Financial Analysis puzzles, solved step by step

Puzzles: 100
Traced to a firm: 47
Topics: 13
Hard: 30

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003A company earns a 20% after-tax return on the capital it invests and wants operating profit to grow 15% a year. What share of operating profit must it reinvest each year? What changes if the return on capital is 10%?Ratio and margin riddlesHardEquity researchCorporate finance
Try it first
At a 20% return on capital, how much of each year's operating profit has to go back into the business to grow 15%?
Show the worked solution
It must reinvest 75% of operating profit at a 20% return, and 150% at a 10% return. Growth equals return on capital times the share of profit reinvested, so the reinvestment rate is 15% divided by the return. At 20% that leaves Rs 25 of every Rs 100 free. At 10% the company must invest Rs 150 for every Rs 100 it earns, raising Rs 50 from lenders or shareholders to keep growing.
Where does growth in operating profit come from?
Picture a tailor who earns 20% a year on every rupee of sewing machines she owns. If she wants next year's profit 15% higher, she needs 15% more machines' worth of profit, and each rupee of machines earns only 20 paise. Growth equals the return on new capital multiplied by the share of profit put back in. That share is the reinvestment rate, and it is the price the company pays today for tomorrow's growth.
The relationship
$g = \text{ROC} \times b \quad\Rightarrow\quad b = \frac{g}{\text{ROC}} = \frac{15\%}{20\%} = 75\%$
g growth in operating profit, 15% a year
ROC after-tax return on the new capital invested
b the share of after-tax operating profit reinvested
What it says in wordsDivide the growth you want by the return you earn, and that is the share of profit you must plough back.
To grow operating profit 15% a year, a company earning 30% on capital reinvests 50% of its profit, one earning 20% reinvests 75%, one earning 15% reinvests all of it, and one earning 10% must invest 150%, raising the gap from outside.
What does the 10% case tell an analyst?
Below a 15% return, 15% growth cannot be paid for out of profit at all. At 10%, for every Rs 100 earned the company must invest Rs 150, so Rs 50 comes from new debt or new shares every year and free cash flow is negative. The table shows how the free cash left for owners changes with the return on capital.
Return on capital Reinvestment rate Free cash per Rs 100 of profit
30% 50% 50
20% 75% 25
15% 100% 0
10% 150% (50)
Same 15% growth in every row. Free cash is operating profit less reinvestment; a bracket means cash must be raised from outside.
Growth at a low return can still be worth having if the return beats the cost of capital, and it destroys value if it does not. Say the limitation: the formula assumes new capital earns what old capital earns. Price increases and efficiency gains are growth that needs no reinvestment, so real companies sometimes beat it.
Where candidates lose it
The quick wrong answer is 15%, treating the growth rate as the share you reinvest. That ignores the return: the same growth costs very different amounts of capital at 10% and at 30%.
The second loss is stopping at 150% without saying what it means. A reinvestment rate above 100% is a funding need, and the interviewer wants to hear that growth at a low return on capital burns cash.
What the interviewer asks next
At a 12% cost of capital, is 15% growth at a 10% return good or bad for shareholders?
How would you estimate return on capital for a company from its annual report?
What happens to the valuation if growth falls to 5% with the return held at 20%?

Return on capital	Reinvestment rate	Free cash per Rs 100 of profit
30%	50%	50
20%	75%	25
15%	100%	0
10%	150%	(50)

006A product's price rises 25%, its volume falls 4% and the currency it is sold in adds 10% when translated back. Without paper, what is the combined change in revenue?Mental mathsCoreLeveraged financePrivate equity

Try it first

Say a number before you work it out properly.

Show the worked solution

Revenue rises 32%. Growth rates multiply, they do not add. Pair the price and volume factors first because they cancel neatly: 1.25 times 0.96 is exactly 1.20. Then 1.20 times 1.10 is 1.32. Adding 25, minus 4 and 10 gives 31%, which misses the cross terms, the growth earned on growth, worth one point here.

Why do the three changes multiply instead of add?

