Mutual Fund Mastery puzzles, solved step by step

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.

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 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.

52%, the same as on any other day. If each day's move is independent, the five falls carry no information about tomorrow. A run of five falls was unlikely before it began, 0.48 to the fifth power or about 2.5%, but that probability belonged to days that are now finished. Believing a rise is due is the gambler's fallacy.

Why does the streak not make a rise more likely?

A captain who has lost five tosses in a row walks out for the sixth feeling owed a win. The coin has no record of the first five and no sense of fairness to restore; it is still 50:50. Independence means past outcomes do not enter the calculation for the next one, so tomorrow's chance of a rise is the 52% it always was. The feeling that things must even out is real, and it has a name: the gambler's fallacy. It is strongest exactly when a streak is long, which is when it does the most damage to decisions.

Where does the tiny probability come from, and why is it the wrong number?

It answers a different question. Standing on Monday morning, the chance of five falls in a row was 0.48 multiplied by itself five times, about 2.5%, and the chance of six in a row was about 1.2%, roughly one in 82. Once five falls have happened, the only uncertainty left is tomorrow, and for independent days the chance of a rise given the streak equals the chance of a rise on any day. Six falls in a row is rare only when viewed from the start; viewed from Friday evening, it needs just one more fall.

Is the independence assumption true for real markets?

Not exactly, and saying so earns credit. Day-to-day direction in a broad index is very hard to predict from the previous days, but the size of moves does cluster: a run of falls often comes with bigger swings, so tomorrow's move may be larger even if its direction is a near coin toss. For an adviser the danger is not the arithmetic but the conversation: a client who believes a bounce is due will add money for the wrong reason, or hold out for a rebound the odds do not promise. A streak on its own is not a reason to act; any case for investing has to rest on the client's plan and horizon, not on the last five days.

The true average was 10.2% a year, and a survivor-only database overstates it by 1.8 points. Seventy funds at 12% and thirty at 6% average to 0.7 x 12% plus 0.3 x 6%, which is 10.2%. A database that drops merged and closed funds shows 12%, because the funds that disappeared were mostly the weak ones, and the gap compounds every year.

Why do dead funds disappear from the numbers?

Think of a coaching centre that advertises the average marks of students who stayed to the final exam. The ones who struggled and left are not in the average, so the centre looks better than its teaching. A fund database that lists only live schemes does the same: the funds that did badly were merged or shut, their records left the table, and the category average rose without anyone earning it. Fund houses merge weak schemes into stronger ones as a matter of routine, so the losers vanish quietly rather than with a headline. The name for this is survivorship bias.

How much does the gap matter over ten years?

Compound it. Rs 1 lakh at 12% for ten years becomes about Rs 3.11 lakh; at 10.2% it becomes about Rs 2.64 lakh. A chart built on survivors shows roughly Rs 46,000 more per lakh than the category delivered to the average investor who chose a fund ten years ago. Compounding a category average is itself approximate, because each fund compounds on its own path, but the direction and the size of the gap are right.

The bias leaks into rankings too. A fund that looks top quartile among survivors may be only average once the vanished funds are put back, because the bottom of the table has been cut off. Before comparing a fund with its category, ask whether the category figure includes funds that no longer exist. Good research databases keep dead funds in; if yours does not, treat its averages as a ceiling rather than a middle.

Allocation cost 0.70% and selection added 0.75%, a net active return of about plus 0.05%. The IT overweight is 10 points in a sector that trailed the benchmark by 7 points: 0.10 x (5% minus 12%) is minus 0.7%. The bank stocks beat their index by 3 points on a 25% weight: 0.25 x 3% is plus 0.75%. Good stock picking almost exactly paid for a poor sector bet.

What is the difference between allocation and selection?

Picture a selector who picks four spinners for a pitch that suits pace, but whose four spinners bowl better than any other spinners in the country would have. Two separate decisions: how many of each kind, and which ones. Allocation measures the first decision, sector weights against the benchmark's weights; selection measures the second, how the stocks chosen inside a sector did against that sector. Splitting them tells a fund research team whether a manager's skill lies in calling sectors or in picking stocks, which matters more than the total when deciding what to trust next.

Why is allocation measured against the whole benchmark rather than against zero?

