Financial Analysis puzzles, solved step by step

Equity falls from Rs 2,400 crore to Rs 800 crore, a 66.7% drop on a 20% fall in EV. Debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore; at 8x it is Rs 6,400 crore. Debt keeps its claim, so the equity slice absorbs the whole Rs 1,600 crore. At 7x leverage, equity moves 3.3 times as fast as EV, and a 7x multiple wipes it out.

Why does equity fall so much further than the business?

Think of a flat bought for Rs 1 crore with a Rs 70 lakh home loan. If the flat's price drops 20% to Rs 80 lakh, the bank still wants its Rs 70 lakh, so the owner's stake falls from Rs 30 lakh to Rs 10 lakh, a two-thirds loss. Debt is a fixed claim in rupees, so any change in enterprise value lands entirely on the equity that sits on top. Here debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore and equity is Rs 2,400 crore, only 30% of the value.

So what does 7x leverage tell you about sensitivity?

Leverage of 7x EBITDA means the debt alone is worth 7x EBITDA. Whenever the valuation multiple sits close to the leverage multiple, equity is a thin slice of the gap between them, and small changes in the multiple swing it hard. At 10x the gap is 3 turns; each turn of multiple is Rs 800 crore, a third of the equity. Software companies can carry this because their cash flows recur and their multiples are high, but the same 7x on a business valued at 8x leaves one turn of cushion. Say the limitation: the sum assumes debt is worth its face value, while in distress the debt itself would trade below par and soak up some of the loss.

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.

3/32 is 9.375%, 5/16 is 31.25% and 7/64 is 10.9375%. Anchor on 1/8 = 12.5% and halve: 1/16 is 6.25%, 1/32 is 3.125% and 1/64 is 1.5625%. Then multiply the rung by the numerator, or step from a nearby eighth: 7/64 is 1/8 less 1/64, which is 12.5 minus 1.5625.

Why should powers of two be easy?

Think of cutting a pizza: halve it, halve the halves, halve again, and you have eighths without measuring anything. Fractions with 2, 4, 8, 16, 32 or 64 at the bottom work the same way. Every power-of-two fraction is 12.5% halved some number of times, so you only need one anchor and the habit of halving. 1/8 is 12.5%, 1/16 is 6.25%, 1/32 is 3.125% and 1/64 is 1.5625%. The digits 125, 625, 3125, 15625 repeat in every problem of this kind.

When should you multiply up, and when should you step down from an eighth?

Small numerators multiply cleanly: 3 x 3.125 is 9.375 and 5 x 6.25 is 31.25. When the numerator sits one below a multiple of the anchor, subtract instead: 7/64 is 8/64 less 1/64. That turns a seven-times table of 1.5625 into one subtraction, 12.5 less 1.5625, which is 10.9375%. Checking 3/32 the same way, 4/32 is 12.5%, less 3.125% is 9.375%, which matches. Two routes agreeing is the habit that catches a slipped digit.

Where does this show up on a desk? Some bond prices are quoted in thirty-seconds of a point, so 3/32 of a point is 0.09375. Ownership splits and waterfall percentages often come in halves and quarters of halves. And an interviewer who asks this is really watching how fast you find a structure instead of reaching for longhand division.

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.

Market capitalisation is Rs 2,000 crore. With no other liabilities, assets equal debt plus equity. A D/E of 0.5 means debt is half of equity, so assets are 1.5 times equity and book equity is Rs 1,500 crore over 1.5, which is Rs 1,000 crore, with Rs 500 crore of debt. At 2.0x book, the market values that equity at Rs 2,000 crore.

How does a debt-to-equity ratio split the balance sheet?

Think of a Rs 15 lakh car bought with a loan that is half the size of your down payment. If the loan is half the down payment, the car cost one and a half down payments, so the down payment was Rs 10 lakh and the loan Rs 5 lakh. A D/E of 0.5 does not mean half the assets are debt; it means debt is half of equity, so assets are 1.5 times equity. Rs 1,500 crore over 1.5 gives book equity of Rs 1,000 crore and debt of Rs 500 crore.

What assumptions sit under the answer, and what could you add?

The question rules out other liabilities, such as payables and provisions, and that matters: a real balance sheet has plenty, and then assets are not just debt plus equity. Say the identity you are using, assets equal liabilities plus equity, before you lean on it. You can also go one step further than asked. With Rs 500 crore of debt and no cash given, enterprise value is about Rs 2,500 crore. And a P/B of 2.0 says the market thinks the assets earn more than their cost of capital; read it next to return on equity rather than on its own.

The new margin is 13.75%. Take revenue of 100, so cost is 90 and profit 10. Revenue grows to 120 and cost to 90 x 1.15 = 103.5, leaving profit of 16.5. The margin is 16.5 over 120, which is 13.75%. Profit itself grows 65%, because a 5 point gap between growth rates is large next to a 10% margin.

