Financial Analysis puzzles, solved step by step

Light rope A at both ends and rope B at one end, together. When A burns out, 30 minutes have passed; light B's other end, and B finishes 15 minutes later, at 45 minutes. Lighting both ends halves any rope's remaining burn time whatever its speed profile, because two flames use the time up together. B has 30 minutes of burn left at the half hour, and two flames clear that in 15.

Why does lighting both ends give exactly half, when the rope burns unevenly?

Picture two friends eating one plate of biryani from opposite sides. One side is all rice and goes fast, the other is full of bones and goes slowly. They will meet somewhere off centre, but if the plate takes one person 60 minutes, two people eating at the same moment finish it in 30, because between them they are always doing two minutes of eating every minute. The rope's total burn time is fixed at 60 minutes; two flames consume it twice as fast, so it lasts 30 minutes regardless of where the slow sections are.

That is the whole trick. Measuring by length fails, which is why cutting a rope in half or marking its midpoint gets you nothing: the midpoint of the string is not the midpoint of the time. Measuring by burn time is exact.

Where do the last 15 minutes come from?

Rope B was lit at one end at the same moment as A. When A dies at minute 30, B has burned for 30 minutes, so it holds exactly 30 minutes of burn time, although nobody knows how long the remaining piece is. Lighting B's unlit end at that moment halves its remaining 30 minutes to 15, so B goes out at 30 plus 15, or 45 minutes. Both ropes are doing the same job: A is a 30 minute timer, and B is a 60 minute timer that you shorten halfway through.

Say the steps in order and say the start condition: all three ends must be lit at the same instant. An interviewer listening to a phone screen is checking that the sequence is clean and that you explain why it works, not just that you have heard the riddle before.

What else can two ropes measure?

With one rope you can time 60 or 30 minutes. With two you can add 15, 45, 75 and 90: 90 by burning one rope at both ends and then the other at one end, 75 by burning A at both ends, then B at one end for 30 minutes and lighting its second end. Each rope gives you only halving moves on whatever time it has left, so the times you can build are sums of 60, 30 and 15. You cannot get 20 minutes with these ropes, and saying why is a strong close.

A's lenders are in trouble and B's are merely uncomfortable; both sets of shareholders take a far bigger hit than revenue. A's EBIT swings from Rs 100 crore to minus Rs 40 crore and cannot cover Rs 50 crore of interest. B's falls to Rs 60 crore and covers it 1.2 times. Profit before tax falls 280% at A and 80% at B: operating and financial leverage multiply.

Why do identical profits hide very different risks?

Take two tea stalls that each clear Rs 10,000 a month. One rents a shop for Rs 60,000 a month and buys cheap; the other rents a cart for Rs 10,000 and pays more per cup. In a good month they look the same. In a bad month the shop's rent is still due while the cart's costs fall with its sales. Fixed costs do not shrink when revenue does, so the higher the fixed share of costs, the more of every lost rupee of revenue comes straight out of profit. That sensitivity is operating leverage.

Work out each company's variable cost. A spends Rs 900 crore to make Rs 100 crore of EBIT; Rs 600 crore of that is fixed, so Rs 300 crore, or 30% of revenue, is variable. B's Rs 900 crore is Rs 100 crore fixed and Rs 800 crore variable, 80% of revenue. So a lost rupee of revenue costs A 70 paise of EBIT and B only 20 paise. Revenue falls by Rs 200 crore: A loses Rs 140 crore of EBIT, B loses Rs 40 crore.

What does each lender actually face?

Each lender is owed Rs 50 crore of interest a year, 10% on Rs 500 crore. B's lender is paid out of operating profit with Rs 10 crore to spare: thin, and a rating analyst would note coverage dropping from 2.0x to 1.2x, but no default. A's lender is not paid from operations at all. A must find Rs 90 crore of cash, Rs 50 crore of interest plus the Rs 40 crore operating loss, from its cash balance, an asset sale or new borrowing, so its lender's safety now depends on liquidity, not earnings. A's break-even revenue to cover interest is Rs 929 crore, just 7% below today; B's is Rs 750 crore, 25% below.

Why do shareholders and lenders feel operating leverage differently?

Run revenue up 20% as well as down. A's profit before tax jumps from Rs 50 crore to Rs 190 crore and B's to Rs 90 crore. A's shareholders get the steep line both ways, and their loss is capped at the value of their shares. The lender gets the same Rs 50 crore in the good year and the base year, and loses only in the bad year, so operating leverage hands the upside to equity and the downside to debt. That is why credit analysts ask about the fixed cost base before they ask about margins, and why lenders to high fixed cost businesses ask for lower debt or tighter covenants.

