Financial Analysis puzzles, solved step by step

About Rs 66.67 a unit, not Rs 75. The first Rs 10,000 buys 100 units at Rs 100 and the second buys 200 units at Rs 50, so Rs 20,000 buys 300 units. Rs 75 is the average of the two prices, which would be your cost only if you had bought equal numbers of units. Equal rupee amounts buy more units when the price is low, so the average cost is the harmonic mean of the prices.

Why does the cheap purchase count for more?

Think of filling your scooter with Rs 500 of petrol every week. In a week when petrol is cheap, Rs 500 buys more litres; when it is dear, fewer. At the end of the month the litres you own lean towards the cheap weeks. A fixed rupee amount buys more units when the price is low, so the low price carries more weight in your average cost than the high one. Here Rs 10,000 buys 100 units at Rs 100 but 200 units at Rs 50, so the Rs 50 price carries two thirds of the weight.

When is Rs 75 the right answer?

When you buy equal numbers of units rather than equal rupee amounts. Buy 100 units at Rs 100 and 100 units at Rs 50 and you spend Rs 15,000 on 200 units, which is exactly Rs 75 a unit. So the right average depends on what you held fixed: units fixed gives the simple average of prices, money fixed gives the harmonic mean. This is the arithmetic behind the idea of rupee cost averagingInvesting the same rupee amount at regular intervals, so that more units are bought when prices are low and fewer when they are high..

Does the lower average cost mean you made money?

No, and saying so is the mark of a careful answer. A lower average cost is arithmetic about what you paid, not a return. If the NAV stays at Rs 50, your 300 units are worth Rs 15,000 against Rs 20,000 invested, a 25% loss. Your break-even NAV is Rs 66.67, not Rs 75, which is the useful fact. The effect also needs prices to vary: at a constant price every average is the same.

About 5,100 people, as a range of roughly 3,400 to 5,100. Take a floor plate of 25,000 square feet. After lifts, stairs and services about 80% is usable, 20,000 square feet. At 100 square feet a seat that is 200 seats, and at 85% occupancy about 170 people a floor. Thirty floors gives about 5,100. Roomier offices at 150 square feet a seat bring it down to about 3,400.

Why start from one floor?

Think of estimating the people in a cinema: you count the seats in one row, then the rows. A tower is the same: thirty copies of one floor, so estimate one floor carefully and multiply, because every assumption is easier to defend on a single floor you can picture. Define the answer too: people who work in the building, meaning assigned staff, not visitors and not the count present on any one day.

What goes into one floor?

Four assumptions, each said out loud. A floor plate of about 25,000 square feet, typical for a large commercial tower. About 20% of that is the coreThe central block of a floor holding lifts, stairs, toilets, shafts and service rooms, which cannot be used for desks., leaving 20,000 usable. At 100 square feet a seat, including aisles and meeting rooms, the floor holds 200 seats, and with 85% of seats assigned that is about 170 people. The rest are vacant desks, new hires not yet in, or space held for growth.

Which assumption would you flex, and how would you check it?

Seat density moves the answer most. A dense back office runs near 100 square feet a seat; a bank's front office with larger desks and more meeting rooms nearer 150. At 150 square feet a seat the tower holds about 3,400 people, so give the answer as about 3,400 to 5,100. Then trim for floors that hold no desks: if the lobby and a plant floor take two of the thirty, the dense case falls to about 4,760. A check from another side: assume a dozen lifts, each carrying about 15 people a trip on a round trip of about 3 minutes. That moves about 12 x 15 x 20, or 3,600 people an hour, which fits a building of a few thousand people arriving over a morning, not tens of thousands.

Without a provision, net income falls Rs 37.5 lakh; with a full provision already booked, it does not move. With no provision, the Rs 50 lakh is an expense, tax falls Rs 12.5 lakh, and net income falls 37.5. Cash rises 12.5 from lower tax, receivables fall 50 and retained earnings fall 37.5. With a provision booked last year, the write-off only removes the debt and the provision together, so profit and cash are unchanged.

