Financial Analysis puzzles, solved step by step

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.

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.

200 km. The car takes 100 / 50 = 2 hours to reach B, and the bird flies without stopping for exactly those 2 hours at 100 km/h. Distance is speed times time, so the bird covers 200 km. How many trips it makes and where each turn happens do not matter; only the time in the air does.

Why is the zigzag the wrong thing to count?

Picture a dog running back and forth between you and your front gate while you walk home. You could chart every lap, or you could notice that the dog runs for exactly as long as you walk. When something moves at a constant speed for a known time, its distance is speed times time, and the route it takes is irrelevant. The puzzle is built to pull you into the laps.

Here the clock belongs to the car. It has 100 km to cover at 50 km/h, so the whole episode lasts 2 hours. The bird flies at 100 km/h throughout, so it flies 200 km. That is the whole answer, and it takes one sentence.

How do you check it the long way if the interviewer asks?

Sum the laps. On the first leg the bird and the car close at 150 km/h across 100 km, so they meet after 40 minutes, 33.3 km from A. The bird has flown 66.7 km and flies the same back to B, a round trip of 133.3 km. By then the car is 33.3 km from B, so the next round trip is 44.4 km. Each round trip is one third of the one before, so the laps form a geometric series that sums to 133.3 / (1 - 1/3) = 200 km.

Why would a finance interviewer ask this?

It tests whether you look for the quantity that controls a problem before you start calculating. Most analysis questions hide one controlling number under a lot of detail, and the skill is finding it first. In a model review that might be the single assumption the valuation hangs on; in a variance analysis it might be one volume line. Give the two hour answer first, then offer the series as a check. Speed plus a second method is exactly what the interviewer is listening for.

Revenue up 10% adds the most, Rs 4 crore, against Rs 3 crore for the COGS cut and Rs 0.75 crore for EBITDA up 5%. New revenue brings only its gross margin: 10% of Rs 40 crore of gross profit. The COGS cut saves 5% of Rs 60 crore. The two levers tie when 10% of gross profit equals 5% of COGS, at a gross margin of 33.3%; below that, cutting COGS wins.

Why is a revenue increase worth less than it sounds?

A tea stall that sells 10% more cups also buys 10% more milk and tea leaves. When costs move with sales, extra revenue adds only its gross margin to profit, not the whole rupee. Here each extra rupee of sales brings 40 paise of gross profit, so Rs 10 crore of new revenue adds Rs 4 crore. The Rs 25 crore of other costs is assumed fixed; if part of it rose with sales too, the revenue lever would shrink further.

The COGS cut is simpler: 5% of Rs 60 crore is Rs 3 crore straight to EBITDA. EBITDA up 5% is the decoy: 5% of a Rs 15 crore base is only Rs 0.75 crore, because a percentage of a small number is a small number.

At what gross margin does the answer flip?

Write both gains per rupee of revenue. The revenue lever adds 10% times the gross margin; the COGS lever adds 5% times the cost ratio, which is one minus the gross margin. They tie when 0.10 x GM = 0.05 x (1 - GM), which gives a gross margin of one third. Above 33.3%, growing sales does more; below it, as in grocery or commodity processing, the cost cut does more. At the tie each lever adds Rs 3.33 crore on Rs 100 crore of revenue.

What does the interviewer want to hear beyond the number?

Ask for the margin structure before answering, because the right lever depends on it. Then add the practical view: a 5% cost cut is often more within management's control than 10% more sales, which may need price cuts or marketing spend that eat into the gain. Finally, EBITDA up 5% can never beat the revenue lever, since EBITDA can never exceed gross profit; it beats the COGS cut only when EBITDA is larger than COGS, as in some software businesses. If half the other costs were variable, the revenue lever would fall to Rs 2.75 crore and the COGS cut would win.

The company's own cash repaid Rs 400 crore of debt, and every rupee repaid moved a rupee of the unchanged Rs 1,000 crore value from lenders to the fund. Equity went in at Rs 400 crore and came out at Rs 800 crore: a MOIC of 2.0x and, over five years, an IRR of about 14.9%. No growth and no change in multiple were needed.