The limitation: real fixed costs are not perfectly fixed. A could cut overheads, defer maintenance or renegotiate rent within a few quarters, so the first year is the dangerous one. Say that, and the answer moves from a formula to a judgement.

About 1.49% a month, or 19.4% a year effective. The real price is the Rs 57,000 cash price, so choosing the EMI means borrowing Rs 57,000 and repaying Rs 60,000 in six monthly instalments. The rate that makes six payments of Rs 10,000 worth exactly Rs 57,000 today is 1.49% a month. The forgone Rs 3,000 discount is the interest, charged on a balance that shrinks every month.

Where is the interest, if the EMI says no-cost?

A friend offers to sell you his bike for Rs 57,000 today, or for Rs 60,000 paid over six months. Nobody calls the second option interest-free; it is a loan of Rs 57,000 with Rs 3,000 of interest. A shop doing the same thing is no different. When a cash discount is available, the discount you give up by paying in instalments is the interest on the loan. The 'no-cost' label means the instalments add up to the sticker price, not that money has no time value.

So the question becomes: at what monthly rate is a stream of six Rs 10,000 payments, starting in one month, worth Rs 57,000 today? That is the internal rate of return of the loan.

How do you solve it without a spreadsheet?

Guess and check. The annuity factor must be 57,000 / 10,000 = 5.70. At 1% a month, six payments are worth 5.80 times one payment; at 2% they are worth 5.60. Halfway gives about 1.5%, and 1.5% gives 5.697, a whisker under 5.70, so the answer is just below: 1.49%. Then compound, do not multiply: 1.0149 to the power 12 is 1.194, so the effective annual rate is 19.4%, against 17.8% if you simply multiply by 12.

A second check uses the average balance. The loan starts at Rs 57,000 and is paid down to zero over six months; the balance at the start of each month averages about Rs 33,700. Rs 3,000 of interest on about Rs 33,700 over six months is 8.9%, or about 17.8% a year simple, which matches 1.49% x 12.

What changes the answer in real life?

Timing matters a great deal. If the first instalment is due today rather than in a month, you really borrow only Rs 47,000 and repay Rs 50,000 over five months, and the hidden rate jumps to 2.10% a month, about 28% a year. Any processing fee, or tax charged on the interest component, raises the true cost further, so always ask what you pay on day one. Check the specific charges and the current tax treatment with the lender. And if there was never a cash discount to forgo, the EMI really is close to free for the buyer, because the retailer, not the buyer, is paying the lender.

112, and Rs 560 crore. For 17.5%, build it from blocks: 10% of 640 is 64, 5% is half of that, 32, and 2.5% is half again, 16; 64 + 32 + 16 = 112. For 12.5%, spot the fraction: it is one eighth, so halve 4,480 three times, to 2,240, 1,120 and 560. Check each with a second route before you say it.

How do you take an awkward percentage of anything in your head?

When a restaurant bill is Rs 640 and you want to tip 17.5%, nobody multiplies by 0.175. You find 10%, which is just moving the decimal point, and then cut pieces off it. Any percentage can be built from 10%, 5% (half of 10%), 1% and their halves, and each block is one easy operation on the last. For 17.5% of 640: 10% is 64, 5% is 32, 2.5% is 16, and the three add to 112.

Check with a second route: 20% is a fifth, 128, and 17.5% is 2.5 points less, so 128 minus 16 is 112. Two routes agreeing is what lets you say a number aloud with confidence in an interview where speed is being watched.

Which percentages are fractions in disguise?

Some percentages are clean fractions, and dividing is faster than building. 12.5% is one eighth, so 12.5% of Rs 4,480 crore is three halvings: 2,240, 1,120, 560. The same idea gives 37.5% as three eighths (560 x 3 = 1,680) and 87.5% as the whole less one eighth (4,480 minus 560 = 3,920). Desk interviewers like these because they turn up constantly in fee splits, stake sizes and tax at round rates.

What makes a mental answer trustworthy under time pressure?

Two habits. First, estimate before computing: 17.5% is a little under a fifth, so the answer must be a little under 128. If your arithmetic produces 1,120 or 11.2, the estimate catches the slipped decimal at once. Second, keep the units: the second answer is Rs 560 crore, not 560. An interviewer marks a right number with units and a sanity check far above a right number mumbled after a long silence. Speak the route as you go, so a slip is visible and recoverable.

