Mutual Fund Mastery puzzles, solved step by step

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

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063Estimate how many full-time mutual fund distributors a district town of 5 lakh people can support. Build it from households, the share that invests, average fund assets and the trail income a distributor needs to make a living.Estimation and market sizingHardIndian AMCsDistribution and sales
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Which is the right place to start the estimate?
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About 20 full-time distributors. Five lakh people in households of four is 1,25,000 households. If 12% invest Rs 3 lakh each, the town holds Rs 450 crore. Suppose half comes through distributors: Rs 225 crore, which at an assumed 0.8% trail pays Rs 180 lakh a year. A distributor needing Rs 9 lakh a year in gross trail means about 20 can live on it.
What actually limits the number of distributors?
Ask how many tailors a colony can support and you do not count the people; you count the stitching work and divide by what keeps one tailor going. A distributor is paid from trail commissionA yearly commission paid to a distributor as a percentage of the client assets they hold in a fund, for as long as the money stays invested., so the town's capacity is the trail pool divided by one livelihood, not the population divided by some ratio of advisers to people. That tells you the order of the chain before you guess a single number: people, then households, then investors, then assets, then the share that pays a distributor, then the trail.
Five lakh people become 1,25,000 households, 15,000 investing households and Rs 450 crore of fund assets, of which Rs 225 crore through distributors pays a trail pool of Rs 180 lakh a year, enough for about 20 full-time distributors at Rs 9 lakh each.
How do you defend each guess, and which one moves the answer most?
Say each assumption with a reason. Four people to a household is a fair Indian average. Twelve percent of households holding funds is a cautious guess for a district town. Rs 3 lakh is a modest balance once systematic plans have run a few years. Half through distributors leaves the rest to direct plans, apps and bank branches. Trail rates vary by scheme and over time, so 0.8% is an assumption to confirm. The answer is a product of guesses, so a change in any one moves it in proportion: halve the trail and the town supports half as many.
Change one input Distributors supported
Base case 20
Trail 0.5% instead of 0.8% 12.5
20% of households invest instead of 12% 33.3
Each needs Rs 18 lakh instead of Rs 9 lakh 10
Each row changes one assumption from the base case and leaves the rest alone.
Close with the sanity check and the limit. Twenty distributors for 15,000 investing households is about 750 families each, which is a full book for one person with a small office. In reality many towns have far more registered distributors than this, because most are part-time, also sell insurance or deposits, or work for a bank, so their fund trail is only part of their income. The estimate sizes the full-time capacity, not the head count on a register.
Where candidates lose it
The common loss is starting from the population and dividing by an invented ratio of advisers to people. That produces a number with no economics behind it, and the interviewer cannot check any step.
The second is giving one number with no sensitivity. Say which assumption you are least sure of, usually the trail rate or the share that invests, and show how the answer moves when it changes.
What the interviewer asks next
How would the answer change if the town's investors move steadily to direct plans?
Estimate the same town's total yearly flow into systematic investment plans.
Why might an AMC still want a branch in a town that supports only 20 full-time distributors?
068Option A turns Rs 1 lakh into Rs 3 lakh in 12 years. Option B turns Rs 1 lakh into Rs 1.5 lakh in 4 years. Which has the higher IRR, which has the higher NPV at a 6% discount rate, and why do the two measures disagree?Performance measurement and returnsHardPIMCOMunich · 2024
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Which statement is right?
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B has the higher IRR, 10.7% against 9.6%, but A has the higher NPV at 6%, Rs 0.49 lakh against Rs 0.19 lakh. IRR measures speed of growth; NPV measures rupees of value at your cost of money. B grows faster but stops after four years, while A keeps compounding above 6% for twelve. The two rankings swap at 9.05%: below that A wins on NPV, above it B does.
What does each measure reward?
Compare a short, well-paid contract with a long, steady job. The contract pays more per month, but it ends quickly and you must find something else. IRR ranks by speed: the rate at which the money grows. NPV ranks by size: how many rupees of value the option creates once everything is discounted at your own cost of money. A's IRR is 3 to the power 1/12, less one, 9.59%. B's is 1.5 to the power 1/4, less one, 10.67%. B is faster.
At a 6% discount rate, A's Rs 3 lakh in year 12 is worth 3 / 1.06 to the 12th, Rs 1.49 lakh today, an NPV of Rs 0.49 lakh. B's Rs 1.5 lakh in year 4 is worth Rs 1.19 lakh, an NPV of Rs 0.19 lakh. A creates more than twice the value, because it earns well above 6% for three times as long.
Option A's NPV starts higher and falls faster as the discount rate rises, crossing B's at 9.05%, so A creates more value at a 6% cost of money while B has the higher IRR, 10.7% against 9.6%.
