Mutual Fund Mastery puzzles, solved step by step

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

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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?
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?
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?
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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?
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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
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What growth rate should go into the valuation?
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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%?
037One investor earns 15% a year for 15 years and then 9% a year for 15 years; another earns 9% first and 15% later. With a lump sum they end with identical money. With a Rs 10,000 monthly SIP they do not. Who ends richer, and why does the order matter only when money arrives over time?Compounding and time valueHardIndian AMCsDistribution and sales
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With the monthly SIP, who ends the 30 years richer?
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The investor who gets 9% first and 15% later ends richer, about Rs 3.62 crore against Rs 2.61 crore. A lump sum sees every year's return, and multiplication does not care about order: 1.15 to the 15 times 1.09 to the 15 is 29.64 either way. An SIP's money arrives over time, so the pot is small early and large late. Whichever return lands on the large late pot decides the result.
Why does order not matter for a lump sum?
A shop that marks a price up 15% and then 9% ends at the same tag as one that marks up 9% and then 15%; both multiply the price by 1.15 and by 1.09. A lump sum is one block of money exposed to every year's return, and because returns multiply, their order cannot change the product. Rs 10 lakh grows by 29.64 times in either order, to Rs 2.96 crore.
A Rs 10 lakh lump sum ends at Rs 2.96 crore in either order, but a Rs 10,000 monthly SIP does not: the 15%-first corpus leads at year 15 with Rs 61.6 lakh against Rs 36.9 lakh, is overtaken in year 24, and ends at Rs 2.61 crore against Rs 3.62 crore.
What breaks the symmetry in an SIP?
Think of a savings jar you add Rs 100 to every week. A bonus that doubles the jar is worth far more in December than in January, because in December the jar holds a year of savings. In an SIP, each instalment only experiences the returns after it is paid, so a return applied when the pot is large moves more rupees than the same return applied when the pot is small. After 15 years of Rs 10,000 a month, Rs 18 lakh has gone in and the pot holds Rs 61.6 lakh or Rs 36.9 lakh. The next 15 years apply to that pot plus Rs 18 lakh more, and it is the 15% that lands on it in one order and the 9% in the other.
The relationship
$\text{corpus} = \sum_{t=1}^{360} m \prod_{s=t}^{360} (1 + i_s)$
m the monthly instalment, Rs 10,000
i_s the return in month s, 15% or 9% a year as a monthly rate
t the month the instalment is paid
What it says in wordsEach instalment grows only by the returns from its own month onward, so late returns touch every instalment and early returns touch only the first few.
What does this mean for a real SIP investor?
For anyone still contributing, the returns in the final years carry the most weight, because that is when the most money is exposed. A poor decade at the start of an SIP mostly costs a little growth on a small pot; the same decade at the end hits the whole accumulated corpus. The investor who saw 15% first was ahead by Rs 24.7 lakh at year 15 and still finished behind. The limits: real returns do not arrive in two neat blocks, and nobody can choose the order. The point is what the order does, which is why a plan near its goal often moves money towards steadier assets.
Where candidates lose it
The trap is answering that order does not matter at all, because candidates remember that multiplication commutes and apply it to the SIP. That holds only when the same money sees every return. Say who is exposed to which year before you answer.
The second trap is picking the 15%-first investor because early growth compounds for longer. That is true per rupee, but there are very few rupees early on. The weight of the pot, not the length of compounding, decides it.
What the interviewer asks next
Which order wins for a retiree who is withdrawing rather than contributing?
If the SIP stopped at year 15 and the money stayed invested, would order matter for the rest?
How would a step-up SIP change the size of the gap between the two orders?

040Two retirees each start with Rs 60 lakh and withdraw Rs 6 lakh at the start of every year. Both see the same ten yearly returns, averaging 6.8%: one meets minus 20% and minus 10% in the first two years, the other in the last two. After ten years one has about Rs 2 lakh left and the other about Rs 39 lakh. Why?Withdrawals and after-tax arithmeticHardWealth and advisoryDistribution and sales

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What explains the gap of about Rs 37 lakh?