Think of a tea stall that sells 100 cups at Rs 10. Raise the price to Rs 12.50 and it now takes Rs 1,250 if it still sells 100 cups. If it then sells 4% fewer cups, those 4% are lost at the new, higher price, not the old one. Each change acts on revenue that the earlier changes have already moved, so the factors multiply. Revenue is price times volume times the exchange rate, and a product of three factors changes by the product of their changes.

The relationship

1+g = (1.25)(0.96)(1.10) = 1.20 \times 1.10 = 1.32

1.25	price factor, a 25% rise
0.96	volume factor, a 4% fall
1.10	currency factor, a 10% gain on translation
g	the combined revenue growth

What it says in wordsTurn every change into a factor, multiply the factors, and subtract one.

How do you do it in your head without fumbling?

Choose the order. Multiplication does not care about order, but your head does. 0.96 is 24 over 25 and 1.25 is 5 over 4, so their product is 120 over 100, exactly 1.20, and the hard step disappears. Ten per cent of 1.20 is 0.12, so the last step is 1.32. Pair the factors the other way and you are stuck with 1.25 times 1.10, then 1.375 times 0.96, which is the same answer by a much longer road.

Pair first	First product	Then multiply by	Ease in your head
Price x volume	1.2000	1.10	Easy: 24/25 x 5/4 is exactly 1.2
Price x currency	1.3750	0.96	Harder: 1.375 less 4% of it
Volume x currency	1.0560	1.25	Harder: 1.056 plus a quarter of it

All three routes give 1.32. The first is the one to say out loud, because every step is a round number.

Revenue of 100 moves to 125, then 120, then 132, so the combined change is 32%. Adding the rates gives 31%, because it leaves out the cross terms, which net to plus one point: 2.5 from price on the currency gain, less 1.0, 0.4 and 0.1 from the other pairings.

Why does the one-point gap matter on a real desk?

On a Rs 2,000 crore revenue line, one point is Rs 20 crore, which is a material miss in a budget bridge. The gap between adding and multiplying grows with the size of the changes, so it is small for 2% moves and large for 25% moves. When a board pack splits revenue growth into price, volume and currency, the cross terms have to sit somewhere, and an analyst should say where they put them rather than leave the bridge one point short.

Where candidates lose it

Saying 31% is the fast loss. It is close enough to sound right, and the interviewer asked precisely because the cross terms are what separates someone who knows growth compounds from someone who adds percentages.

The second loss is getting 32% by a slow route, multiplying 1.25 by 1.10 first and grinding through 1.375 times 0.96. Spend one second choosing the pairing that cancels; the interviewer is watching that choice as much as the answer.

What the interviewer asks next

Price falls 25% and volume rises 25%. Is revenue up, down or flat?
In local currency terms, how much did revenue grow?
How would you show the 32% split into price, volume and currency in a revenue bridge so it adds up?

