Financial Analysis puzzles, solved step by step

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.

Both hold when the two stocks share a slow driver that changes month to month, while their day-to-day moves push in opposite directions. On any one day the shared drift is a small part of each move and the opposing noise wins, so the daily correlation is negative. Over a month the drift adds up 21 times while the noise partly cancels, so the shared part dominates and the correlation turns positive.

How can two stocks move apart by the day but together by the month?

Take two neighbours' electricity meters. On a given day one runs high because guests are staying while the other family is away, so daily readings look opposed. But both bills climb every summer when the air conditioners come on. Correlation is not a fixed property of two assets; it belongs to a horizon, because different forces drive short moves and long moves. Daily returns are dominated by trading noise and stock-specific news; monthly returns by sector and macro trends.

Why does the shared driver win over a month?

Write each daily return as a shared drift plus the stock's own noise. Over 21 days the drift adds up in a straight line, so a month's drift is 21 times a day's and its variance is 21 x 21 = 441 times larger. Noise does not line up day after day, so its variance grows only 21 times. Whatever the two stocks share pulls harder the longer you hold them, because shared moves stack while unshared moves wash out. With daily noise variance of 1, a noise correlation of minus 0.3 and a drift variance of 0.107, the monthly correlation comes out at exactly plus 0.6.

The model also shows how sensitive the result is. Halve the drift variance and the monthly correlation drops from 0.6 to 0.39. Read across the whole year, daily returns would show a correlation of about -0.17: still negative, because on any single day the drift is too small to matter.

What else could cause it, and what would you check?

Two other mechanisms produce the same pattern. A lead-lag, where one stock reacts to shared news a day later, hides co-movement in daily data that monthly data captures. And short-term trading that pushes money from one stock into the other, such as a pairs or index rebalancing flow, creates opposing daily moves inside a shared trend. Then question the evidence. Twelve monthly points give a correlation of 0.6 a standard error of about 0.19, so the true figure could plausibly be anywhere from about 0.2 to near 1. Check other years before building a hedge on it, because a hedge ratio estimated on daily data can have the wrong sign for a position held for months.

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.

Goodwill is Rs 250 crore. Fair value of net assets is book equity of Rs 500 crore plus the Rs 200 crore brand, less a Rs 50 crore deferred tax liability (25% of 200): Rs 650 crore. Price paid less that is 900 - 650 = 250. On the combined balance sheet cash falls Rs 900 crore, the target's assets arrive with the brand and goodwill, a Rs 50 crore liability appears, and the target's equity disappears.

What exactly is goodwill?

Buying a running restaurant for Rs 90 lakh when its kitchen, furniture and stock are worth Rs 50 lakh, and its name alone could be sold for Rs 20 lakh, leaves Rs 20 lakh paid for things you cannot list: the regulars, the location's habit, the team. Goodwill is the plug: price paid less the fair value of every asset and liability you can identify, including the tax that comes with them. So the work is in the identifiable side, and the brand and its tax are the two lines people miss.

Why does the brand create a tax liability?

The brand goes onto the books at Rs 200 crore, but in a share purchase the tax authorities see no new asset: its tax base stays at zero. As the brand is amortised in the accounts, no matching tax deduction arrives, so future tax bills will be higher than the book profits suggest. That future tax is a real liability, 25% of the Rs 200 crore gap, Rs 50 crore, and it is booked on day one. A liability reduces net assets, so it increases goodwill by the same Rs 50 crore. In an asset purchase where the step-up is tax-deductible, there would be no liability and goodwill would be Rs 200 crore; check which structure and which tax rules apply.

What happens to these numbers after the deal?

The target's equity vanishes because the acquirer now owns it; only the acquirer's equity survives, unchanged by a cash deal. Afterwards the brand, if it has a finite life, is amortised, and the deferred tax liability unwinds in step, which softens the hit to net income. Goodwill is not amortised under Ind AS and IFRS; it is tested for impairment at least once a year, so a disappointing acquisition shows up later as a write-down. Confirm the treatment under the standard the company reports in. The interviewer is checking that you can make the balance sheet balance and explain why each new line exists.

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.

WACC is 7.25%. With no growth, a P/E of 10 means an earnings yield of 1 / 10 = 10%, which stands in for the cost of equity. Debt costs 6% before tax and 6% x (1 - 25%) = 4.5% after. Weighted half and half: 0.5 x 10% + 0.5 x 4.5% = 5% + 2.25% = 7.25%.

Why can the P/E stand in for the cost of equity?

Suppose you buy a shop for ten years of its profit, and the profit never grows and is all paid to you. Each year you collect a tenth of what you paid: a 10% return. With no growth and all earnings paid out, the price is earnings divided by the cost of equity, so the earnings yield, one over the P/E, is the return shareholders require. At 10x that is 10%.

The tax step matters because interest is deducted before tax is calculated. Every Rs 100 of interest saves Rs 25 of tax, so lenders cost the company only Rs 75 of every Rs 100 they receive. Dividends earn no such deduction, so equity gets no adjustment.

When does the shortcut break?

When earnings grow. The dividend growth relation gives P/E = payout / (k - g), so with growth the same 10x implies a higher cost of equity. At a 70% payout and 4% growth, k = 0.7 / 10 + 4% = 11%, and WACC becomes 7.75%. With growth, the earnings yield understates the cost of equity, because part of the shareholder's return comes from growth rather than from today's earnings. That is why the question says no growth.

What would an interviewer probe next?

Two things. First, weights: they must be market values, not book values, because WACC is the return investors require on what their claims are worth today. Second, leverage: the 10x P/E belongs to shares in a company already carrying 50% debt, so the 10% already includes the extra risk that debt places on equity. Unlever it before applying it to a company with a different capital structure. The tax shield is only real if the company has profits to deduct interest from, which is the limit to name.

The 12.5% annual loan is cheaper. Twelve per cent compounded monthly is an effective 12.68% a year, against 12.50%. A 12% nominal rate charged monthly means 1% a month, and interest earns interest eleven more times within the year: 1.01 to the power 12, less 1, is 12.68%. Convert every quote to an effective annual rate before comparing.

Why is 12% a year compounded monthly more than 12%?

Leave a credit card balance unpaid and the interest charged in January is itself charged interest in February. The bank's 'monthly rate' grows faster than the same rate charged once a year. A nominal rate tells you how interest is quoted; the effective annual rate tells you what it costs, and only effective rates can be compared. Here 12% a year becomes 1% a month, and after twelve months of compounding Rs 100 has grown to Rs 112.68.

What would the monthly lender need to quote to match?

Run the conversion backwards: the monthly rate that compounds to 12.5% is 1.125 to the power 1/12, less 1, about 0.98% a month, or a nominal 11.84% a year. Any monthly quote above 11.84% is dearer than 12.5% annual. For a quick mental check, the extra from monthly compounding at these rates is roughly half the rate squared: 0.5 x 0.12 x 0.12 is 0.72 points, close to the true 0.68.

What else decides which loan is really cheaper?

The effective rate prices the money, not the whole deal. Processing fees, prepayment penalties and insurance bundled into the loan all add to the cost and must be folded in. Watch for flat-rate quotes too: a 7% flat rate on a three-year loan charges interest on the original amount even as it is repaid, which works out to an effective rate of about 13.6% a year. Confirm how any lender computes its rate before comparing; the limit of the EAR is that it compares like with like only once every cost is in it.

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.

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.