Mutual Fund Mastery puzzles, solved step by step

Puzzles: 100
Traced to a firm: 29
Topics: 13
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070Two people each invest Rs 10,000 a month at 12% a year until they are 60, one starting at 25 and one at 35. The early starter puts in only Rs 12 lakh more, yet ends with about 3.4 times the corpus. Why?Compounding and time valueWarm upIndian AMCsDistribution and sales

Try it first

Of the early starter's corpus at 60, roughly how much comes from the first ten years of saving alone?

Show the worked solution

Because the extra ten years come first, and the earliest rupees compound the longest. At 1% a month, the early starter reaches about Rs 6.50 crore and the late starter about Rs 1.90 crore, 3.4 times less, though the gap in money paid in is only Rs 12 lakh. That Rs 12 lakh, saved from 25 to 35, grows to about Rs 4.60 crore by 60, 71% of the early starter's corpus.

Why is the early starter so far ahead for so little extra?

Plant a mango sapling at 25 and another at 35, and at 60 the first tree is not slightly bigger; it has had ten more seasons of growth on a trunk that kept getting larger. Money saved early does not just add ten more years of deposits; it gives every rupee in those years twenty-five more years of compounding after them. The late starter pays in Rs 30 lakh over 25 years and ends with Rs 1.90 crore. The early starter pays Rs 42 lakh over 35 years and ends with Rs 6.50 crore.

Rs 10,000 a month at 12% grows to Rs 6.50 crore by 60 when started at 25 and to Rs 1.90 crore when started at 35, because the Rs 12 lakh saved in the first ten years alone grows to about Rs 4.60 crore.

How much does the first decade contribute on its own?

Split the early starter's plan in two. From 25 to 35 she saves Rs 12 lakh, which is about Rs 23.2 lakh by 35. From then she saves exactly what the late starter saves, so that part ends at Rs 1.90 crore. The Rs 23.2 lakh from the first decade compounds for 25 more years at 1% a month and becomes about Rs 4.60 crore, 71% of her final corpus. Her first ten years are worth more at 60 than the late starter's entire 25 years of saving.

The relationship

FV = 10{,}000 \times \frac{(1.01)^n - 1}{0.01} \times 1.01 \qquad n = 420: \text{Rs }6.50\text{ cr} \qquad n = 300: \text{Rs }1.90\text{ cr}

10,000	the monthly investment, Rs
0.01	the monthly rate, 12% a year taken as 1% a month
n	the number of monthly instalments, 420 from 25 and 300 from 35
x 1.01	each instalment is invested at the start of the month

What it says in wordsThe future value of a monthly plan grows with the number of months as a power, not in a straight line, so extra months at the start count most.

State the assumptions, because the size of the gap depends on them. A steady 12% every year is a simplification; real equity returns swing, and the order of good and bad years matters for a monthly saver. Inflation shrinks both corpora in today's rupees, though not the ratio. And the gap narrows if the late starter saves more each month; to match the early starter at 60 she would have to invest about 3.4 times as much, around Rs 34,000 a month.

Where candidates lose it

The trap is reasoning in straight lines: ten more years out of thirty-five, so about 40% more money. Compounding makes the earliest deposits the most valuable, not the least, and candidates who scale by years miss the point of the question.

The second miss is quoting the corpus without the assumptions. Say it rests on a constant 12%, monthly compounding and deposits at the start of each month, and that real returns arrive unevenly.

What the interviewer asks next

How much a month would the late starter need to invest to match Rs 6.5 crore at 60?
The early starter stops saving at 35 and never adds another rupee. What does she have at 60?
How does a 10% return instead of 12% change the ratio between the two?

082Without a calculator: at 8% a year, how long does Rs 1 lakh take to double, and how close does the rule of 72 get to the exact answer? Then do the same at 6% and at 12%.Compounding and time valueWarm upIndian AMCsDistribution and sales
Try it first
How far is the rule of 72 from the exact doubling time at 8%?
Show the worked solution
About 9 years at 8%; the exact figure is 9.006 years, so the rule of 72 is out by about 2 days. At 6% the rule gives 12 years against an exact 11.90, and at 12% it gives 6 years against an exact 6.12. In the range where fund returns usually sit, the rule is off by weeks, not years.
Why does 72 divided by the rate give the doubling time?
If the price of a Rs 100 thali rises 8% a year, it costs about Rs 200 in nine years. You did not need a calculator, only the fact that 72 over 8 is 9. The exact doubling time is ln 2 divided by ln(1 + r), and for rates of a few per cent ln(1 + r) is close to r, so the answer is close to 69.3 divided by the rate in per cent. Using 72 instead of 69.3 does two jobs: it nudges the answer up to correct for rates around 8%, and it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is what makes it a mental tool.
The relationship
$t = \frac{\ln 2}{\ln(1+r)} \approx \frac{72}{100\,r} \qquad \frac{0.6931}{\ln 1.08} = 9.006$
t years for money to double
r the annual rate, as a decimal
ln the natural logarithm
What it says in wordsThe exact doubling time is log 2 over log of one plus the rate, and 72 over the rate in per cent is a close, easy stand-in.
The rule of 72 gives 12, 9 and 6 years at 6%, 8% and 12%, against exact doubling times of 11.90, 9.006 and 6.12 years, so in the range fund returns occupy it is wrong by weeks at most.
Where does the rule start to slip?
At both ends, but slowly. At 6% it says 12 years and the truth is 11.90, so it is long by about 38 days. At 12% it says 6 years against 6.12, short by about 42 days. The rule is almost exact near 8% and drifts by a little over a month either side, which is far inside the uncertainty of any return assumption you would feed it. Only at high rates does the drift become worth mentioning: at 24% the rule says 3 years against an exact 3.22.
Two uses in a fund interview. Turn a return into something a client can feel: at 8%, money doubles roughly every nine years, so it quadruples in about eighteen. And run a claim backwards to test it: a fund that says it tripled money in ten years has compounded at about 11.6% a year, because tripling is about 1.6 doublings, one every 6.3 years, and 72 over 6.3 is a little over 11.
Where candidates lose it
The common slip is dividing 100 by the rate, which gives 12.5 years at 8%. That is what simple interest would give, with no interest earned on interest, and it overstates the wait by about three and a half years.
The quieter loss is presenting the rule as exact. Say 9 years, then add that the exact figure is 9.006, so the interviewer hears that you know what the shortcut is and where it stops working.
What the interviewer asks next
How long does money take to triple at 8%?
Inflation runs at 6%. How long before Rs 1 lakh buys half what it buys today?
What number would you use in place of 72 for continuously compounded rates, and why?

Try it first

Why is the early starter so far ahead for so little extra?

How much does the first decade contribute on its own?

Where candidates lose it

What the interviewer asks next

Try it first

Why does 72 divided by the rate give the doubling time?

Where does the rule start to slip?

Where candidates lose it

What the interviewer asks next