008A company's D&A is Rs 80 crore and its capex Rs 200 crore. Revenue is Rs 1,600 crore and growing 10% a year, and fixed asset turnover is 2.0x. Estimate how much of the capex is growth capex and how much is maintenance capex.Working capital and cash riddlesHardEquity researchCorporate FP&A
Try it first
How much of the Rs 200 crore is maintenance capex?
Show the worked solution
Growth capex is about Rs 80 crore and maintenance capex about Rs 120 crore. Revenue grows by Rs 160 crore, and at a fixed asset turnover of 2.0x each rupee of new revenue needs 50 paise of new assets, so growth capex is Rs 80 crore. The rest, Rs 120 crore, replaces worn assets. That is Rs 40 crore more than D&A, because D&A records assets at the older prices paid for them.
Why not just say maintenance capex equals D&A?
Think of a taxi owner who bought a car eight years ago for Rs 6 lakh and has been setting aside Rs 75,000 a year as depreciation. When the car dies, the same model costs Rs 9 lakh, not Rs 6 lakh. D&A spreads the price paid for old assets, while maintenance capex pays today's price to replace them, so with any inflation the two drift apart. D&A is a reasonable floor for maintenance spending, not an estimate of it.
How does fixed asset turnover split the capex?
Fixed asset turnoverRevenue divided by net fixed assets. At 2.0x, each rupee of plant and equipment supports two rupees of yearly revenue. tells you how much plant each rupee of revenue needs. At 2.0x, the company's Rs 1,600 crore of revenue sits on about Rs 800 crore of net fixed assets. If the new revenue needs assets at the same ratio, the Rs 160 crore of growth needs Rs 80 crore of new capacity, and that is growth capex. Whatever is left of the Rs 200 crore went on keeping the existing Rs 1,600 crore of revenue alive.
The relationship
$\text{growth capex} = \frac{\Delta \text{revenue}}{\text{FAT}} = \frac{160}{2.0} = 80 \qquad \text{maintenance} = 200 - 80 = 120$
Δ revenue next year's extra revenue, 10% of 1,600
FAT fixed asset turnover, revenue over net fixed assets
What it says in wordsNew capacity is the new revenue divided by how much revenue each rupee of assets supports; the rest of capex is upkeep.
Treating D&A as maintenance splits the Rs 200 crore into 80 of maintenance and 120 of growth. The turnover method gives the opposite split, 120 of maintenance and 80 of growth, so maintenance runs Rs 40 crore above D&A.
Is Rs 120 crore believable, and what is the catch?
Check it against inflation. If the average asset was bought about eight years ago and equipment prices rose 5% a year, replacing it costs 1.05 to the power 8, about 1.48 times its original price. Rs 80 crore of D&A at today's prices is about Rs 118 crore, close to the Rs 120 crore estimate, so the split hangs together. The catch is that turnover is measured on net book value, which is itself at old prices. New capacity bought at today's prices may need more than 50 paise per rupee of revenue, which would make growth capex larger and maintenance smaller. Give the estimate as a range, and say why it matters: free cash flow before growth spending is Rs 40 crore lower than the D&A shortcut suggests.
Where candidates lose it
The common loss is the shortcut: maintenance equals D&A, so growth capex is 200 less 80, which is 120. It gives exactly the reverse of the turnover answer and overstates how much cash the business could release if it stopped growing.
The second loss is giving 120 and stopping. Say why D&A understates replacement cost, then name the weakness in your own method, the book-value turnover, before the interviewer does.
What the interviewer asks next
If growth stopped tomorrow, how much free cash flow would the business release each year?
How would you estimate maintenance capex from five years of the company's own history?
Why might a company with ageing assets report rising margins while its true earnings power falls?
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?
011You have twelve coins that look identical. One is counterfeit and is either heavier or lighter than the rest; you do not know which. Using a balance scale only three times, find the counterfeit and say whether it is heavy or light.Logic and counting brainteasersWarm upConsulting-style caseBank credit
Try it first
How many coins go on each pan in the first weighing?
Show the worked solution
Weigh four against four with four left off, then use known good coins to mix the suspects. If the first weighing balances, the fake is among the four left off, and two more weighings find it. If it tips, the fake is one of four possibly heavy or four possibly light coins; weighing 1, 2 and 5 against 3, 6 and a good coin splits those eight into groups of three or fewer, and the third weighing names the coin and its direction.
How do you know three weighings can be enough?