What happens when nothing was set aside?

Think of a shopkeeper with a customer who has run up a tab and then left town. The money was counted as owed; now it never will be. Writing off a debt with no provision turns an asset into an expense in one step: receivables fall Rs 50 lakh and a bad debt expense of Rs 50 lakh hits profit. Assuming the write-off is tax deductible, tax falls Rs 12.5 lakh, so net income falls Rs 37.5 lakh. On the cash flow statement, add back the Rs 50 lakh because no cash moved; cash from operations rises Rs 12.5 lakh, the tax saved. The balance sheet balances: assets fall 37.5 (receivables down 50, cash up 12.5) and retained earnings fall 37.5.

Why does the provision case show nothing this year?

Because the loss was already recognised. A provision for doubtful debtsAn amount set aside against receivables the company expects not to collect. It reduces net receivables on the balance sheet and is charged to profit when it is created. is booked when a loss becomes likely, and that is when profit takes the hit. Accounting recognises a loss when it becomes probable, not when the customer finally fails, so the write-off later is a tidy-up between two balance sheet lines. Gross receivables fall Rs 50 lakh, the provision against them falls Rs 50 lakh, and net receivables are unchanged. Under Ind AS 109 and IFRS 9 companies provide for expected credit losses in advance, which is this logic applied to a whole loan book.

What about tax in the provision case?

This is the follow-up that separates good answers. In many tax systems a general provision is not deductible until the debt is actually written off, so the tax saving arrives now even though the expense was booked last year. The books handle that with a deferred tax asset created last year and reversed this year: cash tax falls Rs 12.5 lakh now, with no effect on this year's profit. Treat the rule as something to confirm for the jurisdiction where the company files tax, rather than a fixed fact.

17 minutes. 1 and 2 cross (2 minutes) and 1 brings the torch back (1). Then 5 and 10 cross together (10), and 2 brings the torch back (2). Finally 1 and 2 cross again (2). The obvious plan, where the 1-minute person escorts everyone, takes 19 minutes because the two slowest walkers cost 15 minutes on separate trips but only 10 together.

Why does the obvious plan lose two minutes?

The escort plan sounds efficient: the fastest person walks each one across and runs back with the torch. It costs 2, 1, 5, 1 and 10, which is 19 minutes. The waste is that the 5 and the 10 cross on separate trips, so you pay both their times, 15 minutes, when one shared trip would cost only 10. Think of two slow friends who will hold up any group they walk with; the cheapest thing is to let them hold up the same group, once.

How do you make sure the torch is on the right side?

If 5 and 10 cross first, one of them has to bring the torch back, which costs at least 5 more minutes. So before the slow pair goes, a fast walker must already be waiting on the far side to carry the torch back. That is why 1 and 2 go first and 1 returns: it parks the 2 on the far side. After the slow pair crosses, the 2 brings the torch back, and the fast pair finishes together.

How do you show 17 cannot be beaten?

There must be three forward trips and two returns. The 10 must cross on some forward trip, costing 10. If the 5 crossed on a different forward trip, the forward trips alone would cost at least 10 + 5 + 2, which is 17, and the two returns add at least 2 more, so 5 and 10 must share a trip. The slow pair cannot go first, or one of them walks the torch back. So 1 and 2 go first, costing 2, and one of them returns. After the slow pair crosses, the only fast walker on the far side is whichever of 1 and 2 stayed, so the two returns cost 1 + 2 between them, and the last trip, 1 and 2, costs 2. That gives at least 10 + 2 + 2 + 1 + 2 = 17. A brute-force search over every schedule confirms 17.

About Rs 25 crore of cash leaves. Cost of goods sold is Rs 10 crore a day, so payables are 45 days, Rs 450 crore, and stock is 20 days, Rs 200 crore. With no receivables, net working capital is minus Rs 250 crore: suppliers fund the business. A 10% fall cuts daily cost to Rs 9 crore, payables to Rs 405 crore and stock to Rs 180 crore, so net working capital rises to minus Rs 225 crore. That Rs 25 crore is cash consumed.