Where do the rankings swap, and what does that rate mean?
The NPV curves cross where both options are worth the same: 3 / (1 + r) to the 12th = 1.5 / (1 + r) to the 4th, so (1 + r) to the 8th = 2 and r = 9.05%. That crossover has a plain meaning: B returns Rs 1.5 lakh in year 4, and to match A it would need to double over the next eight years, which takes 9.05% a year. If you can reinvest B's proceeds above that, B is better; below it, A is. IRR silently assumes you can reinvest at B's own 10.7%, which is why it favours the short option.
The relationship
$\frac{3}{(1+r)^{12}} = \frac{1.5}{(1+r)^4} \;\Rightarrow\; (1+r)^8 = 2 \;\Rightarrow\; r \approx 9.05\%$
3 option A's payout in year 12, Rs lakh
1.5 option B's payout in year 4, Rs lakh
r the discount rate at which the two options are worth the same
What it says in wordsThe crossover rate is the return B's proceeds must earn afterwards to catch up with A.
For a fund desk this is the reinvestment question in plain clothes. A client choosing between a short product with a high stated yield and a long one with a lower yield is really asking what the short money will earn when it comes back. The limit: NPV needs a discount rate, and a different rate can flip the answer; at 10%, A's NPV turns negative, about Rs 4,411 below zero, while B's stays positive at about Rs 2,452, so B wins.
Where candidates lose it
The trap is answering that the higher IRR must also have the higher NPV. That holds only when the options have the same size and the same timing; here the timing differs by eight years.
The second miss is picking a winner without naming the discount rate. Say that the answer depends on the cost of money, find the crossover rate, and explain it as the reinvestment rate B must beat.
What the interviewer asks next
B can be repeated three times in a row at the same terms. Does that change the comparison?
Why do private funds report IRR rather than NPV to their investors?
At what discount rate does option A's NPV become zero?
Asked at PIMCO, Real Estate, Munich, 2024 (Wall Street Oasis): just asked a bunch of questions on recent real estate news, as well as a couple of technicals including IRR, NPV
083A fund charges 1.5% a year but has an active share of only 20%: the other 80% of the portfolio effectively is the index. If index exposure is worth 0.2% a year, what are you really paying for the active 20%?Costs and fee dragHardFund research and ratingsIndian AMCs
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What is the effective fee on the active slice?
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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.
Priced at 0.2%, the index-like 80% of the portfolio accounts for only 0.16 points of the 1.5% fee, so the remaining 1.34 points pay for the 20% active slice, an effective 6.7% a year per rupee of active management.
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.
The relationship
$f_{active} = \frac{F - (1 - AS)\,f_{index}}{AS} = \frac{1.5 - 0.8 \times 0.2}{0.2} = 6.7\%$
F the fund's total fee, 1.5%
AS active share, 20%
f_{index} the price of index exposure, 0.2%
f_{active} the effective price of each rupee of active management
What it says in wordsTake the index part's fair cost out of the fee, then spread what is left over the active slice only.
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.
Where candidates lose it
Most candidates take 1.5% as the price, compare it with the index fund's 0.2%, and conclude the fund costs 1.3 points more. That understates the problem, because it spreads the extra cost over money that is not being actively managed at all.
The second loss is reaching 6.7% and not saying what it means. Turn it into a hurdle: the active slice must beat the index by 6.7 points a year just to break even with the cheap alternative.
What the interviewer asks next
At what active share would the effective active fee fall to 2%?
Why might a fund's active share drift down as its assets grow?
What else would you check before calling a fund a closet indexer?
084Two diversified funds each pick 50 stocks at random from the same universe of 100 stocks. How many stocks do you expect them to hold in common, and what does that suggest about a client who holds three large cap funds?Probability and expected valueHardIndian AMCsDistribution and sales
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How many stocks do you expect the two funds to share?
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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.
The relationship
$E[\text{shared}] = N \times \frac{k}{N} \times \frac{k}{N} = 100 \times 0.5 \times 0.5 = 25$
N stocks in the universe, 100
k stocks each fund holds, 50
k/N the chance a given stock is in one fund
What it says in wordsEach stock is in both funds with probability one half times one half, and there are 100 stocks, so expect 25 shared.
In one random draw of two 50-stock funds from 100 stocks, 25 stocks land in both, matching the expected 50 x 50/100, and a third such fund would leave only about 87.5 distinct names across 150 holdings.
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.
Where candidates lose it
The instinct is that two independent managers should have little in common, so candidates guess 5 or 10. They forget that each fund covers half the universe, which makes sharing the norm, not the exception.
