Financial Analysis puzzles, solved step by step

Both are true, but say 2 percentage points first. A margin is already a percentage, so a change in it is measured in points: 8% to 10% is up 2 points, or 200 basis points, and up 25% relative to where it started. The NPA ratio fell 1 percentage point, or 100 basis points, which is a 25% relative fall. Use points for a rate and per cent for an amount.

Why can one change honestly be called both 2 and 25?

Think of a student whose exam score goes from 40% to 50%. She scored 10 more marks out of 100, so she is up 10 points. Her score is also a quarter higher than before, up 25%. A percentage point measures the gap between two rates; a per cent measures that gap against the starting rate. Neither is wrong. The confusion comes from writing the same % sign for both, so the reader cannot tell which one you meant.

Which label should lead, and when does the relative one matter?

Lead with points, because that is the convention in results notes, credit reports and board packs. A basis pointOne hundredth of a percentage point. 100 basis points equal 1 percentage point. is one hundredth of a point, which helps when moves are small: margin up 200 bps. The relative change matters when the rate drives an amount. On flat revenue of Rs 1,000 crore, EBITDA goes from Rs 80 crore to Rs 100 crore, which is a 25% rise in profit. So the 2 points and the 25% are describing two different things: the margin, and the profit it produces.

For the NPA ratio, a falling ratio is good news, so say it plainly: gross NPA down 1 percentage point to 3%, or down 100 basis points. You can add that the ratio is a quarter lower. Say the limit too: a lower ratio can also come from loans growing faster than bad loans, so check the rupee amount of NPAs before calling it an improvement in asset quality.

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.

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.

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.

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.

The real raise is 3.77%, so the 4% shortcut overstates it by about a quarter of a point. Real growth divides growth factors: 1.10 / 1.06 = 1.0377. The shortcut subtracts the rates, which ignores that the raise is itself paid in money worth 6% less. At low rates the gap is small; at a 30% raise with 20% inflation the shortcut says 10% against a true 8.33%.

Why divide rather than subtract?

Measure your pay in lunches instead of rupees. Last year a thali cost Rs 100 and you earned Rs 100 a day: one thali. This year the thali costs Rs 106 and you earn Rs 110, so you can buy 110 / 106 = 1.0377 thalis. Your real pay rose 3.77%, because real growth is the ratio of two growth factors, not the difference between two rates. The relation is usually credited to the economist Irving Fisher.

The second form shows exactly where the shortcut goes wrong. The 4% gap is right in the numerator; it just has to be shrunk by dividing by 1.06. The error is small when inflation is small and grows as inflation rises.

When does the shortcut become a real error?

Three places. When rates are high, as the chart shows. When the rate compounds: over ten years, 4% a year grows purchasing power by 48.0% but the true 3.77% grows it by 44.8%, a gap of about three points. And when a budget is being judged: an FP&A analyst splitting revenue growth into price and volume uses exactly this division, and subtracting instead misstates the volume growth the business really delivered. For quick talk in a meeting, 4% is fine; in a model, divide.

No: 20% then 10% off is 28% off, not 30%; buy two get one free is a 33.3% discount per unit. Successive discounts multiply, because the second applies to the already reduced price: 0.80 x 0.90 = 0.72, so you pay 72% of the original. Buy two get one free means paying for two units and taking three, so each unit costs two thirds of its price, a third off, but only if you wanted three.

Why do discounts not simply add?

A Rs 1,000 kurta at 20% off costs Rs 800. The cashier then takes 10% off what you are about to pay, which is Rs 800, so the second discount is worth Rs 80, not Rs 100. You pay Rs 720. Each successive discount is taken off a price that the earlier discounts have already reduced, so discounts combine by multiplying what is left: 0.80 x 0.90 = 0.72, or 28% off. The order does not matter, 10% then 20% also gives 0.72, but the base does.

The shortfall from simple addition is the product of the two discounts: 20% x 10% = 2 points, the discount on the discount. It is small here and grows with the size of the cuts: 50% then 50% off is 75% off, not 100%.

What is buy two, get one free really worth?

You pay for two units and walk out with three, so each unit costs two thirds of the list price. Buy two get one free is a 33.3% discount per unit, deeper than the flat 30%, but only on the condition that you buy three. Buy one, get the second at half price is milder: one and a half prices for two units, 25% off each. If you needed only one, the effective discount of either offer is zero.

Where does this show up at work?

Everywhere rates are applied one after another. A distributor margin taken on a price that already carries a retailer margin, a trade discount followed by a cash discount, a fee charged after another fee: all compound. An FP&A analyst who adds layered discounts to estimate net price will overstate the discount and understate revenue per unit. The same arithmetic runs in reverse for mark-ups: a 20% price rise followed by a 20% cut leaves you 4% below where you started, because 1.2 x 0.8 = 0.96.