Mutual Fund Mastery puzzles, solved step by step

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

Showing 1–10 of 30 · filtered from 100Clear filters

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?
013A parent needs Rs 50 lakh in 12 years for a child's education and assumes 11% a year. What monthly SIP does that take? And why does waiting three years before starting raise the SIP by about 62%, rather than the 25% most people guess?Compounding and time valueHardIndian AMCsDistribution and sales
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Starting three years late, how much larger is the monthly SIP?
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About Rs 16,691 a month now, and about Rs 27,048 after a three-year wait, 62% more. At 11% a year, taken as 0.917% a month and paid at the start of each month, 144 payments of Rs 16,691 grow to Rs 50 lakh. With 108 payments the SIP must be Rs 27,048. The early years carry the most growth, so losing them costs far more than their share of time.
How do you get the starting SIP?
A monthly SIP is a stream of equal payments, each growing until the goal date. The first rupee grows for 144 months, the last for one. Add up what one rupee a month becomes, and the required SIP is the goal divided by that sum. At 11% a year, taken as 11 over 12, about 0.917% a month, one rupee paid at the start of each month for 144 months grows to about 300. Rs 50 lakh divided by 299.6 is Rs 16,691.
The relationship
$\text{SIP} = \frac{FV}{\dfrac{(1+i)^n - 1}{i}\,(1+i)} = \frac{50{,}00{,}000}{\dfrac{1.00917^{144}-1}{0.00917}\times 1.00917} \approx \text{Rs } 16{,}691$
FV the goal, Rs 50 lakh
i the monthly rate, 11% divided by 12
n the number of monthly payments, 144
(1+i) one extra month of growth because each payment is made at the start of the month
What it says in wordsThe SIP is the goal divided by what a rupee a month grows into over the period.
Why is the cost of waiting not proportional to the wait?
Think of planting trees for shade in twelve years. A tree planted in year one is huge by the deadline; one planted in year ten is a sapling. Skip the first three years of planting and you cannot make it up by planting a few extra saplings later. The payments a delay removes are the ones that would have compounded the longest, so the later SIP must replace both the missing payments and the growth they would have earned. Starting now, Rs 24.0 lakh paid in becomes Rs 50 lakh; starting late, Rs 29.2 lakh has to be paid in, because growth contributes Rs 20.8 lakh instead of Rs 26.0 lakh.
A three-year delay raises the monthly SIP from Rs 16,691 to Rs 27,048, 62% more, because growth supplies Rs 20.8 lakh of the goal instead of Rs 26.0 lakh and the parent must pay the difference.
The two guesses are worth naming. Twenty-five per cent comes from three years being a quarter of twelve. Thirty-three per cent comes from spreading the same total contributions over nine years instead of twelve, Rs 22,255 a month. Both ignore growth. The true figure, Rs 27,048, sits well above both.
Say the limits. Eleven per cent is an assumption, not a forecast, and equity returns arrive unevenly; a plan built on one smooth rate should be checked against lower rates and reviewed as the date nears. Monthly compounding at 11 over 12 slightly overstates an 11% annual rate; using the exact monthly equivalent raises the SIP to about Rs 17,300, and the ratio of about 1.6 survives either way.
Where candidates lose it
The common slip is linear thinking: three years of twelve, so 25% more. It treats the SIP as a savings jar with no growth. The interviewer is testing whether you can see that the first payments do the most work.
The second slip is computing the goal as an ordinary annuity in one place and an annuity due in another, or mixing the 11 over 12 monthly rate with the exact monthly rate. Pick one convention, state it, and keep it for both SIPs; the ratio barely changes.
What the interviewer asks next
What lump sum invested today would replace the SIP entirely?
If returns came in at 9% instead of 11%, how short would the original SIP fall?
How would a 10% yearly step-up in the SIP change the starting amount?
014A bond has a modified duration of 7 and a convexity of 80. Yields rise by 150 basis points. Estimate the percentage price change with duration alone, then with convexity added.Bond maths and durationHardFixed income desksIndian AMCs
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With convexity added, what is the estimated price change?
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About -10.5% with duration alone and about -9.6% with convexity. Duration gives -7 times 0.015, which is -10.5%. Convexity adds one half times 80 times 0.015 squared, or +0.90%. The price-yield curve bends upward, so a straight-line duration estimate overstates losses when yields rise and understates gains when they fall, and the gap grows with the square of the move.
Why is duration alone not enough for a big move?
