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%?
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?
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
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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?
026A company trades at 15x earnings and 6x EBITDA. Market cap is Rs 300 crore, net debt is Rs 180 crore, interest expense is Rs 18 crore and depreciation and amortisation is Rs 30 crore. What is the implied effective tax rate, and what does it suggest about the company?Valuation and multiples riddlesHardEquity researchTransaction advisory
Try it first
Before you calculate: which two numbers must you find first?
Show the worked solution
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.
Market cap of 300 at 15x gives net income of 20, and adding net debt of 180 gives EV of 480 and EBITDA of 80 at 6x. EBITDA less D&A of 30 and interest of 18 leaves pre-tax profit of 32, so a tax charge of 12 is a 37.5% rate.
The relationship
$t = \frac{\text{PBT} - \text{NI}}{\text{PBT}} = \frac{(480/6 - 30 - 18) - 300/15}{480/6 - 30 - 18} = \frac{12}{32} = 37.5\%$
PBT pre-tax profit: EBITDA less D&A less interest
NI net income: market cap over the P/E
480 enterprise value: market cap 300 plus net debt 180
What it says in wordsBuild pre-tax profit from the EV multiple, build net income from the P/E, and the tax rate is the share of pre-tax profit that did not survive.
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.
Where candidates lose it
Candidates reach EBITDA of 80, subtract tax from somewhere and forget one of the two lines between EBITDA and pre-tax profit, usually D&A. Walk the income statement in order, one line per step, and the missing line has nowhere to hide.
The second loss is stopping at 37.5%. The interviewer asked what it suggests. One sentence on non-deductible costs or loss-making units turns arithmetic into analysis.
What the interviewer asks next
If the company also held Rs 40 crore of cash earning 5%, how does the implied rate change?
Which items typically push an effective tax rate above the statutory rate?
What happens to the implied rate if the P/E rises to 20x with everything else fixed?
029A bowl holds 100 noodles. You pick two free ends at random and tie them together, and keep doing this until no free ends are left. What is the expected number of loops in the bowl at the end?Probability and expected valueHardConsulting-style caseKPO research support
Try it first
Before you compute: roughly how many loops do you expect from 100 noodles?
Show the worked solution
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.
Early ties almost never close a loop: with 100 strands the chance is 1 in 199, and the expected count is only 1.15 after 90 ties. The last three ties add 0.2, 0.33 and 1, bringing the total to 3.28 loops.
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.
The relationship
$E[\text{loops}] = \sum_{k=1}^{n} \frac{1}{2k-1} = 1 + \tfrac13 + \tfrac15 + \cdots + \tfrac{1}{199} \approx 3.28$
n the number of noodles, 100
k strands left before a tie
1/(2k-1) the chance the second end chosen belongs to the same strand
What it says in wordsAdd, tie by tie, the chance that the tie closes a loop.
Where candidates lose it
Candidates try to picture how the strands grow and get lost in cases. The problem is only tractable when you notice that each tie does exactly one of two things and that the strand count falls by one either way.
The second loss is getting the chance wrong as 1 over 2k. You have already picked one end, so the pool for the second pick is 2k - 1, not 2k.
What the interviewer asks next
With just two noodles, what is the chance of ending with two loops?
Roughly how many loops would you expect from 10,000 noodles?
Where else does linearity of expectation save you from tracking dependence?
067You have twelve coins that look identical. One is either heavier or lighter than the rest, you do not know which. Using a balance scale only three times, find the odd coin and say whether it is heavy or light.Logic and counting brainteasersHardConsulting-style caseCorporate finance
Try it first
What is the key to fitting the puzzle into three weighings?
Show the worked solution
Weigh coins 1 to 4 against 5 to 8. If they balance, the odd coin is among 9 to 12 and two weighings against known good coins find it. If one side is heavy, eight cases remain, and a second weighing that swaps some coins across the pans cuts them to three or fewer. It works because three weighings give 27 outcomes and there are only 24 cases.
Why can three weighings be enough?
Picture a guessing game where every answer is higher, lower or spot on. A question that leaves you with one of three equal groups does three times the work of a yes or no. A balance has three outcomes, left heavy, right heavy or level, so three weighings can tell apart 3 x 3 x 3 = 27 situations, and this puzzle has only 24: any of 12 coins, heavy or light. The plan works only if every weighing splits the remaining cases as evenly as possible into three.
