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.

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.

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.

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%.

1,596 and 4,891. Both pairs sit equally either side of a round number, so each product is that round number squared less the gap squared. 38 x 42 = 40 squared less 2 squared = 1,600 - 4 = 1,596. 67 x 73 = 70 squared less 3 squared = 4,900 - 9 = 4,891. Two easy steps replace four partial products.

Why does a pair either side of a round number multiply so neatly?

Picture a square courtyard 40 tiles by 40. Rebuild it 42 tiles wide but only 38 deep. You lose a row of 40 tiles along the bottom and gain a column of 38 down the side. The column is two tiles shorter than the row, and the missing piece is a 2 by 2 corner. Any two numbers equally either side of a middle number multiply to the middle squared less the gap squared, because (a - b)(a + b) = a squared less b squared.

So the work is two squares you already know. For 67 x 73 the middle is 70 and the gap is 3: 4,900 less 9 is 4,891. A quick check catches slips: 7 x 3 ends in 1, and so does 4,891; 8 x 2 ends in 6, and so does 1,596. Both products sit just under the square, as they must.

What do you do when the pair is not centred on a round number?

Find the middle anyway, or nudge one number. 38 x 44 has a middle of 41 and a gap of 3: 1,681 less 9 is 1,672. Or reuse what you have: 38 x 42 is 1,596, add two more 38s and get 1,672. When the sum is odd, as in 37 x 42, the middle is 39.5, which is messy; adjust instead: 38 x 42 less one 42 is 1,554. The skill is choosing the anchor you can square instantly, then correcting by a small, easy amount.

Why does a finance interviewer test this?

Interviewers use mental maths to see whether you can keep a number straight under pressure, which is what a live model review or a client call demands. They watch the method as much as the answer: a candidate who says 'that is 40 squared less 4' sounds in control. The limit is honest too: the trick only helps when a pair straddles a square you know. For 38 x 57, round and correct, or split 57 into 50 and 7, rather than forcing the pattern.

The cash never runs out. Burn reaches zero in month 17, by which point Rs 68 crore has gone, so the balance bottoms at Rs 52 crore. The burns are 8, 7.5, 7 and so on down to 0.5 in month 16: sixteen payments averaging (8 + 0.5) / 2 = Rs 4.25 crore, Rs 68 crore in all. Quoting a 15-month runway from 120 / 8 ignores the improvement.

Why is 120 / 8 the wrong runway?

A student who spends Rs 8,000 this month and cuts back by Rs 500 every month does not run through savings at Rs 8,000 a month; each month costs less than the last. A runway is cash divided by burn only when the burn is constant; when the burn changes by a fixed amount each month, the total spent is an arithmetic series, and you sum it. Here the burn falls to zero after sixteen steps, so the question is whether the sum of those sixteen burns is more or less than Rs 120 crore.

How do you sum the burn quickly?

Pair the first and last months, as the schoolboy Gauss did: 8 + 0.5 = 8.5, 7.5 + 1 = 8.5, and so on. There are sixteen burns, so eight pairs of 8.5, which is Rs 68 crore. Equivalently, sixteen months at the average burn of 4.25. Month 17's burn is zero, so nothing more is spent. Rs 120 crore less Rs 68 crore leaves Rs 52 crore at the lowest point, reached at the end of month 16.

What would you warn the founder about?

The answer rests entirely on the slope of the improvement. If the burn fell by Rs 0.25 crore a month instead of 0.5, the burn would reach zero only in month 33, and the cumulative burn would pass Rs 120 crore in month 23: the company runs out of money. Halving the pace of improvement turns a comfortable floor into a cash-out, so test the slope before trusting the floor. The second warning is that Rs 52 crore is a forecast floor, not a cushion: a lender or a board would want headroom above it for a bad quarter.

Volume must rise by a third, 33.3%, just to stand still. On a Rs 100 price, contribution is Rs 40 a unit. Cut the price 10% and variable cost stays at Rs 60, so contribution falls to Rs 30, a quarter lower. Selling 40 / 30 = 1.333 times as many units keeps total contribution flat. The general rule is the cut divided by the margin less the cut: 10 / (40 - 10).

Why does a 10% price cut need far more than 10% more volume?

