Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
Hard: 30

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007Tarvela AMC's assets under management rose from Rs 40,000 crore to Rs 52,000 crore over a period in which the industry grew from Rs 5 lakh crore to Rs 7 lakh crore. Did Tarvela's market share rise?Logic and numeracy brainteasersWarm upIndian AMCsGlobal asset managers
Try it first
Quick call: what happened to Tarvela's share?
Show the worked solution
No. Tarvela's share fell from 8.0% to about 7.4%. Its AUM grew 30% while the industry grew 40%. Share is 40,000 over 5,00,000 before and 52,000 over 7,00,000 after. To hold 8% Tarvela needed Rs 56,000 crore, so it is Rs 4,000 crore short of standing still. Growing slower than your market means losing share while growing.
Why can a business grow and still shrink?
A child who grows 3 cm in a year while every classmate grows 5 cm is taller than last year and lower in the line-up. Market share is a ratio, so it rises only when your growth beats the market's, and a large rupee gain means nothing on its own. Tarvela's Rs 12,000 crore gain sounds big until you set it next to the industry's Rs 2 lakh crore.
Tarvela AMC grew its assets 30% while the industry grew 40%, so its market share fell from 8.0% to 7.4%, and holding 8% would have needed Rs 56,000 crore rather than Rs 52,000 crore.
What is the fastest way to answer without dividing big numbers?
Compare growth factors. New share equals old share times Tarvela's growth factor over the industry's: 8% times 1.30 over 1.40. 1.3 over 1.4 is about 0.93, so the share falls by about 7% of itself, from 8.0% to 7.43%. You never need to divide 52,000 by 7,00,000 unless asked for the decimal.
The relationship
$s_1 = s_0 \times \frac{1 + g_{AMC}}{1 + g_{ind}} = 8.0\% \times \frac{1.30}{1.40} = 7.43\%$
s_0, s_1 market share before and after
g_{AMC} Tarvela's AUM growth, 30%
g_{ind} the industry's AUM growth, 40%
What it says in wordsShare moves by the ratio of your growth factor to the market's.
Add the analyst's question. AUM growth has two parts, market movement and net flows. If Tarvela is heavier in a segment that rose less, it could lose share with no problem in its sales. Splitting growth into the part the market gave and the part investors gave is the next thing an interviewer will ask for.
Where candidates lose it
The trap is answering yes because Rs 12,000 crore of growth sounds impressive. The interviewer has deliberately given a big absolute gain in a market growing faster, to see whether you reach for the ratio.
The second slip is subtracting growth rates, 30 minus 40, and saying share fell by 10 points. Share fell by about 0.57 points, from 8.0% to 7.43%; it is the ratio 1.3 over 1.4 that matters.
What the interviewer asks next
What growth did Tarvela need to gain half a point of share?
How would you split Tarvela's growth into market movement and net inflows?
Why might an AMC accept losing share in one category?
057A fund cuts its expense ratio from 1.5% to 1.2%. Is that a 0.3% cut, a 30 basis point cut or a 20% cut, and which description will a client misunderstand?Logic and numeracy brainteasersWarm upIndian AMCsGlobal asset managers
Try it first
Which descriptions of the change are correct?
Show the worked solution
It is a 0.3 percentage point cut, which is 30 basis points, and a 20% relative cut in the fee; "a 0.3% cut" is the sloppy one. Basis points remove the ambiguity. The 20% is true but invites a client to imagine returns rising 20%; on Rs 10 lakh the saving is Rs 3,000 a year, and a net return of 8.5% becomes 8.8%.
Why are there two different answers that are both right?
A shop raises the price of milk from Rs 50 to Rs 60. It went up by Rs 10, and it went up by 20%. Nobody confuses those, because rupees and percent look different. With rates, the change and the base are both percentages, so "percent" can mean the gap between two rates or the change relative to the old rate, and you must say which. The gap between 1.5% and 1.2% is 0.3 percentage pointsThe plain difference between two percentages. From 1.5% to 1.2% is a fall of 0.3 percentage points.. Relative to the old fee, 0.3 is a fifth of 1.5, a 20% cut.
The same fee cut from 1.5% to 1.2% is 0.3 percentage points, 30 basis points or a 20% relative cut, and on Rs 10 lakh it saves Rs 3,000 a year while lifting a net return from 8.5% to 8.8%.
Which description will a client get wrong, and what should you say instead?
"A 0.3% cut" fails one way: a careful reader can take it as 0.3% of the fee, which would leave 1.5% x 0.997 = 1.4955%, a cut a hundred times too small. "A 20% cut" fails the other way: it is true of the fee, but a client hears 20% and imagines the investment doing 20% better. The fee is a slice of a slice; a 20% smaller slice lifts a net return of 8.5% to 8.8%, only 3.5% better in relative terms. The clean form for a desk is basis pointsHundredths of a percentage point. 30 basis points equal 0.30 percentage points.; the clean form for a client is rupees: Rs 15,000 a year on Rs 10 lakh becomes Rs 12,000.