Think of estimating how far a ball thrown upward will fall back by looking only at its speed at the start. For a moment the straight line is fine; over a longer stretch the curve of its path takes over. Duration is the slope of the price-yield curve at today's yield, so it is exact for tiny moves and drifts further from the true price the larger the move. The curve bends upward for an ordinary bond, so it always sits above the tangent.
For a 150 basis point rise, the duration tangent puts the price at 89.5 while the curve with convexity puts it at 90.4, so duration alone overstates the loss by 0.9 points, and the gap widens as the move grows.
How do you compute both estimates quickly?
Write the move as a decimal first, 0.015. The duration term is minus duration times the move, -7 times 0.015, which is -10.5%; the convexity term is half the convexity times the move squared, 40 times 0.000225, which is +0.90%. Add them: -9.60%. For a 150 basis point fall the same two terms give +10.5% plus 0.90%, a gain of 11.40%, so convexity helps in both directions.
The relationship
$\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2 = -7(0.015) + \tfrac{1}{2}(80)(0.015)^2 = -10.5\% + 0.9\% = -9.6\%$
D_{mod} modified duration, 7
C convexity, 80
\Delta y the change in yield as a decimal, 0.015
What it says in wordsThe price change is the straight-line duration estimate plus a correction for the curve that grows with the square of the move.
The squared term is why size matters. At 50 basis points the convexity term is only +0.10%, easy to ignore; at 300 basis points it is +3.6%, too large to leave out. The limitation is that this is still a two-term approximation of a curve; for very large moves, or for bonds with call features whose convexity can turn negative, a desk reprices the bond in full rather than trusting the formula.
Where candidates lose it
The common slip is subtracting the convexity term because yields went up, giving -11.4%. For an ordinary bond the convexity term is positive in both directions; the squared move has no sign.
The other slip is a units error: using 1.5 instead of 0.015, which makes the convexity term 90 and the answer absurd, or forgetting the one half. Convert basis points to a decimal before anything else and say the half out loud.
What the interviewer asks next
Why is convexity valuable to a bondholder, and what does the market charge for it?
Why can a callable bond show negative convexity when yields fall?
Which has more convexity at the same duration, a bullet bond or a barbell of short and long bonds?
016A value stock has a 5% dividend yield and 5% growth; a growth stock has a 1% yield and 9% growth. Both are priced at a 10% required return. If the required return rises to 10.5%, which price falls more, by how much, and what does that say about each stock's duration?Valuation riddlesHardBlackRockNew York · 2026
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Required return goes from 10% to 10.5%. How far does the growth stock fall?
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The growth stock falls about 33% and the value stock about 9%. With a constant-growth model, price is dividend over required return minus growth. The value stock goes from 5 over 0.05, 100, to 5 over 0.055, 90.9. The growth stock goes from 1 over 0.01, 100, to 1 over 0.015, 66.7. Their implied durations are 20 and 100 years.
Why can a stock have a duration at all?
Think of two people selling their future income for cash today. One will earn a steady amount starting now; the other will earn little for years and a lot later. If the lender's interest rate rises, the second one's offer falls much more, because every rupee of it has to be discounted over a longer wait. A stock is a stream of future cash, and the further out the cash sits, the more its price moves when the discount rate moves: that sensitivity is its duration. In the constant-growth modela valuation that prices a share as the dividend expected next year divided by the required return minus a steady growth rate, price is D over (r minus g), and duration works out to 1 over (r minus g).
When the required return rises from 10% to 10.5%, the value stock falls from 100 to 90.9 while the growth stock falls from 100 to 66.7, because their implied durations are 20 and 100 years.
How do you work the two falls quickly?
Watch the gap, r minus g, not r. For the value stock the gap goes from 5 points to 5.5 points, ten per cent wider, so the price falls by 1 minus 5 over 5.5, which is 9.1%. For the growth stock the gap goes from 1 point to 1.5 points, fifty per cent wider, so the price falls by a third, 33.3%. The same half-point move is a small change to a wide gap and a large change to a narrow one.
The relationship
$P = \frac{D}{r-g},\qquad -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g}: \quad \frac{1}{0.05} = 20, \quad \frac{1}{0.01} = 100$
D next year's dividend per 100 of price: 5 for value, 1 for growth
r the required return, 10% rising to 10.5%
