Financial Analysis puzzles, solved step by step

The competitor keeps more: Rs 25 against Rs 20 on every Rs 100 of sales. Cost plus 25% means a price of 1.25 times cost, so on Rs 100 of sales cost is Rs 80 and the margin is 20%. A 25% gross margin needs cost of Rs 75 on a price of Rs 100, which is a markup of 25 over 75, or 33.3%.

Why does the same 25% mean different money?

Think of a vegetable seller who buys tomatoes for Rs 80 a kilo and adds Rs 20. To him, that is 25% on what he paid. To his customer, Rs 20 out of the Rs 100 she paid is 20%. Same Rs 20, two percentages. Markup divides profit by cost; margin divides it by price, and price is always the bigger number, so a margin is always smaller than the markup that produced it. The distributor's 25% markup is a 20% margin; the competitor's 25% margin is a 33.3% markup.

Where does this mix-up cost real money?

In pricing conversations, sales teams often quote markups because the numbers sound larger, while finance teams report margins. If a target says 25% margin and the price list is built at cost plus 25%, every sale falls 5 rupees short per Rs 100, a fifth of the profit the plan assumed. Discounts make it worse: a 10% discount off a cost-plus-25% price leaves a price of Rs 90 on cost of Rs 80, an 11.1% margin, so half the profit per unit is gone. Say the conversion formula out loud whenever someone gives you a percentage of profit, and ask which base it is on.

Net income rises Rs 48 crore, cash rises Rs 88 crore, and the balance sheet balances at plus Rs 48 crore. The gain of Rs 60 crore less Rs 12 crore tax gives net income of Rs 48 crore. Cash from operations starts at 48 and subtracts the Rs 60 crore gain, leaving minus 12; investing shows the full Rs 100 crore. Cash is up Rs 88 crore, land down Rs 40 crore.

Why subtract a gain you actually made?

Think of selling your old scooter for Rs 50,000 when you had it on your books at Rs 20,000. You did receive Rs 50,000 in cash, all of it, on the day of the sale. If your diary records a Rs 30,000 profit under daily earnings and also Rs 50,000 under scooter sold, you have counted Rs 30,000 twice. The cash flow statement puts the full sale proceeds in investing, so the gain sitting inside net income has to be taken out of operations, or part of the sale would be counted twice.

How does each statement move, line by line?

Income statement: a gain on sale of Rs 60 crore, tax of 20% on it, Rs 12 crore, so net income is up Rs 48 crore. Cash flow statement: start from net income of 48, subtract the non-operating gain of 60, and cash from operations is minus 12; investing shows plus 100; cash is up 88. Balance sheet: cash up 88, land down 40, so assets are up 48; retained earnings up 48. The check that matters is that the change in assets equals the change in equity, Rs 48 crore each side.

State the assumption: the tax is paid in cash in the same year. If it were deferred, operations would show zero, a tax liability would sit on the balance sheet and cash would be up the full Rs 100 crore. Say also why the tax stays in operations: accounting rules generally classify income taxes as operating unless they are specifically tied to an investing activity, and practice varies, so confirm the treatment the company uses.

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.

The true CAGR is 14.9%, and the second series compounds at 12.5%, not the 15% its average suggests. Doubling in five years means (200/100) to the power one-fifth, less one: 14.87%. For the yearly series, multiply the growth factors: 1.5 x 0.8 x 1.3 x 1.1 x 1.05 = 1.8018, whose fifth root gives 12.50%. A simple average ignores both compounding and volatility.

Why is doubling in five years not 20% a year?

Think of a savings account that pays interest on interest. To double your money in five years you need less than 20% a year, because each year's growth is earned on a larger balance. Total growth divided by the number of years ignores compounding, so it always overstates the annual rate for a gain. The rate that compounds 100 to 200 in five years is the fifth root of 2 less one, 14.87%. Compounding at the claimed 20% would have produced 248.8.

Why does an uneven path lose so much to its average?

A fall hurts more than an equal rise helps, because the rise then starts from a smaller base: 100 up 50% is 150, down 20% is 120, not 130. The gap between the average and the compound rate grows with how much the yearly rates vary; roughly, CAGR is the average less half the variance of the rates. Here the rates have a variance of 0.056, so the shortcut gives 15% less 2.8 points, about 12.2%, close to the exact 12.5%. Use this when a pitch quotes average growth: ask for the start and end values and compute the compound rate yourself.