How can a business be funded by its suppliers?

Think of a school canteen where parents pay for the term in advance and the canteen pays its vegetable seller at the end of each month. The canteen holds other people's money for weeks. A supermarket sells for cash at the till, keeps stock for 20 days and pays suppliers after 45, so it collects from customers about 25 days before it pays for the goods. Its net working capital is negative: suppliers are lending it money, free of interest, all year.

Why does shrinking cost this business cash?

Run it as two movements. Stock falls by Rs 20 crore, which frees cash, but supplier credit falls by Rs 45 crore, which the supermarket has to pay out, so the net is Rs 25 crore out. Each day of lower sales means paying suppliers for last month's larger purchases while this month's till receipts are smaller. The mirror image is why such businesses love growth: a 10% rise in sales would release about Rs 25 crore.

What makes the real number worse?

The answer assumes the days stay fixed, and in a downturn they often do not. Suppliers who see sales fall may shorten credit; if payable days drop from 45 to 40, payables fall to Rs 360 crore and the cash outflow grows to about Rs 70 crore. On top of the working capital effect, lower sales also cut profit. Say both, then name the lesson a lender draws: a business funded by its suppliers can look cash-rich while it grows and turn cash-hungry quickly when it shrinks.

The implied tax rate is 37.5%. Market cap of 300 at 15x earnings gives net income of 20. Adding net debt of 180 gives EV of 480, so EBITDA at 6x is 80. Less D&A of 30 is EBIT of 50; less interest of 18 is pre-tax profit of 32. Tax is 32 minus 20, which is 12, and 12 over 32 is 37.5%, high enough to ask why.

Where do you start when six numbers arrive at once?

Think of working out a friend's take-home pay from what they spend and what they save. You do not start with their job title; you start with the two numbers that pin the answer. A tax rate is tax divided by pre-tax profit, so every other number in the question is a road to net income or to pre-tax profit. Say that first. Net income is one step: 300 over 15 is 20. Pre-tax profit needs EBITDA, which needs EV, which needs the net debt you were handed.

What does a 37.5% rate tell you, and what have you assumed?

You assumed there are no minority interests, no associates and no interest income on cash, so EV is just market cap plus net debt and every rupee of pre-tax profit belongs to shareholders. Say that. Then read the number. An effective rate well above the statutory rate usually means some costs are not tax deductible, some losses sit in units that cannot use them, or there are one-off tax charges. Compare it with the statutory rate the company actually pays, which you should confirm for the year in question. A second check: interest of 18 on net debt of 180 is 10%, which is plausible, so the inputs hang together.

Better, by about 7.3%. Take a price of 100 with variable cost 70, so each unit contributes 30. After the rise the price is 105 and each unit contributes 35. Selling 92 units instead of 100 gives 92 x 35 = 3,220 against 3,000 before. Volume could fall as far as 14.3% before the price rise stopped paying.

Why does 5% on price beat 8% on volume?

Picture a tea stall that sells a cup for 10 rupees with 7 rupees of milk, leaves and sugar in it. Raise the price to 10.50 and the stall's profit per cup goes from 3 to 3.50, a sixth more. A price rise falls straight to contribution, so its effect is measured against the margin, not against the price. The volume loss, by contrast, removes whole cups and their contribution at the same rate as the lost units, 8%.

How much volume could you lose before the price rise hurts?

Set new contribution equal to old: units times 35 must equal 100 times 30, so units can fall to 3,000 over 35, which is 85.7. The breakeven volume loss for a price rise is the price change divided by the new margin: 5 over 35, or 14.3%. That is why thin-margin businesses gain so much from price and lose so much from discounts: at a 30% margin, a 5% discount needs volume to rise by 5 over 25, a full 20%, just to stand still.

Say the limit too. The sum treats variable cost per unit as fixed and ignores any fixed costs that could be cut when volume falls, and it says nothing about customers who leave and do not come back next year.

