Mutual Fund Mastery puzzles, solved step by step

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

Showing 1–10 of 100

001A bank fixed deposit pays 7% a year. Your interest is taxed at a 30% slab, assumed here for the arithmetic, and inflation runs at 6%. What is your real return after tax, and roughly how many years until the deposit has lost 10% of its purchasing power?Compounding and time valueCoreIndian AMCsDistribution and sales
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Before you calculate: what does the deposit earn in real, after-tax terms?
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About -1.0% a year, so the deposit loses about 10% of its purchasing power in roughly 10 years. Tax at 30% turns 7% into 4.9%. Inflation of 6% then shrinks what that buys: 1.049 divided by 1.06, less 1, is -1.04%. Compounding that loss, Rs 1 lakh buys what Rs 90,094 buys today after ten years.
Why does the order of tax and inflation matter?
Think of a salary rise that matches price rises but pushes you into a higher tax bracket. On paper you kept pace; in the shop you did not. Tax is charged on the whole nominal interest, including the part that only makes up for inflation, so the investor pays tax on money that is not a real gain. That is why you take tax off first, on the 7%, and only then compare what is left with inflation.
Tax of 30% on 7% is 2.1 points, leaving 4.9%. Inflation at 6% is bigger than 4.9%, so the deposit is already behind before any compounding. The quick answer is 4.9 minus 6, about minus 1.1%; the exact answer uses the ratio, because both rates compound.
A 7% deposit taxed at an assumed 30% slab keeps 4.9%, and 6% inflation turns that into a real return of about -1.0% a year, so Rs 1 lakh held in the deposit buys about 10% less after roughly 10 years.
The relationship
$r_{\text{real}} = \frac{1 + i(1-t)}{1 + \pi} - 1 = \frac{1.049}{1.06} - 1 \approx -1.04\%$
i the nominal deposit rate, 7%
t the assumed tax slab, 30%
\pi inflation, 6%
What it says in wordsGrow the money at the after-tax rate, shrink its buying power at the inflation rate, and the ratio is the real return.
How do you get from minus 1% a year to ten years?
Losing 1% a year compounds, but slowly. Ten years of losing about 1.04% a year leaves 0.9896 to the power 10, about 0.90, so roughly 10% of purchasing power is gone in about 10 years. The exact figure is the log of 0.9 over the log of 0.9896, which is 10.1 years. A rule of thumb works too: a 1% annual loss takes about 70 years to halve the money, so about a seventh of that for a tenth.
Say the limitation. The 30% slab and 6% inflation are assumptions for this arithmetic, and your own slab and the inflation you actually face may differ; confirm the current tax rules before using a slab in advice. The point survives any sensible inputs: a deposit is safe in rupees and can still lose ground in what those rupees buy.
Where candidates lose it
The fast wrong answer is plus 1%: seven minus six. It forgets that tax is charged on the nominal 7%, including the 6% that only replaces lost buying power. Candidates who say it have shown the interviewer they would mis-sell a deposit to a client in a high bracket.
The second trap is getting minus 1% and then answering the time question linearly, ten years at 1% is exactly 10%. It is close here, but say that it compounds and give the log form; the interviewer is checking you know why it is close.
What the interviewer asks next
What deposit rate would just keep a 30% taxpayer level with 6% inflation?
How does the answer change for an investor with no taxable income?
Why might a debt fund held for several years be compared with a deposit on an after-tax basis, and what would you check first?
002Fund A has 20% annual volatility and fund B has 12%. Their returns have a correlation of 0.3. What is the volatility of a portfolio that is 60% A and 40% B, and why is it below the 16.8% weighted average?Statistics, correlation and diversificationCoreFund research and ratingsGlobal asset managers
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Pick the portfolio's volatility before you work it.
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About 14.2%, against a weighted average of 16.8%. Variance is 0.6 squared times 20% squared, plus 0.4 squared times 12% squared, plus twice 0.6 times 0.4 times 0.3 times 20% times 12%, which sums to 0.0202. Its square root is 14.2%. The 2.6 point gap exists only because the correlation is below 1.
