Mutual Fund Mastery puzzles, solved step by step

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

Showing 81–90 of 100

081Scheme A, with a NAV of Rs 42.00, is merged into scheme B, whose NAV is Rs 18.00. How many units of B does an investor holding 1,000 units of A receive, and has anyone gained or lost in the swap?NAV, units and fund mechanicsWarm upFund operationsRegistrars and transfer agents
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How many units of scheme B does the investor receive?
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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.
The investor's 1,000 units of A become 2,333.333 units of B, but both holdings are worth Rs 42,000, because the swap ratio of 2.333 is set by the two NAVs on the merger date.
The relationship
$u_B = u_A \times \frac{NAV_A}{NAV_B} = 1{,}000 \times \frac{42}{18} = 2{,}333.333$
u_A, u_B units held in scheme A before and scheme B after
NAV_A, NAV_B each scheme's net asset value per unit on the merger date
What it says in wordsThe new unit count is the old count times the ratio of the two NAVs, which keeps the rupee value unchanged.
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.
Where candidates lose it
The fast wrong answer is 1,000 units, as if a merger were a change of name. It would leave the investor with Rs 18,000 in place of Rs 42,000. The next wrong answer inverts the ratio and gives about 429 units. Both come from working with unit counts instead of rupees.
State the rupee value first, Rs 42,000, and divide by the new NAV. Then add the sentence that shows judgement: the swap is fair on the day, and the real question is what the investor now owns and what it costs.
What the interviewer asks next
Scheme B's expense ratio is 0.5 points higher. Roughly what does that cost on Rs 42,000 over ten years?
In what circumstances could the merger be a taxable event for the investor, and what would you check?
The investor had a monthly systematic investment plan into scheme A. What should happen to it?
082Without a calculator: at 8% a year, how long does Rs 1 lakh take to double, and how close does the rule of 72 get to the exact answer? Then do the same at 6% and at 12%.Compounding and time valueWarm upIndian AMCsDistribution and sales
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How far is the rule of 72 from the exact doubling time at 8%?
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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.
The relationship
$t = \frac{\ln 2}{\ln(1+r)} \approx \frac{72}{100\,r} \qquad \frac{0.6931}{\ln 1.08} = 9.006$
t years for money to double
r the annual rate, as a decimal
ln the natural logarithm
What it says in wordsThe exact doubling time is log 2 over log of one plus the rate, and 72 over the rate in per cent is a close, easy stand-in.
The rule of 72 gives 12, 9 and 6 years at 6%, 8% and 12%, against exact doubling times of 11.90, 9.006 and 6.12 years, so in the range fund returns occupy it is wrong by weeks at most.
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.
Where candidates lose it
The common slip is dividing 100 by the rate, which gives 12.5 years at 8%. That is what simple interest would give, with no interest earned on interest, and it overstates the wait by about three and a half years.
The quieter loss is presenting the rule as exact. Say 9 years, then add that the exact figure is 9.006, so the interviewer hears that you know what the shortcut is and where it stops working.
What the interviewer asks next
How long does money take to triple at 8%?
Inflation runs at 6%. How long before Rs 1 lakh buys half what it buys today?
What number would you use in place of 72 for continuously compounded rates, and why?
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?
085A fund's monthly returns have a standard deviation of 5%. What is that as an annual volatility, and is a month of minus 12% a rare event?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs
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What is the fund's annual volatility?
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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.
The relationship
$\sigma_{year} = \sigma_{month}\sqrt{12} = 5\% \times 3.464 = 17.3\% \qquad z = \frac{-12\%}{5\%} = -2.4$
\sigma_{month} the standard deviation of monthly returns, 5%
\sqrt{12} the square root of the number of months in a year
z how many monthly standard deviations the fall is from an average month
What it says in wordsScale volatility up by the square root of time, and measure a single month against the monthly figure, not the annual one.
With a monthly standard deviation of 5%, a minus 12% month lies 2.4 deviations below the centre, and the normal curve puts only 0.82% of months beyond it, about one in 122, while the same fund's annual volatility is 17.3%.
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.
Where candidates lose it
The first loss is scaling the wrong way. Multiplying by 12 gives 60%, which describes a far wilder fund than this one, and it comes from treating risk as if it added up like returns. Say variance adds, then take the square root.