The relationship
$1.5\% - 1.2\% = 0.3\text{ pp} = 30\text{ bps} \qquad \frac{1.5 - 1.2}{1.5} = 20\%$
pp percentage points, the plain gap between two rates
bps basis points, one hundredth of a percentage point
1.5 the old fee, the base for the relative change
What it says in wordsSubtract for points, divide by the old rate for percent change, and name which one you mean.
The same confusion runs through everything a fund desk reports: a yield moving from 7.0% to 7.5% is 50 basis points, not 0.5%, and a fund outperforming by 2% might mean two points or a fiftieth more. The limit worth stating: neither unit tells the client the money, so pair any rate change with the rupee effect on their actual holding.
Where candidates lose it
The trap is picking one answer and defending it. The interviewer is not after 0.3 or 20; they want to hear that these are two different quantities and that the unit must be named every time.
The second miss is not taking the client's view. Saying 20% to a client is legally true and practically misleading, because it invites them to apply the 20% to their returns. Translate into rupees on their holding.
What the interviewer asks next
A bond yield rises from 6.8% to 7.2%. Describe the change in three correct ways.
A fund's return beat the index's 10% by 2%. List the two things this could mean.
Why do desks quote spreads in basis points rather than percent?
074A gold fund's vault holds nine coins that look identical, but one is slightly lighter than the others. Using a balance scale only twice, how do you find the light coin?Logic and numeracy brainteasersCoreIndian AMCsGlobal asset managers
Try it first
What should the first weighing be?
Show the worked solution
Weigh three coins against three. The pan that rises holds the light coin; if they balance, it is among the three left aside. Then weigh two of those three suspects against each other: the rising pan holds it, and a balance means it is the third. Each weighing has three possible outcomes, so two weighings can separate 3 x 3 = 9 coins.
Why split into three groups rather than two?
A shopkeeper asking a customer "yes, no, or not sure?" learns more from one answer than one asking only "yes or no?". A balance scale gives three answers, left pan rises, right pan rises, or the pans balance, so the best weighing splits the suspects into three equal groups and lets the scale tell you which group holds the light coin. Put coins 1 to 3 on the left and 4 to 6 on the right. If the left pan rises, the light coin is 1, 2 or 3. If the right pan rises, it is 4, 5 or 6. If they balance, it is 7, 8 or 9.
Weighing three coins against three leaves three suspects whatever the scale shows, and weighing two of those three against each other identifies the light coin, so two weighings cover all nine coins.
How do you know two weighings is the minimum, and how far does this go?
Count outcomes. One weighing has 3 outcomes, two have 3 x 3 = 9, and there are 9 possible light coins, so two weighings are just enough. The number of weighings needed is the smallest w with 3 to the power w at least the number of coins: 9 coins need 2, 27 coins need 3. One weighing can only handle 3 coins, so you cannot do 9 in one. Splitting into halves wastes the balance outcome and needs three weighings for nine coins.
The relationship
$3^w \ge N \;\Rightarrow\; w = \lceil \log_3 N \rceil, \qquad N = 9:\; w = 2$
w the number of weighings
N the number of coins, one of them light
3 the outcomes of one weighing: left rises, right rises, balance
What it says in wordsEach weighing can at most divide the suspects by three, so the weighings needed grow with the logarithm to base three of the number of coins.
The desk lesson is about information, not coins: a test with three outcomes is worth more than one with two, if you design it to use all three. The limit of the puzzle is that you are told the odd coin is lighter. If it could be heavier or lighter, each coin has two possible states, there are more cases to separate, and the strategy needs more care, which is the classic twelve-coin follow-up.
Where candidates lose it
The trap is splitting in halves: four against four, then two against two, then one against one. It works but takes three weighings, and the interviewer asked for two. Candidates who think in halves have missed that a balance is an answer too.
The second miss is getting the method but not the reason. Say that a weighing has three outcomes and two weighings give nine, which is why nine coins is the most two weighings can handle.
What the interviewer asks next
How many coins can you handle with three weighings?