Why is the mix not just the average of the two volatilities?
Two friends walking home on a windy night: if they stumble at exactly the same moments, holding hands does not steady them. If their stumbles come at different moments, each one's lean is partly caught by the other. Only the part of the two funds' swings that happens together adds up in full; the rest partly cancels, so the portfolio is calmer than the average of its parts. Correlation of 0.3 says most of their swings are not shared.
A 60/40 mix of a 20% and a 12% volatility fund has 14.2% volatility at a correlation of 0.3, below the 16.8% weighted average, and it would only reach 16.8% if the two funds moved in perfect step.
How do you work it quickly on paper?
Square the volatilities first, because variances are what add. Fund A's weighted variance is 0.36 times 0.04, which is 0.0144. Fund B's is 0.16 times 0.0144, which is 0.0023. The cross term is where correlation lives: 2 times 0.6 times 0.4 times 0.3 times 0.20 times 0.12 is 0.0035. The total is 0.0202, and the square root of 0.02 is about 0.141, so 14.2% is the answer.
The relationship
$\sigma_p = \sqrt{w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\,\sigma_A\sigma_B} = \sqrt{0.0202} \approx 14.2\%$
w_A, w_B the weights, 0.6 and 0.4
\sigma_A, \sigma_B the funds' volatilities, 20% and 12%
\rho the correlation between them, 0.3
What it says in wordsPortfolio variance is each fund's own variance, weighted, plus a shared term scaled by how closely the two move together.
Give the two edges to show you see the shape. At a correlation of 1 the formula collapses to the weighted average, 16.8%. At zero the cross term vanishes and the mix is 12.9%. At minus 1 it drops to 7.2%. The limitation is that correlations measured in calm years often rise in a selloff, so the benefit you computed can shrink exactly when it is wanted.
Where candidates lose it
Candidates answer 16.8% because averaging feels natural and the weights are right there. It is only true at a correlation of 1, and saying it tells the interviewer you do not see where diversification comes from.
The second loss is mixing units: adding volatilities in one term and variances in another, or forgetting the factor of 2 on the cross term. Write the formula once, square everything first, and take one square root at the end.
What the interviewer asks next
What correlation would make the 60/40 mix exactly as volatile as fund B on its own?
Which weight in A gives the lowest possible volatility at a correlation of 0.3?
Why might this calculation understate risk in a market crash?
003A 5-year bond pays an 8% annual coupon and trades at par, so its yield is 8%. Without a calculator, bracket its modified duration, then give the exact figure.Bond maths and durationCorePIMCOLos Angeles · 2024
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Where does the modified duration sit?
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Modified duration is about 3.99. Bracket first: a coupon bond's Macaulay duration sits below its 5-year maturity but not far, because the principal dominates, so somewhere in the low fours; dividing by 1.08 takes it just under 4. Exactly, the Macaulay duration is 4.312 years, and 4.312 divided by 1.08 is 3.993: a 1 point rise in yield cuts the price by roughly 4%.
How do you bracket it before doing any arithmetic?
Picture a see-saw with one heavy child at the far end and four small children spread along the plank. The balance point sits close to the heavy child but is pulled in a little by the others. A bond's Macaulay duration is the balance point of its discounted cash flows, so it can never exceed maturity and sits close to it when the final payment dominates. A zero coupon 5-year bond sits exactly at 5; this one pays 8 a year along the way, so it lands a little inside.
The final payment of 108 is worth 73.5 today, about 74% of the price of 100. The four coupons carry the rest at years 1 to 4. A balance point roughly three quarters of the way at 5 and a quarter spread between 1 and 4 lands a bit above 4, and dividing by 1 plus the yield lands just under 4.
The bond's discounted cash flows are 7.41, 6.86, 6.35, 5.88 and 73.50, which balance at 4.31 years, so its Macaulay duration sits well inside the 5-year maturity and its modified duration is 3.99.