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Because withdrawals turn the order of returns into a permanent difference. Without withdrawals both would end at the same Rs 108.1 lakh, since returns multiply in any order. But the first retiree takes Rs 6 lakh out of a pot that has just fallen, to Rs 33.5 lakh after two years, so later good years work on a small base and she ends with about Rs 2.0 lakh. The second takes withdrawals from a pot that has grown, and ends with about Rs 39.0 lakh.

Why would order not matter without withdrawals?

Returns multiply, and multiplication does not care about order. Rs 60 lakh left untouched through these ten years ends at Rs 108.1 lakh whichever year comes first. The withdrawals are what break the symmetry: a fixed rupee amount taken out of the pot is a bigger share of a small pot than of a large one.

What exactly happens in the first two years?

Think of a shopkeeper who must pay rent of Rs 6,000 on the first of every month by selling stock. If prices have just crashed, he sells many more items to raise the same Rs 6,000, and those items are gone when prices recover. A retiree withdrawing a fixed sum after a fall sells more units at low prices, and those units never take part in the recovery. The first retiree's Rs 54 lakh, after the first withdrawal, meets minus 20% and becomes Rs 43.2 lakh; Rs 37.2 lakh then meets minus 10% and becomes Rs 33.5 lakh. The second retiree's Rs 54 lakh meets plus 20% first and is Rs 64.8 lakh after year one.

Both retirees withdraw Rs 6 lakh at the start of each year and see the same returns averaging 6.8%. The one who meets minus 20% and minus 10% first is down to Rs 33.5 lakh after two years and ends with Rs 2.0 lakh, while the one who meets them last ends with Rs 39.0 lakh.

Year	Return, bad first	Corpus, Rs lakh	Return, bad last	Corpus, Rs lakh
1	-20%	43.2	+20%	64.8
2	-10%	33.5	+13%	66.4
3	+18%	32.4	+5%	63.5
4	+15%	30.4	+7%	61.5
5	+11%	27.1	+9%	60.5
6	+9%	23.0	+11%	60.5
7	+7%	18.2	+15%	62.6
8	+5%	12.8	+18%	66.8
9	+13%	7.6	-10%	54.8
10	+20%	2.0	-20%	39.0

Year by year: Rs 6 lakh is withdrawn at the start of each year, then that year's return applies to what is left. The returns are the same set in reverse order.

The relationship

V_{t} = (V_{t-1} - W)(1 + r_t), \qquad V_0 = 60,\; W = 6

V_t	the corpus at the end of year t, Rs lakh
W	the withdrawal at the start of each year, Rs 6 lakh
r_t	the return in year t

What it says in wordsEach year the withdrawal comes out first and the return applies only to what is left, so a fall before a withdrawal makes the withdrawal bite harder.

What does an adviser do with this?

The years just before and just after withdrawals begin carry the most weight; the risk is called sequence risk, and it is managed, not predicted. Common responses are holding two or three years of withdrawals in a steadier, short-term bucket so that a bad year does not force selling equity at a low, or trimming withdrawals after a fall. Note the mirror image of an SIP: for a saver adding money, bad years early are the kinder order; for a retiree taking money out, bad years early are the cruel one. The returns here are illustrative, and a 10% withdrawal rate is high by most planning conventions, which is what makes the gap so stark.

Where candidates lose it

The trap is answering that the average return decides the outcome, or insisting the order cannot matter because returns multiply. Both are true only for money that sits untouched. The question tells you withdrawals happen; build from there.

The second loss is explaining it as bad luck in general. The interviewer wants the mechanism in one sentence: fixed withdrawals after a fall sell more units at low prices, and those units miss the recovery.

What the interviewer asks next

How large a cash bucket would have let the first retiree avoid selling equity in years one and two?
If withdrawals were 4% of the current corpus instead of a fixed Rs 6 lakh, would the order still matter?
How would you explain sequence risk to a client in two sentences?