The second loss is stopping at 25 without the portfolio point. The question is really about clients holding three or four funds from the same category who believe they are diversified; say what the overlap does to that belief, with the 87.5 distinct names as the number.
What the interviewer asks next
If each fund picks only 20 of the 100 stocks, what overlap do you expect?
How would you measure overlap by portfolio weight rather than by count?
A client holds four large cap funds. What would you look at first?
086A client needs her money in exactly three years. Why does a bond fund with a duration of about three years protect her whether interest rates rise or fall? Show it with Rs 1 crore and yields jumping from 7% to 8% on the first day.Bond maths and durationHardPIMCOLondon · 2022
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Yields jump to 8% on day one and the fund's value falls. Where does she stand at year three?
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Because a fall in price and a rise in reinvestment income cancel at a horizon equal to the duration. If yields jump to 8%, Rs 1 crore at a three-year duration drops about Rs 2.74 lakh on day one, then compounds at 8% instead of 7%. By year three it is worth Rs 122.53 lakh against Rs 122.50 lakh had nothing moved. If yields fall to 6%, the gain today offsets the lower reinvestment rate in the same way.
Why do rising rates both hurt and help a bond investor?
Think of a tenant who has locked in a lease. If market rents rise, the lease itself is worth less to sell on, because a new buyer could get a better deal elsewhere, but any new space the tenant takes from now on costs more too. A bond investor faces the mirror image. When yields rise, bonds already held fall in price, but every coupon and every maturing bond is reinvested at the new, higher yield. One effect is immediate and the other builds over time. Which one wins depends only on how long you hold.
Make it concrete with a fund that has a duration of exactly three years. Picture its Rs 100 lakh as two holdings of Rs 50 lakh each at 7%: paper that matures in one year, and paper that matures in five. The value-weighted average of one and five is three, so the fund's Macaulay durationThe average time until a bond portfolio pays back its cash, with each payment weighted by its share of present value. is 3.0 years. Left alone at 7%, both halves grow to Rs 61.25 lakh by year three, Rs 122.50 lakh in all.
A jump to 8% knocks Rs 2.74 lakh off the fund on day one, but reinvesting at 8% wins it back by about year 3.0, so at her three-year date she holds Rs 122.53 lakh against a plan of Rs 122.50 lakh; a fall to 6% gives the mirror image.
What happens to each holding when yields jump to 8%?
The one-year paper matures at Rs 53.50 lakh and is rolled for two more years at 8%, reaching Rs 62.40 lakh, Rs 1.15 lakh more than the plan. The five-year paper still has two years to run at year three and must be sold at an 8% yield, fetching Rs 60.12 lakh, Rs 1.13 lakh less than the plan. One leg carries reinvestment risk and the other carries price risk, and a duration equal to the horizon sets them against each other in equal size. Held alone, each leg would fail: all short paper with yields falling to 6% gives only Rs 119.10 lakh, and all five-year paper with yields at 8% gives only Rs 120.25 lakh.
The relationship
$V_3 = P(y)\,(1+y)^3 \qquad P(8\%) = \frac{53.50}{1.08} + \frac{70.13}{1.08^5} = 97.26 \;\Rightarrow\; 97.26 \times 1.08^3 = 122.53$
P(y) the fund's value today at yield y, Rs lakh
53.50, 70.13 the amounts the one-year and five-year paper pay at maturity
V_3 the value at year three if all cash is reinvested at y
What it says in wordsWhatever the new yield, the fund's value at year three is today's repriced value grown at that yield, and at a horizon equal to the duration the two changes offset.
Where does this protection stop working?
In three places, and naming them is what the interviewer is after. Her horizon shrinks by exactly a year every year, but the fund's duration does not keep pace on its own: maturing paper has to be reinvested and coupons and rate moves shift the average, so the fund has to be rebalanced to stay matched. The offset assumes all yields move together; if short yields rise while long yields fall, the two legs no longer cancel. And the match only protects someone whose date is fixed: an open-ended fund whose manager keeps duration at three years forever is matched to nobody's date in particular. A product that runs down towards a fixed maturity close to her date does the matching more naturally. The small surplus either way, about Rs 0.02 lakh, comes from convexity and is a bonus, not the point.
Where candidates lose it
The common answer is that a shorter fund is safer, so she should use a liquid fund. That removes price risk but leaves her fully exposed to falling rates: if yields drop to 6% and stay there, three years of rolling short paper leaves her at Rs 119.10 lakh, about Rs 3.4 lakh short of plan.
The other loss is saying duration is a measure of price sensitivity and stopping there. This question is about the second meaning of duration, a time horizon at which price risk and reinvestment risk balance. Say both meanings, then show the cancellation with one number.
What the interviewer asks next
After one year, what has happened to the fund's duration and to her horizon, and what should change?
Short yields rise 1% and long yields fall 1% on the same day. Does the protection hold?
Why does the matched fund end slightly above plan whichever way yields move?