Count the possible answers before touching a coin. Any of 12 coins could be the fake, and it could be heavy or light, so there are 24 answers. A balance has three outcomes, left down, right down or level, so three weighings give 3 x 3 x 3 = 27 outcomes. Twenty four answers fit inside twenty seven outcomes, but only just, so every weighing must split the remaining possibilities into three nearly equal groups. Think of the game of twenty questions: a question with three possible answers is only worth asking if all three answers are about equally likely.
Why four against four, and what happens next?
Six against six can never balance, which throws away one of the three outcomes. Four against four with four left off splits the 24 answers into 8, 8 and 8. If it balances, the eight coins on the scale are proven good, and you can use them as reference weights in the next weighing. Weigh three of the suspects, 9, 10 and 11, against three good coins. Level means coin 12 is fake and one last weighing against a good coin says heavy or light. A tip tells you both that the fake is among 9 to 11 and which way it is off, so weighing 9 against 10 finishes it.
Weighing four coins against four splits the 24 possible answers into three groups of 8. Each branch then uses a second weighing that mixes suspects with known good coins, so every one of the 24 answers ends at a single coin and a heavy or light verdict after three weighings.
If the first weighing tips with the left side heavy, the fake is one of 1 to 4 and heavy, or one of 5 to 8 and light. Now move coins between pans. Weigh 1, 2 and 5 against 3, 6 and good coin 9: each outcome leaves at most three suspects, each with a known direction. Level leaves 4 heavy, 7 light or 8 light, and weighing 7 against 8 settles it. Left heavy leaves 1 heavy, 2 heavy or 6 light; weigh 1 against 2. Left light leaves 3 heavy or 5 light; weigh 3 against a good coin.
The relationship
$2N \le 3^{n} - 3 \quad\Rightarrow\quad N \le \frac{3^{3}-3}{2} = 12$
N number of coins
n number of weighings, here 3
2N possible answers: each coin, heavy or light
What it says in wordsThree weighings can handle at most twelve coins when you must also say heavy or light, so twelve is the hardest version that still works.
Where candidates lose it
The usual loss is opening with six against six, or splitting into halves out of habit. The scale tips, you have twelve candidates and two weighings left, and nine outcomes cannot cover them.
The second loss is keeping the suspects in fixed groups after the first weighing. The solution only works because you move coins between pans and bring in coins already proven good. Say that idea out loud even if the details of a branch take you a moment.
What the interviewer asks next
What if you only had to find the fake, not say whether it is heavy or light?
With four weighings, how many coins could you handle?
If you are told the fake is heavier, how many coins can three weighings handle?
012A supplier lists a part at Rs 100. It offers two schemes: 5% off every unit if you buy 1,000 or more, or 10% off only the units above 1,000. At what volume do the two schemes cost the same? And why might a buyer who needs 990 units order 1,000?Pricing, costing and unit economicsHardCorporate FP&ACost accounting
Try it first
At what order size do the two schemes cost exactly the same?
Show the worked solution
The schemes cost the same at 2,000 units, Rs 1,90,000 each. The all-units deal costs 95 times the quantity; the incremental deal costs Rs 1,00,000 for the first 1,000 and Rs 90 after that, which equals 95Q only at 2,000. Below that, all-units is cheaper. It also creates a cliff: 990 units cost Rs 99,000 but 1,000 cost Rs 95,000, so anyone ordering more than 950 should round up to 1,000.
How do you set the two schemes side by side?
Think of two mobile data plans, one with a flat lower price on everything once you cross a usage level and one that only discounts the extra data. They feel similar, but they reward different amounts of use. Write each scheme as total cost against quantity, then find where the two formulas are equal. All-units: 95Q once Q reaches 1,000. Incremental: 1,00,000 for the first 1,000 units, plus 90 for each unit beyond.
The relationship
$95Q = 100{,}000 + 90(Q - 1{,}000) \;\Rightarrow\; 5Q = 10{,}000 \;\Rightarrow\; Q = 2{,}000$
Q units ordered, at least 1,000
95 all-units price after 5% off
90 incremental price on units above 1,000
What it says in wordsThe incremental deal saves Rs 5 a unit more on every unit above 1,000, and needs 2,000 units in total to make up the Rs 5,000 head start the all-units deal gives at 1,000.
A quicker way to see it: at 1,000 units the all-units scheme is Rs 5,000 cheaper. Each extra unit costs Rs 95 under all-units and Rs 90 under incremental, so the incremental deal claws back Rs 5 a unit. It needs another 1,000 units to recover Rs 5,000, so the schemes meet at 2,000. Above that the incremental scheme wins: at 3,000 units it costs Rs 2,80,000 against Rs 2,85,000.
