Private Wealth Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 3
Topics: 13
Hard: 30

Showing 1–8 of 8 · filtered from 100Clear filters

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?Compounding and doublingCoreWealth 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 relationship
$W_{48} = \frac{16}{2^{(64-48)/8}} = \frac{16}{4} = 4 \qquad r = 2^{1/8} - 1 \approx 9.05\%$
W_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?
014One deposit pays 12% a year compounded monthly. Another pays 12.5% a year compounded annually. Which pays more, and by how much on Rs 1 lakh over a year?Compounding and doublingHardIndian wealth management
Try it first
Which ends the year ahead?
Show the worked solution
The 12% monthly deposit pays more: an effective 12.68% against 12.50%. One per cent a month, compounded twelve times, is 1.01 to the 12th, which is 1.1268. On Rs 1 lakh that is Rs 1,12,683 after a year against Rs 1,12,500, Rs 183 more. The gap is small, but the method is the point: convert every quoted rate to an effective annual rate before comparing.
Why is 12% monthly not 12%?
Imagine a savings box where interest is dropped in every month rather than at year end. From the second month on, the interest already in the box also earns. A quoted rate with a compounding frequency is a label, not the return; the return is the effective annual rate, which rises with every extra compounding. For 12% compounded monthly, that effective annual rateThe rate that, compounded once a year, gives the same result as the quoted rate with its compounding frequency. is 12.683%.
Rs 1 lakh at 1% a month climbs in twelve steps to Rs 1,12,683, while 12.5% compounded annually jumps once to Rs 1,12,500, so the monthly deposit ends Rs 183 ahead on an effective 12.68%.
How do you do 1.01 to the 12th in your head?
Use the binomial shortcut. 1.01 to the 12th is roughly 1 plus 12 times 0.01 plus 66 times 0.0001, the number of pairs of months times the interest on interest. That is 1 plus 0.12 plus 0.0066, about 1.1266, within a hair of the exact 1.1268. The 0.66 point of extra return is the interest on interest, and it is what closes most of the gap to 12.5%.
The relationship
$\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 12.68\% \;>\; 12.50\%$
0.12/12 the monthly rate, 1%
12 the number of compounding periods in a year
EAR the effective annual rate
What it says in wordsCompound the periodic rate for a year and subtract one to get the rate you can compare.
Give the two numbers that frame it. Compounded continuously, 12% would give 12.75%, the ceiling for a 12% quote. And a monthly deposit would need to quote only 11.84% to match 12.5% annual. The limitation for a client: tax, premature withdrawal terms and the credit of the issuer usually matter more than Rs 183.
Where candidates lose it
The trap is comparing the quoted numbers, 12% against 12.5%, and picking the annual deposit. The candidate has compared two labels written in different units.
The opposite loss is overselling the result. Rs 183 on Rs 1 lakh is a small gap; say so, and say that the method, converting to effective rates, is what the interviewer wanted.
What the interviewer asks next
What would 12% compounded quarterly give as an effective annual rate?
What monthly-compounded rate would exactly match 12.5% annual?
A loan quotes 1.5% a month. What is the effective annual rate?
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?Compounding and doublingCoreMutual 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 relationship
$FV = P \cdot \frac{(1+i)^n - 1}{i}\cdot(1+i) = 10{,}000 \times \frac{1.01^{240}-1}{0.01} \times 1.01$
P 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?

040A client can run a Rs 10,000 monthly SIP that rises 10% every year, or a flat Rs 15,000 monthly SIP. Both run 15 years at 12% a year, treated as 1% a month. Which ends bigger, and in which year does the step-up overtake on the monthly instalment?Compounding and doublingHardMutual fund distributionIndian wealth management

Try it first

The step-up ends bigger. Where does most of its advantage come from?

Show the worked solution

The step-up ends bigger, Rs 86.8 lakh against Rs 75.7 lakh, but it wins late and by saving more. Its instalment first beats Rs 15,000 in year 6, at Rs 16,105, and its corpus overtakes only in year 11. Over 15 years it puts in Rs 38.1 lakh against Rs 27.0 lakh; the growth on each is almost identical, about Rs 48.7 lakh.