A samosa seller charges Rs 20 and spends Rs 12 on ingredients, so each samosa leaves Rs 8. Cutting the price by Rs 2 does not cut the ingredient bill; it cuts the Rs 8 to Rs 6. A price cut comes entirely out of contribution, because variable cost does not fall with the price, so the volume needed to make it back is set by the margin, not the price. In the puzzle the 10% cut takes a quarter of the contribution, so volume must rise by 33.3%.

What is the general rule, and how fast does it bite?

The table shows why low-margin businesses fear discounting. At a 20% margin a 10% cut needs volume to double; at 60% it needs only 20% more. The rule also runs the other way: a 10% price rise lifts contribution to Rs 50 a unit, so volume can fall by up to 20% before total contribution drops. Price rises are usually safer than they feel and price cuts riskier than they look.

What would you check before backing the cut?

Three things. First, whether demand will really grow by a third, which depends on how sensitive buyers are to price and whether rivals match the cut. Second, capacity: a third more units may need overtime or a new shift, which turns fixed costs into step costs. Third, the existing customers who would have paid Rs 100 anyway now pay Rs 90, which is where most of the loss sits. The limit of the rule is that it holds contribution flat; it says nothing about cash tied up in the extra stock and receivables.

Rs 180. The 18% is charged on the pre-tax price, so the MRP is the base times 1.18. The base is 1,180 / 1.18 = Rs 1,000 and the tax is Rs 180. As a share of the inclusive price, tax is 18 / 118, about 15.25%, not 18%. Taking 18% of Rs 1,180 charges tax on the tax and gives a wrong Rs 212.40.

Why is the tax not 18% of the price you pay?

A restaurant bill of Rs 1,100 that includes a 10% service charge contains Rs 100 of service, not Rs 110: the 10% was added to the Rs 1,000 of food, not to the final total. A rate applied to a base and then included in the price is rate over one plus rate of the inclusive price, never the rate itself. For 18% that share is 18 / 118, about 15.25%. The base is the starting point, and the inclusive price is the base grown by 18%.

How do you strip tax out of any inclusive price?

The rates in the table are for illustration. GST rates vary by product and change over time, so confirm the current rate for the item before you rely on a figure.

Where else does the same slip turn up?

Everywhere a percentage of one base gets quoted against another. A 25% markup on cost is only a 20% margin on price, because 25 / 125 = 20%. A fund that charges 2% on assets is not taking 2% of the return. An analyst who multiplies an inclusive revenue line by the GST rate overstates the tax the company passes on to the government. Before applying any percentage, ask what it is a percentage of, and convert if the base you hold is a different one. The test is quick: 1,000 plus 18% of 1,000 must give back the MRP.

About 32 lakh litres a day, inside a plausible range of roughly 18 to 47 lakh. Assume about 1.3 crore people in four-person homes, 32.5 lakh households. If 80% buy packaged milk at about a litre a day, that is 26 lakh litres. Tea stalls, restaurants and sweet shops add perhaps a quarter more. Each input is an assumption to state aloud and test.

What is the natural unit for a daily staple?

Think about your own home: someone buys one or two packets every morning, and the number depends on how many people share the kitchen, not on who drinks what. For something bought every day by almost every home, the household is the unit, because the buying decision and the quantity are set per kitchen. Start from people, convert to homes, decide what share buys the product, and multiply by the daily quantity. Then add the buyers who are not homes.

Every number in the chain is an assumption, not a fact, and should be said that way. The population depends on where you draw the boundary: the city proper and the wider metropolitan region differ by a large factor, so confirm the latest census or municipal estimate and say which one you are using. The 80% packaged share leaves room for loose milk from local dairies, which is still common in parts of the city.

How do you cross-check it without reusing the same assumptions?

Use a check that rests on different inputs. Per head, 32.5 lakh litres across 1.3 crore people is about 250 ml a day: two or three cups of tea at about 50 ml of milk each plus a glass for a child. That is believable. From the supply side, if a road tanker carries about 15,000 litres, which is an assumption to state, the city needs about 217 tankers a day. A good estimate survives a check built from a different direction; if the per-head number had come out at two litres, you would know a step was wrong.

What would sharpen the estimate most?

The packaged share is the biggest swing, so it is the first thing to research, followed by the trade uplift, which is hard to observe because tea stalls and sweet shops often buy loose milk in bulk. Seasonality matters too: festival weeks lift demand for sweets and milk. The limit to say: this is a demand estimate on an average day, and a dairy planning supply would also need the peak day and the share each brand holds.