Because the extra 10% in IT had to come from somewhere, and the alternative was the benchmark itself. An overweight in a sector that makes money still costs you if that sector made less than everything else you could have held. Say the assumptions behind the split out loud: the bank weight matches the benchmark's, so banks carry no allocation effect; the IT stocks held matched the IT index, so IT carries no selection effect; and the 10 points taken from other sectors came from sectors that earned the benchmark's 12%.

One detail an interviewer may probe. Using the fund's 25% weight in the selection term folds in what the BrinsonThe Brinson method, named after the authors who set it out in the 1980s, splits a fund active return into allocation, selection and an interaction term. framework calls the interaction effect; the textbook version uses the benchmark's weight and reports interaction separately. With the bank weights equal here, both give the same 0.75%.

2,333.333 units, and nobody gains or loses on the day. The holding is worth 1,000 x Rs 42, or Rs 42,000. Divided by B's NAV of Rs 18, that is 2,333.333 units, and 2,333.333 x Rs 18 is Rs 42,000 again. The swap ratio, 42 over 18, is 2.333 units of B per unit of A. A merger changes the unit count and the scheme, not the value held.

Why does a lower NAV not make scheme B cheaper?

Change a Rs 500 note into Rs 100 notes and you get five of them. You now hold more notes, but you are no richer. A NAV is the value of one unit, so the number of units you hold only means something when multiplied by the NAV; a merger at NAV swaps notes of one size for notes of another. Scheme B's Rs 18 NAV only says its units started at a lower price or have grown less since launch. It says nothing about whether B is cheap, good or bad.

So can a merger leave an investor worse off?

Not on the day of the swap, when both schemes are valued at their closing NAVs. What a merger can change is everything after the swap: a different portfolio, a different expense ratio, a different level of risk, and possibly a tax event. If scheme B charges more, holds riskier bonds or follows another strategy, the investor's future returns change even though the swap itself was fair. That is why investors in a scheme being merged are generally offered a window to exit without an exit load; confirm the current rules on how that window works before relying on it.

The three decimals are not decoration. Registrars in India commonly allot units to three decimal places, so the answer is 2,333.333 and not a rounded 2,333. Rounding down to whole units would take about Rs 6 from this investor, and across lakhs of folios that adds up, so fractional units exist precisely to keep the swap exact.

About 9 years at 8%; the exact figure is 9.006 years, so the rule of 72 is out by about 2 days. At 6% the rule gives 12 years against an exact 11.90, and at 12% it gives 6 years against an exact 6.12. In the range where fund returns usually sit, the rule is off by weeks, not years.

Why does 72 divided by the rate give the doubling time?

If the price of a Rs 100 thali rises 8% a year, it costs about Rs 200 in nine years. You did not need a calculator, only the fact that 72 over 8 is 9. The exact doubling time is ln 2 divided by ln(1 + r), and for rates of a few per cent ln(1 + r) is close to r, so the answer is close to 69.3 divided by the rate in per cent. Using 72 instead of 69.3 does two jobs: it nudges the answer up to correct for rates around 8%, and it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is what makes it a mental tool.

Where does the rule start to slip?

At both ends, but slowly. At 6% it says 12 years and the truth is 11.90, so it is long by about 38 days. At 12% it says 6 years against 6.12, short by about 42 days. The rule is almost exact near 8% and drifts by a little over a month either side, which is far inside the uncertainty of any return assumption you would feed it. Only at high rates does the drift become worth mentioning: at 24% the rule says 3 years against an exact 3.22.

Two uses in a fund interview. Turn a return into something a client can feel: at 8%, money doubles roughly every nine years, so it quadruples in about eighteen. And run a claim backwards to test it: a fund that says it tripled money in ten years has compounded at about 11.6% a year, because tripling is about 1.6 doublings, one every 6.3 years, and 72 over 6.3 is a little over 11.

About 6.7% a year for each rupee of genuine active management. The 80% that tracks the index is worth 0.2%, so it accounts for 0.8 x 0.2%, or 0.16 points, of the fee. The remaining 1.34 points pay for the 20% that differs from the index, and 1.34 divided by 0.20 is 6.7%. The active slice must beat the index by 6.7 points a year just to earn its fee.

What is active share, and why does it change the price?

Suppose a caterer charges a premium price per plate, but four of the five dishes are the same ones any local dhaba serves. You are paying the premium almost entirely for the fifth dish. Active shareThe share of a fund portfolio, by weight, that differs from its benchmark index. Zero means an exact copy of the index; 100% means no overlap at all. is the part of a fund's portfolio that differs from its benchmark; the rest is the index in disguise, and you could buy that part for an index fund's price. An active share of 20% means only a fifth of the money is doing anything the index would not.