Is there a shortcut for the exact figure?
For a bond priced at par there is a closed form. At par, Macaulay duration equals (1 + y) over y, times one minus the discount factor at maturity: 13.5 times (1 minus 0.6806), which is 4.312 years. Divide by 1.08 for modified duration, 3.993. Saying you know the par shortcut, then checking it against the bracket, is a strong answer in the room.
The relationship
$D_{mod} = \frac{D_{mac}}{1+y}, \qquad D_{mac}^{par} = \frac{1+y}{y}\left[1 - \frac{1}{(1+y)^n}\right] = 13.5 \times 0.3194 = 4.312$
y the yield, 8%, equal to the coupon because the bond is at par
n years to maturity, 5
D_{mac} Macaulay duration, the balance point in years
What it says in wordsAt par the balance point has a closed form, and modified duration is that balance point divided by one plus the yield.
Say what the number is for. A modified duration of 3.99 means a 1 percentage point rise in yield costs roughly 3.99% of price, and a 0.25 point rise roughly 1%. The estimate is linear, so it drifts for large moves; convexity handles that.
Where candidates lose it
Answering 5 is the common slip: it treats the bond as a zero coupon bond and ignores the coupons that come back early. The second is giving the Macaulay figure, 4.31, when the question asks for modified duration.
Candidates also freeze without a calculator. The interviewer wants the bracket said out loud first: below 5, above 4 for Macaulay, divide by 1.08. The exact figure is a bonus.
What the interviewer asks next
What is the duration of a 5-year zero coupon bond at an 8% yield?
If the coupon were 4% with the yield still 8%, would duration rise or fall?
Estimate the price change for a 50 basis point fall in yield.
Asked at PIMCO, Product & Strategy, Los Angeles, 2024 (Wall Street Oasis): Lots of random bond math questions -- duration of this bond with x coupon sold at par
004A fund's NAV goes from 10 to 15 in year one and falls to 12 by the end of year two. An investor put in Rs 1 lakh at the start and another Rs 5 lakh after year one. The fund returned about 9.5% a year. What did the investor earn?Performance measurement and returnsHardFund research and ratingsIndian AMCs
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The fund made money over the two years. Did this investor?
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He lost about 11.6% a year while the fund made 9.5% a year. Rs 1 lakh bought 10,000 units at 10 and Rs 5 lakh bought 33,333 units at 15. At NAV 12 his 43,333 units are worth Rs 5.2 lakh against Rs 6 lakh put in. The rate that makes those cash flows balance, his internal rate of return, is -11.6% a year.
How can the fund and the investor disagree on the return?
Picture a shop that sells ten mangoes at a profit on Monday, then stocks up on a hundred on Tuesday just before the price drops. The shop's price record looks fine; the owner's till does not. The fund's return measures what one rupee did if it stayed the whole time; the investor's return weights each period by how much of his money was there. Here only Rs 1 lakh enjoyed the rise from 10 to 15, and Rs 6.5 lakh suffered the fall to 12.
The NAV rose from 10 to 15 and fell to 12, a fund return of 9.5% a year, but the investor put Rs 5 of his Rs 6 lakh in at 15, so his holding ends at Rs 5.2 lakh and his money-weighted return is -11.6% a year.
How do you compute the investor's figure without a calculator?
Count units, then value them. Rs 1 lakh at 10 buys 10,000 units; Rs 5 lakh at 15 buys 33,333. That is 43,333 units, worth Rs 520,000 at 12. The money-weighted return is the single rate r that makes the two payments, grown at r, equal the final value. With x as 1 plus r, x squared plus 5x equals 5.2, and the positive root is 0.8838, so r is -11.6%.
