Financial Analysis puzzles, solved step by step

Puzzles: 100
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009A company's EBITDA margin moves from 8% to 10%. Is that up 2% or up 25%? And a bank's gross NPA ratio falls from 4% to 3%: how would you describe that change in a results note?Percentages and averagesWarm upCorporate FP&ARating agencies
Try it first
Which phrase belongs in a results note about the margin moving from 8% to 10%?
Show the worked solution
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.
The margin moving from 8% to 10% is a rise of 2 percentage points, or 200 basis points, and 25% in relative terms. The NPA ratio falling from 4% to 3% is a fall of 1 percentage point, or 100 basis points, and a 25% relative fall.
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.
The relationship
$\Delta_{\text{pp}} = 10\% - 8\% = 2 \text{ pp} \qquad \Delta_{\text{rel}} = \frac{10\% - 8\%}{8\%} = 25\%$
pp percentage points, the plain gap between two rates
rel the relative change, the gap divided by the starting rate
What it says in wordsSubtract for the change in points; divide that by the start for the change in per cent.
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.
Where candidates lose it
The loss is writing margin up 2% in a results note or saying it in an interview. A reader who takes you literally hears a move from 8% to 8.16%, and a credit reader will mark it as sloppy at once.
The second loss is choosing the bigger number to sound better: up 25% sounds stronger than up 2 points. Interviewers listen for whether you pick the label that fits, then offer the other as context.
What the interviewer asks next
Interest rates rise from 6.50% to 6.75%. How many basis points is that, and what is the relative change?
A bank's gross NPA ratio falls but its NPAs in rupees rise. How is that possible?
Margin is flat at 10% while revenue grows 20%. What happened to EBITDA?
065A pack carries a maximum retail price of Rs 1,180, inclusive of 18% GST. How much of that price is tax: Rs 212.40 or Rs 180?Percentages and averagesWarm upCorporate FP&ABig Four
Try it first
How much of the Rs 1,180 is tax?
Show the worked solution
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%.
The Rs 1,180 price is a Rs 1,000 base plus Rs 180 of tax, 18% of the base but only 15.25% of the whole, while 18% of the full price, Rs 212.40, reaches into the base because it charges tax on the tax.
How do you strip tax out of any inclusive price?
The relationship
$\text{base} = \frac{1{,}180}{1.18} = 1{,}000 \qquad \text{tax} = 1{,}180 \times \frac{0.18}{1.18} = 180$
1,180 the inclusive price, MRP
0.18 the tax rate, applied to the base
0.18 / 1.18 the tax as a share of the inclusive price, about 15.25%
What it says in wordsDivide an inclusive price by one plus the rate to get the base; the tax is the rate over one plus the rate, times the inclusive price.
Rate on base Share of inclusive price Tax in a Rs 1,180 price
5% 4.76% 56.19
12% 10.71% 126.43
18% 15.25% 180.00
28% 21.88% 258.12
The higher the rate, the wider the gap between the rate on the base and the tax's share of the inclusive price, so the error from multiplying the inclusive price by the rate grows with the rate.
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.
Where candidates lose it
The fast answer multiplies the MRP by 18% and says Rs 212.40. It sounds precise to two decimals and is wrong by Rs 32.40, because the rate applies to the price before tax, not after.
The other slip is subtracting 18% from the MRP to get the base, 1,180 less 212.40 = 967.60. Undoing a percentage increase needs a division by 1.18, not a subtraction of 18%.
What the interviewer asks next
A shop offers 20% off the MRP of Rs 1,180. How much tax is in the discounted price?
A product's cost is Rs 800 and it is sold at a 25% markup. What is the margin on price?
Revenue in a set of accounts is reported inclusive of an 18% indirect tax at Rs 590 crore. What is net revenue?
078Your salary rises 10% in a year when inflation is 6%. What is your real pay rise, and how close is the shortcut of 10 minus 6, or 4%?Percentages and averagesWarm upCorporate FP&ABusiness finance
Try it first
Is the exact real raise above, equal to, or below 4%?
Show the worked solution
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 relationship