Why is the answer not simply 15%?

Think of a tea stall that takes in Rs 100 a day and spends Rs 90. If takings rise 20% and spending rises 15%, the extra takings are Rs 20 but the extra spending is Rs 13.50, because 15% is taken on 90, not on 100. Growth rates act on their own bases, so a margin moves by the gap in rupees, not by the gap in percentage points. Work in rupees on a base of 100 and the trap disappears: profit goes from 10 to 16.5 on revenue of 120.

Why does a thin margin make the move so large?

Profit is the small gap between two big numbers, so a modest difference in their growth rates is a big change to the gap. The thinner the starting margin, the more operating leverage a growth gap carries: here profit grows 65% on 20% revenue growth. Run the same 20% and 15% from a 30% margin and the new margin is 32.9%, a smaller relative move, because cost is a smaller share of revenue. Say the limit too: the formula assumes all cost grows at 15%. In practice fixed costs grow slowly and variable costs track volume, so you would split the cost line before trusting the answer.

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.

Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. The 4 already holds 3 litres, so it takes only 1 more. That is four moves. A longer route fills the 4 first and ends with 2 litres in the 4-litre bottle after six moves.

How do you avoid getting lost halfway through?

Think of following a recipe that says pour half the milk into the other pan: if you do not write down how much is in each pan, two steps later you have no idea. Track the pair of levels after every single move, written as (3-litre, 4-litre), and the puzzle becomes a short walk through a few states instead of a juggling act. There are only three kinds of move: fill a bottle from the tap, empty a bottle, or pour one into the other until the source is empty or the target is full.

Say the states out loud as you go: (0, 0), (3, 0), (0, 3), (3, 3), (2, 4). The interviewer can check each one in a second, and if you slip, you can see where.

Which amounts can these two bottles measure at all?

Every pour moves water in steps of 3 and 4, so the amounts you can isolate are the combinations of 3 and 4 with whole numbers in front: 4 minus 3 is 1, 3 times 2 minus 4 is 2, and so on. Because 3 and 4 share no common factor bigger than 1, these bottles can measure every whole number of litres from 1 to 4, and up to 7 if you count both bottles together. With a 4-litre and a 6-litre bottle you could never get an odd number, since every amount would be a multiple of 2. Saying that rule turns a party trick into reasoning.

About 15.4%. Picture 10,000 invoices. 100 are fraudulent and the screen flags 90 of them. 9,900 are clean and the screen flags 5% of them, 495. The flagged pile holds 585 invoices, of which 90 are fraud, so a flag means fraud only 90 times in 585. The rare base rate swamps the screen's accuracy.

Why is a 90% accurate screen right only 15% of the time?

Think of a smoke alarm that never misses a fire but also goes off whenever someone makes toast. In a house where fires are rare and toast is daily, almost every alarm is toast. When the thing you are hunting is rare, even a small false positive rate on the large innocent pile produces more false alarms than true hits. Here 5% of 9,900 clean invoices is 495 false flags, against only 90 true ones.

What would make the screen useful, and how would an auditor use it?

Cutting the false positive rate does far more than raising the catch rate. At a 0.5% false positive rate the flagged pile would be 90 frauds and about 49.5 clean invoices, and a flag would mean fraud 65% of the time. Raising the catch rate from 90% to 100% would only move the answer from 15.4% to about 16.8%. In practice a screen like this is a triage tool: it shrinks 10,000 invoices to 585 for a human to review, and that review is where the 495 false alarms are cleared. Say the limit too: the 1% base rate is itself an estimate, and the answer is only as good as it is.

The right DSO is about 61 days; the junior set a quarter's revenue against a full year of days. DSO is receivables over revenue times the days in the same period. Either annualise revenue, 450 x 4 = 1,800, giving 300 / 1,800 x 365 = 60.8 days, or keep the quarter and multiply by its 91.25 days. The 243 is exactly four times too large.

What does DSO actually measure?

Think of a tailor who sells Rs 1,000 of clothes a day and is owed Rs 30,000 by customers. Customers are, on average, 30 days behind. DSO is receivables divided by revenue per day, so the revenue and the day count must come from the same period. Rs 450 crore over a quarter is about Rs 4.93 crore a day, and Rs 300 crore of receivables is about 61 days of that.

How would you have caught 243 days before it went out?

Run a plausibility check before any number leaves your desk. 243 days says the average customer pays eight months after the sale, which almost no ordinary business tolerates, so the number should have triggered a second look on sight. A second check: compare with last quarter's DSO computed the same way. A sudden fourfold jump is almost always a formula or period error, not a collections crisis. Say the limitation of the corrected figure too: quarter-end receivables against average daily sales can swing with seasonality, so analysts often use average receivables over the period, and a strong last month of the quarter will push DSO up even when customers pay on time.