The all-units scheme drops the average price from Rs 100 to Rs 95 the moment an order reaches 1,000 units, while the incremental scheme lowers it gradually. The incremental average only reaches Rs 95 at 2,000 units, which is where the two schemes cost the same.
Why would anyone order more than they need?
Because the all-units scheme makes buying more cost less. 990 units at Rs 100 cost Rs 99,000, while 1,000 units at Rs 95 cost Rs 95,000, so ten extra parts arrive with Rs 4,000 back in your pocket. The same logic holds for any order above 950 units, because 1,000 units cost Rs 95,000 and anything from 951 to 999 costs more. A buyer with somewhere to store the spares should always round up.
Under the all-units scheme, 990 units cost Rs 99,000 but 1,000 units cost only Rs 95,000. Any order between 951 and 999 units costs more than rounding up to 1,000, which is the cliff an all-units discount creates.
Say the limit from both sides of the deal. Extra units carry storage, handling and the risk they are never used, so the saving is only real if the spares have a use. For the supplier, the cliff means giving away Rs 4,000 on a sale that would have happened anyway, which is why incremental schemes are common where buyers order close to a threshold.
Where candidates lose it
The common loss is answering 1,000, the point where the discounts start, instead of finding where the costs meet. Writing both costs as formulas takes ten seconds and makes 2,000 obvious.
The second loss is missing the cliff. Interviewers add the 990 question to see whether you notice that an all-units discount makes total cost fall as quantity rises, which is the opposite of what a cost curve normally does.
What the interviewer asks next
What is the smallest order at which rounding up to 1,000 saves money?
If storing each spare part costs Rs 3 a year, does the 990 buyer still round up?
Which scheme would you offer as the supplier, and why?
017A vendor offers a game: a fair coin is tossed until it shows heads. If heads comes on the first toss you get Rs 100, on the second toss Rs 200, and the prize doubles with every toss after that. The game costs Rs 1,000 to play, but the vendor has only Rs 10 lakh to pay out. What is the game really worth, and would you play?Probability and expected valueCoreEquity capital marketsConsulting-style case
Try it first
With the vendor's Rs 10 lakh limit, what is the game worth on average?
Show the worked solution
About Rs 761, so at Rs 1,000 the game is not worth playing. Each possible toss adds Rs 50 of expected value: a prize that doubles times a chance that halves. Prizes fit under the Rs 10 lakh cap for 14 tosses, which is Rs 700, and all longer runs are paid Rs 10 lakh, which adds only about Rs 61. The infinite value of the textbook game rests entirely on prizes no vendor can pay.
Why does every toss add exactly Rs 50?
Imagine a lottery stall where a ticket wins Rs 200 one time in four, or Rs 400 one time in eight. Each prize is worth the same on average, Rs 50, because the prize doubles exactly as the chance halves. In this game, heads first appearing on toss k has probability one over 2 to the k, and pays 100 times 2 to the k minus 1, so every possible toss contributes Rs 50 of expected value. With no limit there are infinitely many tosses, so the expected value is infinite. That is the famous St Petersburg paradox, and it is why nobody sensible pays a fortune to play.
The relationship
$E = \sum_{k=1}^{\infty} \frac{1}{2^{k}}\,\min(100 \cdot 2^{k-1},\ 10^{6}) = 14 \times 50 + \frac{10^{6}}{2^{14}} \approx 761$
k the toss on which the first head appears
1/2^k the chance the first head comes on toss k
100 x 2^(k-1) the promised prize
10^6 the vendor's limit, Rs 10 lakh
What it says in wordsFourteen tosses each add Rs 50; every longer run pays the capped Rs 10 lakh, and the chance of reaching toss 15 is one in 16,384.
What does the Rs 10 lakh limit do to the value?
The prize on toss 14 is Rs 100 times 2 to the 13, which is Rs 8,19,200; on toss 15 it would be Rs 16,38,400, more than the vendor holds. So only 14 tosses pay in full, worth Rs 700, and every longer run pays Rs 10 lakh, which happens one time in 16,384 and adds about Rs 61. The game is worth about Rs 761. Paying Rs 1,000 means losing about Rs 239 a game on average.
Each of the first 14 tosses adds Rs 50 of expected value, reaching Rs 700, and the capped tail adds only about Rs 61, so the game is worth about Rs 761. Without the cap the value would keep rising Rs 50 a toss, but the capped game never reaches the Rs 1,000 price.
How rich would the vendor need to be for Rs 1,000 to be fair?