Year one: profit falls Rs 15 crore, inventory falls Rs 20 crore, cash rises Rs 5 crore from tax saved. Year two: gross margin is flattered. The scrapped stock sits on the books at zero, so the Rs 5 crore sale is pure gross profit, lifting margin from 30.0% to 30.9% on these figures. The write-down moved profit between years, because it wrote off more than the stock turned out to be worth.

How does a write-down move through the three statements?

Suppose you bought Rs 20,000 of winter jackets for your shop two years ago, and they are now unsellable. Admitting it does not take any money out of your till today; the money left when you bought them. An inventory write-down is a non-cash expense that recognises a loss of value already suffered, so its only cash effect is the tax it saves.

Check the balance sheet: assets are down Rs 15 crore, Rs 20 crore of stock gone and Rs 5 crore of cash added, and retained earnings are down by the same Rs 15 crore. It balances, which is the test an interviewer listens for.

Why does next year's gross margin look better than normal?

The stock now has a book value of zero. When it is sold for Rs 5 crore, revenue rises Rs 5 crore and cost of goods sold rises by nothing, because there is no cost left to expense. On a normal Rs 400 crore of sales at 30%, the write-down year shows 25.0% and the scrap year 30.9%. Writing down more than the stock is finally worth pulls a loss into this year and pushes an equal gain into the next.

Over both years nothing is lost or gained by the timing. Pre-tax, the total is minus Rs 20 crore plus Rs 5 crore, or minus Rs 15 crore: the true economic loss on stock that cost Rs 20 crore and fetched Rs 5 crore. Accounting rules generally require inventory to be carried at the lower of cost and net realisable value. If the company expected Rs 5 crore of scrap value, the honest write-down was Rs 15 crore, and the sale would have produced no gain at all.

What does an analyst do with this?

Treat both effects as one event. Strip the write-down out of year one's gross margin and the scrap gain out of year two's, and you see the business's real run-rate margin of 30%. A large write-down followed by margins above the old normal is a pattern worth asking management about, because over-provisioning creates a reserve that can be released into later profits. That is not proof of anything; obsolete stock genuinely varies in what it fetches. It is a question to ask.

In the year it happens, the capex cut: it adds the full Rs 10 crore to free cash flow, while more revenue or lower cost of goods sold adds Rs 7.5 crore after 25% tax. Capex is a direct cash item. If the change repeats every year, lower capex means lower depreciation and more tax, so all three converge on Rs 7.5 crore a year, and an exit multiple on EBITDA favours revenue and cost.

Why does capex hit cash harder than revenue or cost?

If you earn Rs 10,000 more, you keep what is left after income tax. If you decide not to buy a Rs 10,000 laptop, you keep all Rs 10,000, because buying it was never a deduction in that year. Revenue and cost changes reach free cash flow after tax; a capex change reaches it in full, because capex is subtracted below the tax line. So with a 25% tax rate, Rs 10 crore more revenue or Rs 10 crore less cost of goods sold adds Rs 7.5 crore, and Rs 10 crore less capex adds Rs 10 crore.

One assumption sits under the revenue answer: that the extra Rs 10 crore comes with no extra cost and no extra working capital. In practice more revenue usually needs more receivables and stock, which takes a little more away. Say so in one clause.

What changes when the change repeats every year?

Capex becomes depreciation. Spend Rs 10 crore less every year on assets depreciated over five years, and depreciation falls by Rs 2 crore more each year until it is Rs 10 crore lower, which raises tax by Rs 2.5 crore. In steady state a recurring capex cut adds Rs 10 crore of cash less Rs 2.5 crore of lost tax shield, Rs 7.5 crore a year: exactly the same as the revenue and cost changes. The capex cut's edge is timing, worth something in present value, but not a permanent advantage.

Even a one-off cut keeps only part of its edge. Discounted at 10%, Rs 10 crore saved next year is worth Rs 9.09 crore, but the depreciation shield given up over the following five years is worth Rs 1.72 crore, leaving Rs 7.37 crore against Rs 6.82 crore for a one-year revenue gain. Still ahead, by much less than Rs 2.5 crore.

How does the terminal value change the ranking?

If the terminal value uses an exit multiple of EBITDA, a permanent Rs 10 crore of extra revenue or lower cost raises terminal EBITDA by Rs 10 crore and the terminal value by Rs 80 crore at 8x, while a capex cut does not touch EBITDA and adds nothing through the multiple. With an exit multiple, recurring revenue and cost changes beat capex by a wide margin; with a perpetuity growth terminal, all three are equal in steady state. Name which terminal method you are assuming before you rank them. And add the business limit: capex cut without consequence is rare, because the assets usually drive future revenue.