The second is judging the minus 12% month against the annual 17.3% and calling it ordinary. Put the move and the volatility on the same time scale before comparing them, and the month turns out to be a 2.4 deviation event.
What the interviewer asks next
What is this fund's volatility over a single week?
The fund's worst month in ten years was minus 20%. What does that tell you about the normal assumption?
Why might reported annual volatility understate the risk of a bad month?
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?
090Fund A's returns have a correlation of 0.9 with the index. Fund A's volatility is 24% a year and the index's is 16%. When the index moves 1%, how much does fund A typically move, and why is the correlation not the answer?Statistics, correlation and diversificationCoreFund research and ratingsGlobal asset managers
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When the index moves 1%, fund A typically moves about:
Show the worked solution
About 1.35%: beta is the correlation times the ratio of the two volatilities. Correlation of 0.9 says the fund's moves line up closely with the index; the ratio 24 over 16 says the fund's moves are 1.5 times as large. Together, 0.9 x 1.5 is 1.35. Correlation measures how tightly the points hug the line, beta measures how steep the line is, and a fund can have a high correlation with a beta below one.
What is the difference between how tight and how steep?
Two friends walking a dog on a lead: the lead's length sets how far the dog can stray, and the dog's pace sets how far it travels for each step you take. A short lead says nothing about whether the dog walks faster or slower than you. Correlation is the lead, how closely the fund's moves track the index's; beta is the pace, how far the fund moves for each 1% the index moves. They are related, but they answer different questions, and confusing them is the commonest slip in fund statistics.
The relationship
$\beta = \rho \times \frac{\sigma_{fund}}{\sigma_{index}} = 0.9 \times \frac{24\%}{16\%} = 1.35$
\beta the slope: the fund's typical move for a 1% index move
\rho correlation, how tightly fund and index move together, 0.9
\sigma annual volatility of the fund, 24%, and of the index, 16%
What it says in wordsBeta is correlation stretched by how much more the fund swings than the index.
Fund A, correlation 0.9, and fund B, correlation 0.6, share the same slope of 1.35 against the index, but fund A's points hug the line while fund B's scatter widely, which is the difference between beta and correlation.
Can two funds have the same beta and very different correlations?
Yes, and the figure shows it. Fund B has a correlation of only 0.6 but a volatility of 36%, so its beta is 0.6 x 36 / 16, also 1.35. Both funds move 1.35% for a 1% index move on average, but fund B's actual months scatter far from that line, because most of its swings come from things the index does not explain. The square of correlation measures the share explained: 81% of fund A's variance comes from the index against 36% of fund B's. For a fund researcher that changes the reading of beta itself: fund A's 1.35 is a reliable guide to a typical month, fund B's is an average around which a single month can land almost anywhere.
One more asymmetry worth having ready. Correlation is the same whichever way round you put the two series, but the slope is not: regressing the index on the fund gives 0.9 x 16 / 24, or 0.6. Beta always needs a direction, and for a fund it is the fund's return explained by the index. Both figures are estimates from past returns and shift with the window chosen, so quote them with the period they came from.
Where candidates lose it
The fastest wrong answer is 0.9%, read straight off the correlation. It sounds right because both numbers describe how fund and index relate, but correlation is capped at one and carries no information about size of moves, so it cannot be a slope.
The next slip is 1.5%, the volatility ratio, which forgets that 10% of the fund's movement has nothing to do with the index. Give the formula, then say in a sentence that correlation is tightness and beta is steepness.
What the interviewer asks next
What would fund A's beta be if its correlation with the index fell to 0.5?
Can a fund have a negative correlation and still lose money when the index falls?
Why might a fund's measured beta change when you switch from daily to monthly returns?

u_A, u_B	units held in scheme A before and scheme B after
NAV_A, NAV_B	each scheme's net asset value per unit on the merger date

t	years for money to double
r	the annual rate, as a decimal
ln	the natural logarithm

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

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Why does a small overdraw turn into a countdown?

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What does this leave out?

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What is the difference between how tight and how steep?

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