Asked at PIMCO, Sales, London, 2022 (Wall Street Oasis): Typically the product interview was toughest with questions regarding applications of duration
087The street values a refining company at 6 times its current EBITDA of Rs 1,500 crore. You think mid-cycle EBITDA, at normal refining margins, is Rs 1,050 crore. What multiple of mid-cycle EBITDA is the street really paying, and what does that say about your variant view?Valuation riddlesHardFranklin TempletonSan Mateo · 2024
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What multiple of mid-cycle EBITDA does the street's price imply?
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About 8.6 times mid-cycle EBITDA, so the stock is cheap only if today's margins last. Six times Rs 1,500 crore is an enterprise value of Rs 9,000 crore. On your normal earnings of Rs 1,050 crore that is 8.6x. If 6x is a fair multiple of normal earnings, the business is worth about Rs 6,300 crore, 30% less. Your variant view is really a view on earnings, not on the multiple.
Why can a low multiple be expensive?
A cricketer who scores 600 runs in a dream season is not a 600-run player; his normal year is nearer 400. Pay for 600 every year and you have overpaid, however modest the fee looks against this season's scorecard. A multiple is a price divided by one year's earnings, so when that year is a cyclical peak, the multiple looks cheap precisely because the earnings are temporarily high. Refiners are a textbook case: their profit swings with the gap between crude oil costs and product prices, which moves in cycles.
The street's Rs 9,000 crore is 6.0 times peak EBITDA of Rs 1,500 crore but 8.6 times mid-cycle EBITDA of Rs 1,050 crore, and at 6 times normal earnings the business would be worth Rs 6,300 crore, 30% less.
How do you turn the gap into a variant view?
Hold the price still and change only the earnings. The street pays Rs 9,000 crore of enterprise value. Divided by your mid-cycle Rs 1,050 crore, that is 8.6 times normal earnings, a premium multiple hiding behind a discount one. Suppose refiners have usually traded near 6 times normal earnings. Then the price you can defend is 6 x 1,050, or Rs 6,300 crore, and the street is implicitly assuming that today's Rs 1,500 crore is the new normal. Your disagreement is a 30% lower view of sustainable EBITDA.
The relationship
$\frac{EV}{EBITDA_{mid}} = \frac{6 \times 1{,}500}{1{,}050} = \frac{9{,}000}{1{,}050} = 8.6\text{x}$
EV enterprise value the street's price implies, Rs crore
1,500 current EBITDA, Rs crore
EBITDA_{mid} your estimate of EBITDA at normal margins, Rs crore
What it says in wordsKeep the price the street pays, swap in normal earnings, and see what multiple you are really being asked to pay.
Why does the equity fall further than the enterprise value?
Because debt does not shrink when margins do. With an assumed Rs 3,000 crore of net debt, the street's equity value is Rs 6,000 crore; at Rs 6,300 crore of enterprise value it is Rs 3,300 crore. A 30% fall in the value of the business becomes a 45% fall for shareholders, because all of the loss lands on the equity slice. Say the weak point of your own view too: mid-cycle is an estimate of an average over a cycle whose length and depth nobody knows, and if capacity is genuinely scarce for years, today's margins may be closer to normal than you think. The defence is to show the history of margins and say what would make you wrong.
Where candidates lose it
Candidates argue about the multiple: refiners deserve 5x, or 7x, or the sector average. That misses the point of the question, which is that the street and you can agree on 6x and still disagree by Rs 2,700 crore, because you are applying it to different earnings.
The second loss is calling the stock cheap at 6x without asking which year's EBITDA sits underneath. For a cyclical business, always ask whether the denominator is a peak, a trough or a normal year before reading the multiple at all.
What the interviewer asks next
At the bottom of the cycle EBITDA is Rs 600 crore and the stock trades at 15x. Is it expensive?
How would you estimate mid-cycle EBITDA for a refiner, and what history would you need?
What would make you abandon the variant view?
Asked at Franklin Templeton, Oil & Gas, San Mateo, 2024 (Wall Street Oasis): Why did I have a variant view of the multiple I applied to a refiner company relative to street expectations.
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
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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?
089A retiree has Rs 40 lakh invested at an assumed 8% a year and withdraws Rs 30,000 a month. How long does the money last, and what monthly withdrawal would last forever?Withdrawals and after-tax arithmeticHardWealth and advisoryDistribution and sales
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Roughly how long does Rs 40 lakh last at Rs 30,000 a month?
Show the worked solution
About 27.6 years, and about Rs 26,700 a month would last forever. At 8% a year, about 0.67% a month, Rs 40 lakh earns Rs 26,667 a month. Taking Rs 30,000 eats about Rs 3,333 of capital in month one, and the gap widens as the corpus shrinks, so the money runs out after about 331 months. Withdraw only the interest and the corpus never falls.
Why does a small overdraw turn into a countdown?