The relationship
$1\cdot x^2 + 5\cdot x = 5.2 \;\Rightarrow\; x = \frac{-5 + \sqrt{25 + 20.8}}{2} = 0.884,\quad r = x - 1 \approx -11.6\%$
1 and 5 the payments in Rs lakh at the start and after year one
x one plus the investor's annual return
5.2 the holding's value in Rs lakh at the end of year two
What it says in wordsGrow each payment at one unknown rate to the end date and solve for the rate that matches what the holding is worth.
The fund's own figure is simple: 12 over 10 is 1.2 in two years, and the square root of 1.2 is 1.0954, so 9.54% a year. That is the number a factsheet shows, because the manager does not choose when investors arrive. The investor's IRR is the number his account statement should show, and it is the one his experience matches.
Where candidates lose it
Most candidates answer 9.5% or a smaller positive number, because they assume the investor must share the fund's result. The interviewer is testing whether you know two returns exist and which one belongs to whom.
The second trap is dividing the total loss by the total invested, Rs 0.8 lakh on Rs 6 lakh. That ignores that Rs 1 lakh was in for two years and Rs 5 lakh for one; the IRR handles the timing.
What the interviewer asks next
Which of the two numbers should a fund manager be judged on, and why?
If the Rs 5 lakh had gone in at the start instead, what would the investor have earned?
Why do investors in a fund often earn less than the fund's reported return over long periods?
005An active large cap fund charges 1.8% a year and an index fund on the same index charges 0.2%. How much must the active manager beat the index by, before costs, just to tie? And on Rs 10 lakh over 15 years with the index returning 12% a year, what does zero skill cost the investor?Costs and fee dragCoreFranklin TempletonSan Mateo · 2017
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Over 15 years, how big is the wealth gap if the active manager has no skill?
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The manager must beat the index by 1.6 points a year before costs just to tie, and with zero skill the investor ends about Rs 10.4 lakh behind. Both funds earn the index's 12% before costs. Net, that is 11.8% against 10.2%. Rs 10 lakh compounds to Rs 53.3 lakh in the index fund and Rs 42.9 lakh in the active fund, a gap of about 19% of the final wealth.
Why is the break-even the whole cost gap and not the active fee?
Two taxis to the same station, one charging Rs 18 per km and one Rs 2. The expensive one only wins if it is a much shorter route. The investor's alternative is not zero cost, it is the index fund, so the active manager has to earn back the difference in costs, 1.6 points a year, before adding anything. Beating the index by 1% before costs sounds like skill and still leaves the investor 0.6 points a year behind the cheaper fund.
Rs 10 lakh compounding at 11.8% after costs reaches Rs 53.3 lakh in 15 years, against Rs 42.9 lakh at 10.2%, so a 1.6 point cost gap becomes a Rs 10.4 lakh gap, about a fifth of the final wealth.
Why does 1.6 points a year become a fifth of the money?
Because the fee is charged on the balance every year, and the rupees it takes would themselves have compounded. A cost gap compounds exactly like a return gap: the ratio of final wealth is (1.102 over 1.118) to the 15th power, about 0.806, so the active investor keeps about 81% of what the index investor has. The longer the horizon, the larger that share becomes; over 30 years it would be over a third.
The relationship
$10(1.118)^{15} - 10(1.102)^{15} = 53.3 - 42.9 = 10.4 \text{ lakh}$
1.118 one plus the index fund's return after its 0.2% cost
1.102 one plus the active fund's return after its 1.8% cost, assuming no skill
15 years held
What it says in wordsCompound the Rs 10 lakh at each net return and subtract; the gap is the price of paying for skill that did not show up.
The limitation: the 12% index return is an assumption for the arithmetic, and some active managers do beat their index after costs. The question is not whether active management can win. It is how large the hurdle is, and 1.6 points a year, every year, is a high bar to clear consistently.
Where candidates lose it
The common wrong answer to the first part is 1.8%, the active fund's fee. The investor's real choice is the index fund, which also costs something, so the hurdle is the gap, 1.6 points.