Why does the step-up lag for so long?

Two students save for a trip: one puts in Rs 150 a week from the start, the other begins at Rs 100 and adds 10% every term. The steady saver is ahead for most of the year because her money arrived first. The step-up pays in less than the flat SIP for the first five years, and money put in early is the money that compounds longest, so its corpus trails until year 11. Its year-5 instalment is still only Rs 14,641.

The step-up instalment passes the flat Rs 15,000 in year 6 and reaches Rs 37,975 by year 15, but its corpus overtakes only in year 11. At year 15 it holds Rs 86.8 lakh against Rs 75.7 lakh, with almost identical growth of about Rs 48.7 lakh on each.

	Flat Rs 15,000	Step-up from Rs 10,000
Total put in	Rs 27.0 lakh	Rs 38.1 lakh
Corpus at year 5	Rs 12.4 lakh	Rs 9.8 lakh
Corpus at year 10	Rs 34.9 lakh	Rs 33.7 lakh
Corpus at year 15	Rs 75.7 lakh	Rs 86.8 lakh
Growth at year 15	Rs 48.7 lakh	Rs 48.7 lakh

Both SIPs invested at the start of each month at 1% a month; the step-up rises 10% at the start of each year.

So is the step-up the better plan?

It is a different plan, not a smarter one. The step-up ends ahead because it asks the client to save Rs 11.1 lakh more; per rupee saved, the flat SIP compounds better because its money arrives earlier. Where the step-up earns its place is fit: it matches a salary that rises each year, so the client can afford it without strain in year one. The honest comparison for a client is between what each plan asks of his budget in each year, not between two final numbers.

Limits: 1% every month is smooth, real returns are not, and the step-up puts its biggest instalments in the last years, so a bad market late in the plan hurts it more. The 10% step-up also assumes the client's income actually rises that fast.

Where candidates lose it

Most candidates say the step-up wins and credit compounding. The step-up does win, but compounding favours the flat SIP rupee for rupee; the extra corpus is almost exactly the extra saving. That is the insight the interviewer is fishing for.

The other loss is the crossover year. The instalment crosses in year {P40['cross_contrib']}, when 10,000 x 1.1 to the power 5 first tops 15,000, but the corpus takes until year {P40['cross_corpus']}. Say both, and say why they differ.

What the interviewer asks next

What flat SIP would match the step-up's corpus at year 15?
How does the answer change over 25 years instead of 15?
The client's salary rises 5% a year, not 10%. Which plan fits him better, and why?

