Financial Analysis puzzles, solved step by step

About Rs 43. Each use ties up a Rs 10 note for a year, which at 10% costs you Rs 11 by year end, plus Rs 2 of electricity, and pays Rs 20. That is Rs 7 a year for ten years, and Rs 7 times the ten-year annuity factor of 6.1446 is Rs 43.01. If the electricity is paid at the start of each year the net is Rs 6.80 and the value Rs 41.78.

What is one use of the machine actually worth?

Think of lending a friend Rs 10 for a year and getting Rs 20 back. You doubled your note, but you also went a year without it, and that year had a price: whatever the money would have earned in the bank. The machine does not create Rs 10 a year; it creates Rs 20 at year end in exchange for Rs 10 today, and Rs 10 today is worth Rs 11 at year end at 10%. Take off Rs 2 of electricity and each use leaves Rs 7, received one year after you feed it the note.

Why does the timing of the electricity change the answer?

The question does not say when you pay for power. If you pay at the end of the year, the net is the Rs 7 above. If you pay at the start, alongside the note, the Rs 2 also costs you a year of interest, Rs 2.20 by year end, and the net drops to Rs 6.80, worth Rs 41.78. State the timing assumption out loud, because an interviewer who wrote this puzzle is listening for whether you notice that cash flows need a date. The other common answer, Rs 49.16, discounts Rs 8 a year and is wrong for a reason, not a rounding: it treats the Rs 10 note as free to borrow.

One check makes the answer believable. You could reproduce the machine with a bank loan: borrow Rs 10 at 10%, run the machine, repay Rs 11 and the Rs 2 of power, keep Rs 7. A buyer will pay up to the present value of that stream, and no more, because the bank can supply the money at 10% anyway.

Consolidated net income is Rs 250 crore, Rs 230 crore is attributable to the parent, and non-controlling interest on the balance sheet is Rs 120 crore. Control brings in all of the subsidiary's Rs 50 crore profit. The outside holders' 40%, Rs 20 crore, is shown as profit attributable to non-controlling interest. On the balance sheet, their 40% of Rs 300 crore of book equity sits as a separate line in equity.

Why take in all of a company you only partly own?

Think of a family that controls a shop it co-owns with a cousin. The family runs it, banks its takings and pays its bills, so to describe what the family controls you count the whole shop, then note that a share of the profit belongs to the cousin. Control, not ownership share, decides consolidation, so the parent adds 100% of the subsidiary's revenue, costs, assets and debt, then shows the minority's share of profit and equity on separate lines. That is why consolidated net income is 200 plus 50, Rs 250 crore, and not 200 plus 30.

Where does non-controlling interest show up on each statement?

On the income statement, net income of Rs 250 crore is split into Rs 230 crore for the parent's shareholders and Rs 20 crore for non-controlling interest. EPS uses the Rs 230 crore. On the balance sheet, non-controlling interest of Rs 120 crore sits inside total equity, next to the parent's own equity; if the subsidiary keeps its Rs 50 crore profit for the year, the line grows by Rs 20 crore. On the cash flow statement nothing is deducted, because the minority share of profit is not a cash payment; only dividends paid to the outside holders leave as cash, in financing. Non-controlling interest is a claim on the group by other shareholders, which is why an EV bridge adds it back alongside debt.

State your assumption: the minority is measured at its share of the subsidiary's book equity, with no fair-value uplift or goodwill allocated to it on acquisition. Under the full goodwill method the line would be larger. And note the boundary case: had the parent owned 40% without control, it would use the equity method, report the same Rs 230 crore of net income, but show none of the subsidiary's revenue or debt.

About 106 a year, on stated assumptions. Assume 15 crore domestic trips a year. A 180-seat jet, 85% full, flying 5 sectors a day for 350 days carries about 2.68 lakh passengers, so the fleet is about 560 jets. At 8% traffic growth it needs 1,209 in ten years, 65 more a year, plus about 41 retirements a year on a 20-year life.

How do you turn passengers into aircraft?

Think of a school deciding how many buses to buy. It counts the children who need a seat, divides by how many one bus can carry in a day, then adds buses for next year's bigger intake and for the old ones being scrapped. Aircraft demand is the same: fleet size is passenger trips divided by what one jet carries in a year, and new orders are fleet growth plus replacement. State each number as an assumption, because the interviewer cares about the structure first and the inputs second.

Start with traffic. Assume about 15 crore domestic passenger trips a year, and tell the interviewer you would check the current figure in the aviation regulator's monthly traffic data. A narrow-body has about 180 seats; assume 85% of them are filled, 5 sectors a day and 350 flying days. That is 267,750 passengers a jet a year, so today's fleet is about 560 jets.

Which assumption moves the answer most, and what did you leave out?

Growth carries more of the answer than replacement: at 8% a year the fleet more than doubles in ten years, adding 65 jets a year, while retirements on a 20-year life add 41. The number most worth defending is passengers per jet, because a sixth sector a day would cut the fleet by a sixth and every later number with it. Then name what you left out: international short-haul routes also fly narrow-bodies, airlines hold spare aircraft for maintenance, and engine problems can ground jets for months. Each pushes the true need above 106. Also say that orders and deliveries differ: airlines order years ahead, so order books can be far larger than a decade's need.

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.

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.

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.

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.

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.

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.