The common wrong answer to the second part is simple interest: 1.6% of Rs 10 lakh for 15 years, Rs 2.4 lakh. Fees come out of a growing balance and compound; say that sentence and then give the Rs 10.4 lakh.
What the interviewer asks next
What gross alpha does the active manager need for the investor to end Rs 5 lakh ahead of the index fund?
The index fund also lags its index by 0.3% a year through tracking difference. How does that change the hurdle?
How would you explain this gap to a client in one sentence without recommending either fund?
Asked at Franklin Templeton, Risk Management, San Mateo, 2017 (Wall Street Oasis): explain the difference between actively managed and passively managed mutual funds
006A broad equity index trades at 22 times earnings, an earnings yield of about 4.5%, while the 10-year government bond yields 7%. How much earnings growth does equity need just to match the bond, and which of the two is cheaper?Valuation riddlesCorePIMCOSan Diego · 2026
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Roughly what yearly earnings growth makes the index match the bond, before any extra reward for risk?
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About 2.5% a year of earnings growth just to tie, and about 5.5% if you demand an assumed 3 point premium for equity risk. Equity's expected return is roughly its earnings yield plus growth. At 22 times earnings the yield is 4.5%, 2.5 points below the bond. On starting yield the bond is cheaper; equity is cheaper only if you expect earnings growth comfortably above 5.5% a year.
How do you put a stock index and a bond on the same scale?
Think of two shops for sale. One pays its owner a fixed Rs 7 for every Rs 100 of price, forever. The other pays Rs 4.5 today, but its takings rise every year. Turn the P/E upside down to get an earnings yield, then add the growth the earnings will carry, and you have a number you can hold against the bond's yield. One over 22 is 4.55%, so the index starts 2.5 points behind the bond.
At 22 times earnings the index yields 4.5%, so it needs about 2.5% yearly earnings growth to match a 7% bond and about 5.5% once an assumed 3 point equity risk premium is added.
The relationship
$E[r_{eq}] \approx \frac{E}{P} + g \;\Rightarrow\; g = 7.0\% - 4.5\% = 2.5\%$
E/P the earnings yield, one over the P/E of 22
g the long-run growth rate of earnings
7.0% the government bond yield
What it says in wordsEquity's rough expected return is what the earnings pay now plus how fast they grow, so the growth needed is the bond yield minus the earnings yield.
So which is cheaper?
Answer with a condition, not a verdict. On starting yield the bond is cheaper, and equity is cheaper only if earnings can grow faster than about 5.5% a year for a long time. The 3 point premium is an assumption for this arithmetic; you should say you would set it from your own view of equity risk. Then compare the required growth with a sensible estimate of nominal earnings growth in that economy, and state which side of the line you think it falls.
Name the limitation that marks you out. The bond yield is nominal, while earnings rise with inflation, so comparing 4.5% directly with 7% mixes a real yield with a nominal one. This comparison, often called the Fed model, is a quick screen, not a valuation. It also ignores that some earnings are reinvested, and the payout and the return on that reinvestment both shape the growth you can expect.
Where candidates lose it
The common slip is comparing 4.5% with 7% and declaring bonds cheaper, full stop. That treats equity like a bond with a fixed coupon and throws away the growth that is the whole reason to own it.
The opposite slip is saying equity is cheaper because it grows, without putting a number on how much growth is already needed. The interviewer wants the 2.5% said out loud, the premium on top, and a view on whether that growth is achievable.
What the interviewer asks next
If the bond yield falls to 6%, what P/E gives the same required growth?
How would you adjust the comparison for inflation?
Why might an index with a lower earnings yield still be priced fairly?
Asked at PIMCO, Debt Capital Markets, San Diego, 2026 (Wall Street Oasis): Which is cheaper us bonds or us equities
007Tarvela AMC's assets under management rose from Rs 40,000 crore to Rs 52,000 crore over a period in which the industry grew from Rs 5 lakh crore to Rs 7 lakh crore. Did Tarvela's market share rise?Logic and numeracy brainteasersWarm upIndian AMCsGlobal asset managers
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Quick call: what happened to Tarvela's share?