That rules out six against six: one side always tips, so the balance outcome is wasted and twelve cases are left for two weighings that can separate nine. Four against four is right. If the left pan is heavy, the odd coin is one of 1 to 4 and heavy, or one of 5 to 8 and light: eight cases. If the pans balance, coins 1 to 8 are good and the odd coin is among 9 to 12, heavy or light: eight cases again.
Four against four splits the 24 cases into three groups of eight, the second weighing splits each eight into groups of three or fewer, and the third weighing settles each group, because three weighings give 27 outcomes for 24 cases.
How does the unbalanced branch get from eight cases to three?
Suppose the left pan was heavy. Weigh 1, 2, 5 against 3, 6 and coin 9, which is known to be good. Coins 4, 7 and 8 sit out. If the left is heavy again, the culprit is 1 or 2 heavy, or 6 light; weigh 1 against 2, and if they balance it is 6. If the right is now heavy, it is 3 heavy or 5 light; weigh 3 against a good coin. If they balance, it is 4 heavy or 7 or 8 light; weigh 7 against 8, and the lighter one is the culprit, or 4 if they balance. The trick is that some coins stay, some swap pans and some leave, so each outcome points to a different group.
If the first weighing balanced, weigh 9, 10, 11 against good coins 1, 2, 3. A heavy left means one of 9 to 11 is heavy, and 9 against 10 finds it. A light left means one is light, found the same way. A balance means coin 12 is odd, and weighing it against any good coin says whether it is heavy or light.
Why would a finance interviewer ask this?
It tests whether you count the possibilities before you start moving coins, which is the habit behind good diligence: design each question so that every possible answer tells you something new. The interviewer is happy with a clear first move and the counting argument, even if you need a minute for the unbalanced branch. The limit worth adding: with thirteen coins you can still find the odd coin in three weighings, but not always say whether it is heavy or light.
Where candidates lose it
Most candidates start with six against six, because halving feels efficient. It throws away the balance outcome, and from twelve cases left with two weighings there is no way back.
The second loss comes in the unbalanced branch: weighing the suspect coins against each other without mixing in known good coins, so two outcomes point to the same group. Use the good coins from the first weighing as references and move coins between pans on purpose.
What the interviewer asks next
What is the largest number of coins you can handle in three weighings if you only need to find the odd one?
You are told the odd coin is heavier. How many coins can three weighings handle?
How would you explain the counting argument to someone without using the word information?
073You pay Rs 100 to roll a fair die and receive Rs 30 times the face. Should you play? Now suppose that after seeing the roll you may pay Rs 20 to roll once more and take the second result instead. What is your rule, and what is the game worth?Probability and expected valueHardConsulting-style caseTreasury
Try it first
With the Rs 20 re-roll available, which rolls do you re-roll?
Show the worked solution
Play: one roll pays 30 x 3.5 = Rs 105 on average against a Rs 100 fee, an edge of Rs 5. With the re-roll, keep a 3 or better and re-roll a 1 or 2; the game is worth Rs 118.33, an edge of Rs 18.33. A fresh roll is worth 105 less the Rs 20 fee, 85, so re-roll only a result paying less than 85: 30 and 60 qualify, 90 does not.
Why is the first game worth playing?
A fair die averages 3.5, so the average payout is 30 x 3.5 = Rs 105, more than the Rs 100 you pay. On expected value the game pays Rs 5 a play, so a player who can repeat it should play. Half the time you lose money, since a 1, 2 or 3 pays 30, 60 or 90, and the other half you win 120, 150 or 180; the wins are larger than the losses, which is where the edge sits.
How does the re-roll change the rule?
Think of returning a shirt. You keep it if what you have is worth more than what the shop will hand you after the restocking fee. Re-roll only when the roll in hand is worth less than a fresh roll net of its fee, which is 105 - 20 = Rs 85. A 1 pays 30 and a 2 pays 60, both below 85: re-roll. A 3 pays 90, above 85: keep it, even though 3 is below the average face. The fee is what makes a 3 worth keeping.
Faces 3 to 6 are kept for 90, 120, 150 and 180, while faces 1 and 2 lead to a Rs 20 re-roll worth 85 net, so the game is worth 118.33 against 105 without the option, an edge of 18.33 over the Rs 100 fee.
The relationship
$E = \tfrac{1}{6}(90 + 120 + 150 + 180) + \tfrac{2}{6}(105 - 20) = 90 + 28.33 = 118.33$
90 to 180 the payouts kept on a 3, 4, 5 or 6
105 - 20 a fresh roll's average payout less the re-roll fee
2/6 the chance of a 1 or a 2, the rolls you re-roll
What it says in wordsThe game is worth the chance of keeping times the average kept, plus the chance of re-rolling times the fresh roll's value after the fee.