About 11.1%. Before, Rs 50 buys 100 g, Rs 0.500 a gram. After, Rs 50 buys 90 g, Rs 0.556 a gram. The rise is 100 over 90 less one, which is 10 over 90, or 11.1%. In general a size cut of x at the same price is a price rise of x divided by (1 minus x), always a little more than the cut.

Why is a 10% smaller pack more than a 10% price rise?

Think of sharing the same Rs 50 bill among 10 friends instead of 11. Each pays more, and the increase is measured against the new, smaller headcount. Price per gram is price divided by grams, so cutting the grams by 10% divides the price by 0.9, which raises it by 1/0.9 less one, 11.1%. The percentage fall in grams is measured on 100; the percentage rise in price per gram is measured on what is left, 90.

How would a finance team use this?

Inflation in raw materials pushes consumer companies to protect a price point that shoppers remember, such as Rs 10 or Rs 50, and adjust grams instead. To compare price changes across products, convert every pack-size change into a price per unit, or a volume decline will be misread as stable pricing. In a revenue bridge, the same rupee sales on fewer grams show up as a price gain, not as flat sales. Say the limitation: price per gram is the right comparison only if quality and the mix of the pack stay the same.

It under-forecasts by Rs 20 lakh every month. On a straight-line trend, the average of the last three months equals the middle month, so the average lags the latest month by one month. The forecast is for the following month, one more month ahead. Two months of Rs 10 lakh growth is Rs 20 lakh. For a window of n months, the miss is the monthly growth times (n + 1) / 2.

Why does an average lag behind a trend?

Think of judging how fast a child is growing by averaging her height over the last three birthdays. The average describes her at the middle birthday, not today, and certainly not next year. A moving average describes the middle of its window, so on a rising trend it always sits below the latest value and further below the next one. With months 4, 5 and 6 at 130, 140 and 150, the average is 140, the month 5 figure, while month 7 will be 160.

What does that mean for the window you choose?

A longer window smooths noise better but lags more: a 12-month average on the same trend would miss by 10 x 6.5 = Rs 65 lakh. Moving averages trade noise against lag, so they suit flat, noisy series and fail on trending ones. The fix on a trend is to model the trend itself: add the slope back, or use a method that tracks level and slope separately, such as Holt's linear exponential smoothing. Say the limit both ways: if sales are seasonal, a three-month average also mixes high and low months, and if the trend turns, any trend-adjusted method overshoots for a while.

Unlevered free cash flow is Rs 1,00,000 a year and the NPV is about Rs 1,43,308. Sales are Rs 4,50,000, goods Rs 2,70,000 and rent Rs 60,000, so EBITDA is Rs 1,20,000. Less Rs 40,000 of depreciation is EBIT of Rs 80,000; after 25% tax, NOPAT is Rs 60,000. Adding back depreciation gives Rs 1,00,000. Five years of that at 14% is worth Rs 3,43,308, less the Rs 2 lakh machine.

How do you get from sales to unlevered free cash flow?

Think of a tea stall owner counting what she can take home each year: takings, less tea and milk, less rent, less the tax man's share. The stall's old kettle wearing out is a cost on paper, but no cash leaves her purse for it each year. Unlevered free cash flow is the cash the asset throws off before any financing: operating profit after tax, plus non-cash charges, less the capital spending and working capital it needs. Here: sales of Rs 4,50,000 less goods of Rs 2,70,000 and rent of Rs 60,000 is EBITDA of Rs 1,20,000. Depreciation of Rs 40,000 gives EBIT of Rs 80,000, tax takes Rs 20,000, and NOPAT is Rs 60,000.

Why does depreciation matter only through tax?

Depreciation is subtracted to reach EBIT and added back to reach cash flow, so on its own it washes out. Its only cash effect is the tax it saves: Rs 40,000 of depreciation at 25% cuts tax by Rs 10,000 a year. That is why taxing EBITDA directly is wrong: it would give Rs 90,000 a year and lose the shield. The answer leans on assumptions you should say out loud: no working capital for stock in the machine, no repairs, no salvage value, a steady 50 sales a day, and a 14% rate that reflects the risk of the location. The NPV is positive by a wide margin, so the decision is most sensitive to daily volume, the one number nobody can check in advance.

1/7 is 0.142857 repeating, 5/7 of 910 is 650, and 3/7 is 42.86%. 910 divides cleanly by 7 to give 130, and five of those is 650. For any seventh, the six digits 142857 repeat in the same order; only the starting digit changes. 3/7 is about 0.43, so it starts at the 4: 0.428571, or 42.86%.