Value grows very slowly with the vendor's wealth, because each doubling of the bankroll adds just one more Rs 50 toss. A vendor holding Rs 1,000 crore would make the game worth only about Rs 1,425, and the game reaches Rs 1,000 only if the vendor can pay about Rs 2.6 crore. Say the limit too: even a fair price ignores how much risk you can stomach, since almost every game pays Rs 100 or Rs 200. That is why economists use this game to show people value money by its usefulness to them, not by its face amount.
Where candidates lose it
The common loss is reciting "infinite expected value" and stopping, or worse, saying you would pay any price. The interviewer added the vendor's limit precisely to see whether you notice that infinite value depends on payouts that cannot happen.
The second loss is getting the cap wrong by assuming the tail adds a lot. Work out the last toss that pays in full, count Rs 50 per toss, and add the small capped tail; it takes thirty seconds.
What the interviewer asks next
What would you pay if the vendor could pay out Rs 100 crore?
The vendor offers to play the game ten times in a row for Rs 7,000. Does that change your answer?
How does this game relate to valuing a company with a tiny chance of an enormous outcome?
019A plant's output grows 10% every month. Roughly how many months until output doubles, and how long until it is eight times today's level? Use the rule of 72, then check it against the exact answer.Compounding and time valueWarm upConsulting-style caseCorporate FP&A
Try it first
How long until output is eight times today's level?
Show the worked solution
Output doubles in about 7.2 months and reaches eight times in about 21.6 months, by the rule of 72. Divide 72 by the growth rate in per cent: 72 over 10 is 7.2 months per doubling, and eight times is three doublings. The exact answer, the log of 2 over the log of 1.1, is 7.27 months, or 21.8 months for eight times, so the rule is within a fifth of a month.
Why does eight times take three doublings and not 70 months?
Think of a rumour that doubles the number of people who know it every week. One becomes two, then four, then eight in three weeks, not eight weeks. Compounding growth multiplies, so the question to ask is how many doublings fit in, not how many 10% steps add up to 700%. Two times two times two is eight, so eight times today's output is three doublings away. Dividing 700% by 10% a month treats each month's growth as if it were earned only on the starting level.
The relationship
$t_{2x} = \frac{\ln 2}{\ln 1.10} = \frac{0.693}{0.0953} = 7.27 \qquad \text{rule of 72: } \frac{72}{10} = 7.2$
ln 2 the natural log of 2, about 0.693
ln 1.10 the natural log of one month's growth factor, about 0.0953
72 a convenient number close to 100 times ln 2, with many divisors
What it says in wordsDoubling time is the log of two divided by the log of one period's growth factor; the rule of 72 approximates that with one division.
At 10% a month, output reaches twice today's level after 7.27 months, four times after 14.5 and eight times after 21.8. The rule of 72 marks at 7.2, 14.4 and 21.6 months sit just short of each, so the shortcut is close enough to say out loud.
Where does the rule of 72 work, and where does it slip?
The exact doubling time uses the log of 1 plus the rate, which is a little less than the rate itself, and 72 is chosen to compensate for typical rates. The rule is within about 1% for rates from 5% to 10% a period and drifts at the extremes: at 2% it overstates, and at 50% it understates. At 10% the gap is only 0.07 months. A check in whole months helps too: 1.1 to the 7th is 1.95 and to the 8th is 2.14, so output passes double during month 8.
Growth per period Rule of 72 Exact Gap
2% 36.00 35.00 +2.8%
5% 14.40 14.21 +1.4%
10% 7.20 7.27 -1.0%
25% 2.88 3.11 -7.3%
50% 1.44 1.71 -15.8%
Periods to double, with the rule's error as a share of the exact answer. It is within about 1% at 5% and 10% a period, and drifts further at very low and very high rates.
Say the limit in context. Output growing 10% a month for two years is a startup or a ramp-up, not a steady state; a plant hits capacity, demand or cash constraints long before it has grown nearly tenfold. The interviewer wants the arithmetic, then one sentence of business sense.
Where candidates lose it
The common loss is answering about 70 months for eight times, treating the growth as simple rather than compound. It is the same instinct that underestimates how quickly compounding builds.
The second loss is reaching for a calculator-style log in your head and stalling. Give the rule of 72 answer first, then say the exact number is a little longer, about 7.3 months.
What the interviewer asks next
Using the same idea, how long does it take for prices to halve in value at 6% inflation?
Why do some people use 69 or 70 instead of 72?
Output grows 10% a month for a year. What is the annual growth rate?