g the steady growth rate: 5% for value, 9% for growth
What it says in wordsPrice is the dividend over the gap between required return and growth, and the narrower the gap, the more a change in the required return moves the price.
The duration numbers also show their own limit. Twenty years times half a point predicts a 10% fall, close to the true 9.1%. A hundred years times half a point predicts 50%, well off the true 33.3%, because a straight-line estimate fails when the move is large next to the gap. And the model assumes growth runs forever at 9% against a 10% required return, which is fragile; real growth stocks are valued with a high-growth phase that fades, but the direction of the answer holds.
Where candidates lose it
The common slip is answering that both fall about the same because both are priced at 100 at the same required return. Equal prices hide very different timing of the cash, and timing is what rate sensitivity measures.
The second slip is applying duration in a straight line and saying the growth stock falls 50%. Mention duration, then show the exact repricing: when r minus g is only 1 point, the curve matters as much as the slope.
What the interviewer asks next
Why did long-duration growth stocks fall hardest when interest rates rose sharply?
What happens to the growth stock's price if expected growth falls from 9% to 8.5% at a 10% required return?
How would you estimate the duration of an index rather than one stock?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks MSE Forecasting equation
017Two funds' daily returns are negatively correlated within any given month, yet their yearly returns are positively correlated. How can both be true?Statistics, correlation and diversificationHardSquarepoint CapitalMontreal · 2024
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Which explanation fits?
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A shared driver that moves slowly lifts or sinks both funds together across years, while short-term noise pushes them in opposite directions day to day. Within a month the slow driver barely changes, so daily correlation reflects only the opposing noise. Across years it dominates. With daily noise of 0.8% at -0.5 correlation and a shared yearly drift of 20% standard deviation, yearly correlation comes out at about +0.57.
What kind of situation produces this?
Think of two ice cream stalls on the same beach. On any given day, a customer who buys from one does not buy from the other, so their daily sales move against each other. Across years, both do well in hot summers and badly in wet ones. Correlation is not one fixed number between two things; it depends on which driver dominates at the horizon you measure, and different drivers dominate at different horizons. For funds, the slow driver might be the economy's earnings cycle, which both portfolios share; the fast one might be money rotating between their two styles day to day.
In a simulated month the two funds move in opposite directions on 16 of 21 days, yet across years the shared drift adds 0.040 of covariance against 0.008 removed by the opposing noise, so yearly returns are positively correlated at about 0.57.
Can you show it with numbers?
Build each fund's yearly return from two parts. A shared drift, the same for both within a year but different from year to year, with a standard deviation of 20%. Plus daily noise of 0.8% a day for each fund, correlated at -0.5 between them, over 250 trading days. Within a month the drift is a constant, so it drops out of any correlation measured around the month's average, and the daily figure is the noise's -0.5. Across years, the drift contributes 0.2 squared, 0.040, to covariance, and the noise contributes -0.5 times 250 times 0.008 squared, -0.008, leaving +0.032.
The relationship
$\rho_{year} = \frac{\sigma_F^2 + \rho_d\, n\, \sigma_d^2}{\sigma_F^2 + n\,\sigma_d^2} = \frac{0.040 - 0.008}{0.040 + 0.016} = \frac{0.032}{0.056} \approx 0.57$
\sigma_F the standard deviation of the shared yearly drift, 20%
\rho_d the correlation of daily noise, -0.5
n trading days in a year, 250
\sigma_d each fund's daily noise, 0.8%
What it says in wordsYearly correlation is the shared drift's variance less the summed opposing noise, divided by each fund's total yearly variance.
Give the condition and a second mechanism. The sign flips only if the shared drift's variance is larger than the summed noise covariance; with a drift of 10% instead of 20%, covariance would be 0.010 minus 0.008 and correlation barely positive. A second route is timing: if one fund's holdings are priced with a lag, its daily moves look unrelated or even opposite to the other's, while over a year the lags wash out. Either way, the lesson for a fund analyst is that a diversification benefit measured on daily data may not exist at the horizon a client actually holds.
Where candidates lose it
The common failure is saying it is impossible, on the belief that correlation is a fixed property of two assets. The interviewer is checking whether you know correlation is horizon dependent and can name what drives each horizon.
