Financial Analysis puzzles, solved step by step

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%.

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.

14% today, rising to about 16.7% after B doubles, with no change in either division's own margin. Group EBITDA is 80 + 60 = Rs 140 crore on Rs 1,000 crore. When B grows to Rs 400 crore at 30%, EBITDA is 80 + 120 = Rs 200 crore on Rs 1,200 crore. The group margin is a revenue-weighted average, so it moves when the weights move.

Why is the group margin not the average of the two?

A cafe earns 30% on coffee and 10% on sandwiches. If it sells mostly sandwiches, its overall margin sits near 10%; if coffee takes off, the overall margin climbs, even though neither item is any more profitable. A group margin is a revenue-weighted average of segment margins, so it moves whenever the mix moves, even if no segment changes. Today B is a fifth of revenue, so the group sits a fifth of the way from 10% to 30%: 14%.

Can the group margin rise while every division gets worse?

Yes. Suppose that while B doubled, A's margin slipped to 9% and B's to 28%. Group EBITDA would be 72 + 112 = Rs 184 crore on Rs 1,200 crore, 15.3%, still above the 14% start. Both divisions got less profitable and the group margin still rose, because the shift towards B outweighed the decline inside each. Statisticians call this pattern Simpson's paradox, and a management commentary that cites only the group margin can hide it.

What should an analyst do with a reported margin gain?

Split it into mix and rate. The mix effect is the margin you would get with the new weights and the old divisional margins, less the starting margin: 16.67% less 14% is 2.67 points. The rate effect is the rest. In the puzzle the rate effect is zero; in the deteriorating version it is -1.33 points, which is the story a reader needs. The limit: segment data is often reported only twice a year and with shared costs allocated by management, so the split is only as clean as the allocation.

Root 50 is about 7.07, and 1.2% a day is about 19% a year. Root 49 is 7, and near a known square the curve is nearly straight, so add the step divided by twice the root: 7 + 1 / 14 = 7.0714 against a true 7.0711. Volatility scales with the square root of time, so multiply by root 250, about 15.81: 1.2 x 15.81 = 19.0%.

How do you get root 50 without a calculator?

Think of walking up a gentle hill: one more step forward lifts you by the slope at the point you are standing, and for a single step the hill may as well be straight. Near a square you know, the root curve is nearly straight, so the root of 49 plus 1 is 7 plus 1 divided by the slope's denominator, twice 7. That gives 7 + 1/14 = 7.0714; the true value is 7.0711, an error of 0.0004. The slope of the root curve at a point is one over twice the root, which is why the denominator is 14 and not 7.

Why does volatility scale with the square root of time?

Toss a coin 100 times and count heads minus tails. The lead is almost never 100; it is typically about 10, the root of 100, because the tosses partly cancel. Independent moves add in variance, not in standard deviation, so the spread after n days is the daily spread times root n. Daily variance is 1.2 squared, 1.44; over 250 days it is 360; the root of 360 is 19.0. Root 250 by the tangent at 256 is 16 - 6/32 = 15.812, close enough to the true 15.811.

When does the root-of-time rule mislead?

The rule assumes each day's move is independent of the last and drawn from the same distribution. Trending markets, where moves follow moves, make the true spread wider than root n suggests; mean-reverting ones make it narrower. Volatility also clusters, so a 1.2% daily figure measured in a calm month understates a stormy one. The day count is a convention: 250, 252 and 365 are all in use, and annualised volatilityThe standard deviation of returns restated to a one-year horizon, usually by multiplying a daily figure by the square root of the number of trading days. quoted on each will differ, so state the one you used. The limit to say: the rule converts a horizon, it does not forecast the year.

From shrinking sales, not better collecting. With days sales outstanding fixed at 73, receivables are a fifth of revenue, so they fell from Rs 200 crore to Rs 150 crore as revenue fell Rs 250 crore, releasing Rs 50 crore. Unchanged days means customers paid no faster. The release is a one-off that reverses when sales recover, and it was bought with a quarter of the revenue.

Why does falling revenue release cash?

A tailor who gives customers two months to pay always has about two months of sales outstanding. If orders halve, the amount owed to him halves too, and for a while he collects old bills faster than he issues new ones: cash arrives. Receivables are revenue times the collection period, so when the period is fixed, receivables move with revenue, and a fall in sales releases cash once, mechanically. Here 73 days is 73 / 365, one fifth of a year, so receivables are a fifth of revenue: 200 on 1,000, 150 on 750. The Rs 50 crore is a fifth of the Rs 250 crore of sales that disappeared.

How do you separate the volume effect from the collections effect?

Split the change in receivables into two pieces. The volume effect is the change in revenue times the old days: (750 - 1,000) x 73 / 365 = -50. The collections effect is the new revenue times the change in days: 750 x (73 - 73) / 365 = 0. Every rupee of the release is volume; the collections effect is exactly zero, and that is the number management's claim rests on. Had days improved to 60, receivables would be Rs 123 crore, a release of 76.7: 50 of volume and 26.7 of genuine improvement, and only that 26.7 would deserve the word collections.

What does the analyst say about the quality of the Rs 50 crore?

Three things. It is one-off: receivables cannot keep falling unless sales keep falling. It reverses: if revenue climbs back to Rs 1,000 crore at 73 days, receivables return to 200 and the Rs 50 crore is reabsorbed. And it was expensive: at a 30% contribution margin, Rs 250 crore of lost revenue costs about Rs 75 crore of contribution every year, more than the cash released once. The limit: this uses year-end revenue and year-end receivables, and a company whose sales fell late in the year can show a days figure that flatters or punishes it; check the quarterly pattern before concluding.