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No. Tarvela's share fell from 8.0% to about 7.4%. Its AUM grew 30% while the industry grew 40%. Share is 40,000 over 5,00,000 before and 52,000 over 7,00,000 after. To hold 8% Tarvela needed Rs 56,000 crore, so it is Rs 4,000 crore short of standing still. Growing slower than your market means losing share while growing.
Why can a business grow and still shrink?
A child who grows 3 cm in a year while every classmate grows 5 cm is taller than last year and lower in the line-up. Market share is a ratio, so it rises only when your growth beats the market's, and a large rupee gain means nothing on its own. Tarvela's Rs 12,000 crore gain sounds big until you set it next to the industry's Rs 2 lakh crore.
Tarvela AMC grew its assets 30% while the industry grew 40%, so its market share fell from 8.0% to 7.4%, and holding 8% would have needed Rs 56,000 crore rather than Rs 52,000 crore.
What is the fastest way to answer without dividing big numbers?
Compare growth factors. New share equals old share times Tarvela's growth factor over the industry's: 8% times 1.30 over 1.40. 1.3 over 1.4 is about 0.93, so the share falls by about 7% of itself, from 8.0% to 7.43%. You never need to divide 52,000 by 7,00,000 unless asked for the decimal.
The relationship
$s_1 = s_0 \times \frac{1 + g_{AMC}}{1 + g_{ind}} = 8.0\% \times \frac{1.30}{1.40} = 7.43\%$
s_0, s_1 market share before and after
g_{AMC} Tarvela's AUM growth, 30%
g_{ind} the industry's AUM growth, 40%
What it says in wordsShare moves by the ratio of your growth factor to the market's.
Add the analyst's question. AUM growth has two parts, market movement and net flows. If Tarvela is heavier in a segment that rose less, it could lose share with no problem in its sales. Splitting growth into the part the market gave and the part investors gave is the next thing an interviewer will ask for.
Where candidates lose it
The trap is answering yes because Rs 12,000 crore of growth sounds impressive. The interviewer has deliberately given a big absolute gain in a market growing faster, to see whether you reach for the ratio.
The second slip is subtracting growth rates, 30 minus 40, and saying share fell by 10 points. Share fell by about 0.57 points, from 8.0% to 7.43%; it is the ratio 1.3 over 1.4 that matters.
What the interviewer asks next
What growth did Tarvela need to gain half a point of share?
How would you split Tarvela's growth into market movement and net inflows?
Why might an AMC accept losing share in one category?
008Estimate the monthly SIP inflow into mutual funds from a single city of 1 crore people. State each assumption as you go.Estimation and market sizingCoreAllianceBernsteinNew York · 2021
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Which assumption will move your answer the most?
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About Rs 150 crore a month, with a range of roughly Rs 100 to 200 crore. One crore people at four per household is 25 lakh households. Assume 15% have at least one SIP, 3.75 lakh households, with 1.6 SIPs each, 6 lakh SIPs. At an average Rs 2,500 per SIP that is Rs 150 crore a month, about Rs 1,800 crore a year. Penetration is the assumption to test first.
Where do you start, people or money?
Think of estimating how much a housing society spends on milk. You would count flats, then flats that buy from the dairy, then litres per flat, then price. A sizing answer is judged on the chain: each step one assumption, stated with a number and a reason, so the interviewer can challenge any link without the whole answer falling over. Start with people because they are the one number given, and move to households because SIP decisions are made at home.
One crore people become 25 lakh households, 3.75 lakh investing households at an assumed 15% penetration, 6 lakh SIPs and about Rs 150 crore a month at a Rs 2,500 average, and moving penetration between 10% and 20% moves the answer between Rs 100 and Rs 200 crore.
How do you defend each assumption?
