Financial Analysis puzzles, solved step by step

Play: one roll pays 30 x 3.5 = Rs 105 on average against a Rs 100 fee, an edge of Rs 5. With the re-roll, keep a 3 or better and re-roll a 1 or 2; the game is worth Rs 118.33, an edge of Rs 18.33. A fresh roll is worth 105 less the Rs 20 fee, 85, so re-roll only a result paying less than 85: 30 and 60 qualify, 90 does not.

Why is the first game worth playing?

A fair die averages 3.5, so the average payout is 30 x 3.5 = Rs 105, more than the Rs 100 you pay. On expected value the game pays Rs 5 a play, so a player who can repeat it should play. Half the time you lose money, since a 1, 2 or 3 pays 30, 60 or 90, and the other half you win 120, 150 or 180; the wins are larger than the losses, which is where the edge sits.

How does the re-roll change the rule?

Think of returning a shirt. You keep it if what you have is worth more than what the shop will hand you after the restocking fee. Re-roll only when the roll in hand is worth less than a fresh roll net of its fee, which is 105 - 20 = Rs 85. A 1 pays 30 and a 2 pays 60, both below 85: re-roll. A 3 pays 90, above 85: keep it, even though 3 is below the average face. The fee is what makes a 3 worth keeping.

What does the option itself cost and earn?

The re-roll lifts the game from 105 to 118.33, so the option is worth Rs 13.33 on average, and it also cuts the chance of losing money from a half to a third, because a bad first roll gets a second chance. Re-rolling a 3 as well would give 117.50, worse by 0.83; a free re-roll would make re-rolling the 3 right and lift the game to 127.5. The threshold moves with the fee: below a fee of Rs 15 the 3 is worth re-rolling, above it the 3 is kept. The limit to say: the worst path, a 1 or 2 followed by a 1, costs Rs 90, and a player who cannot afford that should not value the game by its average.

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.

Roughly Rs 70,000 to 75,000 crore a year, on the assumptions below. Start from an assumed 1.8 crore two-wheelers sold a year, assume a little over half are bought on credit, and lend about three quarters of the price. Splitting the market into commuter motorcycles, scooters and premium bikes gives about 99 lakh loans averaging Rs 73,600, or Rs 72,832 crore. Every input is an assumption to state and then confirm.

What is the structure before any number?

To estimate what a college canteen sells on credit, you would count meals, the share put on a tab, and the average tab. Loans work the same way. A finance market size is units times financing penetration times ticket size, and saying that structure first lets the interviewer follow every number after it. Here the units are new two-wheelers sold in a year, penetration is the share bought on a loan, and the ticket is the price less the down payment.

Assume about 1.8 crore two-wheelers are sold in India a year. Treat that as an assumption to confirm against the industry body's current sales data, never a fact to quote. Prices range from a basic commuter bike to a premium motorcycle, so one average price is fragile; split the market three ways instead.

How do you check it a second way?

Run it top-down in one line: 1.8 crore units x 55% financed x 75% of an average Rs 1,00,000 price is Rs 74,250 crore. Two routes landing within about 2% of each other is the check, and the segment split earns its place by showing where the uncertainty lives. The softest input is the financed share: every 5 points on it moves the answer by about Rs 6,621 crore. Then a feel check: 99 lakh loans a year is about 27,123 loans a day across the country, which is plausible for a mass market sold through thousands of dealers.

Say what the number is not. It is a yearly flow of new loans. The loan book outstanding at any time is a stock: with loans running about two and a half years and repaid evenly, the average loan is half outstanding for that period, so the book is roughly 1.25 times a year's disbursement, about Rs 91,041 crore. Used-vehicle loans are excluded.

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.

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.

The zero falls more: about 8.8% against about 6.4% for the coupon bond. Price sensitivity follows duration, the present-value-weighted average time to each cash flow. The zero's only cash flow is at year 10, so its duration is 10 years. The coupon bond returns part of its value early, so its duration is 7.25 years. Modified duration predicts falls of 9.26% and 6.71%; convexity makes the actual falls slightly smaller.

Why does the timing of cash flows decide the sensitivity?

Two friends each owe you Rs 1,000 in total. One will repay it all in ten years; the other pays Rs 80 a year and the rest at the end. If prices start rising faster and money loses value faster, which IOU loses more value? The one where every rupee is ten years away. The second friend's early payments are already in your hands and can be reinvested at the new higher rates. A bond's price sensitivity to yield depends on when its value arrives on average, not on its final maturity date. That average, weighted by present value, is the Macaulay duration.