049A Rs 10,000 monthly SIP runs for 20 years at 12% a year, taken as 1% a month. What share of the final corpus appears only in the last five years? Most people guess about a quarter.Compounding and time valueHardIndian AMCsDistribution and sales
Try it first
What share of the 20-year corpus is added in years 16 to 20?
Show the worked solution
About half: 49.5%. At 1% a month the corpus is about Rs 50.5 lakh after 15 years and Rs 99.9 lakh after 20, so the final five years add Rs 49.5 lakh. Only Rs 6 lakh of that is new instalments; about Rs 41.2 lakh is growth on the pot already built. An SIP's corpus is back-loaded, which is why stopping early costs far more than the missed instalments.
Why is the guess of a quarter so far off?
A quarter assumes the corpus grows by the same amount each year, like a piggy bank filled with the same coins every month. Compounding is not a piggy bank. Each year's growth is a percentage of a pot that is already larger than the year before, so the rupee gains get bigger every year and the corpus is heavily back-loaded. After 15 years of Rs 10,000 a month the pot holds about Rs 50.5 lakh, and five more years at 1% a month multiply that by 1.01 to the 60th, about 1.82, before a single new instalment is counted.
The SIP corpus reaches Rs 23.2 lakh at year 10 and Rs 50.5 lakh at year 15, then adds Rs 49.5 lakh in the last five years to end at Rs 99.9 lakh; of that addition, Rs 41.2 lakh is growth on the year-15 pot and only Rs 6 lakh is new instalments.
Where exactly does the last five years' money come from?
Split the Rs 49.5 lakh three ways. The 60 new instalments put in Rs 6 lakh. Those instalments earn about Rs 2.2 lakh of growth by year 20. The year-15 pot of Rs 50.5 lakh earns about Rs 41.2 lakh. The engine of the final years is the money already invested, not the money still to come. For contrast, the first ten years build only about Rs 23.2 lakh, 23% of the final corpus, though half the instalments are paid in them.
The relationship
$\text{share}_{16\text{-}20} = 1 - \frac{FV_{180}}{FV_{240}}, \qquad FV_n = m\,\frac{(1.01)^n - 1}{0.01}\,(1.01)$
m the monthly instalment, Rs 10,000
FV_n the corpus after n monthly instalments paid at the start of each month
1.01 one plus the 1% monthly return
What it says in wordsThe last five years' share is one minus the year-15 corpus divided by the year-20 corpus.
What does this mean for an investor thinking of stopping at year 15?
It depends what stopping means. Redeeming at year 15 gives up about Rs 49.5 lakh, half the eventual corpus, to save Rs 6 lakh of instalments. Pausing the instalments but staying invested is far cheaper: the year-15 pot alone grows to about Rs 91.7 lakh, so the cost is about Rs 8.2 lakh. The 12% is an assumption for the arithmetic; real returns vary year to year, and a bad last five years would cut the back-loaded gain sharply, which is the sequence risk of an SIP.
Where candidates lose it
The trap is answering a quarter because five years is a quarter of twenty. It treats the corpus as if it grew in a straight line. Name the compounding before you name a number.
The second loss is crediting the last five years' gain to the last five years' instalments. Only Rs 6 lakh of the roughly Rs 49 lakh is new money; most of it is growth on the pot built in the first fifteen years.
What the interviewer asks next
What share of the corpus appears in the last five years of a 30-year SIP at the same return?
At 8% a year instead of 12%, is the corpus more or less back-loaded?
Why does starting an SIP five years earlier matter more than adding five years at the end?

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

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

\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

Try it first

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

Try it first

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

Try it first

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

Try it first

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

Try it first

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

Try it first

Why does it take so long?

What does this mean for how funds are judged?

Where candidates lose it

What the interviewer asks next

Try it first

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

Try it first

Why does order not matter for a lump sum?

What breaks the symmetry in an SIP?

What does this mean for a real SIP investor?

Where candidates lose it

What the interviewer asks next

Try it first

Why would order not matter without withdrawals?

What exactly happens in the first two years?

What does an adviser do with this?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the guess of a quarter so far off?

Where exactly does the last five years' money come from?

What does this mean for an investor thinking of stopping at year 15?

Where candidates lose it

What the interviewer asks next