Give a reason in one breath for each. Four per household reflects a mix of nuclear and joint families in a large city. Fifteen per cent penetration is the weakest link: it assumes a minority of salaried and business households invest monthly through funds, and you should say you would check it against published folio or SIP account data for the city. Families with an SIP often run more than one, one per goal or per earner, hence 1.6. A Rs 2,500 average blends small starter SIPs with larger ones.
Then check the result a second way. Rs 150 crore a month is Rs 150 per person, or Rs 4,000 per investing household. If an investing household in this city earns around Rs 50,000 a month, that is about 8% of income going into SIPs, which is plausible. A figure that implied half of income would tell you an assumption has gone wrong.
The relationship
$\text{Inflow} = \frac{10^7}{4} \times 0.15 \times 1.6 \times 2{,}500 = \text{Rs } 150 \text{ crore a month}$
10^7 / 4 households, one crore people at four each
0.15 the assumed share of households with an SIP
1.6 and 2,500 SIPs per investing household and the average SIP in rupees
What it says in wordsMultiply down the chain, one stated assumption per step, and the product is the monthly flow.
Where candidates lose it
The common failure is jumping to a number without the chain, or quoting a national SIP figure from memory and scaling it by population. The interviewer cannot test a number with no steps, and a remembered figure may be wrong or out of date.
The second failure is precision theatre: carrying decimals through assumptions that are only good to one significant figure. Give Rs 150 crore, the range, and which link you would check first.
What the interviewer asks next
How would the answer change for a city of the same size with a much younger population?
How would you size the stoppage rate, the SIPs that end each month?
Build the same estimate top-down from a national figure. What would you need to look up?
Asked at AllianceBernstein, Investment Banking, New York, 2021 (Wall Street Oasis): Case study market sizing question
009A thousand fund managers have no skill at all: each has a 50% chance of beating the index in any year, independently. How many beat it five years in a row by luck, and what is the chance that at least one beats it ten years running?Probability and expected valueCoreFund research and ratingsIndian AMCs
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Guess the chance that at least one of the thousand posts a ten-year streak.
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About 31 managers beat the index five years running by luck alone, and there is about a 62% chance that at least one beats it ten years running. Each year halves the survivors: 1,000, 500, 250, 125, 62.5, 31.25. For ten years, one manager's chance is 1 in 1,024, so the chance none of the thousand does it is (1023/1024) to the 1,000th, about 38%.
Why do perfect records appear even when nobody is skilled?
Ask a stadium of a thousand people to toss a coin and sit down on tails. After five rounds about 31 are still standing, and each of them has a perfect record. A streak that is rare for one person is expected somewhere in a large enough crowd, so the size of the starting group matters as much as the length of the streak. The five-year count is just 1,000 halved five times.
Starting from 1,000 managers with no skill, halving each year leaves about 31 with a perfect five-year record, and the chance that at least one of the 1,000 posts a ten-year streak is about 62%.
How do you get the ten-year figure without a calculator?
Go through the complement: work out the chance that nobody does it. Each manager fails with probability 1023 over 1024. For many small independent chances, (1 minus 1/n) to the power n is close to 1 over e, about 0.37, and here the power is 1,000 against 1,024, so the chance nobody does it is about 0.38. That leaves about 62% for at least one ten-year streak.
The relationship
$P(\text{at least one}) = 1 - \left(1 - \tfrac{1}{1024}\right)^{1000} \approx 1 - e^{-0.977} \approx 62\%$
1/1024 one manager's chance of ten wins in a row, one half to the tenth
1000 the number of managers trying
What it says in wordsTake the chance that every manager fails, and one minus that is the chance at least one succeeds.
Say what it means for fund selection. A long record of beating the index is evidence, but weaker evidence than it looks when it is picked from a large universe after the fact. The limitation of the model is that real returns are not coin tosses and some managers do have skill; the puzzle only shows how much luck alone can produce.
Where candidates lose it
The trap in the second part is answering 1 in 1,024 or 0.1%, the chance for one named manager, when the question asks about anyone in the group. It is the same slip as being amazed that someone at a party shares your birthday.
