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.

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.

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.

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.

Equity falls from Rs 2,400 crore to Rs 800 crore, a 66.7% drop on a 20% fall in EV. Debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore; at 8x it is Rs 6,400 crore. Debt keeps its claim, so the equity slice absorbs the whole Rs 1,600 crore. At 7x leverage, equity moves 3.3 times as fast as EV, and a 7x multiple wipes it out.

Why does equity fall so much further than the business?

Think of a flat bought for Rs 1 crore with a Rs 70 lakh home loan. If the flat's price drops 20% to Rs 80 lakh, the bank still wants its Rs 70 lakh, so the owner's stake falls from Rs 30 lakh to Rs 10 lakh, a two-thirds loss. Debt is a fixed claim in rupees, so any change in enterprise value lands entirely on the equity that sits on top. Here debt is 7 x 800 = Rs 5,600 crore. At 10x, EV is Rs 8,000 crore and equity is Rs 2,400 crore, only 30% of the value.

So what does 7x leverage tell you about sensitivity?

Leverage of 7x EBITDA means the debt alone is worth 7x EBITDA. Whenever the valuation multiple sits close to the leverage multiple, equity is a thin slice of the gap between them, and small changes in the multiple swing it hard. At 10x the gap is 3 turns; each turn of multiple is Rs 800 crore, a third of the equity. Software companies can carry this because their cash flows recur and their multiples are high, but the same 7x on a business valued at 8x leaves one turn of cushion. Say the limitation: the sum assumes debt is worth its face value, while in distress the debt itself would trade below par and soak up some of the loss.

Market capitalisation is Rs 2,000 crore. With no other liabilities, assets equal debt plus equity. A D/E of 0.5 means debt is half of equity, so assets are 1.5 times equity and book equity is Rs 1,500 crore over 1.5, which is Rs 1,000 crore, with Rs 500 crore of debt. At 2.0x book, the market values that equity at Rs 2,000 crore.

How does a debt-to-equity ratio split the balance sheet?

Think of a Rs 15 lakh car bought with a loan that is half the size of your down payment. If the loan is half the down payment, the car cost one and a half down payments, so the down payment was Rs 10 lakh and the loan Rs 5 lakh. A D/E of 0.5 does not mean half the assets are debt; it means debt is half of equity, so assets are 1.5 times equity. Rs 1,500 crore over 1.5 gives book equity of Rs 1,000 crore and debt of Rs 500 crore.

What assumptions sit under the answer, and what could you add?

The question rules out other liabilities, such as payables and provisions, and that matters: a real balance sheet has plenty, and then assets are not just debt plus equity. Say the identity you are using, assets equal liabilities plus equity, before you lean on it. You can also go one step further than asked. With Rs 500 crore of debt and no cash given, enterprise value is about Rs 2,500 crore. And a P/B of 2.0 says the market thinks the assets earn more than their cost of capital; read it next to return on equity rather than on its own.

The new margin is 13.75%. Take revenue of 100, so cost is 90 and profit 10. Revenue grows to 120 and cost to 90 x 1.15 = 103.5, leaving profit of 16.5. The margin is 16.5 over 120, which is 13.75%. Profit itself grows 65%, because a 5 point gap between growth rates is large next to a 10% margin.

Why is the answer not simply 15%?

Think of a tea stall that takes in Rs 100 a day and spends Rs 90. If takings rise 20% and spending rises 15%, the extra takings are Rs 20 but the extra spending is Rs 13.50, because 15% is taken on 90, not on 100. Growth rates act on their own bases, so a margin moves by the gap in rupees, not by the gap in percentage points. Work in rupees on a base of 100 and the trap disappears: profit goes from 10 to 16.5 on revenue of 120.

Why does a thin margin make the move so large?

Profit is the small gap between two big numbers, so a modest difference in their growth rates is a big change to the gap. The thinner the starting margin, the more operating leverage a growth gap carries: here profit grows 65% on 20% revenue growth. Run the same 20% and 15% from a 30% margin and the new margin is 32.9%, a smaller relative move, because cost is a smaller share of revenue. Say the limit too: the formula assumes all cost grows at 15%. In practice fixed costs grow slowly and variable costs track volume, so you would split the cost line before trusting the answer.

About 15.4%. Picture 10,000 invoices. 100 are fraudulent and the screen flags 90 of them. 9,900 are clean and the screen flags 5% of them, 495. The flagged pile holds 585 invoices, of which 90 are fraud, so a flag means fraud only 90 times in 585. The rare base rate swamps the screen's accuracy.

Why is a 90% accurate screen right only 15% of the time?

Think of a smoke alarm that never misses a fire but also goes off whenever someone makes toast. In a house where fires are rare and toast is daily, almost every alarm is toast. When the thing you are hunting is rare, even a small false positive rate on the large innocent pile produces more false alarms than true hits. Here 5% of 9,900 clean invoices is 495 false flags, against only 90 true ones.

What would make the screen useful, and how would an auditor use it?

Cutting the false positive rate does far more than raising the catch rate. At a 0.5% false positive rate the flagged pile would be 90 frauds and about 49.5 clean invoices, and a flag would mean fraud 65% of the time. Raising the catch rate from 90% to 100% would only move the answer from 15.4% to about 16.8%. In practice a screen like this is a triage tool: it shrinks 10,000 invoices to 585 for a human to review, and that review is where the 495 false alarms are cleared. Say the limit too: the 1% base rate is itself an estimate, and the answer is only as good as it is.

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.