$1 + r = \frac{1+g}{1+\pi} \quad\Rightarrow\quad r = \frac{g-\pi}{1+\pi} = \frac{0.04}{1.06} = 3.77\%$
g nominal pay growth, 10%
\pi inflation, 6%
r real pay growth
What it says in wordsThe real rate is the shortcut divided by one plus inflation.
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.
At a 10% raise and 6% inflation the shortcut gives 4.00% against an exact 3.77%, but at a 30% raise and 20% inflation it gives 10.00% against 8.33%, so the subtraction shortcut drifts further from the truth as rates rise.
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.
Where candidates lose it
Most candidates say 4% and stop. That is an acceptable first instinct, but the question is asked precisely to see whether you know it is an approximation and which way it errs. Say 3.77% and add that the shortcut runs slightly high.
The other loss is the opposite: dividing correctly but being unable to say why. The one-line reason is that your raise is itself paid in rupees that buy 6% less, so part of the extra 10% is eaten too.
What the interviewer asks next
Your pay rose 5% and inflation was 7%. What happened to your real pay, exactly?
Revenue grew 18% and prices rose 8%. What was volume growth?
Why do lenders quote real interest rates, and how would you compute one?
091A shop offers 20% off, and then an extra 10% off at the till. Is that the same as 30% off? And what discount per unit is 'buy two, get one free'?Percentages and averagesWarm upCorporate FP&ABusiness finance
Try it first
On a Rs 1,000 item, what do you pay after 20% off and then a further 10% off?
Show the worked solution
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 relationship
$1 - d_{\text{total}} = (1 - d_1)(1 - d_2) = 0.80 \times 0.90 = 0.72 \quad\Rightarrow\quad d_{\text{total}} = 28\%$
d_1 the first discount, 20%
d_2 the second discount, 10%
d_total the single discount with the same effect
What it says in wordsMultiply what each discount leaves you paying; the total discount is one minus that product.
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%.
On a Rs 1,000 item, 20% then 10% off leaves Rs 720, a 28% discount, against Rs 700 at a flat 30%, while buy two get one free cuts the price per unit by a third, the largest of the offers if you wanted three units.
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.
Where candidates lose it
The trap is answering yes, 30%, because the two numbers add neatly. The interviewer wants to hear that the second discount works on a smaller base, and the clean proof is 0.8 x 0.9 = 0.72.
The second loss is calling buy two get one free a 50% discount, because one item in two is free. It is one item in three. Count the units you walk out with, not the units you pay for, and add that the offer is only worth anything if you wanted three.
What the interviewer asks next
A price rises 25% and then falls 20%. Where does it end up?
What single discount equals 10%, then 10%, then 10% off?
A distributor buys at 30% below MRP and the retailer at 20% below MRP. What margin, on its own selling price, does the distributor earn?

pp	percentage points, the plain gap between two rates
rel	the relative change, the gap divided by the starting rate

1,180	the inclusive price, MRP
0.18	the tax rate, applied to the base
0.18 / 1.18	the tax as a share of the inclusive price, about 15.25%

Rate on base	Share of inclusive price	Tax in a Rs 1,180 price
5%	4.76%	56.19
12%	10.71%	126.43
18%	15.25%	180.00
28%	21.88%	258.12

g	nominal pay growth, 10%
\pi	inflation, 6%
r	real pay growth

Try it first

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

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

Where candidates lose it

What the interviewer asks next

Try it first

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

How do you strip tax out of any inclusive price?

Where else does the same slip turn up?

Where candidates lose it

What the interviewer asks next

Try it first

Why divide rather than subtract?

When does the shortcut become a real error?

Where candidates lose it

What the interviewer asks next

Try it first

Why do discounts not simply add?

What is buy two, get one free really worth?

Where does this show up at work?

Where candidates lose it

What the interviewer asks next