Financial Analysis puzzles, solved step by step

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.

112, and Rs 560 crore. For 17.5%, build it from blocks: 10% of 640 is 64, 5% is half of that, 32, and 2.5% is half again, 16; 64 + 32 + 16 = 112. For 12.5%, spot the fraction: it is one eighth, so halve 4,480 three times, to 2,240, 1,120 and 560. Check each with a second route before you say it.

How do you take an awkward percentage of anything in your head?

When a restaurant bill is Rs 640 and you want to tip 17.5%, nobody multiplies by 0.175. You find 10%, which is just moving the decimal point, and then cut pieces off it. Any percentage can be built from 10%, 5% (half of 10%), 1% and their halves, and each block is one easy operation on the last. For 17.5% of 640: 10% is 64, 5% is 32, 2.5% is 16, and the three add to 112.

Check with a second route: 20% is a fifth, 128, and 17.5% is 2.5 points less, so 128 minus 16 is 112. Two routes agreeing is what lets you say a number aloud with confidence in an interview where speed is being watched.

Which percentages are fractions in disguise?

Some percentages are clean fractions, and dividing is faster than building. 12.5% is one eighth, so 12.5% of Rs 4,480 crore is three halvings: 2,240, 1,120, 560. The same idea gives 37.5% as three eighths (560 x 3 = 1,680) and 87.5% as the whole less one eighth (4,480 minus 560 = 3,920). Desk interviewers like these because they turn up constantly in fee splits, stake sizes and tax at round rates.

What makes a mental answer trustworthy under time pressure?

Two habits. First, estimate before computing: 17.5% is a little under a fifth, so the answer must be a little under 128. If your arithmetic produces 1,120 or 11.2, the estimate catches the slipped decimal at once. Second, keep the units: the second answer is Rs 560 crore, not 560. An interviewer marks a right number with units and a sanity check far above a right number mumbled after a long silence. Speak the route as you go, so a slip is visible and recoverable.

In the year it happens, the capex cut: it adds the full Rs 10 crore to free cash flow, while more revenue or lower cost of goods sold adds Rs 7.5 crore after 25% tax. Capex is a direct cash item. If the change repeats every year, lower capex means lower depreciation and more tax, so all three converge on Rs 7.5 crore a year, and an exit multiple on EBITDA favours revenue and cost.

Why does capex hit cash harder than revenue or cost?

If you earn Rs 10,000 more, you keep what is left after income tax. If you decide not to buy a Rs 10,000 laptop, you keep all Rs 10,000, because buying it was never a deduction in that year. Revenue and cost changes reach free cash flow after tax; a capex change reaches it in full, because capex is subtracted below the tax line. So with a 25% tax rate, Rs 10 crore more revenue or Rs 10 crore less cost of goods sold adds Rs 7.5 crore, and Rs 10 crore less capex adds Rs 10 crore.

One assumption sits under the revenue answer: that the extra Rs 10 crore comes with no extra cost and no extra working capital. In practice more revenue usually needs more receivables and stock, which takes a little more away. Say so in one clause.

What changes when the change repeats every year?

Capex becomes depreciation. Spend Rs 10 crore less every year on assets depreciated over five years, and depreciation falls by Rs 2 crore more each year until it is Rs 10 crore lower, which raises tax by Rs 2.5 crore. In steady state a recurring capex cut adds Rs 10 crore of cash less Rs 2.5 crore of lost tax shield, Rs 7.5 crore a year: exactly the same as the revenue and cost changes. The capex cut's edge is timing, worth something in present value, but not a permanent advantage.

Even a one-off cut keeps only part of its edge. Discounted at 10%, Rs 10 crore saved next year is worth Rs 9.09 crore, but the depreciation shield given up over the following five years is worth Rs 1.72 crore, leaving Rs 7.37 crore against Rs 6.82 crore for a one-year revenue gain. Still ahead, by much less than Rs 2.5 crore.

How does the terminal value change the ranking?

If the terminal value uses an exit multiple of EBITDA, a permanent Rs 10 crore of extra revenue or lower cost raises terminal EBITDA by Rs 10 crore and the terminal value by Rs 80 crore at 8x, while a capex cut does not touch EBITDA and adds nothing through the multiple. With an exit multiple, recurring revenue and cost changes beat capex by a wide margin; with a perpetuity growth terminal, all three are equal in steady state. Name which terminal method you are assuming before you rank them. And add the business limit: capex cut without consequence is rare, because the assets usually drive future revenue.

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.

