Private Wealth Management puzzles, solved step by step

Puzzles: 100
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Topics: 13
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002A fund advertisement says the scheme returned 150% over the last 10 years. What annual rate of return is that?Returns arithmeticWarm upMutual fund distributionIndian wealth management
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Answer before you calculate.
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About 9.6% a year. A 150% return means Rs 1 became Rs 2.5. The yearly rate is 2.5 to the power one tenth, minus one, which is 9.60%. Dividing 150 by 10 to get 15% ignores compounding: 15% a year for ten years would turn Rs 1 into 4.05, not 2.5.
Why is 15% the wrong annual figure?
Think of a child's height chart. If a child grew 50 centimetres over ten years, you could say 5 centimetres a year, because height adds. Money does not add; it multiplies, and each year's gain earns its own gain the next year. An absolute return over many years must be converted to a compounded yearly rate before it can be compared with anything else. Dividing by the number of years overstates the yearly rate, and the overstatement grows with the length of the period.
Rs 1 lakh growing to Rs 2.5 lakh is the advert's 150%, and built year by year it is about 9.6% a year, each year's bar 1.096 times the last; reading it as 15% a year would imply 4.05 times, not 2.5.
How do you get 9.6% without a calculator?
Use doubling as the anchor. At about 9.6% money doubles in roughly 7.5 years, so in ten years it goes a little past double, which matches 2.5 times. You can also bracket it: 1.10 to the tenth is 2.59, a touch above 2.5, so the answer is a touch below 10%. Saying a bracket out loud, just under 10% because 10% gives 2.59, is as convincing to an interviewer as the exact 9.60%.
The relationship
$r = \left(1 + R\right)^{1/n} - 1 = 2.5^{1/10} - 1 \approx 9.6\%$
R the absolute return over the whole period, here 150% or 1.5
n the number of years, here 10
r the compounded annual rate, often called CAGR
What it says in wordsTurn the total return into a growth multiple, take the n-th root, and subtract one.
Why does a wealth interviewer care? Because a client comparing a 150% ten-year fund with a fixed deposit quoting a yearly rate is comparing two different units. The 9.6% is also before any comparison with a benchmarkThe index or reference portfolio a fund is measured against, so its return can be judged relative to what the market gave., which is the next question worth asking.
Where candidates lose it
Saying 15% is the whole trap, and it comes from treating the advert's number as if it were simple interest. It is the most common error clients themselves make, which is exactly why a wealth desk tests it.
The second loss is getting 9.6% but not being able to say why 15% is wrong. Have the one-line check ready: 15% compounded for ten years is about 4 times, not 2.5.
What the interviewer asks next
The same fund returned 40% in the last three years. What is the annual rate over those three?
What did it return a year over the first seven years?
Why do regulators ask funds to show annualised returns for periods over a year?
028A husband's Rs 1 crore portfolio made 20% this year. His wife's Rs 4 crore portfolio lost 5%. They ask you what the family earned. What do you tell them?Returns arithmeticWarm upWealth management
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Answer inside ten seconds: what did the household earn?
Show the worked solution
The household earned 0%. The husband's 20% on Rs 1 crore is a gain of Rs 20 lakh. The wife's minus 5% on Rs 4 crore is a loss of Rs 20 lakh. Together they started with Rs 5 crore and ended with Rs 5 crore. The simple average of 7.5% is wrong because it gives the small account the same weight as the large one.
Why does the simple average mislead?
Two children score 100 on a 10-mark test and 40 on a 100-mark test. Their average percentage is 70, but they got 50 marks out of 110, which is 45%. A household's return is the change in its total rupees divided by the rupees it started with, so each account counts in proportion to its size. Averaging percentages silently treats a Rs 1 crore account and a Rs 4 crore account as equals.
Drawn to size, the husband's Rs 20 lakh gain and the wife's Rs 20 lakh loss are blocks of exactly the same length, so they cancel. The simple average of the two returns is 7.5%, but the rupee-weighted return on the Rs 5 crore family pool is 0%.
The relationship
$R_{\text{household}} = \frac{1}{5}(20\%) + \frac{4}{5}(-5\%) = 4\% - 4\% = 0\%$
1/5 and 4/5 each account's share of the Rs 5 crore family pool at the start of the year
20% and -5% each account's own return
What it says in wordsThe family's return is each account's return weighted by its share of the family's money.
Why does this matter in a wealth review?