053At 9% a year, roughly how long does Rs 10 lakh take to become Rs 80 lakh? Answer in your head, then check it.Compounding and doublingWarm upIndian wealth management
Try it first
Your first answer, inside ten seconds?
Show the worked solution
About 24 years. Rs 10 lakh to Rs 80 lakh is eight times the money, which is three doublings. At 9% the rule of 72 gives 72 / 9 = 8 years per doubling, so three doublings take about 24 years. The exact figure, the log of 8 over the log of 1.09, is 24.13 years, so the shortcut is off by under two months.
Why count doublings instead of years?
A staircase is easier to climb in your head than a ramp. Compound growth is a ramp, but every doubling takes the same number of years at a fixed rate, so you can turn it into equal steps. First ask how many times the money must double, then multiply by the years one doubling takes. Eight times is 2 x 2 x 2, three doublings, and at 9% each takes about 8 years.
At 9% a year Rs 10 lakh reaches Rs 20 lakh in about 8 years, Rs 40 lakh in about 16 and Rs 80 lakh in about 24, because every doubling takes the same time; the exact path reaches Rs 80 lakh in 24.1 years.
How good is the rule of 72 at 9%?
Very good. The exact doubling time at 9% is the log of 2 over the log of 1.09, which is 8.04 years, against 8.00 from the rule. The rule of 72 is most accurate for rates around 8%, which is why it serves so well for Indian savings and equity assumptions. Three doublings carry the small error three times, which is how 24 becomes 24.13.
The relationship
$t = \frac{\ln 8}{\ln 1.09} = \frac{2.079}{0.0862} \approx 24.13\ \text{years}$
8 the growth multiple needed, 80 lakh over 10 lakh
1.09 one year of growth at 9%
t years needed
What it says in wordsThe years needed are the log of the multiple divided by the log of one year's growth.
Then turn it into a client sentence. Rs 10 lakh at 9% needs a working lifetime, not a decade, to reach Rs 80 lakh, and the last doubling, from Rs 40 lakh to Rs 80 lakh, adds more rupees than the first two combined. That is the argument for starting early, said with numbers rather than slogans.
Where candidates lose it
Candidates divide 80 by 10, get 8, and then try to compound year by year in their head. They lose the room in arithmetic. The move is to see that 8 is 2 cubed before touching the rate.
The second slip is quoting 8 years, which is the time for one doubling, because the rule of 72 is the first thing that comes to mind. Say how many doublings first, then the time for each.
What the interviewer asks next
How long does the same money take to reach Rs 1 crore?
At 12% instead of 9%, how many years does the client save on the way to Rs 80 lakh?
Why is the rule of 72 less accurate at 20% 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.Compounding and doublingCoreIndian 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 relationship
$t_3 = \frac{\ln 3}{\ln 1.08} = \frac{1.0986}{0.0770} \approx 14.27 \qquad \frac{114}{8} = 14.25$
ln 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?
077A client tells you his Rs 5 lakh investment became Rs 20 lakh in 12 years. Without a calculator, what annual growth rate is that?Compounding and doublingWarm upWealth management
Try it first
Your instinct, in five seconds.
Show the worked solution
About 12% a year. Rs 5 lakh to Rs 20 lakh is four times, which is two doublings. Two doublings in 12 years means one every 6 years, and the rule of 72 gives 72 divided by 6, which is 12%. The exact rate is 4 to the power one twelfth, less 1, which is 12.25%.
Why turn the multiple into doublings first?
If someone tells you a town's population went from 5,000 to 20,000, you naturally say it doubled and doubled again. Doublings are easy to count and hard to get wrong. A growth multiple that is a power of two converts straight into a number of doublings, and the rule of 72 turns years per doubling into a rate. Four times is two doublings; eight times would be three.
Rs 5 lakh doubles to Rs 10 lakh in six years and doubles again to Rs 20 lakh by year 12, so the rule of 72 gives 72 / 6 = 12% a year against an exact 12.25%.
How good is the rule of 72 here, and when does it slip?
The relationship
$r = 4^{1/12} - 1 = 12.25\% \qquad \text{rule of 72: } 72/6 = 12\%$
4 the multiple, Rs 20 lakh over Rs 5 lakh
1/12 one twelfth, because the growth happened over 12 years
r the compound annual growth rate
What it says in wordsThe exact rate is the twelfth root of the multiple, less one; the rule of 72 gets within a quarter of a point.
The rule of 72 is most accurate for rates around 8%, and it drifts at the edges: at 12% it undershoots slightly, and at 20% or more it undershoots by more. For interview purposes, 12% with the words "a shade over" is the answer that shows you know it is an approximation. If the multiple is not a clean power of two, say 5 times in 12 years, estimate the doublings: 5 is a bit over two doublings, about 2.3, so a doubling every 5.2 years and roughly 14%.
Then turn it back to the client. A 12% compound rate over 12 years is a good result, but ask what it was in and what the fees and taxes were, because the client quoted a pre-tax figure from memory.
Where candidates lose it
