket. What you would pay for the flat depends on what is left, not on the rent written in the agreement. A cap rate is the yield a buyer wants on net operating income, the cash the building throws off after vacancy and running costs but before any loan payments. Applying it to gross rent values money the owner never receives.

How do you walk from potential rent to value?

Three steps, in order. Gross potential rent is what the building would earn fully let: Rs 20 crore. Take off vacancy first, because operating expenses here are a share of the income actually collected, not of the potential. 10% vacancy leaves effective gross incomeRent the building actually collects after vacancy and bad debts, before operating expenses. of Rs 18 crore. Expenses at 30% of 18 are Rs 5.4 crore, which leaves NOI of Rs 12.6 crore. Then divide by the cap rate.

What makes the exit cap rate the number to argue about?

Value is very sensitive to the cap rate: one point lower, at 7%, the building is worth Rs 180 crore; one point higher, at 9%, Rs 140 crore. That 1 point swing moves value by about Rs 40 crore on a Rs 157.5 crore building. Analysts usually set the exit cap rate a little above today's rate, because the building will be older when it is sold. Say the limit as well: a single cap rate assumes NOI is stable, so a building with large leases expiring soon needs a cash flow model, not one division.

EBITDA is wrong: it should be Rs 80 crore, not Rs 90 crore. Check every subtotal. Revenue less cost of goods sold is 200, so gross profit is right. Gross profit less opex is 80, not 90. EBITDA less D&A is 70, not the 60 shown. EBITDA is the only number in both failed checks, and setting it to 80 makes every line reconcile, including EBIT at 60.

Where do you start when one number in a page is wrong?

Think of a shop's daily cash sheet where the till, the card machine and the total do not agree. You do not stare at the biggest number; you re-add each subtotal and see which ones break. A P&L is a chain of subtractions, so every subtotal can be tested against the lines above it and the lines below it. Write the three identities out loud: gross profit is revenue less cost of goods sold; EBITDA is gross profit less operating expenses; EBIT is EBITDA less D&A.

Why is EBITDA the culprit and not opex or EBIT?

Two checks fail, and they share exactly one line. The wrong number is the one whose single correction fixes every failed check at once. Suppose opex were wrong instead: setting it to 110 makes EBITDA of 90 look right, but 90 less 20 still is not 60, so a second error would be needed. Suppose EBIT were wrong: 70 would fix the bottom check but leave 200 less 120 against 90. Only EBITDA at 80 repairs both. The puzzle says exactly one number is wrong, so that settles it.

What should you say beyond the answer?

State the assumption that made the puzzle solvable: operating expenses here exclude D&A, so EBITDA is gross profit less opex. If opex included depreciation, the chain would read differently. On the job, this test is the first thing a reviewer runs on any summary table, because subtotal errors usually come from a hard-coded number that did not update when the line above changed. Say where you would look next: the cell that feeds EBITDA, and whether margins quoted elsewhere in the pack used the wrong 90. At 90, the EBITDA margin reads 18% instead of 16%.

About Rs 761, so at Rs 1,000 the game is not worth playing. Each possible toss adds Rs 50 of expected value: a prize that doubles times a chance that halves. Prizes fit under the Rs 10 lakh cap for 14 tosses, which is Rs 700, and all longer runs are paid Rs 10 lakh, which adds only about Rs 61. The infinite value of the textbook game rests entirely on prizes no vendor can pay.

Why does every toss add exactly Rs 50?

Imagine a lottery stall where a ticket wins Rs 200 one time in four, or Rs 400 one time in eight. Each prize is worth the same on average, Rs 50, because the prize doubles exactly as the chance halves. In this game, heads first appearing on toss k has probability one over 2 to the k, and pays 100 times 2 to the k minus 1, so every possible toss contributes Rs 50 of expected value. With no limit there are infinitely many tosses, so the expected value is infinite. That is the famous St Petersburg paradox, and it is why nobody sensible pays a fortune to play.

What does the Rs 10 lakh limit do to the value?

The prize on toss 14 is Rs 100 times 2 to the 13, which is Rs 8,19,200; on toss 15 it would be Rs 16,38,400, more than the vendor holds. So only 14 tosses pay in full, worth Rs 700, and every longer run pays Rs 10 lakh, which happens one time in 16,384 and adds about Rs 61. The game is worth about Rs 761. Paying Rs 1,000 means losing about Rs 239 a game on average.

How rich would the vendor need to be for Rs 1,000 to be fair?

Value grows very slowly with the vendor's wealth, because each doubling of the bankroll adds just one more Rs 50 toss. A vendor holding Rs 1,000 crore would make the game worth only about Rs 1,425, and the game reaches Rs 1,000 only if the vendor can pay about Rs 2.6 crore. Say the limit too: even a fair price ignores how much risk you can stomach, since almost every game pays Rs 100 or Rs 200. That is why economists use this game to show people value money by its usefulness to them, not by its face amount.