The chance of at least one default is 18.3%, and the expected number of defaults is 0.2. Work from the complement: each loan survives with probability 98%, so all ten survive with probability 0.98^10 = 81.7%, and at least one default is the rest. The expected number is simply 10 x 2% = 0.2, which holds whether or not the loans are independent. Independence matters for the first answer, not the second.

Why work from the chance that nothing happens?

Ask what the chance is that at least one of ten friends is late for dinner, each being late one time in fifty. Counting the ways someone could be late is messy: one late, two late, any combination. Counting the single way nobody is late is easy: everyone on time. 'At least one' is one minus 'none', and 'none' for independent events is just the single probabilities multiplied. Each loan survives with probability 0.98, so all ten survive with probability 0.98^10.

To do 0.98^10 in your head, use the shortcut that (1 minus x) to the power n is close to 1 minus nx plus a correction of n(n minus 1)/2 times x squared: 1 minus 0.20 plus 45 x 0.0004 = 0.818. The exact figure is 0.8171, so the chance of at least one default is 18.29%. A second check is the Poisson shortcut, 1 minus e to the minus 0.2, which gives 18.1%.

Why is the expected number so much simpler?

Expected values add, always. Each loan contributes 0.02 expected defaults, and ten loans contribute 0.2. The expected number of defaults is 0.2 regardless of how the loans are connected, because the expectation of a sum is the sum of the expectations even when the events are correlated. The distribution behind it is lopsided: 81.7% of the time nothing happens, 16.7% of the time exactly one loan fails, and two or more fail 1.6% of the time.

What does correlation change?

Take the extreme. If all ten borrowers are suppliers to one factory and fail together or survive together, there is a 2% chance that all ten default and a 98% chance that none does. Expected defaults are still 0.2. But the chance of at least one default falls to 2%, and the chance of losing the entire book jumps from practically zero to 2%. Correlation does not change the average loss; it moves probability from many small losses into rare large ones, which is the loss a lender cannot survive. That is why credit portfolio work spends its effort on concentration and correlation rather than on the average default rate, and why a portfolio of ten loans to one industry is riskier than its expected loss suggests.

Cost of goods sold must have grown about 44%. Inventory days are inventory divided by daily cost of goods sold. For days to fall by a sixth while inventory rises a fifth, the denominator must grow by 1.2 x 60 / 50 = 1.44 times. So the larger stock is serving a much larger business, and is leaner relative to it. Had days stayed at 60, the distributor would be holding Rs 28.8 crore more stock in this example.

How can a bigger stock be leaner?

A kirana shop keeps Rs 60,000 of stock and sells Rs 1,000 a day at cost: two months of cover. A year later it keeps Rs 72,000 and sells Rs 1,440 a day: fifty days of cover. The shelf is fuller, yet each rupee of stock waits less time to be sold. Inventory days measure how long stock waits, not how much there is, so a larger rupee stock can still be leaner if what it serves grew faster.

Put rupees on it. Last year: COGS of Rs 730 crore is Rs 2.0 crore a day, and Rs 120 crore of stock is 60 days of it. This year inventory is Rs 144 crore. For that to be 50 days, daily COGS must be Rs 2.88 crore, an annual Rs 1,051.2 crore, up 44%.

What is the cash meaning of the change?

Compare the stock with what it would have been at the old turn. At 60 days, this year's cost of goods sold would need Rs 172.8 crore of inventory. The distributor holds Rs 144 crore. The faster turn means Rs 28.8 crore less cash tied up in stock than the business would otherwise need, even though the balance sheet shows inventory up Rs 24 crore. An analyst who reads only the rupee change calls this a working capital problem; the ratio says it is a working capital improvement.

What would make you distrust the 50 days?

Three checks. Year-end timing: inventory is a snapshot, and a stock-light last week of March can flatter the ratio; average inventory over the year is a better numerator. The denominator: some analysts compute days on revenue, not COGS, and mixing the two across years breaks the comparison. And mix: if the growth came from a fast-moving new product line, the old lines may be turning as slowly as ever. The ratio summarises; it does not explain.