The second failure is waving at small samples. Twelve monthly points are noisy, but noise is not an explanation; the strong answer builds a two-component model and states when the sign flips.
What the interviewer asks next
Using the same model, at what size of shared drift would yearly correlation be exactly zero?
Why might two funds that look like good diversifiers on daily data fail to diversify in a bear market?
How would stale prices in one fund distort its measured volatility as well as its correlation?
Asked at Squarepoint Capital, Hedge Fund, Montreal, 2024 (Wall Street Oasis): correlation can be negative intra-month but positive across a year, how?
020Ten per cent of fund managers are skilled and beat their index in 70% of years; the rest have no skill and beat it in 50% of years. A manager has beaten the index three years running. What is the probability that this manager is skilled?Probability and expected valueHardFund research and ratingsIndian AMCs
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After a three-year streak, how likely is the manager to be skilled?
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About 23%. Picture 1,000 managers. The 100 skilled ones produce 100 times 0.7 cubed, 34.3 three-year streaks. The 900 unskilled produce 900 times 0.5 cubed, 112.5. Of 146.8 streak holders, 34.3 are skilled, 23.4%. The streak more than doubles the odds of skill, from 10%, yet most streak holders are still lucky.
Why does a three-year streak prove so little?
Think of a test for a rare condition that catches most real cases but also flags plenty of healthy people. If the condition is rare, most positive results are false alarms. When the thing you are looking for is rare, even good evidence leaves most positives coming from the larger group, so the starting proportion matters as much as the evidence. Here skill is the rare condition, at 10%, and a three-year streak is a test that unskilled managers pass one time in eight.
Of 1,000 managers, the 100 skilled produce 34.3 three-year streaks and the 900 unskilled produce 112.5, so only about 23% of managers with a streak are actually skilled.
How do you set it up fast in the room?
Use counts, not formulas. Pick 1,000 managers, split them by skill, then split each group by whether it produced the streak, and the answer is one box over the sum of two boxes. Skilled: 100 times 0.343 is 34.3. Unskilled: 900 times 0.125 is 112.5. The answer is 34.3 over 146.8, 23.4%. The same result in odds form: prior odds of 1 to 9, times a likelihood ratiohow many times more likely the evidence is if the manager is skilled than if not of 0.343 over 0.125, about 2.74, gives 2.74 to 9.
The relationship
$P(S\mid \text{streak}) = \frac{0.1 \times 0.7^3}{0.1 \times 0.7^3 + 0.9 \times 0.5^3} = \frac{0.0343}{0.0343 + 0.1125} \approx 23.4\%$
S the manager is skilled
0.1 and 0.9 the shares of skilled and unskilled managers
0.7^3 and 0.5^3 each group's chance of three wins in a row
What it says in wordsThe chance of skill given a streak is the skilled managers' share of all the streaks.
Push it once to show judgement. A ten-year streak changes the picture: 100 times 0.7 to the tenth against 900 times 0.5 to the tenth gives about 76% skilled, because the unskilled pass a ten-year test only about once in a thousand. The limitation is that the 10% and 70% are assumptions, and real years are not independent coin tosses; managers with a style that suits a market phase can string wins together without skill.
Where candidates lose it
The common slip is answering 70%, mixing up the chance a skilled manager wins with the chance a winner is skilled. It is the same inversion that makes people overrate a positive result on a test for a rare condition.
The other slip is ignoring the 10% starting share and answering from the hit rates alone, 0.343 over 0.343 plus 0.125, about 73%. That treats skilled and unskilled managers as equally common, which is the one thing the question told you they are not.
What the interviewer asks next
How many consecutive winning years would push the probability of skill above 50%?
How would the answer change if skilled managers made up 30% of the population?
Why does survivorship in fund databases make this problem worse in practice?
021"Why should I buy your college, and how much would you sell it for?" Value a college with 8,000 students each paying Rs 2 lakh a year in fees, at a 35% operating margin.Estimation and market sizingHardWellington ManagementBoston · 2024
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What is the college's yearly operating profit?
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Roughly Rs 560 to 784 crore, on Rs 56 crore of operating profit at assumed multiples of 10 to 14 times. Fees are 8,000 times Rs 2 lakh, Rs 160 crore; a 35% margin leaves Rs 56 crore. The case for buying is durability: a brand, accreditation and a campus that keep the seats full for decades. The multiple you ask for depends on how sure the buyer can be of that.
