Financial Analysis puzzles, solved step by step

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.

Rs 180. The 18% is charged on the pre-tax price, so the MRP is the base times 1.18. The base is 1,180 / 1.18 = Rs 1,000 and the tax is Rs 180. As a share of the inclusive price, tax is 18 / 118, about 15.25%, not 18%. Taking 18% of Rs 1,180 charges tax on the tax and gives a wrong Rs 212.40.

Why is the tax not 18% of the price you pay?

A restaurant bill of Rs 1,100 that includes a 10% service charge contains Rs 100 of service, not Rs 110: the 10% was added to the Rs 1,000 of food, not to the final total. A rate applied to a base and then included in the price is rate over one plus rate of the inclusive price, never the rate itself. For 18% that share is 18 / 118, about 15.25%. The base is the starting point, and the inclusive price is the base grown by 18%.

How do you strip tax out of any inclusive price?

The rates in the table are for illustration. GST rates vary by product and change over time, so confirm the current rate for the item before you rely on a figure.

Where else does the same slip turn up?

Everywhere a percentage of one base gets quoted against another. A 25% markup on cost is only a 20% margin on price, because 25 / 125 = 20%. A fund that charges 2% on assets is not taking 2% of the return. An analyst who multiplies an inclusive revenue line by the GST rate overstates the tax the company passes on to the government. Before applying any percentage, ask what it is a percentage of, and convert if the base you hold is a different one. The test is quick: 1,000 plus 18% of 1,000 must give back the MRP.

About 32 lakh litres a day, inside a plausible range of roughly 18 to 47 lakh. Assume about 1.3 crore people in four-person homes, 32.5 lakh households. If 80% buy packaged milk at about a litre a day, that is 26 lakh litres. Tea stalls, restaurants and sweet shops add perhaps a quarter more. Each input is an assumption to state aloud and test.

What is the natural unit for a daily staple?

Think about your own home: someone buys one or two packets every morning, and the number depends on how many people share the kitchen, not on who drinks what. For something bought every day by almost every home, the household is the unit, because the buying decision and the quantity are set per kitchen. Start from people, convert to homes, decide what share buys the product, and multiply by the daily quantity. Then add the buyers who are not homes.

Every number in the chain is an assumption, not a fact, and should be said that way. The population depends on where you draw the boundary: the city proper and the wider metropolitan region differ by a large factor, so confirm the latest census or municipal estimate and say which one you are using. The 80% packaged share leaves room for loose milk from local dairies, which is still common in parts of the city.

How do you cross-check it without reusing the same assumptions?

Use a check that rests on different inputs. Per head, 32.5 lakh litres across 1.3 crore people is about 250 ml a day: two or three cups of tea at about 50 ml of milk each plus a glass for a child. That is believable. From the supply side, if a road tanker carries about 15,000 litres, which is an assumption to state, the city needs about 217 tankers a day. A good estimate survives a check built from a different direction; if the per-head number had come out at two litres, you would know a step was wrong.

What would sharpen the estimate most?

The packaged share is the biggest swing, so it is the first thing to research, followed by the trade uplift, which is hard to observe because tea stalls and sweet shops often buy loose milk in bulk. Seasonality matters too: festival weeks lift demand for sweets and milk. The limit to say: this is a demand estimate on an average day, and a dairy planning supply would also need the peak day and the share each brand holds.

Weigh coins 1 to 4 against 5 to 8. If they balance, the odd coin is among 9 to 12 and two weighings against known good coins find it. If one side is heavy, eight cases remain, and a second weighing that swaps some coins across the pans cuts them to three or fewer. It works because three weighings give 27 outcomes and there are only 24 cases.

Why can three weighings be enough?

Picture a guessing game where every answer is higher, lower or spot on. A question that leaves you with one of three equal groups does three times the work of a yes or no. A balance has three outcomes, left heavy, right heavy or level, so three weighings can tell apart 3 x 3 x 3 = 27 situations, and this puzzle has only 24: any of 12 coins, heavy or light. The plan works only if every weighing splits the remaining cases as evenly as possible into three.

That rules out six against six: one side always tips, so the balance outcome is wasted and twelve cases are left for two weighings that can separate nine. Four against four is right. If the left pan is heavy, the odd coin is one of 1 to 4 and heavy, or one of 5 to 8 and light: eight cases. If the pans balance, coins 1 to 8 are good and the odd coin is among 9 to 12, heavy or light: eight cases again.

How does the unbalanced branch get from eight cases to three?

Suppose the left pan was heavy. Weigh 1, 2, 5 against 3, 6 and coin 9, which is known to be good. Coins 4, 7 and 8 sit out. If the left is heavy again, the culprit is 1 or 2 heavy, or 6 light; weigh 1 against 2, and if they balance it is 6. If the right is now heavy, it is 3 heavy or 5 light; weigh 3 against a good coin. If they balance, it is 4 heavy or 7 or 8 light; weigh 7 against 8, and the lighter one is the culprit, or 4 if they balance. The trick is that some coins stay, some swap pans and some leave, so each outcome points to a different group.

If the first weighing balanced, weigh 9, 10, 11 against good coins 1, 2, 3. A heavy left means one of 9 to 11 is heavy, and 9 against 10 finds it. A light left means one is light, found the same way. A balance means coin 12 is odd, and weighing it against any good coin says whether it is heavy or light.

Why would a finance interviewer ask this?

It tests whether you count the possibilities before you start moving coins, which is the habit behind good diligence: design each question so that every possible answer tells you something new. The interviewer is happy with a clear first move and the counting argument, even if you need a minute for the unbalanced branch. The limit worth adding: with thirteen coins you can still find the odd coin in three weighings, but not always say whether it is heavy or light.

14% today, rising to about 16.7% after B doubles, with no change in either division's own margin. Group EBITDA is 80 + 60 = Rs 140 crore on Rs 1,000 crore. When B grows to Rs 400 crore at 30%, EBITDA is 80 + 120 = Rs 200 crore on Rs 1,200 crore. The group margin is a revenue-weighted average, so it moves when the weights move.

Why is the group margin not the average of the two?

A cafe earns 30% on coffee and 10% on sandwiches. If it sells mostly sandwiches, its overall margin sits near 10%; if coffee takes off, the overall margin climbs, even though neither item is any more profitable. A group margin is a revenue-weighted average of segment margins, so it moves whenever the mix moves, even if no segment changes. Today B is a fifth of revenue, so the group sits a fifth of the way from 10% to 30%: 14%.

Can the group margin rise while every division gets worse?

Yes. Suppose that while B doubled, A's margin slipped to 9% and B's to 28%. Group EBITDA would be 72 + 112 = Rs 184 crore on Rs 1,200 crore, 15.3%, still above the 14% start. Both divisions got less profitable and the group margin still rose, because the shift towards B outweighed the decline inside each. Statisticians call this pattern Simpson's paradox, and a management commentary that cites only the group margin can hide it.

What should an analyst do with a reported margin gain?

Split it into mix and rate. The mix effect is the margin you would get with the new weights and the old divisional margins, less the starting margin: 16.67% less 14% is 2.67 points. The rate effect is the rest. In the puzzle the rate effect is zero; in the deteriorating version it is -1.33 points, which is the story a reader needs. The limit: segment data is often reported only twice a year and with shared costs allocated by management, so the split is only as clean as the allocation.

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.