Think of a well that refills by 100 buckets a day. Draw 100 and it lasts forever; draw 110 and the level drops a little, so tomorrow it refills slightly less, and the drop speeds up until the well is dry. A corpus earning interest works the same way: withdraw no more than it earns and it lasts forever; withdraw a little more and each month's shortfall shrinks the base that earns next month's interest. Here she takes Rs 30,000 against Rs 26,667 of interest, only about Rs 3,333 too much at the start, and it still empties the account.
Withdrawing the Rs 26,667 a month the corpus earns keeps Rs 40 lakh intact forever, while Rs 30,000 a month empties it in about 27.6 years and Rs 35,000 in about 18.0, so a few thousand rupees a month decide whether the money is permanent or a countdown.
How do you work out the number of months?
Use the annuity formula and solve for the number of payments. With a monthly rate of 8% divided by 12, the corpus lasts as long as it takes the withdrawals to use up both the capital and the interest it earns along the way. The answer is about 331 months, or 27.6 years, and it is very sensitive to the withdrawal: Rs 28,000 a month lasts about 38 years and Rs 35,000 only about 18.0. The path is not a straight line: after ten years the corpus is still about Rs 33.9 lakh, and after twenty about Rs 20.4 lakh, because the fall speeds up near the end.
The relationship
$n = \frac{-\ln\!\left(1 - \dfrac{P\,i}{W}\right)}{\ln(1+i)} = \frac{-\ln\!\left(1 - \dfrac{26{,}667}{30{,}000}\right)}{\ln(1.00667)} \approx 331 \text{ months}$
P the starting corpus, Rs 40 lakh
i the monthly rate, 8% divided by 12
W the monthly withdrawal, Rs 30,000
P i the first month's interest, Rs 26,667
What it says in wordsThe closer the withdrawal is to the interest earned, the closer the fraction gets to one and the longer the money lasts; at exactly the interest it lasts forever.
What does this leave out?
Three things, each of which shortens the runway. Rs 30,000 today buys less every year, so a real retiree needs a rising withdrawal, not a flat one. The 8% is an assumption, not a promise; a market-linked portfolio delivers it unevenly, and a bad few years early on, while withdrawals continue, does more damage than the same years late. And tax comes out of the return. Even the 'forever' figure only preserves Rs 40 lakh in rupees; after inflation, the capital it protects is shrinking. She is withdrawing 9% of the corpus a year while it earns 8%, and that one comparison tells you the plan has an end date.
Where candidates lose it
The quick answer divides Rs 40 lakh by Rs 3.6 lakh a year and says about 11 years. That treats the corpus as cash in a drawer and throws away the 8% it earns while it waits; the true runway is more than twice as long.
The opposite slip is hearing 8% earned against 9% withdrawn and calling it roughly sustainable. Any withdrawal above the interest has an end date, and the gap compounds against her. Give the 27.6 years, then the Rs 26,700 that would never run out.
What the interviewer asks next
She wants the money to last exactly 25 years. What monthly withdrawal allows that?
Inflation runs at 6%. How would you change the calculation?
Why does a bad market in her first three years of retirement matter more than one in her last three?
092A fund returned 60%, 2% and 3% in three successive years while the median peer returned 25%, 10% and 10%. The fund ranked in the bottom quartile in two of the three years, yet sits in the top quartile on its three-year return. How?Performance measurement and returnsHardFund research and ratingsIndian AMCs
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What are the fund's and the median peer's three-year returns, a year?
Show the worked solution
One outsized year carries the whole period: 18.9% a year against the peers' 14.8%. Compounded, 1.60 x 1.02 x 1.03 is 1.681 for the fund and 1.25 x 1.10 x 1.10 is 1.513 for the median peer. The 35-point lead in year 1 is larger than the 8 and 7 points given back later. A single period's rank says how much was earned, not how consistently, and here the latest two years run at 2.5% a year against 10.0%.
How can losing two years out of three still win the period?
A batter who scores 150 in one innings and 5 and 8 in the next two has a better series total than one who scores 40, 45 and 45, but nobody would call him the more reliable player. A multi-year return adds up the size of each year's result, not the number of years won, so one very large year can outweigh several small losses. Here the fund was 35 points ahead in year one, then 8 and 7 points behind. Compounding the three years keeps most of that first lead intact.
The fund trailed the median peer in two of three years, yet its 60% first year lifts its compounded three-year return to 18.9% a year against 14.8%, while the last two years alone show 2.5% against 10.0%.
The relationship
$\text{CAGR} = \left(1.60 \times 1.02 \times 1.03\right)^{1/3} - 1 = 1.681^{1/3} - 1 = 18.9\%$
1.60, 1.02, 1.03 the fund's growth factor in each year
1/3 the cube root, which turns three years of growth into a yearly rate
CAGR compound annual growth rate
What it says in wordsMultiply the yearly growth factors, then take the cube root to turn the three-year growth into an average yearly rate.
What would a fund researcher look at instead?
Consistency, measured several ways. Shift the start date by one year and the same fund drops from top to bottom: over the last two years it made 2.5% a year against the peers' 10.0%. That is the case for rolling returns, which compute the return over every possible window of a given length and show how often the fund beat its peers, not just whether it did over one window ending today. A researcher would also ask what produced the 60% year: one concentrated bet that paid off, a sector that ran, or something repeatable.