What is the interviewer really testing?
Think of selling the tea stall outside a busy railway station. The buyer is not paying for the kettle; he is paying for the queue that shows up every morning and the confidence it will keep showing up. Any institution can be valued as a stream of cash plus a judgement about how long and how reliably that stream lasts, and the question has two halves because those are the two halves of a valuation. "Why should I buy" asks for the durability story; "how much" asks for the numbers.
Fees of Rs 160 crore less Rs 104 crore of costs leave Rs 56 crore of operating profit, which at assumed multiples of 10 to 14 times values the college at roughly Rs 560 to 784 crore.
How do you get from fees to a value?
Revenue is students times fees: 8,000 times Rs 2 lakh is Rs 160 crore a year. A 35% operating margin leaves Rs 56 crore, and the multiple you put on that profit is where the durability argument turns into a number. At 10 times it is Rs 560 crore, at 14 times Rs 784 crore. These multiples are assumptions for the exercise; say you would set them against what comparable education businesses have changed hands for, and against a cash flow valuation.
The relationship
$V = m \times (N \times F \times M) = m \times (8{,}000 \times 2 \text{ lakh} \times 0.35) = m \times 56 \text{ crore}$
N students, 8,000
F yearly fee per student, Rs 2 lakh
M the operating margin, 35%
m the multiple of operating profit, assumed 10 to 14
What it says in wordsProfit is students times fee times margin, and value is that profit times a multiple that reflects how durable it is.
Then sell the durability and test it. The selling points are a waiting list larger than the intake, accreditation that a new entrant would take years to earn, land owned rather than leased, and alumni who send their children. The tests are the risks: if enrolment falls 10% while costs stay fixed, operating profit drops from Rs 56 crore to about Rs 40 crore, because every rupee of lost fees falls straight to profit. A per-seat check helps too: Rs 560 crore over 8,000 seats is Rs 7 lakh a seat, three and a half years of fees.
Name one structural limit. Many colleges are run by trusts or societies that cannot distribute profit, so in practice a buyer may be acquiring a management contract, the land or a related company rather than the college itself; the cash a buyer can actually take out may be smaller than the operating profit. Asking who can receive the cash shows the interviewer you think like an owner.
Where candidates lose it
The common failure is answering only one half: a heartfelt speech about the college with no number, or a quick multiple with no reason a buyer should believe the profit lasts. The interviewer asked both questions on purpose.
The second failure is multiplying revenue instead of profit, or quoting a multiple as if it were a market fact. Build revenue, then profit, then state the multiple as your assumption and the range it gives.
What the interviewer asks next
What would a buyer pay if fees are capped by a regulator and costs rise 6% a year?
How would you value the college with a discounted cash flow instead of a multiple?
Which single number would you most want to verify before agreeing a price?
Asked at Wellington Management, Investment Research, Boston, 2024 (Wall Street Oasis): Why should I buy your College and how much would you sell it for?
029An active equity fund has an R-squared of 0.97 and a beta of 1.0 against its index, which has 16% volatility. Roughly how much active risk is the fund taking each year, and what does that suggest about its 1.8% expense ratio?Statistics, correlation and diversificationHardFund research and ratingsGlobal asset managers
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Before you work it: roughly how large is the fund's active risk, its tracking error?
Show the worked solution
About 2.8% a year of active risk, which leaves little room to earn back a 1.8% fee. An R-squared of 0.97 means 97% of the fund's variance moves with the index. With a beta of 1.0, the index part is 16% squared; total variance is that over 0.97, and the remaining 3% is the fund's own, about 2.8% a year as a volatility. To beat the index after a 1.8% fee, the manager needs an information ratio of about 0.64, which few sustain.
What does an R-squared of 0.97 actually say?
Picture a cover band that plays a famous album note for note, with one improvised solo per concert. Most of what you hear is the original. An R-squared of 0.97 says 97% of the fund's ups and downs are the index's ups and downs; only 3% of its variance is the manager doing something different. The fee, though, is charged on 100% of the money. That mismatch is what the interviewer wants you to find.