What does the option itself cost and earn?
The re-roll lifts the game from 105 to 118.33, so the option is worth Rs 13.33 on average, and it also cuts the chance of losing money from a half to a third, because a bad first roll gets a second chance. Re-rolling a 3 as well would give 117.50, worse by 0.83; a free re-roll would make re-rolling the 3 right and lift the game to 127.5. The threshold moves with the fee: below a fee of Rs 15 the 3 is worth re-rolling, above it the 3 is kept. The limit to say: the worst path, a 1 or 2 followed by a 1, costs Rs 90, and a player who cannot afford that should not value the game by its average.
Where candidates lose it
The common loss is re-rolling anything below the 3.5 average, including a 3. The bar is not the average face; it is the value of a fresh roll after its fee, and the Rs 20 fee pulls that bar below 90. Re-rolling a 3 gives up almost a rupee a play.
The quieter slip is forgetting to charge the fee at all, which puts the game at 127.5 and the threshold at 3. State the net value of a re-roll, 85, before deciding anything.
What the interviewer asks next
The re-roll fee is Rs 10. Which rolls do you now re-roll, and what is the game worth?
You may re-roll twice, Rs 20 each time. What is the game worth and how does the first-roll bar change?
The payout is Rs 30 times the face squared. Should you still play at Rs 100, and does the re-roll rule change?
083A phone sells for Rs 57,000 cash today, or on a 'no-cost EMI' of Rs 10,000 a month for six months, the first instalment due a month from now. What interest rate is hidden in the EMI?Compounding and time valueHardBank creditCorporate FP&A
Try it first
Roughly what annual rate is hidden in the EMI?
Show the worked solution
About 1.49% a month, or 19.4% a year effective. The real price is the Rs 57,000 cash price, so choosing the EMI means borrowing Rs 57,000 and repaying Rs 60,000 in six monthly instalments. The rate that makes six payments of Rs 10,000 worth exactly Rs 57,000 today is 1.49% a month. The forgone Rs 3,000 discount is the interest, charged on a balance that shrinks every month.
Where is the interest, if the EMI says no-cost?
A friend offers to sell you his bike for Rs 57,000 today, or for Rs 60,000 paid over six months. Nobody calls the second option interest-free; it is a loan of Rs 57,000 with Rs 3,000 of interest. A shop doing the same thing is no different. When a cash discount is available, the discount you give up by paying in instalments is the interest on the loan. The 'no-cost' label means the instalments add up to the sticker price, not that money has no time value.
So the question becomes: at what monthly rate is a stream of six Rs 10,000 payments, starting in one month, worth Rs 57,000 today? That is the internal rate of return of the loan.
The relationship
$57{,}000 = \sum_{k=1}^{6} \frac{10{,}000}{(1+r)^k} = 10{,}000 \times \frac{1-(1+r)^{-6}}{r} \;\Rightarrow\; r \approx 1.49\%$
r the monthly interest rate hidden in the instalments
k the month each Rs 10,000 instalment is paid
What it says in wordsThe hidden rate is the one that discounts the six instalments back to the cash price.
Six Rs 10,000 instalments add to Rs 60,000 at face value, but discounted at 1.49% a month they are worth exactly the Rs 57,000 cash price, so the Rs 3,000 discount given up is interest at 19.4% a year effective.
How do you solve it without a spreadsheet?
Guess and check. The annuity factor must be 57,000 / 10,000 = 5.70. At 1% a month, six payments are worth 5.80 times one payment; at 2% they are worth 5.60. Halfway gives about 1.5%, and 1.5% gives 5.697, a whisker under 5.70, so the answer is just below: 1.49%. Then compound, do not multiply: 1.0149 to the power 12 is 1.194, so the effective annual rate is 19.4%, against 17.8% if you simply multiply by 12.
A second check uses the average balance. The loan starts at Rs 57,000 and is paid down to zero over six months; the balance at the start of each month averages about Rs 33,700. Rs 3,000 of interest on about Rs 33,700 over six months is 8.9%, or about 17.8% a year simple, which matches 1.49% x 12.
What changes the answer in real life?
Timing matters a great deal. If the first instalment is due today rather than in a month, you really borrow only Rs 47,000 and repay Rs 50,000 over five months, and the hidden rate jumps to 2.10% a month, about 28% a year. Any processing fee, or tax charged on the interest component, raises the true cost further, so always ask what you pay on day one. Check the specific charges and the current tax treatment with the lender. And if there was never a cash discount to forgo, the EMI really is close to free for the buyer, because the retailer, not the buyer, is paying the lender.
Where candidates lose it
The quick wrong answer is about 5%: Rs 3,000 on Rs 57,000. Some candidates double it to 10.5% for a year and feel careful. Both ignore that the loan is repaid monthly, so the average balance is only a little over half of Rs 57,000, and the rate on what is actually owed is far higher than the headline.