Where did the gain come from if the price did not move?

Think of a flat bought for Rs 1 crore with a Rs 60 lakh home loan. The rent covers the loan payments and brings the loan down to Rs 20 lakh over five years. You sell the flat for exactly Rs 1 crore, yet your stake has grown from Rs 40 lakh to Rs 80 lakh. Enterprise value is a pie shared between lenders and owners, and each rupee of debt repaid from the business's own cash moves a rupee of that pie to the owners. The price never had to rise.

The return arithmetic follows. MOIC, the multiple on invested capital, is Rs 800 crore out over Rs 400 crore in, 2.0x. IRR is the yearly rate that turns 400 into 800 over five years: 2 to the power one fifth, less 1, which is 14.87%. The rule of 72 gives a quick check: doubling in five years is roughly 72 / 5, about 14.4%.

Is that skill or just leverage?

Run the same business with no debt. The fund pays Rs 1,000 crore of equity, the company's cash flow piles up as at least Rs 400 crore of cash, and at exit the fund receives Rs 1,000 crore plus that cash, about Rs 1,400 crore. That is 1.4x and about 7.0% a year, before adding the interest it never paid. Debt did not create the Rs 400 crore; the business earned it. Debt let a smaller cheque claim all of it, which lifts the return from 1.4x to 2.0x.

Leverage cuts both ways, and saying so is the limit worth stating. Had the company sold for only Rs 500 crore, the levered fund would get back Rs 300 crore on Rs 400 crore, 0.75x, a loss, while the unlevered owner would hold 0.9x. The same debt that doubled the good outcome deepened the bad one.

What does the interviewer listen for?

Name the three ways a buyout makes money: EBITDA growth, a higher exit multiple, and debt paydown. This deal used only the third, which is the point of the question. Then add time: the same 2.0x earned over three years instead of five is about 26% a year, which is why funds care about IRR and not only the multiple. A finishing touch is noting that the Rs 400 crore repaid is cash after interest, so the business had to earn more than Rs 400 crore to do it.

About Rs 375.7 crore. The perpetuity formula, 50 / 0.10 = Rs 500 crore, gives a value one period before the first payment, which here is the end of year 3. Discount that three years: 500 / 1.1 cubed = 500 / 1.331 = Rs 375.66 crore. Discounting four years because the cash starts in year 4 is the classic slip and gives Rs 341.5 crore.

Where in time does the perpetuity formula put its value?

Picture someone who will retire in four years and has been promised a pension from that day on. On the eve of the first payment, the pension is worth a neat round sum; four years out, that round sum itself is still in the future. Cash flow divided by the rate gives the value one period before the first payment, never on the day of it. The standard formula assumes the first payment comes one year from the valuation date. If the first payment is at year 4, the formula is standing at year 3.

So value the stream at year 3 and bring that single lump back to today. At year 3 the concession is worth 50 / 0.10 = Rs 500 crore. Three years of discounting at 10% means dividing by 1.1 x 1.1 x 1.1 = 1.331, which leaves Rs 375.66 crore.

How do you check it a second way?

Value a perpetuity that starts next year and subtract the three payments this one does not have. A full perpetuity from year 1 is worth Rs 500 crore. The missing payments in years 1 to 3 are a three-year annuity worth 50 x 2.487 = Rs 124.34 crore. Rs 500 crore less Rs 124.34 crore is Rs 375.66 crore, the same answer by a route that cannot misplace the year. The tempting shortcut, 500 less 150 = 350, fails because the missing payments are worth less than their face.

Why does this matter beyond the puzzle?

It is the same step as the terminal value in a DCF. A growing perpetuity built on year 6 cash flow gives a value at year 5, so it is discounted five years, not six. Discounting one year too many cuts the value by the full discount rate, here about 9%, and in a DCF the terminal value is often most of the answer. The limit to say aloud: real concessions end. If this one ran for 30 years from year 4, it would be worth Rs 354.1 crore, so the 'forever' after year 33 adds only Rs 21.5 crore. Distant cash matters little at 10%.