The step people miss is that R-squared is a share of variance, and variance is volatility squared. So you cannot take 3% of 16%. Work in squares, then come back with a square root. The fund's own volatility, measured against the index, is its tracking errorThe standard deviation of the gap between a fund return and its benchmark return; the size of the fund active bets, measured as a volatility., and when beta is 1.0 it equals this residual risk.
Of the fund's variance, 97% moves with the index and only 3% is its own. Converted back to volatility, the fund's total risk is about 16.2% a year and its own active risk about 2.8%, so a 1.8% fee needs an information ratio of about 0.64 just to break even.
The relationship
$\sigma_{\varepsilon} = \sqrt{\frac{\beta^2 \sigma_m^2}{R^2}\,(1 - R^2)} = \sqrt{\frac{0.0256}{0.97} \times 0.03} \approx 2.8\%$
\sigma_m the index volatility, 16%
\beta the fund's beta, 1.0
R^2 the share of the fund's variance explained by the index, 0.97
\sigma_{\varepsilon} the fund's own volatility, its active risk
What it says in wordsRebuild the fund's total variance from the index part, take the unexplained share, and square-root it back to a volatility.
Why does a 2.8% active risk make a 1.8% fee hard to earn back?
Think of the active risk as the size of the bets. A manager who deviates by about 2.8% a year and wants to beat the index after costs needs gross excess returns above 1.8% a year. That is an information ratio, excess return over active risk, of about 0.64, and managers who hold even 0.5 for a decade are rare. Put simply, the fund charges an active fee for a portfolio that is mostly the index, and the small active part has to work very hard to pay for all of it. Some analysts call the pattern closet indexing.
State the limits. R-squared and beta are estimated from past returns against one chosen index, and a different index can give a different R-squared. A high R-squared is also not proof of low skill; it only says the room for skill to show is small. The fair conclusion is a question for the fund, not a verdict on it.
Where candidates lose it
The common slip is taking 3% of the 16% index volatility and saying the fund's own risk is about 0.5%. R-squared divides variance, not volatility, and the square root turns a 3% share of variance into a much larger share of volatility, about 17% of the fund's total volatility.
The second loss is stopping at the number. The interviewer asked what it suggests about the fee: say that 2.8% of room against a 1.8% fee implies an information ratio of about 0.64 just to break even.
What the interviewer asks next
What R-squared would give the fund 6% of active risk with the same beta and index?
If the beta were 1.2 with the same R-squared, how would the active risk and its meaning change?
Why can a fund with a high R-squared still have a low correlation of its excess returns with the index?
030A fund manager has delivered 2% a year of alpha with a tracking error of 5%. How many years of data do you need before a two-standard-error test says the alpha is unlikely to be luck?Performance measurement and returnsHardFund research and ratingsIndian AMCs
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Your first guess: how many years?
Show the worked solution
About 25 years. The yearly signal is the 2% alpha and the yearly noise is the 5% tracking error, an information ratio of 0.4. Over n years the average alpha's standard error falls with the square root of n, so the t-statistic is 0.4 times root n. Setting that equal to 2 gives root n of 5, or 25 years. Skill of a realistic size takes longer to prove than most careers last.
Why does it take so long?
Think of judging a batsman from a few innings. A player who averages 45 against a league average of 40 is better, but any single innings swings by 30 runs, so a handful of innings cannot separate him from an average player on a good run. A fund manager's alpha is a small average edge buried in large year-to-year noise, and averaging only shrinks the noise with the square root of time. Four times the data halves the noise; it does not quarter it.
Here the edge is 2% a year and the noise, the tracking error, is 5% a year. One year of data is a signal of 2 against noise of 5, a ratio of 0.4, which is the information ratio. After n years the average alpha is still about 2%, but its standard error has fallen to 5% over root n.
The relationship
$t = \frac{\alpha}{\sigma_{TE}/\sqrt{n}} = IR\,\sqrt{n} \quad\Rightarrow\quad n = \left(\frac{2}{0.4}\right)^2 = 25$
\alpha the average alpha, 2% a year
\sigma_{TE} tracking error, 5% a year
IR information ratio, alpha over tracking error, 0.4
n years of data
What it says in wordsThe confidence in an alpha grows with the square root of the years, so the years needed are the square of 2 divided by the information ratio.
The t-statistic of a 2% alpha on 5% tracking error rises only with the square root of time: it is 0.89 after 5 years and 1.26 after 10, and it crosses the two-standard-error line only at 25 years.
What does this mean for how funds are judged?