How big is each fall?

Use modified duration, Macaulay duration divided by one plus the yield, as the percentage price change per point of yield. For the zero, 10 / 1.08 = 9.26; for the coupon bond, 7.25 / 1.08 = 6.71. So a one point rise should cut prices by about 9.3% and 6.7%. Repricing exactly: the zero goes from 46.32 to 42.24, down 8.80%; the coupon bond goes from par, 100.00, to 93.58, down 6.42%.

Why are the actual falls smaller than duration predicts?

Duration is the slope of the price curve at today's yield, a straight-line estimate. The real curve bows outward, so as yields rise each further step hurts a little less, and as yields fall each step helps a little more. That curvature, convexity, makes a plain bond fall less than duration predicts when yields rise and gain more than predicted when they fall. For a one point move the gap is small, under half a point here; for larger moves, add the convexity term or simply reprice the bond.

Two practical notes. Most Indian government bonds pay coupons twice a year, which shortens the duration slightly but leaves the answer unchanged. And the gap between the bonds widens with maturity and narrows as coupons fall: a coupon bond's duration always sits below its maturity, while a zero's always equals it.

A's overall rate is 9.2%, B's is 3.9%, and A is still the better underwriter in each segment. A: 2 defaults among 100 salaried plus 90 among 900 self-employed, 92 in 1,000. B: 27 plus 12, 39 in 1,000. An overall rate is a weighted average of segment rates, and the weights are the mix of the book. A's book is 90% self-employed, where everyone's rate is high, so its average is pulled up by whom it lends to, not by how well it lends.

How can a branch be better at everything and worse in total?

Two surgeons. One takes the hardest cases in the hospital and loses fewer of them than anyone else would; the other takes mostly routine cases. Count deaths per patient and the first surgeon looks worse, because her patients were sicker to begin with. An overall rate is a weighted average of segment rates, and the weights are the mix of the book, so a branch that lends mostly to the riskier segment carries a higher average even when it is better in every segment. A's rate is 0.1 x 2% + 0.9 x 10% = 9.2%; B's is 0.9 x 3% + 0.1 x 12% = 3.9%.

How do you compare the two branches fairly?

Give them the same customers. Apply each branch's segment rates to one common mix and the mix stops voting. At B's mix, 90% salaried, A's rates give 0.9 x 2% + 0.1 x 10% = 2.8% against B's actual 3.9%. At A's mix, B's rates give 0.1 x 3% + 0.9 x 12% = 11.1% against A's actual 9.2%. At a 50:50 mix, A is 6.0% and B is 7.5%. On any common mix A is lower, so A underwrites better; its higher total is the price of the book it chose to build. This reversal, where a comparison flips when groups are combined, is known as Simpson's paradox.

What should the analyst say beyond the arithmetic?

Three things. First, both statements are true at once: A underwrites better, and A's book loses more money, because capital is charged on actual losses and A chose a riskier book. Whether that was a good choice depends on the pricing: if A earns enough extra yield on self-employed loans to cover the extra 8 points of defaults, its strategy can be sound. Second, ask why the mixes differ, since location, product and targets all shape who walks in. Before judging two rates, ask whether the two populations are the same; if they are not, standardise to one mix or compare segment by segment.

Third, say the limitation. Each segment is itself a mix: of loan sizes, vintages and cities. A's salaried borrowers might be larger-ticket government employees and B's might be gig workers on salary slips, and a finer split could flip the comparison back. Standardising to the segments you can see removes the mix you know about, never the mix you do not. The honest close is that A looks the better underwriter on this cut, and you would want the segment rates by vintage before signing that view.

About 10.4% a year, forever, with the company paying out 48% of its earnings. A forward P/E equals the payout ratio divided by the cost of equity less growth, and with a fixed ROE the payout is set by growth: payout = 1 minus g/ROE. So 30 = (1 minus g/0.20) / (0.12 minus g). Multiply out: 3.6 minus 30g = 1 minus 5g, so 25g = 2.6 and g = 10.4%. Check: a dividend yield of 48% / 30 = 1.6% plus 10.4% growth is the 12% cost of equity.

Where does a multiple hide a growth forecast?

If someone offers to sell you a shop for thirty years of its current profit, they are not quoting a price; they are telling you how fast they expect the profit to grow. The multiple is the forecast. Under the Gordon growth model a share is worth next year's dividend over (r minus g), so dividing by next year's earnings gives P/E = payout / (r minus g), and a given P/E can be solved for g. The one thing people forget is that the payout is not free to choose: growth has to be paid for with retained profit.

