Financial Analysis puzzles, solved step by step

Put the lease liability inside EV whenever the rent is outside EBITDA. After capitalisation the leaser's EBITDA is Rs 100 crore, like the owner's. Its shares and debt are worth Rs 723 crore, so dividing those alone gives 7.2x and makes it look cheaper. Add the Rs 77 crore lease and it is 8.0x, the owner's multiple. The numerator and denominator must describe the same claims.

Why does a lease behave like debt?

Think of two families in identical flats. One bought with a home loan; the other rents on a ten year agreement it cannot walk away from. Both owe fixed payments for years. A long lease is a loan from the landlord, repaid in rent, so the rent contains both the use of the asset and the financing of it. Under IFRS 16, and Ind AS 116 in India, the leaser now shows that promise as a lease liabilityThe present value of the rent the company is contractually committed to pay over the lease term, carried on the balance sheet like a borrowing.: Rs 12 crore a year for ten years, discounted at 9%, is about Rs 77 crore.

What changes in the numbers, and what does not?

Capitalisation moves the rent out of operating costs. It comes back as depreciation on a right-of-use asset and interest on the lease, both below EBITDA. So the leaser's EBITDA jumps by the full Rs 12 crore while nothing about its stores, customers or cash has changed. The same move puts about Rs 77 crore of lease on the balance sheet. The two changes are a pair, and a fair multiple has to use both halves or neither.

Where does this bite in real comparables work?

Data providers and peer tables do not always treat leases the same way, and a peer set can mix companies reporting under different standards. Before trusting a multiple, check whether its EV includes lease liabilities and whether its EBITDA is before or after rent, then make every company in the table match. A retailer, airline or restaurant chain that leases most of its sites can look 11% cheaper than an owner purely from this mismatch, which is the whole of the gap in this example.

Say the limitation too. The capitalised figure depends on the discount rate and the lease term the company chose, so two leasers with the same rent can carry different liabilities. Lease-adjusted multiples are better, not exact.

X carries more debt and owns a stake in an associate. Debt of Rs 750 crore at 8% costs 5.6% after tax, less than the 6.7% its operations earn on their value, so levering lowers the P/E. The associate adds Rs 22 crore to net income but nothing to EBITDA, while its Rs 300 crore value sits inside EV. Strip it out and X is 7.0x, like its peers.

Why can two multiples on the same company disagree?

Picture a house with a mortgage. Its price compared with the rent it earns is one ratio; your equity in it compared with the rent left after the mortgage payment is another. The two only agree if there is no loan. EV/EBITDA looks at the whole business before financing, while P/E looks at the shareholders' slice after interest, tax and anything non-operating. Every gap between them is explained by something that sits between EBITDA and net income, or between EV and equity value.

What numbers make both facts true?

Give both companies the same operations: EBITDA Rs 150 crore, depreciation Rs 50 crore, operating profit Rs 100 crore, tax 30%. The peer has no debt, earns Rs 70 crore and is worth Rs 1,050 crore: 15.0x earnings and 7.0x EBITDA. X borrows Rs 750 crore at 8%, so interest of Rs 60 crore leaves Rs 28 crore from operations, and it books Rs 22 crore as its share of an associateA company in which the group holds a significant minority stake, usually 20% to 50%. The group books its share of that company profit below operating profit, never in revenue or EBITDA.'s profit. Net income of Rs 50 crore at 12x is equity of Rs 600 crore, and with the debt that is an EV of Rs 1,350 crore, 9.0x EBITDA.

Which reason moves which multiple?

Separate them, because the follow-up always asks. Debt lowers the P/E when its after-tax cost is below the earnings yield of the operations. Here debt costs 8% times 0.7, which is 5.6%, while the operations earn 70 on 1,050, or 6.7%. Swapping expensive equity for cheaper debt leaves the operating slice at about 10.7x earnings. The associate does the other job: its Rs 300 crore of value sits inside EV while its profit sits below EBITDA, which lifts EV/EBITDA from 7.0x to 9.0x.

Name a third candidate if you have time: a lower tax rate than peers raises net income without touching EBITDA, so it also lowers P/E alone. Say what you would check to choose between them: the notes on debt, associates and the effective tax rate.

Flip twice: heads then tails counts as heads, tails then heads counts as tails, and anything else is thrown away and flipped again. The two mixed orders each have probability 2/9, so they are equally likely whatever the bias. A pair succeeds 4/9 of the time, so each fair result takes 9/4 pairs, or 4.5 flips. Knowing the bias is exactly 1/3 lets you cut that to 2.25.

Why are heads-tails and tails-heads always equally likely?

Picture two friends flipping the same lopsided coin, one after the other. The chance the first gets heads and the second tails is the heads chance times the tails chance. The chance of the reverse is the tails chance times the heads chance. Multiplication does not care about order, so the two mixed outcomes are exactly equally likely, whatever the bias. That symmetry is the whole trick, known as the von Neumann method. Here each mixed pair has probability 1/3 times 2/3, which is 2/9.

How do you get the average of 4.5 flips?

Each pair either works or does not, independently of the last. Waiting for a success that happens with probability q takes 1/q tries on average, the same reason a die takes six rolls on average to show a six. A pair works with probability 4/9, so you need 9/4 pairs, and two flips a pair makes 4.5 flips. The method pays for its fairness with waste: 5 pairs in 9 are thrown away.

Can you do better if you know the bias exactly?

Yes, and this is usually the follow-up. With p exactly 1/3, tails-tails has probability 4/9, the same as the two mixed pairs together. Call tails-tails one side and either mixed pair the other, and only heads-heads, 1/9 of pairs, is wasted, so each fair result costs 2 times 9/8, or 2.25 flips. The von Neumann method is still the better answer when nobody tells you the bias, because it works for any p. The limit for any scheme is set by how much randomness one flip carries: about 0.92 of a fair bit here, so no method can beat roughly 1.09 flips per fair result on average.

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.

Anywhere from about -0.75 to 0.95. Read each correlation as the cosine of an angle between two arrows. X is 78.5 degrees from Y, and Z is 60 degrees from Y. Z can swing to the same side as X, leaving them 18.5 degrees apart, or to the other side, 138.5 degrees apart. The cosines of those angles are 0.9485 and -0.7485.

Why can a correlation be drawn as an angle?

Think of three people walking away from the same lamp post. Knowing how far apart the first two point, and how far apart the second and third point, limits how far apart the first and third can point, but only loosely. If you standardise each variable, its correlation with another is the cosine of the angle between them, so correlations must behave like angles in space. A correlation of 0.2 is an angle of 78.5 degrees; 0.5 is 60 degrees.

Where does the formula come from, and when does it bite?

The three-by-three correlation matrix must not imply a negative variance for any mix of X, Y and Z, which means its determinant cannot go below zero. Solving that condition gives the formula above, the cosine rule for the difference and sum of two angles. The constraint is loose when the known correlations are weak and tight when they are strong. With 0.2 and 0.5, almost the whole scale stays open. With 0.9 and 0.9, X and Z must correlate at least 0.62: two things each closely tied to Y cannot drift far from each other. Say the practical use: a risk model that fills in missing correlations by hand can break this rule and produce a matrix that is not valid.

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.