One more point about averaging. The simple average of the fund's three years is 21.7%, well above the 18.9% it actually compounded at; volatile return streams always show a bigger gap between the two. Quote compounded returns for anything an investor actually experienced, and treat a period return as one data point, not a verdict. The quartile labels here are illustrative: they assume a peer group in which 2% and 3% fell in the bottom quarter and 18.9% in the top.
Where candidates lose it
The tempting wrong answer is that the numbers must be inconsistent, since a fund that trails in most years should not be able to come out ahead over the period. Candidates who say this have counted wins instead of adding up sizes, and missed that one 60% year outweighs two years of trailing by 7 or 8 points.
The second loss is getting 18.9% right and stopping. The question invites the point that period returns hide consistency; say that rolling returns, and starting one year later, tell a very different story.
What the interviewer asks next
In what order would the three years have to come for the fund to look worst on a three-year view?
How would you measure consistency across ten years of monthly data?
A fund house advertises its fund's five-year rank. What would you ask before believing it says anything about skill?
093Passive funds tracking an index hold an assumed Rs 50,000 crore. A stock is added to the index at a weight of 1.2%, and it trades about Rs 250 crore a day. How much must the passive funds buy, and how many days of the stock's trading is that?Estimation and market sizingHardPassive and index teamsIndian AMCs
Try it first
How many days of the stock's normal trading does the passive buying equal?
Show the worked solution
About Rs 600 crore, or 2.4 days of the stock's entire trading. Passive funds must hold the stock at its index weight: 1.2% of Rs 50,000 crore is Rs 600 crore. Against Rs 250 crore of daily trading, that is 2.4 days of every share traded, and they want to own it by the day the change takes effect. If they took only a quarter of each day's volume, the buying would take about 9.6 days.
Why is index buying forced rather than optional?
Picture a school that announces every student must own a particular textbook by Monday, and the only shop stocks a few dozen copies a day. The shopkeeper knows exactly how many will be needed and by when. An index fund has no view on the stock; its job is to hold every stock at its index weight, so when a stock enters the index, every passive fund tracking it must buy, in a known amount, by a known date. That predictability is what makes inclusion a trading event, not just a news item.
Passive funds holding Rs 50,000 crore must buy Rs 600 crore of a stock entering at a 1.2% weight, which is 2.4 days of all its trading at Rs 250 crore a day, and about 9.6 days if they take only a quarter of each day's volume.
How do you size it, and why measure it in days?
The rupee amount is just the weight times the passive money: 0.012 x Rs 50,000 crore is Rs 600 crore. On its own that number means little, because Rs 600 crore is trivial in one stock and enormous in another. Dividing by daily traded value turns it into a measure of difficulty: 2.4 days means the funds must buy every share traded for more than two full days, which nobody can do without pushing the price up. A desk usually assumes a buyer can take only a fraction of the day's volume quietly; at a quarter, the job takes about 9.6 days.
The relationship
$\text{Days} = \frac{w \times AUM}{ADV} = \frac{0.012 \times 50{,}000}{250} = 2.4$
w the stock's weight in the index, 1.2%
AUM money in funds tracking the index, Rs 50,000 crore, an assumption
ADV average daily traded value of the stock, Rs 250 crore
What it says in wordsForced buying in rupees, divided by what trades in a day, gives the number of days of all trading the funds need.
Who positions for it, and what does the estimate leave out?
Traders who expect the inclusion buy ahead of the effective date and sell to the index funds on it, so part of the price move usually happens before the passive funds trade at all. The passive funds also pay for the purchase by selling a slice of every other stock, about Rs 600 crore spread across the rest of the index, which barely registers in each. The estimate leaves out active funds that hug the same benchmark and buy for similar reasons, which makes the real demand larger, and it assumes the weight is set by the stock's full market value; index providers commonly weight by the shares freely available for trading, so check the actual weight before relying on the size.
Where candidates lose it
Candidates compute Rs 600 crore and stop. The rupee figure is the easy half; the interviewer wants it set against liquidity, because Rs 600 crore tells you nothing until you know whether the stock trades Rs 25 crore a day or Rs 2,500 crore.
The second slip is assuming the funds can simply take all the volume. Say that a buyer can only take part of each day's trading without moving the price, and that traders front-run the known demand.
What the interviewer asks next
The stock is removed from the index instead. Who is on the other side of the passive selling?
Passive assets double over five years. What happens to the days-of-volume figure for the same stock?
Why might a stock rise after the inclusion is announced but fall back after the effective date?