134,547. Split 301 into 300 and 1. Three hundred times 447 is 447 x 3 with two zeros: 1,341, then 134,100. Add one more 447: 134,547. Check before you say it: 300 x 450 is 135,000, so the answer should sit a little below that; the last digit must be 1 x 7, a 7; and the digit sums agree, since 301 reduces to 4, 447 to 6, 4 x 6 = 24 reduces to 6, and 1+3+4+5+4+7 = 24 also reduces to 6.

Why split 301 rather than attack both numbers?

If a shop sells 301 items at Rs 447 each, nobody in the shop multiplies 301 by 447 digit by digit. They price 300 items and add one more. Break the awkward factor into a round number and a small remainder, multiply the round part first, and the remainder is one easy addition. 447 x 3 is 1,200 plus 141, which is 1,341; two zeros make it 134,100; plus 447 is 134,547. Say each intermediate number aloud as you reach it, so the interviewer can follow and a slip is caught before it compounds.

The split can go the other way too. 447 is 450 less 3, so 301 x 447 = 301 x 450 minus 301 x 3. The first is 300 x 450 plus 450, 135,450; the second is 903; the difference is 134,547. Having two routes is what turns a hopeful answer into a checked one.

How do you check an answer you cannot write down?

Three checks, each a few seconds. The estimate: 300 x 450 is 135,000, and you used one factor slightly above and one slightly below, so the answer should be close to it, which 134,547 is. The last digit: 1 x 7 ends in 7, and so does the answer. The digit sums, an old bookkeeper's test: reduce each number to a single digit by adding its digits, 301 gives 4 and 447 gives 15 then 6; the product 24 reduces to 6, and the answer's digits 1+3+4+5+4+7 = 24 also reduce to 6. A wrong answer rarely survives all three checks, and running them aloud shows the interviewer a habit, not a lucky number. The digit-sum test has a blind spot: it misses errors that are exact multiples of 9, including two digits swapped, which is why the estimate sits beside it.

What is the interviewer marking?

Pace, an audible route and a check, in that order. A right answer after a long silence scores below a right answer reached in three spoken steps, because the desk wants to hear how you handle a number, not whether you can. The limitation is that this trick suits factors near a round number. For something like 347 x 683, say that you would estimate, 350 x 700 less a bit, give about 237,000, and label it an estimate; pretending to exact arithmetic you cannot check is the real failure.

36 ways for 10, 45 for 11, and (n minus 1)(n minus 2) / 2 in general. Write 10 as a row of ten ones. Three positive parts means two dividers in two different gaps between the ones, and there are nine gaps, so the count is C(9, 2) = 36. For 11 there are ten gaps: C(10, 2) = 45. For any n it is C(n minus 1, 2). If order did not matter, 10 has only 8 splits, and if zero were allowed it would have 66, so say which version you are answering.

How do you turn 'three numbers add to 10' into something you can count?

Ten sweets in a row, three children, each must get at least one. You hand them out by cutting the row in two places, and every pair of cuts is one way to share. Each ordered triple is one choice of two distinct gaps out of the nine between ten objects, so counting triples is the same as counting pairs of gaps: 9 choose 2. That is 9 x 8 / 2 = 36. The same picture gives 11 at once: eleven sweets have ten gaps, and 10 x 9 / 2 = 45.

Check the small cases before trusting the formula. A total of 3 can only be 1 + 1 + 1, one way, and C(2, 2) = 1. A total of 4 gives 1 + 1 + 2 in three orders, and C(3, 2) = 3. A total of 5 gives six, C(4, 2). The counts 1, 3, 6, 10, 15 are the triangular numbers, which is what (n minus 1)(n minus 2) / 2 produces.

What changes if the interviewer meant something else?

Three variants are common, and the right first move is to ask or to state your reading. If zero is allowed, the dividers may share a gap or sit at the ends, and the count becomes C(n + 2, 2), which is 66 for 10. If order does not matter, you are counting partitions of 10 into three parts: 1+1+8, 1+2+7, 1+3+6, 1+4+5, 2+2+6, 2+3+5, 2+4+4, 3+3+4, 8 in all, and there is no neat formula, so you list them from the smallest part upwards. Say which version you are answering, then answer it; the count is easy once the question is fixed, and most lost marks here come from solving a different question than the one asked. If negative numbers were allowed the count would be infinite, which is worth one sentence to show you noticed.

Why would a trading desk ask this?

Counting outcomes is the floor under every probability question, and the follow-up about 11 tests whether you hold a method or a memorised answer. The natural next step is three dice: how many of the 216 rolls add to 10? Start from 36 ordered triples and remove those with a part above 6: an 8 with two 1s gives 3 orders, a 7 with 1 and 2 gives 6, so 36 minus 9 is 27, and the chance is 27/216, 12.5%. The limitation of the gap method is exactly that it has no upper bound on any part, so a capped problem needs the subtraction.