021You invest Rs 10,000 in a fund when its NAV is Rs 100, and another Rs 10,000 when the NAV is Rs 50. What is your average cost per unit, and why is it not Rs 75?Percentages and averagesCoreCorporate FP&ABusiness finance
Try it first
What is your average cost per unit?
Show the worked solution
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.
The relationship
$\text{average cost} = \frac{\text{total spent}}{\text{total units}} = \frac{20{,}000}{100 + 200} = 66.67 = \frac{2}{1/100 + 1/50}$
total spent Rs 20,000 across both purchases
total units 100 units at NAV 100 plus 200 at NAV 50
2 / (1/100 + 1/50) the harmonic mean of the two prices
What it says in wordsAverage cost is money spent over units bought, which for equal rupee amounts is the harmonic mean of the prices.
Rs 10,000 buys 100 units at a NAV of 100 and 200 units at a NAV of 50, so Rs 20,000 buys 300 units at an average cost of Rs 66.67. That harmonic mean sits below both the geometric mean of 70.7 and the simple average price of 75.
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..
What you hold fixed Units bought Spent, Rs Average cost, Rs
Rs 10,000 each time 300 20,000 66.67
100 units each time 200 15,000 75.00
The same two prices give two different average costs, depending on whether the rupee amount or the number of units was held constant.
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.
Where candidates lose it
The common loss is answering Rs 75 by averaging the two prices. It ignores that the purchases bought different numbers of units, and it is the same slip people make when they average two speeds for a journey.
The second loss is going too far the other way and describing the lower average cost as a gain. Give Rs 66.67, say break-even is at that NAV, and stop short of claiming anything about returns.
What the interviewer asks next
You invest Rs 10,000 at NAV 100, then 50, then 100 again. What is your average cost?
What NAV do you need to break even on the two purchases?
Why is the harmonic mean of two different positive numbers always below their simple average?
023A customer who owes your company Rs 50 lakh goes bankrupt and will pay nothing. Walk the write-off through the three statements at a 25% tax rate, first when no provision was held against the debt, then when a full provision was booked last year.Accounting flow riddlesWarm upBig FourCorporate FP&A
Try it first
A full provision for this debt was booked last year. How much does this year's net income fall when the debt is written off?
Show the worked solution
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.
With no provision held, writing off Rs 50 lakh cuts net income by Rs 37.5 lakh after tax and raises cash by Rs 12.5 lakh of tax saved. With a full provision booked last year, the write-off removes the debt and the provision together, so neither profit nor cash moves this year.
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.
The relationship
$\Delta NI = -50 \times (1 - 0.25) = -37.5 \qquad \Delta\text{Cash} = +50 \times 0.25 = +12.5$
50 the debt written off, Rs lakh
0.25 tax rate
ΔNI change in net income with no provision held
What it says in wordsWithout a provision, profit falls by the after-tax loss and cash rises by the tax saved.
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.
Where candidates lose it
The common loss is putting the Rs 50 lakh through profit again in the provision case, which counts the same loss twice: once when the provision was made and again on write-off.
The second loss is in the no-provision case: saying cash falls by 50. No cash moved when the customer failed; the cash was never received. The only cash effect is the tax saved.
What the interviewer asks next
The provision booked last year was only Rs 30 lakh. Walk the write-off through now.
The customer later pays Rs 10 lakh after all. What happens in each statement?
How would a rising provision show up in a company's cash flow statement?

g	growth in operating profit, 15% a year
ROC	after-tax return on the new capital invested
b	the share of after-tax operating profit reinvested

Δ revenue	next year's extra revenue, 10% of 1,600
FAT	fixed asset turnover, revenue over net fixed assets

N	number of coins
n	number of weighings, here 3
2N	possible answers: each coin, heavy or light

Q	units ordered, at least 1,000
95	all-units price after 5% off
90	incremental price on units above 1,000

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What does the 10% case tell an analyst?

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Why does the cheap purchase count for more?

When is Rs 75 the right answer?

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What happens when nothing was set aside?

Why does the provision case show nothing this year?

What about tax in the provision case?

Where candidates lose it

What the interviewer asks next