Consolidated reporting is one of the first things a family office client asks for, and it is where this error shows up. If the adviser's report leads with the husband's 20%, the family feels rich while its total wealth has not moved at all. The same trap appears when a relationship manager quotes the average return across a client's funds instead of the return on the client's money.
One limit: this weighting assumes no money moved in or out during the year. If the wife added Rs 1 crore in March, you would need a time-weighted or money-weighted calculation, which is a different question.
Where candidates lose it
Saying 7.5% is the whole trap, and it is said fast because both numbers are in front of you. The interviewer wants to hear you ask how big each account is before you combine anything.
Give the rupee answer, then name the rule in one line: weight returns by money, not by account. That line is what shows you would build a consolidated report correctly.
What the interviewer asks next
What if the wife's account had been Rs 2 crore instead?
The husband added Rs 50 lakh halfway through the year. How does that change the calculation?
How would you present this result to a couple who each think their own account did better?
054A client invested Rs 1 lakh six months ago and it is now worth Rs 1.1 lakh. He says that is a 20% annual return. What is the correct annualised figure, and why is annualising six months risky?Returns arithmeticWarm upMutual fund distributionIndian wealth management
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What is 10% in six months, annualised?
Show the worked solution
Annualised, 10% in six months is 21%, not 20%, and neither is a return he has earned. Repeating 10% on the larger Rs 1.1 lakh base gives Rs 1.21 lakh in a year. But annualising assumes the next six months repeat the last six. If they give back 10% instead, the year ends at Rs 99,000, a 1% loss. Short windows are mostly noise.
Why is it 21% and not 20%?
A cricketer who scores 50 in the first ten overs is not guaranteed 250 in fifty, and even the projection has to use the right arithmetic. Annualising a half-year return means compounding it, because the second half would grow the bigger balance. Rs 1 lakh becomes Rs 1.1 lakh, and another 10% on Rs 1.1 lakh is Rs 11,000, not Rs 10,000, which is where the extra point comes from.
The relationship
$r_{\text{annual}} = (1 + r_{\text{half}})^2 - 1 = 1.10^2 - 1 = 21\%$
r_half the return over six months, 10%
2 the number of six-month periods in a year
What it says in wordsCompound the period return as many times as the period fits into a year, then subtract one.
Ten per cent in six months annualises to 21% because the second half compounds on Rs 1.1 lakh, but the year can still end anywhere from plus 21% to minus 1% depending on the next six months, so an annualised short return is a projection, not a result.
Why is annualising a short window risky?
Annualising quietly assumes the next six months will look like the last six. Over short windows, most of a fund's return is noise, and scaling noise up to a year makes it look like a trend. Look at the bottom of the figure: the same first half ends the year at plus 21%, plus 10% or minus 1%, depending on a second half nobody has seen. This is why fund documents commonly show returns for periods under a year as absolute figures rather than annualised ones; confirm the current SEBI presentation rules before you quote them.
In the room, give both halves of the answer: the corrected 21%, then the caution. A client who hears 21% a year will plan around it. A client who hears 10% so far, which would be 21% if it repeated, which it may not, has the number and its limits.
Where candidates lose it
The trap has two layers. The first is agreeing with 20%, doubling instead of compounding. It is a small error, but it tells the interviewer you add returns that should be multiplied.
The larger miss is stopping at 21%. The question asks why annualising is risky, and a candidate who does not say it projects noise has answered only the arithmetic half.
What the interviewer asks next
A fund made 3% in one month. What is that annualised, and would you ever quote it?
The client's Rs 1.1 lakh drops to Rs 1 lakh over the next six months. What was his return for the year?
Why do fund documents usually show returns under a year as absolute numbers?

R	the absolute return over the whole period, here 150% or 1.5
n	the number of years, here 10
r	the compounded annual rate, often called CAGR

1/5 and 4/5	each account's share of the Rs 5 crore family pool at the start of the year
20% and -5%	each account's own return

r_half	the return over six months, 10%
2	the number of six-month periods in a year

Try it first

Why is 15% the wrong annual figure?

How do you get 9.6% without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the simple average mislead?

Why does this matter in a wealth review?

Where candidates lose it

What the interviewer asks next

Try it first

Why is it 21% and not 20%?

Why is annualising a short window risky?

Where candidates lose it

What the interviewer asks next