Private Wealth Management puzzles, solved step by step
- Puzzles
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- Traced to a firm
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- Topics
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- Hard
- 30
001A client's portfolio doubles every 8 years and is worth Rs 16 crore when she turns 64. At what age was it worth Rs 4 crore, and what share of the final Rs 16 crore arrived in the last 8 years alone?Wealth management
Try it first
Before you count back: what share of the final Rs 16 crore was added between 56 and 64?
Show the worked solution
She had Rs 4 crore at 48, and half the final wealth arrived in the last 8 years. Doubling forward means halving backward: Rs 16 crore at 64, Rs 8 crore at 56, Rs 4 crore at 48. Everything between Rs 8 crore and Rs 16 crore, a full Rs 8 crore, was added after 56. Doubling every 8 years is a return of about 9% a year.
Why count backwards instead of forwards?
Think of a mango tree whose fruit count doubles each season. If you know the count in the last season, the season before is simply half of it; you never need to know how many mangoes it started with. Under steady doubling, each step back in time halves the amount, so the answer needs only the final value and the doubling period. Two steps of 8 years back from 64 is 48, and two halvings of Rs 16 crore is Rs 4 crore.
Doubling every 8 years takes the portfolio from Rs 1 crore at 32 to Rs 16 crore at 64, so it was Rs 4 crore at 48 and Rs 8 crore at 56, and the last 8 years alone added Rs 8 crore, half the final wealth. What does the share of the last 8 years tell a client?
It tells her where the money in compounding really comes from. In any doubling process the last doubling adds as many rupees as all the earlier ones put together. The rupees added at each step were 1, 2, 4 and 8 crore after the first crore: the last step is bigger than the three before it combined, which add to 7. That is why an interruption late in the plan, a withdrawal at 56 or a panic sale in a bad year, costs far more than the same interruption at 35.
The relationshipW_48 portfolio value at age 48, Rs crore (64-48)/8 the number of doublings between 48 and 64, here 2 r the yearly return that doubles money in 8 years What it says in wordsDivide the final value by two once for every doubling period you step back; the rule of 72 gives 72 over 8, about 9% a year.Check the rate with the rule of 72: 72 divided by 8 is 9, and the exact figure is 9.05% a year. Say both, then say the limitation: real portfolios do not double on a timetable, so the sequence of good and bad years matters as much as the average. The clean doubling is a teaching model, not a forecast for any client.
Where candidates lose it
The common slip is to divide the time evenly and say the portfolio was Rs 4 crore at a quarter of the way along, or that each of the four doublings added a quarter of the wealth. Both treat compounding as a straight line and give an answer that sounds careful but is wrong.
The second loss is answering 48 and stopping. The interviewer wants the client point: half the money came in the last 8 years, which is why staying invested late in the plan matters most.
What the interviewer asks next
- At what age was the portfolio Rs 1 crore, and what return does that imply?
- If she withdraws Rs 2 crore at 56, what is the portfolio worth at 64?
- What if the portfolio doubles every 9 years instead? What is it worth at 64 starting from Rs 4 crore at 48?
027A client runs a Rs 10,000 monthly SIP for 20 years and earns 1% a month. Roughly how big is the corpus at the end, and what share of it is his own money against growth?Mutual fund distributionIndian wealth management
Try it first
Quick instinct: roughly what share of the final corpus is growth rather than his own contributions?
Show the worked solution
About Rs 1 crore, of which only Rs 24 lakh is his own money. 240 instalments of Rs 10,000 make Rs 24 lakh. At 1% a month, invested at the start of each month, the corpus is Rs 99,91,479, about Rs 99.9 lakh. Growth is Rs 75.9 lakh, roughly 76% of the final pile. In a long SIP the compounding, not the saving, builds most of the corpus.
How do you get close to Rs 1 crore without a calculator?
Start with the growth factor. At 1% a month money doubles in about 70 months, the rule of 70. Twenty years is 240 months, about 3.4 doublings, so each rupee put in on day one grows roughly ten to eleven times; the exact figure is 10.89. The corpus of a level SIP is the instalment times (growth factor minus 1) divided by the monthly rate. That is 10,000 x (10.89 minus 1) / 0.01, close to Rs 99 lakh, and a little more if each instalment goes in at the start of the month.
