Mutual Fund Mastery puzzles, solved step by step

Puzzles
100
Traced to a firm
29
Topics
13
Hard
30
Topic
All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–3 of 3 · filtered from 100Clear filters
  1. 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

    Try it first

    Starting three years late, how much larger is the monthly SIP?

    Show the worked solution

    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
    SIP=FV(1+i)n−1i (1+i)=50,00,0001.00917144−10.00917×1.00917≈Rs 16,691\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
    FVthe goal, Rs 50 lakh
    ithe monthly rate, 11% divided by 12
    nthe 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.

    Waiting three years costs 62% more a month, not 25%16,691Start now20,864Guess+25%22,255Same totalover 9 years27,048Wait 3 yearsMonthly SIP needed, RsWhere the Rs 50 lakh comes from, Rs lakhpaid 24.0growth 26.0Start nowpaid 29.2growth 20.8Wait 3 years11% a year assumed, paid at the start of each month
    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?
  2. 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

    Try it first

    With the monthly SIP, who ends the 30 years richer?

    Show the worked solution

    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.

    Same returns, swapped order: a lump sum does not care, an SIP doesRs 10 lakh lump sum, 30 years2.96 cr15% first2.96 cr9% firstidentical: 1.15^15 x 1.09^15= 29.64 in either order1 cr2 cr3 cr4 cryr 0yr 10yr 15yr 20yr 30returns swap62 lakh37 lakh9% first ends at Rs 3.62 crore15% first ends at Rs 2.61 croreRs 10,000 a month SIP corpus
    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
    corpus=∑t=1360m∏s=t360(1+is)\text{corpus} = \sum_{t=1}^{360} m \prod_{s=t}^{360} (1 + i_s)
    mthe monthly instalment, Rs 10,000
    i_sthe return in month s, 15% or 9% a year as a monthly rate
    tthe 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?
  3. 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.

    Half the corpus appears in the last quarter of the time255075100yr 0yr 5yr 10yr 15yr 20Corpus, Rs lakh23.250.599.9last 5 yearsWhat the last five years addInstalments: 6.0New-money growth: 2.2Growth on theyear-15 pot: 41.2Total Rs 49.5 lakh= 49.5% of the corpus
    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
    share16-20=1−FV180FV240,FVn=m (1.01)n−10.01 (1.01)\text{share}_{16\text{-}20} = 1 - \frac{FV_{180}}{FV_{240}}, \qquad FV_n = m\,\frac{(1.01)^n - 1}{0.01}\,(1.01)
    mthe monthly instalment, Rs 10,000
    FV_nthe corpus after n monthly instalments paid at the start of each month
    1.01one 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?