Most track records are far too short to separate skill from luck at conventional confidence, so a strong five-year record is weak evidence on its own. At 5 years the t-statistic here is only 0.89. That is why fund researchers lean on the process, the consistency of the style and the source of the returns, not the headline number alone. The same arithmetic runs the other way: a manager with an information ratio of 1.0 would need only 4 years, and one at 0.5 would need 16.
Say the assumptions. The test treats yearly alphas as independent draws from a stable process with a constant edge. Real managers change style, teams leave, and markets change, so a 25-year record rarely describes one unchanged process. The number is a sense of scale, not a rule.
Where candidates lose it
Candidates often answer 2 or 3 years, reasoning that 2% beats zero every year on average. They are thinking about the edge and forgetting the noise around it. The ratio of the two, the information ratio, is what sets the clock.
The other loss is the square root. Candidates who see that noise shrinks with more data sometimes divide by n instead of root n and answer 6 or 7 years. Say root n out loud and the 25 follows.
What the interviewer asks next
With monthly data instead of yearly, does the answer change?
What information ratio would let you reach t = 2 in 10 years?
Why might a fund with a 10-year record and a t of 1.3 still be worth backing?
031A mutual fund distributor's book of Rs 50 crore earns a 0.75% trail commission each year. Markets add 10% a year to the book, and 8% of the book leaves each year as clients redeem or move. At a 12% discount rate, roughly what is the book worth?Costs and fee dragHardDistribution and salesIndian AMCs
Try it first
What growth rate should go into the valuation?
Show the worked solution
About Rs 3.5 crore. Year one's trail is 0.75% of Rs 50 crore, Rs 0.375 crore. The book grows 10% from markets and loses 8% to attrition, so it nets 1.10 x 0.92, about 1.2% growth a year, and the trail grows with it. A growing perpetuity at 12% values that at 0.375 divided by (12% minus 1.2%), about Rs 3.47 crore, roughly seven times one year's income.
Why is the book worth so little next to the Rs 50 crore it holds?
A distributor does not own the Rs 50 crore; the clients do. What the distributor owns is a stream of income, like the owner of a rented flat owns the rent, not the tenant's savings. The book is worth the present value of its trail income, and that income is only 0.75% of the assets each year. So the right starting point is Rs 37.5 lakh a year, and the question is how that stream grows and how fast it leaks away.
The Rs 50 crore book grows 10% from markets and loses 8% to attrition, netting about 1.2% a year, while 0.75% flows off as trail. Valued as a growing perpetuity at 12% the book is worth about Rs 3.47 crore, and a 4-point change in attrition moves that value about as much as a 4-point change in market growth.
How do market growth and attrition combine?
They multiply. A client who stays sees the book grow 10%; then 8% of the grown book leaves. The net growth is 1.10 times 0.92 less 1, about 1.2%, not 10% minus 8%. That growth feeds a growing perpetuityA stream of payments that continues forever and grows at a constant rate; its value is the first payment divided by the discount rate less the growth rate., which is the standard way to value a stream that neither ends nor stays flat.
The relationship
$V = \frac{C_1}{r - g} = \frac{0.375}{0.12 - 0.012} \approx \text{Rs } 3.47 \text{ crore}$
C_1 year one's trail, 0.75% of Rs 50 crore, Rs 0.375 crore
r the discount rate, 12%
g net growth of the book, 1.10 x 0.92 less 1, about 1.2%
What it says in wordsThe book is worth one year's trail divided by the gap between the discount rate and the book's net growth.
Which lever matters more, markets or client retention?
About equally, which is the point the interviewer wants. Cutting attrition from 8% to 4% lifts the value to about Rs 5.86 crore; lifting market growth from 10% to 14% lifts it to about Rs 5.27 crore. A distributor cannot control the market but can control service and retention, and retention moves the value as much as the market does. The limits: the formula needs growth below 12%, a trail rate is set by the fund house and can change, and the 10% market growth is an assumption, not a forecast.
Where candidates lose it
The first trap is valuing the assets, not the income, and quoting a number near Rs 50 crore. The distributor's claim is a 0.75% slice each year, and the value must start from Rs 37.5 lakh of income.
The second is subtracting attrition from growth: 10% minus 8% gives 2% and a value of Rs 3.75 crore. The two compound, so the net growth is 1.2%. A small slip in g matters, because g sits in the denominator next to r.
What the interviewer asks next
What attrition rate would make the book worth Rs 5 crore?
Trail is cut to 0.5%. What is the book worth, and what does the distributor's business depend on now?
Why might a buyer of this book use a higher discount rate than 12%?