Anywhere from about -0.75 to 0.95. Read each correlation as the cosine of an angle between two arrows. X is 78.5 degrees from Y, and Z is 60 degrees from Y. Z can swing to the same side as X, leaving them 18.5 degrees apart, or to the other side, 138.5 degrees apart. The cosines of those angles are 0.9485 and -0.7485.

Why can a correlation be drawn as an angle?

Think of three people walking away from the same lamp post. Knowing how far apart the first two point, and how far apart the second and third point, limits how far apart the first and third can point, but only loosely. If you standardise each variable, its correlation with another is the cosine of the angle between them, so correlations must behave like angles in space. A correlation of 0.2 is an angle of 78.5 degrees; 0.5 is 60 degrees.

Where does the formula come from, and when does it bite?

The three-by-three correlation matrix must not imply a negative variance for any mix of X, Y and Z, which means its determinant cannot go below zero. Solving that condition gives the formula above, the cosine rule for the difference and sum of two angles. The constraint is loose when the known correlations are weak and tight when they are strong. With 0.2 and 0.5, almost the whole scale stays open. With 0.9 and 0.9, X and Z must correlate at least 0.62: two things each closely tied to Y cannot drift far from each other. Say the practical use: a risk model that fills in missing correlations by hand can break this rule and produce a matrix that is not valid.

About 3.28 loops. Every tie cuts the number of strands by one, so there are exactly 100 ties. With k strands left, the second end you grab is one of 2k - 1 free ends, and only one of those closes a loop, so that tie adds 1 over (2k - 1) to the expected count. Summing 1 + 1/3 + 1/5 + ... + 1/199 gives 3.2843.

How do you avoid tracking the whole bowl?

Think of shaking hands at a party where each person has two hands. You do not need the seating plan to know that each handshake joins two hands. Whatever the bowl looks like, every tie reduces the count of strands by exactly one, and a strand is either an open piece or a closed loop that has left the game. So you can ignore the tangled history and ask one question per tie: does this tie close a loop, yes or no?

Pick any free end first; which one does not matter. With k strands there are 2k free ends, so 2k - 1 remain for the second pick, and exactly one of them is the other end of the same strand. That tie closes a loop with chance 1 over (2k - 1). Otherwise two strands fuse into one longer strand.

Why can you add the step expectations when the steps depend on each other?

Whether an early tie closes a loop changes nothing about how many strands remain, because every tie removes one strand either way. Expected values add even when the events are linked, so the total is simply the sum of the per-tie chances. This is linearity of expectation, and it is what makes the puzzle a two-minute answer instead of a simulation. For large n the sum is close to half the natural log of n plus about 0.98; for 100 noodles that gives 3.28, a useful sanity check.

Cover is 3.3x, falling to 1.7x. Take EBITDA of 100. Net debt of 300 at 10% costs 30 of interest, so EBITDA covers it 3.33 times. At 5x and 12%, net debt is 500 and interest is 60, so cover is 1.67x. Cover is one over leverage times the rate, and 0.30 doubling to 0.60 halves it.

Why is there no need for an actual EBITDA figure?

Think of a household whose loan EMIs are some multiple of monthly salary. Whether the salary is 50,000 or 5 lakh, the share of salary that goes on interest depends only on how many months of salary it borrowed and at what rate. Interest cover is EBITDA over interest, and interest is leverage times EBITDA times the rate, so EBITDA cancels and cover is one over leverage times the rate. Pick EBITDA of 100 only to make the arithmetic visible.

Why do leverage and rates tend to move together, and what does 1.7x mean?

A lender charges more to a borrower who owes more, so the second scenario is not a coincidence; it is how a stretched balance sheet usually looks. Because leverage and rate multiply, a company that adds debt while rates rise loses cover much faster than either change alone suggests. Cover of 1.7x also flatters the position: EBITDA is before tax, capex and working capital, so once maintenance capex is paid the cash left to service interest can be well under 1.7 times the bill. Say that limitation; credit analysts look at cover after capex for exactly this reason.