The second loss is multiplying the monthly rate by 12 and calling it the annual rate. 17.8% is a nominal rate; money compounding monthly at 1.49% grows 19.4% in a year. Give both and name which is which.
What the interviewer asks next
If the first instalment is paid at purchase, what is the hidden rate? (About 2.1% a month.)
The cash discount is only Rs 1,000. What is the hidden rate now, roughly?
Who pays the lender when the retailer offers no cash discount at all, and why would the retailer do that?
090A company has net income of Rs 300 crore and 10 crore shares. It also has Rs 500 crore of 6% convertible bonds, convertible at Rs 250 a share. The tax rate is 25%. What is basic EPS, what is diluted EPS on the if-converted method, and what happens to each statement if the bonds actually convert?Accounting flow riddlesHardEquity researchBig Four
Try it first
How many new shares does conversion create?
Show the worked solution
Basic EPS is Rs 30.0; diluted EPS is Rs 26.88, about 10% lower. The if-converted method assumes conversion: add back the after-tax interest saved, Rs 30 crore x 0.75 = Rs 22.5 crore, and add 2 crore shares. That gives Rs 322.5 crore over 12 crore shares. On actual conversion, debt falls Rs 500 crore and equity rises Rs 500 crore, interest expense stops, and no cash changes hands.
What does the if-converted method assume?
Imagine a friend lent your shop Rs 5 lakh and can choose, any time, to swap the loan for a share of the business. To be honest with your other partners about what each share is worth, you would show the numbers as if the friend has already swapped: no more interest to pay, but one more partner splitting the profit. Diluted EPS shows earnings per share as if every convertible security that would reduce EPS had already converted, adding back the interest it would save and adding the shares it would create.
The interest is Rs 500 crore x 6% = Rs 30 crore a year. It was tax-deductible, so removing it raises net income by only Rs 30 crore x (1 minus 25%) = Rs 22.5 crore. Conversion creates Rs 500 crore / Rs 250 = 2 crore shares. Diluted EPS is (300 + 22.5) / (10 + 2) = Rs 26.875, which rounds to Rs 26.88.
The relationship
$\text{Diluted EPS} = \frac{\text{NI} + I(1-t)}{N + \frac{F}{P_c}} = \frac{300 + 30 \times 0.75}{10 + \frac{500}{250}} = \frac{322.5}{12} = 26.875$
NI net income, Rs 300 crore
I annual interest on the convertible, Rs 30 crore
N existing shares, 10 crore
F / P_c face value over conversion price, the new shares
What it says in wordsAdd the after-tax interest saved to earnings and the conversion shares to the share count.
Conversion lifts earnings by 7.5% but the share count by 20%, because each new share brings only Rs 11.25 of saved after-tax interest against Rs 30 earned by each existing share, so diluted EPS falls to Rs 26.88.
How do you know whether a convertible dilutes?
Compare what each new share brings with what each existing share earns. Each conversion share brings Rs 22.5 crore / 2 crore = Rs 11.25 of earnings, far below basic EPS of Rs 30, so including the convertible lowers EPS and it must be counted. If basic EPS were below Rs 11.25, conversion would raise EPS; that security is anti-dilutive and is left out of diluted EPS. The test is per security, so a company with several convertibles ranks them from most to least dilutive.
What happens to each statement on actual conversion?
Balance sheet: borrowings fall Rs 500 crore and equity rises Rs 500 crore, split between share capital at face value and securities premium. Total assets do not move; the right-hand side changes its mix and leverage falls sharply. Income statement: from the conversion date, the Rs 30 crore of interest stops, so profit rises Rs 22.5 crore after tax. Cash flow statement: nothing at conversion, because no cash changes hands; it is disclosed as a non-cash transaction. Afterwards, the Rs 30 crore of cash interest is no longer paid. Reported basic EPS after conversion should land close to the diluted EPS already shown, which is why analysts value on the diluted figure.
Where candidates lose it
The common error is adding the full Rs 30 crore of interest back to earnings. Interest was tax-deductible, so saving it also loses the tax shield: only Rs 22.5 crore reaches net income. Using Rs 30 crore gives Rs 27.50 instead of Rs 26.88.
The second loss is the statement walk. Candidates say cash falls by Rs 500 crore to repay the bond, but conversion repays nothing in cash: the bondholders take shares instead. Debt becomes equity, the balance sheet total stays put, and the cash flow statement shows nothing on the day.
What the interviewer asks next
At what level of net income would this convertible become anti-dilutive?
The company also has options on 1 crore shares at Rs 200, and the share price is Rs 300. How do they enter diluted EPS?
Why might a company prefer issuing a convertible to straight debt, and what does it give up?