Lifetime value is about Rs 11,667, LTV to CAC is about 1.9x, and payback is about 17 months. Each customer brings Rs 350 of gross profit a month, 70% of Rs 500. At 3% churn the average customer stays 1 / 0.03 = 33.3 months, so LTV is Rs 350 x 33.3. The Rs 6,000 acquisition cost divided by Rs 350 a month is paid back in 17.1 months.

Why is the average customer life one over churn?

Think of a hostel where 3 residents in every 100 move out each month and are replaced. A resident faces a 3% chance of leaving every month, so on average the wait before leaving is 1 / 0.03 months. With a constant monthly churn rate, the expected customer life is one divided by that rate: at 3%, 33.3 months. Lifetime value is the gross profit the customer brings each month times that life.

Use gross profit, not revenue. The Rs 150 a month it costs to serve the customer is spent whether or not the customer was worth acquiring, so it cannot pay back the acquisition cost. Rs 350 x 33.33 months is Rs 11,667, and against a CACCustomer acquisition cost: marketing, sales and onboarding spend divided by the number of customers it won. of Rs 6,000 that is 1.94x.

Why are there two payback answers?

Payback is the month in which cumulative gross profit covers the Rs 6,000 spent on day zero. For a customer who stays, that is 6,000 / 350 = 17.1 months, the standard answer. Across a whole cohort payback is slower, because some customers leave before they have repaid their share of the acquisition cost. Weight each month by the chance the customer is still there and the average customer crosses Rs 6,000 in month 23.7. Give 17 months and name the cohort figure in one sentence.

Is 1.9x good enough?

Many subscription investors use a rule of thumb of 3x or better with payback inside about a year; treat those as conventions, not laws. At 1.9x this business recovers its acquisition cost with a thin cushion, and the ratio flatters it, because LTV here is undiscounted and ignores any cost of retaining customers. Churn is the strongest lever: cutting it from 3% to 2% lifts LTV by half to Rs 17,500, while a 10% price rise at the same churn only reaches Rs 12,833.

Not on this evidence: most of the drop is regression to the mean. A branch that tops the table usually had skill plus a good year. The good year does not repeat, so the same branches fall back toward the average whatever the manager does. Here only 2 points of a 20 point lead survived, so most of last year's lead was luck. Judge the head against similar branches that kept their managers, not against their own lucky year.

Why do the best performers fall back with no cause at all?

Think of a class test. The student who topped it probably knows the subject and also had a good day: the questions suited her and two guesses landed. In the next test the knowledge stays but the luck is drawn again, so her score is likely to be lower while staying above average. Any result that is part skill and part luck will, when it is extreme, usually be followed by a less extreme one. Francis Galton called this regression toward the mean when he compared the heights of parents and their children.

How much of the 30% was luck?

Measure the lead, not the level. The top five were 20 points above average last year and 2 points above this year. Keeping 2 points of a 20 point lead means about a tenth of last year's outperformance was persistent, and nine tenths was noise that happened to land on those five branches. Look at the bottom of the chart as well: the weakest five rose from -10.0% to 8.3% with no new manager, which is the same drift running the other way.

How would you test whether the new head made a difference?

Build a comparison. Take branches that were about as strong last year but kept their managers, and see how they grew this year. If they also fell to around 12%, the new head is doing exactly what chance predicts. If they held near 20%, the head has a real question to answer. The fair benchmark for a group picked for being extreme is a similar group picked the same way, never its own best year. The same trap sits in sales incentives, fund selection and bonus pools, where praising last year's winners and punishing last year's losers both appear to work.

12% a year at a 40% payout, and 6% at 70%. With no new equity and fixed leverage, the only fuel for growth is retained profit. Equity grows by ROE times the share of profit kept: 20% x 60% = 12%. Debt grows in step to hold the ratio, so assets, and at a steady asset turnover sales, can grow 12%. Keep only 30% and the ceiling halves to 6%.

Why is retained profit the only fuel?

A family shop that refuses outside partners and refuses to borrow more than it already does relative to its size can grow only on the profit it leaves in the till. With new equity ruled out and leverage fixed, equity can grow only by retained profit, and debt can grow only as fast as equity. So the whole balance sheet is capped at the rate equity compounds.

Put numbers on it. Opening equity of Rs 100 crore earns Rs 20 crore at a 20% ROE. Rs 8 crore is paid out and Rs 12 crore kept, so equity closes at Rs 112 crore, 12% higher. Next year's profit on Rs 112 crore is Rs 22.4 crore, also 12% higher, and the pattern repeats.

What assumptions are hiding inside the formula?

The formula, often taught through Robert Higgins' sustainable growth rate, assumes the 20% ROE holds on every new rupee of capital. Split ROE the DuPont way and it could be a 5% net margin x 2.0 asset turnover x 2.0 assets to equity. Growth above 12% therefore needs one of five things: a better margin, faster asset turnover, more leverage, new equity, or a lower payout. That list is the useful part of the answer, because it is exactly the set of levers a finance team argues about when a plan grows faster than its funding.

The limitation is worth one sentence: ROE rarely stays flat as a company grows, because new projects are usually less profitable than the best existing ones. The 12% is a ceiling under today's economics, not a forecast.