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

FV	the goal, Rs 50 lakh
i	the monthly rate, 11% divided by 12
n	the number of monthly payments, 144
(1+i)	one extra month of growth because each payment is made at the start of the month

D_{mod}	modified duration, 7
C	convexity, 80
\Delta y	the change in yield as a decimal, 0.015

D	next year's dividend per 100 of price: 5 for value, 1 for growth
r	the required return, 10% rising to 10.5%
g	the steady growth rate: 5% for value, 9% for growth

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How can the fund and the investor disagree on the return?

How do you compute the investor's figure without a calculator?

Where candidates lose it

What the interviewer asks next

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How do you get the starting SIP?

Why is the cost of waiting not proportional to the wait?

Where candidates lose it

What the interviewer asks next

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Why is duration alone not enough for a big move?

How do you compute both estimates quickly?

Where candidates lose it

What the interviewer asks next

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Why can a stock have a duration at all?

How do you work the two falls quickly?

Where candidates lose it

What the interviewer asks next

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What kind of situation produces this?

Can you show it with numbers?

Where candidates lose it

What the interviewer asks next

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Why does a three-year streak prove so little?

How do you set it up fast in the room?

Where candidates lose it

What the interviewer asks next

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What is the interviewer really testing?

How do you get from fees to a value?

Where candidates lose it

What the interviewer asks next

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What does an R-squared of 0.97 actually say?

Why does a 2.8% active risk make a 1.8% fee hard to earn back?

Where candidates lose it

What the interviewer asks next

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Why does it take so long?

What does this mean for how funds are judged?

Where candidates lose it

What the interviewer asks next

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Why is the book worth so little next to the Rs 50 crore it holds?

How do market growth and attrition combine?

Which lever matters more, markets or client retention?

Where candidates lose it

What the interviewer asks next