Twelve coins, one odd, and you do not know whether it is heavier or lighter. Can you find it in three weighings?
You have a digital scale that shows exact weights instead. How many weighings do you need for nine coins?
088Five funds, A to E, are ranked by one-year return with no ties. C beat A but trailed E. A was neither first nor last. B finished immediately behind A. B beat D. What is the order from first to last, and which of the clues did you not actually need?Logic and numeracy brainteasersHardIndian AMCsGlobal asset managers
Try it first
Which fund finished first?
Show the worked solution
E, C, A, B, D, and two of the clues were not needed. C beat A but trailed E gives E ahead of C ahead of A. B behind A extends the chain, and B beat D adds D at the end: one line of five, so the order is fixed. A being neither first nor last, and B being immediately rather than merely behind A, are both already true of that chain.
Where do you start with a ranking puzzle?
Seating guests at a dinner, you place the couple who must sit together before the guest who will sit anywhere. Start with the clues that relate two items directly, and join them into chains; a chain that runs through every item fixes the whole order at once. Here every 'beat' clue shares a fund with another clue: E over C, C over A, A over B, B over D. Each clue hands the next one a fund to hang on, so the chain builds without any guessing.
Joining the three ordering clues makes one chain of all five funds, E ahead of C ahead of A ahead of B ahead of D, so the positions are fixed before the clue about A's position or the word immediately is ever used.
How do you know the order is the only one?
Because the chain is complete. Every fund sits on one line of 'finished ahead of' links, so there is no fund whose position is free to move. When a chain touches all five items, uniqueness is proved, not hoped for. If any fund had hung off the chain, say D with only 'B beat D' and no other link, you would have had to test each place it could go. Checking uniqueness out loud is what separates a candidate who got lucky from one who reasoned.
Why would an interviewer include clues you do not need?
Two pieces of the puzzle are redundant. A being neither first nor last follows from A sitting third in the chain. And the word immediately adds nothing: B merely behind A is enough, because once E and C are ahead of A and D is behind B, there is no fund left that could sit between A and B. Spotting redundancy shows that you checked what each clue does rather than ticking them off. Interviewers also use extra clues to tempt you into starting from a weak one; A's 'neither first nor last' narrows A to three places, which feels like progress and fixes nothing.
The fund-desk version of the same habit is reading a factsheet. A list of statements about a fund, its rank, its category, its risk grade, often contains claims that follow from each other, and the analyst's job is to find the few that carry information.
Where candidates lose it
The common loss is starting with 'A was neither first nor last', trying A in second, third and fourth place, and running a case for each. It works eventually, but it is slow and looks like guesswork, and under time pressure candidates drop a case and give a wrong order.
The second loss is giving the right order without proving it is the only one. Say the chain touches all five funds, so nothing can move, then name the two clues you never used.
What the interviewer asks next
Remove the clue that B beat D. How many orders are now possible?
Which single clue, if removed, would leave the order unchanged and still fully determined?
Six funds now, with F known only to have beaten C. Where can F go?

s_0, s_1	market share before and after
g_{AMC}	Tarvela's AUM growth, 30%
g_{ind}	the industry's AUM growth, 40%

pp	percentage points, the plain gap between two rates
bps	basis points, one hundredth of a percentage point
1.5	the old fee, the base for the relative change

w	the number of weighings
N	the number of coins, one of them light
3	the outcomes of one weighing: left rises, right rises, balance

Try it first

Why can a business grow and still shrink?

What is the fastest way to answer without dividing big numbers?

Where candidates lose it

What the interviewer asks next

Try it first

Why are there two different answers that are both right?

Which description will a client get wrong, and what should you say instead?

Where candidates lose it

What the interviewer asks next

Try it first

Why split into three groups rather than two?

How do you know two weighings is the minimum, and how far does this go?

Where candidates lose it

What the interviewer asks next

Try it first

Where do you start with a ranking puzzle?

How do you know the order is the only one?

Why would an interviewer include clues you do not need?

Where candidates lose it

What the interviewer asks next