Its beta is zero, yet its expected return is minus 8%. Beta measures movement with the market, and a slot machine's payouts are random and unrelated to the market. CAPM would give a zero-beta asset the risk-free rate, say 7% a year, so the machine falls short by at least 15 points. There is no contradiction: CAPM prices assets bought as investments, and a slot machine is bought as entertainment.

How can something so risky have a beta of zero?

Think of an umbrella seller and an ice cream seller in the same town. Each has a volatile income, but whether it rains has nothing to do with the stock market. BetaHow much an asset tends to move when the market moves, measured as its covariance with the market divided by the variance of the market. measures how an asset moves with the market, not how much it moves, so a gamble driven by a random number generator has a beta of zero. The slot machine is about as volatile as anything in a town, but all of that risk is the kind a diversified owner can spread away, and in this case the casino does exactly that across thousands of players.

Why does CAPM not give it the risk-free rate?

CAPM says expected return equals the risk-free rate plus beta times the equity risk premium. With beta of zero that is just the risk-free rate, 7% a year in this example. The machine instead returns minus 8% on every stake, and a stake lasts seconds, so over a year of play the gap is far wider than the 15 points the two headline numbers suggest. In CAPM language that is a large negative alphaThe return an asset earns above or below what its beta implies under CAPM..

So is CAPM wrong?

No, it is answering a different question. CAPM describes the prices of assets that diversified investors hold to earn a return, where anyone could sell an overpriced asset short. Nobody plays a slot machine for return; players pay 8% of each stake for entertainment, the way a cinema ticket has a negative return. And you cannot short a single slot machine to collect the edge. The only way to take the other side is to own the casino, which needs licences, buildings and capital, and the casino's return on that capital is what an investor would compare with CAPM. Say that and the interviewer hears that you know where a model applies, not just its formula.

Output doubles in about 7.2 months and reaches eight times in about 21.6 months, by the rule of 72. Divide 72 by the growth rate in per cent: 72 over 10 is 7.2 months per doubling, and eight times is three doublings. The exact answer, the log of 2 over the log of 1.1, is 7.27 months, or 21.8 months for eight times, so the rule is within a fifth of a month.

Why does eight times take three doublings and not 70 months?

Think of a rumour that doubles the number of people who know it every week. One becomes two, then four, then eight in three weeks, not eight weeks. Compounding growth multiplies, so the question to ask is how many doublings fit in, not how many 10% steps add up to 700%. Two times two times two is eight, so eight times today's output is three doublings away. Dividing 700% by 10% a month treats each month's growth as if it were earned only on the starting level.

Where does the rule of 72 work, and where does it slip?

The exact doubling time uses the log of 1 plus the rate, which is a little less than the rate itself, and 72 is chosen to compensate for typical rates. The rule is within about 1% for rates from 5% to 10% a period and drifts at the extremes: at 2% it overstates, and at 50% it understates. At 10% the gap is only 0.07 months. A check in whole months helps too: 1.1 to the 7th is 1.95 and to the 8th is 2.14, so output passes double during month 8.

Say the limit in context. Output growing 10% a month for two years is a startup or a ramp-up, not a steady state; a plant hits capacity, demand or cash constraints long before it has grown nearly tenfold. The interviewer wants the arithmetic, then one sentence of business sense.

531,441. Split the power: 9 to the 6th is 9 cubed, squared, and 9 cubed is 729. Square 729 as (730 minus 1) squared: 730 squared is 532,900, less twice 730, which is 1,460, plus 1, gives 531,441. Check it: an even power of 9 ends in 1, the digits add to 18, and 0.9 to the 6th is about 0.53, so the size is right.

How do you break a big power into steps you can hold in your head?

Think of carrying a heavy suitcase up six floors: you rest on the landings. Split the exponent into pieces you know, 6 = 3 x 2, so 9 to the 6th is 9 cubed, squared. Most people know 9 squared is 81 and 9 cubed is 729. That turns the question into one square, 729 times 729, which is still awkward until you move it next to a round number.

Why square 730 instead of 729?

Because 730 squared is easy: 73 squared is 5,329, so 730 squared is 532,900. The identity (a minus 1) squared equals a squared minus 2a plus 1 lets you square any number just below a round one with two easy corrections. Twice 730 is 1,460. So 532,900 less 1,460 is 531,440, and adding 1 gives 531,441. Each step is small enough to say out loud while you do it.

How do you check it before you say it?

Three checks take seconds. The last digit of an even power of 9 is always 1, the digit sum of any power of 9 is divisible by 9, and 9 to the 6th is 0.9 to the 6th times a million. Here the digits 5, 3, 1, 4, 4, 1 add to 18, and 0.9 to the 6th is about 0.53, so the answer must be about 530 thousand. That last check catches the common slip of being off by a factor of ten. A second route also helps: 81 squared is 6,561, and 6,561 times 81 is 524,880 plus 6,561, again 531,441.