How do you split the fee between the two parts?

Price each part at what it would cost on its own. The index-like 80% could be held through an index fund at 0.2%, so it deserves 0.8 x 0.2%, or 0.16 points of the fee. Everything left, 1.34 points, is the price of the active 20%, and expressed per rupee of active exposure that is 6.7% a year.

What would the active slice have to deliver?

To merely match a cheap index fund after costs, the active 20% must beat the index by 6.7 points a year on its own money. Beating an index by that margin year after year is rare, which is why a high fee on a low active share is the first combination fund researchers flag. Compare a genuinely active fund with 80% active share charging the same 1.5%: its index part uses 0.04 points, and its active part costs about 1.8% per rupee. Same headline fee, well under a third of the effective price.

Say the limitation too. Active share measures difference, not skill: a fund can be very different from its index and still lose. The point is narrower and firmer. A fund that is barely different from its index cannot justify an active fee, because most of what you pay for is available for a fraction of the price.

25 stocks, half of each fund, give or take about 2.5. Each of fund B's 50 picks has a 50 in 100 chance of being among fund A's, so the expected overlap is 50 x 0.5 = 25. Two funds fishing in the same pond share half their holdings by chance alone. With three such funds, about 12.5 stocks sit in all three, and together they hold only about 87.5 different names.

Why is the overlap so large with no coordination?

Two friends each order five dishes from a menu of ten without talking to each other. Each dish you pick has a 50% chance of being on your friend's list, so you expect to share about two and a half dishes. When each fund holds a large share of the same universe, overlap is not a coincidence to be explained; it is the default the arithmetic produces. The tool is linearity of expectation: go stock by stock, add up the chance that each one is in both funds, and the dependence between picks never needs to be modelled.

How firm is 25, and what changes with three funds?

The count follows a hypergeometric distributionThe distribution of how many marked items you get when you draw a fixed number without replacement from a pool that holds a fixed number of marked items. with a standard deviation of about 2.5, so most random pairs share between 20 and 30 stocks. With three funds each holding half the universe, a stock is in all three with probability one in eight, about 12.5 stocks, and in none with probability one in eight too. Three such funds do not give three times the diversification: 150 holdings collapse to about 87.5 distinct stocks, and 50 of those are held by two or three of the funds at once.

Real funds are not random, and the difference cuts one way. A large cap mandate points every manager at the same biggest companies, and benchmark weights pull portfolios further together, so real overlap between large cap funds tends to sit above this random baseline rather than below it. Use the random case as the floor: if two funds overlap much more than chance would give, the second fund adds little except a second fee.

About 17.3% a year, and a minus 12% month is a 2.4 standard deviation event: rare, not freakish. Annual volatility is the monthly figure times the square root of 12: 5% x 3.46 is 17.3%. A minus 12% month is 12 divided by 5, or 2.4 monthly standard deviations. A normal curve puts that at about 0.8% of months, roughly once a decade, and real markets produce such months more often.

Why does volatility scale with the square root of time?

Take ten steps where each one goes left or right on a coin toss. You rarely end ten steps from where you began; lefts and rights cancel, and a typical distance is only about three steps, the square root of ten. Monthly returns behave the same way: good and bad months partly cancel, so variance adds up over time and the standard deviation grows only with the square root of the number of months. That is why the conversion uses the square root of 12, about 3.46, and not 12.

So how rare is a minus 12% month?

Measure it in monthly standard deviations. A minus 12% month is 2.4 monthly standard deviations below an average month taken as zero, which a normal curve puts at about 0.8% of months, roughly one in 122, or once a decade. If the fund's average month is plus 1%, the fall is 2.6 deviations and a normal curve makes it a little rarer, 0.47%. Against the 17.3% annual figure the fall sounds small, and that is the confusion to avoid: judge a month on a monthly scale.

Now the limitation. Equity returns have fatter tails than the normal curve: large falls cluster in crises and turn up more often than the bell shape allows. Treat the normal figure as a floor on how often such a month arrives, not as a promise that it will be rare. For a client the useful sentence is plain: a fund with annual volatility near 17% will, from time to time, lose more than a tenth of its value in a single month.

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.

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.