Why should sevenths be memorable when they look messy?

Think of a clock face with six numbers on it instead of twelve. Whatever hour you start from, you read the same numbers in the same order. Sevenths work like that: every k/7 is the cycle 1, 4, 2, 8, 5, 7 read round the ring from a different starting digit. You only need to learn one string, 142857, and a way to pick the start: k/7 is roughly k x 0.14, so 2/7 starts at 2 (0.285714), 3/7 at 4 (0.428571) and 6/7 at 8 (0.857142).

When should you divide first and when should you use the decimal?

For 5/7 of 910, check whether 7 divides 910: 7 x 130 is 910, so divide first and multiply second, and the answer is a clean 650. When the number you are taking a fraction of is a multiple of the denominator, divide first; reach for the decimal only when it is not. For 5/7 of 1,000, there is no clean division, so use the cycle: 5/7 starts at the 7, 0.714285, and the answer is about 714.3. A check that works on both: 5/7 is a little over 70%, and 70% of 910 is 637, so 650 sits in the right place.

Interviewers use sevenths because the decimals look random and catch people who have only learned halves, quarters and eighths. The cycle gives you six decimal places in a second, which is more precision than you will ever need to say out loud.

Segment C, up 2.5 percentage points, from 35.0% to 37.5%. Total revenue rises from Rs 1,000 crore to Rs 1,120 crore, 12% growth. C grew 20%, and 420 of 1,120 is 37.5%. A grew 15% but its 460 of 1,120 is 41.1%, only 1.1 points above 40.0%. B shrank 4% and lost 3.6 points, falling to 21.4%.

Why is growth not the same as gaining share?

Think of a household where every earner got a raise this year. Whose share of the family income went up most? Not the one with the biggest raise in rupees, and not necessarily the one with a big raise at all, but the one whose raise beat the household's overall rise by the widest margin. A segment gains share only by growing faster than the total, so the first number to compute is total growth, here 12%. A grew 15%, three points faster than the total; C grew 20%, eight points faster; B shrank.

How do you answer this fast in a timed test?

Sum each column first: 1,000 and 1,120. Compare every segment's growth with 12%, and only A and C can have gained. Then use the formula on those two: C is 0.35 x 8/112, which is 2.5 points; A is 0.40 x 3/112, about 1.07. Shares always add to 100%, so the gains and losses must net to zero, which gives a free check: 2.50 + 1.07 - 3.57 = 0. Watch the units in the answer options: C's share rose 2.5 percentage points, which is a 7.1% rise in its share. Timed numerical tests often offer both, and only one matches the question's wording.

About 2,520 ATMs, on stated assumptions. Of one crore people, 70 lakh are adults and 60% of them, 42 lakh, use ATMs about three times a month: 4.2 lakh withdrawals a day. If 12% fall in the busiest hour, that is 50,400 an hour. A machine handles 30 an hour at two minutes each; plan for two-thirds busy, 20 an hour, and you need about 2,520.

Where do you start, demand or supply?

Think of a canteen deciding how many counters to open. It does not divide the day's meals by 24 hours; it counts the lunch rush and opens enough counters for that. Size a service network to its peak load, so the estimate runs from people to withdrawals to the busiest hour, and only then to machines. Start with demand: 70% of one crore are adults, 70 lakh; assume 60% of them use ATMs for cash, 42 lakh, and that each withdraws three times a month. That is 1.26 crore withdrawals a month, or 4.2 lakh a day.

How much does the peak assumption change the answer, and how do you check it?

Machines stand idle at 3 a.m. and have queues at 6 p.m. If you spread 4.2 lakh withdrawals evenly over 24 hours, you would size for 17,500 an hour and buy 875 machines. The peak-hour share is the assumption that moves the answer most: at 12% of the day in one hour, the city needs nearly three times what the average implies. The same logic explains the planning margin: a machine busy 100% of the time has a queue that never clears, so plan for two-thirds use. Then sanity check the result: 2,520 machines for one crore people is about 25 per lakh, a ratio you can compare with the published ATMs-per-lakh figure for the city, which you should look up rather than quote from memory.

Say what you left out. Salary days bring a bigger rush at the start of the month; digital payments are shrinking cash use, so the user share and the withdrawals per month are both falling; and machines sit where people work and shop, so a city with a dense business district needs more than population alone suggests.