Change one input	Distributors supported
Base case	20
Trail 0.5% instead of 0.8%	12.5
20% of households invest instead of 12%	33.3
Each needs Rs 18 lakh instead of Rs 9 lakh	10

3	option A's payout in year 12, Rs lakh
1.5	option B's payout in year 4, Rs lakh
r	the discount rate at which the two options are worth the same

F	the fund's total fee, 1.5%
AS	active share, 20%
f_{index}	the price of index exposure, 0.2%
f_{active}	the effective price of each rupee of active management

N	stocks in the universe, 100
k	stocks each fund holds, 50
k/N	the chance a given stock is in one fund

Try it first

What actually limits the number of distributors?

How do you defend each guess, and which one moves the answer most?

Where candidates lose it

What the interviewer asks next

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What does each measure reward?

Where do the rankings swap, and what does that rate mean?

Where candidates lose it

What the interviewer asks next

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

How do you split the fee between the two parts?

What would the active slice have to deliver?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the overlap so large with no coordination?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why do rising rates both hurt and help a bond investor?

What happens to each holding when yields jump to 8%?

Where does this protection stop working?

Where candidates lose it

What the interviewer asks next

Try it first

Why can a low multiple be expensive?

How do you turn the gap into a variant view?

Why does the equity fall further than the enterprise value?

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

Try it first

Why does a small overdraw turn into a countdown?

How do you work out the number of months?

What does this leave out?

Where candidates lose it

What the interviewer asks next

Try it first

How can losing two years out of three still win the period?

What would a fund researcher look at instead?

Where candidates lose it

What the interviewer asks next

Try it first

Why is index buying forced rather than optional?

How do you size it, and why measure it in days?

Who positions for it, and what does the estimate leave out?

Where candidates lose it

What the interviewer asks next