With a fixed return on reinvested profit, growth equals ROE times the retention rate, so retention is g/ROE and payout is 1 minus g/ROE. At a 20% ROE, growing at 10% means keeping half the earnings. Put that into the multiple: 30 = (1 minus 5g) / (0.12 minus g). Cross-multiply: 3.6 minus 30g = 1 minus 5g, so 2.6 = 25g and g = 10.4%. The implied payout is 1 minus 0.104/0.20 = 48%, and the implied dividend yield is 48% of a 3.33% earnings yield, 1.6%.

Why does the ROE matter as much as the growth rate?

Because the ROE sets how much growth costs. At a 15% ROE the same 30x needs 11.1% growth, since a bigger share of earnings has to be retained to fund each point of it. At an ROE equal to the cost of equity, 12%, the multiple is 1/r = 8.3x whatever the growth rate, because every retained rupee earns exactly what shareholders could earn elsewhere. Growth adds value only when the reinvested profit earns more than the cost of equity; the multiple prices the spread between ROE and r as much as it prices g. The curve is also steep near the answer: at 10% the multiple is 25x, at 11% it is 45x, so a 30x stock is one point of growth away from either 25x or 45x.

Now say the limitation. 10.4% growth forever, only 1.6 points below the discount rate, is not a forecast anyone would defend; no company outgrows the economy indefinitely. The honest reading is that 30x prices a long period of fast growth that will fade, and a two-stage model, say 10.4% for a decade and a lower rate after, is the next thing to build. The one-line solve is still worth doing, because it converts a multiple into a sentence you can argue with: the market expects this company to compound earnings at about 10% for a very long time while earning 20% on what it retains.

Six tosses for HH on average, four for HT. For HH, a tail after a head sends you back to the start with nothing. For HT, once you hold a head, every further head keeps you one toss from finishing, so nothing is lost. Solve HH with two unknowns: from the start E0 = 1 + E0/2 + E1/2, and after a head E1 = 1 + E0/2, which gives E1 = 4 and E0 = 6. For HT, E1 = 1 + E1/2 = 2 and E0 = 4.

Why is waiting for two heads different from waiting for a head then a tail?

Suppose you will go trekking after two dry days in a row. One wet day after a dry one puts you back to zero: you need two fresh dry days. Now suppose instead you are waiting for a dry day followed by a wet one. Extra dry days cost nothing; you stay ready, and the first wet day finishes the job. A failed attempt at HH throws away the progress you had, while a failed attempt at HT keeps it, so HH pays for its failures twice: the wasted toss and the lost head. That asymmetry is the whole answer; the algebra just prices it.

Write the states. E0 is the expected tosses still needed from nothing, E1 from one head in hand. From nothing, one toss is spent and you land back at E0 with a tail or at E1 with a head: E0 = 1 + E0/2 + E1/2, so E0 = 2 + E1. For HH, from one head a head finishes and a tail drops you to E0: E1 = 1 + E0/2. Substitute: E1 = 1 + (2 + E1)/2, so E1/2 = 2, E1 = 4 and E0 = 6. For HT, from one head a tail finishes and a head keeps you at E1: E1 = 1 + E1/2, so E1 = 2 and E0 = 4.

Is there a rule for longer patterns?

Yes, and it is quick enough for the room. For each length k at which the pattern's first k symbols equal its last k symbols, add 2 to the power k. HH matches itself at k = 1 (H and H) and at k = 2 (the whole thing): 2 + 4 = 6. HT matches itself only at k = 2: 4. HHH gives 2 + 4 + 8 = 14; HTH gives 2 + 8 = 10; HTT gives just 8. Patterns that overlap with themselves take longer to appear from a fresh start, because their occurrences come in clusters, and the same long-run frequency spread into clusters means longer gaps between them. Every pattern of length n shows up once in 2 to the n positions on average; what differs is how bunched those appearances are.

The average hides a wide spread. The chance that HH has not appeared after ten tosses is 14.1%, against 1.1% for HT, so one game in seven is still running at toss ten when you wait for HH. And the figures assume a fair, independent coin: with heads at 60%, the HH wait falls to 1/p + 1/p squared, about 4.4 tosses. The finance link is any rule that needs consecutive results, such as a bonus or a covenant test that requires two good quarters in a row: one bad quarter resets the count, so the rule is harder to satisfy than its two-in-four odds suggest.