The fast wrong answer is 25%: a 300% gain split evenly over 12 years. It treats the growth as a straight line and overstates the rate by more than double.
The second slip is reaching for a calculator or saying "about 10%" without a method. Say two doublings, six years each, 72 over 6: the method is what the interviewer is listening for.
What the interviewer asks next
The same Rs 5 lakh became Rs 40 lakh in 18 years. What rate is that?
At 12%, how long does it take Rs 20 lakh to reach Rs 1 crore?
The client says 12% beat the market. What do you ask him next?
090Two sisters invest at 10% a year. Sister A puts in Rs 1 lakh a year from age 25 to 34 and then stops. Sister B puts in Rs 1 lakh a year from 35 to 60. Payments are made at the start of each year. Who has more at 60, and by how much?Compounding and doublingHardIndian wealth management
Try it first
Who has more at 60?
Show the worked solution
Sister A, with about Rs 190 lakh against Rs 109 lakh, ahead by Rs 81 lakh. A pays in only Rs 10 lakh, but her pot is Rs 17.5 lakh at 35 and then compounds untouched for 25 years, about 10.8 times. B pays in Rs 26 lakh, yet her last payments have little time to grow. The early years carry the compounding.
How do you get the answer without a spreadsheet?
Work A in two stages. Ten payments of Rs 1 lakh at 10% build to about Rs 15.9 lakh by the last payment at 34, and to Rs 17.5 lakh a year later at 35. From 35 to 60 that pot compounds for 25 years. At 10% money doubles about every 7.2 years, so 25 years is about three and a half doublings, roughly 11 times, which takes Rs 17.5 lakh to about Rs 190 lakh.
The relationship
$A_{60} = 1.1^{26} \cdot \frac{1.1^{10} - 1}{0.1} \approx 189.9 \qquad B_{60} = \frac{1.1^{26} - 1}{0.1} \approx 109.2$
(1.1^10 - 1) / 0.1 ten yearly payments of Rs 1 lakh added up with their growth, about 15.9 lakh
1.1^26 growth from the last payment at 34 to age 60
B_60 26 payments from 35 to 60, the last earning nothing
What it says in wordsA's small early pot grows for 26 more years; B's larger total has much less time in the market.
Sister A stops paying at 34 with Rs 17.5 lakh at 35 and still ends at Rs 189.9 lakh at 60, well above Sister B's Rs 109.2 lakh from 26 payments, because A's money has decades longer to compound.
Why does B never catch up, even with 26 payments?
Picture a snowball rolled from the top of a long slope against a bigger pile of snow thrown on halfway down. The first has more distance to gather snow. At 10% a rupee invested at 25 is worth about 28 rupees at 60, while a rupee invested at 50 is worth about 2.6, so a decade of early payments outweighs a quarter century of later ones. For B to match A she would need to pay about Rs 1.74 lakh a year from 35.
Then add the honest limits. A steady 10% is an illustration, not a forecast; a real sequence of returns will differ, and fees and taxes reduce both pots. The ratio between the sisters, though, comes from time in the market, and that holds at any positive rate: the lower the rate, the smaller A's lead.
Where candidates lose it
Most candidates answer B because she paid in more than twice as much. They count rupees and forget the time each rupee has to grow, which is exactly the mistake a wealth client makes when he delays starting.
The second loss is freezing because the sum looks like it needs a spreadsheet. Split A into two stages and use doublings for the second: an estimate within a few lakh is what the interviewer wants.
What the interviewer asks next
At what rate of return would the two sisters end level?
How much would B need to invest each year to match A at 60?
What does this puzzle say about an employer's retirement contribution that starts at 25?

W_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

0.12/12	the monthly rate, 1%
12	the number of compounding periods in a year
EAR	the effective annual rate

ln 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

(1.1^10 - 1) / 0.1	ten yearly payments of Rs 1 lakh added up with their growth, about 15.9 lakh
1.1^26	growth from the last payment at 34 to age 60
B_60	26 payments from 35 to 60, the last earning nothing

Try it first

Why count backwards instead of forwards?

What does the share of the last 8 years tell a client?

Where candidates lose it

What the interviewer asks next

Try it first

Why is 12% monthly not 12%?

How do you do 1.01 to the 12th in your head?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get close to Rs 1 crore without a calculator?

Why does most of the growth arrive in the last few years?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the step-up lag for so long?

So is the step-up the better plan?

Where candidates lose it

What the interviewer asks next

Try it first

Why count doublings instead of years?

How good is the rule of 72 at 9%?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a tripling rule exist at all?

Where does the shortcut stop being good enough?

Where candidates lose it

What the interviewer asks next

Try it first

Why turn the multiple into doublings first?

How good is the rule of 72 here, and when does it slip?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get the answer without a spreadsheet?

Why does B never catch up, even with 26 payments?

Where candidates lose it

What the interviewer asks next