The other slip is adding the chances, 1,000 times 1 in 1,024, to get 98%. That double counts the cases where two or more managers succeed; the complement avoids it.
What the interviewer asks next
How many managers would you expect with exactly four wins out of five years?
If one manager in the group truly beats the index 60% of years, how likely is a ten-year streak for them?
Why does survivorship, funds closing after bad years, make published track records look better still?
010A Rs 1,000 crore equity fund has a 1.5% annual expense ratio. How much is charged each day, where does the charge show up, and why does an investor never see a fee deducted from their account?NAV, units and fund mechanicsCoreFund operationsRegistrars and transfer agents
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Where does the daily expense actually come out?
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About Rs 4.1 lakh a day, taken inside the NAV. 1.5% of Rs 1,000 crore is Rs 15 crore a year; divided by 365 it is about Rs 4.11 lakh a day. The fund books that as a liability before striking the NAV, so each unit is worth a little less, about 0.21 paise a day at an NAV of 50. The investor never sees a deduction because the cost is already in the price.
If nothing is deducted, how is the fee paid?
Think of a restaurant that folds the service charge into the menu price instead of adding it to the bill. You pay it with every dish; you just never see a line saying so. A fund's expenses are charged to the scheme itself every day, so they reduce net assets and the NAV, and the investor pays through a slightly lower price per unit rather than a visible deduction. The number of units you hold never changes because of the fee.
A 1.5% expense ratio on Rs 1,000 crore is Rs 15 crore a year, about Rs 4.1 lakh accrued each day as a scheme liability, so the NAV every investor sees is already net of that day's cost.
How small is it per unit, and why does that matter?
At an NAV of 50, one day's share is 50 times 1.5% over 365, about Rs 0.0021, or 0.21 paise. Nobody notices a fifth of a paise a day, which is exactly why the expense ratio has to be read off the factsheet rather than felt in the account. Across a year it is 1.5% of the money, and it compounds as a drag like any other cost.
The relationship
$\text{Daily accrual} = \frac{1{,}000 \times 0.015}{365} = 0.0411 \text{ crore} \approx \text{Rs } 4.1 \text{ lakh}$
1,000 the scheme's net assets in Rs crore
0.015 the annual expense ratio
365 days over which the annual charge is spread
What it says in wordsSpread the yearly percentage across the days and charge that slice to the scheme each day.
Two practical points complete the answer. Because assets change every day, the accrual is recomputed on each day's net assets rather than fixed at Rs 4.1 lakh. And the published returns of a fund are already after this cost, so a fair comparison with an index must use the fund's NAV returns against the index's total return. Expense ratio limits are set by regulation and change; confirm the current SEBI framework before quoting one.
Where candidates lose it
The common wrong picture is that the fee is billed once a year or taken by cancelling units, as a bank might debit a charge. Candidates who say it in a fund operations interview show they have not seen how the NAV is struck.
The second slip is dividing by 250 trading days. The charge accrues on every calendar day, so divide by 365 and say why.
What the interviewer asks next
How would the daily accrual change if the fund's assets doubled over the year?
Why do the direct and regular plans of the same scheme have different NAVs?
Where on a factsheet or annual report would you find the expense ratio?

i	the nominal deposit rate, 7%
t	the assumed tax slab, 30%
\pi	inflation, 6%

w_A, w_B	the weights, 0.6 and 0.4
\sigma_A, \sigma_B	the funds' volatilities, 20% and 12%
\rho	the correlation between them, 0.3

y	the yield, 8%, equal to the coupon because the bond is at par
n	years to maturity, 5
D_{mac}	Macaulay duration, the balance point in years

1 and 5	the payments in Rs lakh at the start and after year one
x	one plus the investor's annual return
5.2	the holding's value in Rs lakh at the end of year two

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If nothing is deducted, how is the fee paid?

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