The relationshipP the monthly instalment, Rs 10,000 i the monthly return, 1% n the number of instalments, 240 (1+i) the extra month of growth because each instalment is invested at the start of the month What it says in wordsEvery instalment grows for the months it has left, and the corpus is the sum of all those grown instalments.Rs 24 lakh of contributions becomes a corpus of about Rs 99.9 lakh, so growth supplies Rs 75.9 lakh. The growth share of the pile climbs from 27% at year 5 to 48% at year 10 and 76% at year 20, because the compounding arrives late. Why does most of the growth arrive in the last few years?
Think of a mango tree planted every month. The saplings from the last year bear nothing yet; the trees from year one are fully grown. At year 10 the pile is about Rs 23.2 lakh; the next ten years add Rs 76.7 lakh, 3.3 times as much, although the client saves exactly the same amount in each decade. That is why stopping a SIP in year 12 to fund a car costs far more than the instalments skipped.
The limitation: 1% every month is a smooth assumption. Real equity returns arrive unevenly, and the order in which good and bad years land changes the final number. Say the corpus is an illustration of the mechanism, not a projection for the client.
Where candidates lose it
Candidates either multiply 24 lakh by a rough growth factor, treating every rupee as if it compounded for 20 years, or they give up and say the answer needs a spreadsheet. The first overstates the corpus several times over; the second fails the numeracy the question is testing.
Say the contributions first, then the growth factor from the rule of 70, then the annuity formula. Ending with the growth share is what turns the arithmetic into an adviser's point.
What the interviewer asks next
- How much bigger is the corpus if the SIP runs 25 years instead of 20?
- What monthly SIP gives Rs 1 crore in 15 years at the same rate?
- Why is 1% a month slightly more than 12% a year?
065How long does money take to triple at 8% a year? Use the rule of 114, then check it against the exact answer.Indian wealth management
Try it first
Where does the number 114 come from?
Show the worked solution
About 14.25 years by the rule, 14.27 years exactly. The rule of 114 says divide 114 by the rate: 114 / 8 = 14.25. The exact answer is log 3 over log 1.08, which is 14.27, about a week longer. The rule works because tripling takes log 3 over log 2, about 1.585 times, as long as doubling, and 72 x 1.585 is about 114.
Why does a tripling rule exist at all?
If a train takes 9 hours to cover some distance, it takes about 14 hours to cover 1.585 times that distance at the same speed. Compound growth works the same way on a log scale. The time to reach any multiple is the log of that multiple divided by the log of one year's growth, so every multiple's rule is the doubling rule scaled by log of the multiple over log 2. For tripling that scale is 1.585, and 72 x 1.585 is 114.1, rounded to 114.
The relationshipln 3 the log of the growth multiple, tripling ln 1.08 the log of one year's growth at 8% 114 72 times ln 3 / ln 2, rounded What it says in wordsYears to triple are the log of three over the log of one year's growth; the rule of 114 approximates that division.At 8% the rule of 72 gives 9.00 years to double against an exact 9.01, and the rule of 114 gives 14.25 years to triple against an exact 14.27; the tripling rule is within about a week at 8% and drifts to -7 weeks at 10% and +25 weeks at 4%. Where does the shortcut stop being good enough?
Both rules are tuned to rates near 8%. Between 6% and 10% the rule of 114 is within about two months of the exact answer, which is fine for a client conversation; at 4% it is off by about half a year. At low rates it says too long, at high rates too short, as the bottom of the figure shows. When the rate is far from 8%, say the rule, then say which way it errs.
The client version: at 8%, money roughly triples in 14 years and roughly doubles in 9, so a 35 year old's savings can triple before retirement at 49 and nearly multiply ninefold by 63. Two rules, one sentence, no calculator.
Where candidates lose it
The trap is not knowing the rule and trying to compound 8% year by year in your head until the money triples. It takes too long and usually drifts by a year or more.
The second miss is knowing 114 as a memorised number without knowing why. The interviewer who asks for quadrupling next expects you to say 144 at once, because it is just two doublings.
What the interviewer asks next
- What is the rule for quadrupling, and why is it obvious?
- Using the rules, how long does money take to grow sixfold at 8%?
- Why is the rule of 69.3 exact for continuous compounding?