Depreciation and amortisation rose from Rs 60 crore to Rs 90 crore, up Rs 30 crore or half again. D&A is exactly the gap between EBITDA and EBIT: 180 minus 120 last year, 200 minus 110 this year. The usual causes are a burst of capex now being depreciated, an acquisition bringing amortisation of intangibles, or an accounting change, such as lease accounting, that moves a cash cost out of operating expenses and into depreciation.

How do you find D&A from two margins?

If your take-home pay rises but your savings fall, something between the two lines got bigger: rent, EMI, school fees. You find it by looking at the gap, not the lines. EBITDA and EBIT differ by exactly one thing, depreciation and amortisation, so when the two margins move in opposite directions, D&A has grown by the sum of the two moves. Here, 2 points up and 1 point down is 3 points of Rs 1,000 crore of revenue, or Rs 30 crore.

Convert to rupees to be sure. EBITDA is Rs 180 crore last year and Rs 200 crore this year; EBIT is Rs 120 crore and Rs 110 crore. D&A is Rs 60 crore, then Rs 90 crore. Revenue is flat, so D&A went from 6% to 9% of revenue.

What could explain a 50% jump in D&A on flat revenue?

Three stories, each with a different meaning. First, a capex burst: a new plant or system has come into use and is being depreciated, but has not yet lifted revenue. That is a timing story, and the question is when revenue arrives. Second, an acquisition: buying a business brings customer lists and brands onto the balance sheet, and amortising them raises D&A without any new cash spend. Third, an accounting change. Under lease accounting standards such as Ind AS 116, rent stops being an operating expense and becomes depreciation plus interest, which raises EBITDA and D&A together without changing the cash the company pays.

The lease story fits the EBITDA rise neatly, since rent leaving operating expenses lifts EBITDA directly. But EBIT usually edges up under that change too, because part of the old rent now sits in interest, below EBIT. EBIT falling here suggests at least some genuine new depreciation as well.

Which margin should an analyst trust?

Neither on its own. EBITDA ignores the cost of the assets that produce the profit, so a company can raise its EBITDA margin simply by buying capacity or by reclassifying rent. EBIT charges for those assets, but through accounting depreciation that may not match their real wear. The honest check is cash: EBITDA less capex less lease payments, compared across both years, tells you whether the business really earns more. Ask management which of the three stories is true before praising the higher EBITDA margin.

Basic EPS is Rs 30.0; diluted EPS is Rs 26.88, about 10% lower. The if-converted method assumes conversion: add back the after-tax interest saved, Rs 30 crore x 0.75 = Rs 22.5 crore, and add 2 crore shares. That gives Rs 322.5 crore over 12 crore shares. On actual conversion, debt falls Rs 500 crore and equity rises Rs 500 crore, interest expense stops, and no cash changes hands.

What does the if-converted method assume?

Imagine a friend lent your shop Rs 5 lakh and can choose, any time, to swap the loan for a share of the business. To be honest with your other partners about what each share is worth, you would show the numbers as if the friend has already swapped: no more interest to pay, but one more partner splitting the profit. Diluted EPS shows earnings per share as if every convertible security that would reduce EPS had already converted, adding back the interest it would save and adding the shares it would create.

The interest is Rs 500 crore x 6% = Rs 30 crore a year. It was tax-deductible, so removing it raises net income by only Rs 30 crore x (1 minus 25%) = Rs 22.5 crore. Conversion creates Rs 500 crore / Rs 250 = 2 crore shares. Diluted EPS is (300 + 22.5) / (10 + 2) = Rs 26.875, which rounds to Rs 26.88.

How do you know whether a convertible dilutes?

Compare what each new share brings with what each existing share earns. Each conversion share brings Rs 22.5 crore / 2 crore = Rs 11.25 of earnings, far below basic EPS of Rs 30, so including the convertible lowers EPS and it must be counted. If basic EPS were below Rs 11.25, conversion would raise EPS; that security is anti-dilutive and is left out of diluted EPS. The test is per security, so a company with several convertibles ranks them from most to least dilutive.

What happens to each statement on actual conversion?

Balance sheet: borrowings fall Rs 500 crore and equity rises Rs 500 crore, split between share capital at face value and securities premium. Total assets do not move; the right-hand side changes its mix and leverage falls sharply. Income statement: from the conversion date, the Rs 30 crore of interest stops, so profit rises Rs 22.5 crore after tax. Cash flow statement: nothing at conversion, because no cash changes hands; it is disclosed as a non-cash transaction. Afterwards, the Rs 30 crore of cash interest is no longer paid. Reported basic EPS after conversion should land close to the diluted EPS already shown, which is why analysts value on the diluted figure.