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

096Branch A has a lower default rate than branch B among salaried borrowers, and a lower rate among self-employed borrowers, yet a higher overall default rate. A lent to 100 salaried borrowers who defaulted at 2% and 900 self-employed who defaulted at 10%. B lent to 900 salaried at 3% and 100 self-employed at 12%. Show how this is possible, and say which branch underwrites better.Data and statistics intuitionHardBank creditRating agencies

Try it first

Before you compute: which branch has the higher overall default rate?

Show the worked solution

A's overall rate is 9.2%, B's is 3.9%, and A is still the better underwriter in each segment. A: 2 defaults among 100 salaried plus 90 among 900 self-employed, 92 in 1,000. B: 27 plus 12, 39 in 1,000. An overall rate is a weighted average of segment rates, and the weights are the mix of the book. A's book is 90% self-employed, where everyone's rate is high, so its average is pulled up by whom it lends to, not by how well it lends.

How can a branch be better at everything and worse in total?

Two surgeons. One takes the hardest cases in the hospital and loses fewer of them than anyone else would; the other takes mostly routine cases. Count deaths per patient and the first surgeon looks worse, because her patients were sicker to begin with. An overall rate is a weighted average of segment rates, and the weights are the mix of the book, so a branch that lends mostly to the riskier segment carries a higher average even when it is better in every segment. A's rate is 0.1 x 2% + 0.9 x 10% = 9.2%; B's is 0.9 x 3% + 0.1 x 12% = 3.9%.

Segment	Branch A	Branch B	Lower rate
Salaried	100 loans, 2%, 2 defaults	900 loans, 3%, 27 defaults	A
Self-employed	900 loans, 10%, 90 defaults	100 loans, 12%, 12 defaults	A
Whole book	1,000 loans, 9.2%, 92 defaults	1,000 loans, 3.9%, 39 defaults	B

A beats B in each segment and loses on the whole book because its mix is nine tenths self-employed.

Drawn with width as the number of loans and height as the default rate, A's book is one wide bar at 10% and a sliver at 2%, averaging 9.2%, while B's is one wide bar at 3% and a sliver at 12%, averaging 3.9%, so A is better in both segments and worse overall.

How do you compare the two branches fairly?

Give them the same customers. Apply each branch's segment rates to one common mix and the mix stops voting. At B's mix, 90% salaried, A's rates give 0.9 x 2% + 0.1 x 10% = 2.8% against B's actual 3.9%. At A's mix, B's rates give 0.1 x 3% + 0.9 x 12% = 11.1% against A's actual 9.2%. At a 50:50 mix, A is 6.0% and B is 7.5%. On any common mix A is lower, so A underwrites better; its higher total is the price of the book it chose to build. This reversal, where a comparison flips when groups are combined, is known as Simpson's paradox.

The relationship

\text{overall rate} = \sum_i w_i\, r_i \qquad A: 0.1(2\%) + 0.9(10\%) = 9.2\% \qquad B: 0.9(3\%) + 0.1(12\%) = 3.9\%

w_i	the share of the book in segment i, the mix
r_i	the default rate within segment i

What it says in wordsThe overall default rate is each segment's rate weighted by its share of the loans, so the mix can outweigh the rates.

What should the analyst say beyond the arithmetic?

Three things. First, both statements are true at once: A underwrites better, and A's book loses more money, because capital is charged on actual losses and A chose a riskier book. Whether that was a good choice depends on the pricing: if A earns enough extra yield on self-employed loans to cover the extra 8 points of defaults, its strategy can be sound. Second, ask why the mixes differ, since location, product and targets all shape who walks in. Before judging two rates, ask whether the two populations are the same; if they are not, standardise to one mix or compare segment by segment.

Third, say the limitation. Each segment is itself a mix: of loan sizes, vintages and cities. A's salaried borrowers might be larger-ticket government employees and B's might be gig workers on salary slips, and a finer split could flip the comparison back. Standardising to the segments you can see removes the mix you know about, never the mix you do not. The honest close is that A looks the better underwriter on this cut, and you would want the segment rates by vintage before signing that view.

Where candidates lose it

The common failure is disbelief: candidates assume the overall rates must be wrong, or declare that B is the better underwriter because its total is lower. Do the two weighted averages aloud, 0.1 x 2 + 0.9 x 10 and 0.9 x 3 + 0.1 x 12, and the paradox dissolves into mix.

The second loss is stopping at the explanation. The interviewer wants the fair comparison: put both branches on one mix and A is lower on every one (2.8% against 3.9% at B's mix). Then add that A's book still loses more rupees, which is a question about strategy and pricing, not underwriting skill.

What the interviewer asks next

If A charges self-employed borrowers 4 points more interest than salaried ones, is its riskier book a sound strategy?
What single number would you report to the credit committee to compare the two branches' underwriting?
Give another finance example where combining groups reverses a comparison.

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

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

PBT	pre-tax profit: EBITDA less D&A less interest
NI	net income: market cap over the P/E
480	enterprise value: market cap 300 plus net debt 180

Try it first

Where does growth in operating profit come from?

What does the 10% case tell an analyst?

Where candidates lose it

What the interviewer asks next

Try it first

Why not just say maintenance capex equals D&A?

How does fixed asset turnover split the capex?

Is Rs 120 crore believable, and what is the catch?

Where candidates lose it

What the interviewer asks next

Try it first

How do you set the two schemes side by side?

Why would anyone order more than they need?

Where candidates lose it

What the interviewer asks next

Try it first

Where do you start when six numbers arrive at once?

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

Where candidates lose it

What the interviewer asks next

Try it first

How do you avoid tracking the whole bowl?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why can three weighings be enough?

How does the unbalanced branch get from eight cases to three?

Why would a finance interviewer ask this?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the first game worth playing?

How does the re-roll change the rule?

What does the option itself cost and earn?

Where candidates lose it

What the interviewer asks next

Try it first

Where is the interest, if the EMI says no-cost?

How do you solve it without a spreadsheet?

What changes the answer in real life?

Where candidates lose it

What the interviewer asks next

Try it first

What does the if-converted method assume?

How do you know whether a convertible dilutes?

What happens to each statement on actual conversion?

Where candidates lose it

What the interviewer asks next

Try it first

How can a branch be better at everything and worse in total?

How do you compare the two branches fairly?

What should the analyst say beyond the arithmetic?

Where candidates lose it

What the interviewer asks next