Private Wealth Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 3
- Topics
- 13
- Hard
- 30
021A fund house's factsheet shows its 10 open equity schemes averaging 14% a year since launch. It actually launched 16; the 6 it closed or merged averaged minus 2% a year. What was the average across all 16 schemes it launched?Mutual fund distributionIndian wealth management
Try it first
Pick the average across all 16 schemes.
Show the worked solution
8%, not 14%. Weight each group by how many schemes it holds: 10 schemes at 14% contribute 140, and 6 at minus 2% contribute minus 12. The total, 128, over 16 schemes is 8%. The factsheet's 14% describes only the schemes that survived, which is why the full launch record is the honest measure of the fund house.
Why does the factsheet overstate the record?
Think of a coaching centre that advertises the average score of the students who finished the course, after the weaker ones dropped out. The average is true of the finishers and misleading about the centre. A record that only shows what survived is flattered, because the failures were removed for being failures. That is survivorship biasThe distortion that comes from judging a group only by the members still around, when the ones that failed have dropped out of the data.: the closed schemes did not vanish at random, they closed because they did badly.
The factsheet's 10 open schemes average 14%, but the 6 closed schemes averaged minus 2%, so across all 16 schemes the fund house launched the record is 8%. Why not take the midpoint of 14% and minus 2%?
Because the groups are different sizes. An average of averages is only right when each group holds the same number of items; otherwise each average has to be weighted by its count. The midpoint, 6%, gives the 6 closed schemes as much weight as the 10 open ones and understates the record. Ten at 14 and six at minus 2 gives 8%.
The relationship10, 6 the number of open and closed schemes 14%, -2% each group's average yearly return 16 all schemes the fund house launched What it says in wordsThe average across all schemes is each group's average weighted by how many schemes it holds.Take it to the client conversation. A factsheet does not have to show closed or merged schemes, so the question to ask a fund house is how many schemes it has launched and what happened to the ones that are gone. The same bias sits inside category averages that drop merged funds; the numbers here are an illustration of the method.
Where candidates lose it
The trap is accepting 14% because it is printed on an official document. The candidate who does not ask what is missing has shown the exact blind spot an adviser is paid to cover.
The second loss is the midpoint, 6%. It notices the closed schemes but forgets to weight by count, so it ends up wrong in the other direction.
What the interviewer asks next
- If the closed schemes had been merged into the survivors, how would the reported record change?
- How would you check a fund house's record for survivorship before recommending it?
- Where else in wealth management does survivorship bias show up?
035A client needs Rs 80,000 in cash. He owns two lots of 1,000 shares each of the same stock, now at Rs 80. He bought one lot at Rs 100 and the other at Rs 60. Which lot should he sell to raise the money, and why?Wealth management
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What should decide which lot he sells?
Show the worked solution
Sell the lot bought at Rs 100; the only thing the purchase price changes is tax. Both lots are the same stock at Rs 80, and he keeps 1,000 identical shares either way, so the future is the same. Selling the Rs 60 lot books a Rs 20,000 gain and, at an illustrative 20% rate, Rs 4,000 of tax now. Selling the Rs 100 lot books a Rs 20,000 loss, pays nothing, and can shelter other gains.
Why does the price he paid not matter to the stock?
Two people own identical Rs 80 notes, one found on the road and one earned after a hard week. The notes buy exactly the same things. A share at Rs 80 has one future, and it is the same whether its owner paid Rs 60 or Rs 100. The urge to sell the winner to lock in a profit and to hold the loser until it gets back to cost is called the disposition effect, and it comes from anchoring on a number the market has forgotten.
Selling either lot raises Rs 80,000 and leaves the client holding 1,000 identical shares at Rs 80. Selling the Rs 60 lot triggers Rs 4,000 of illustrative tax now; selling the Rs 100 lot books a Rs 20,000 loss that pays no tax and can shelter other gains. How big is the tax difference, and is it permanent?
At an illustrative 20% on short-term gains, selling lot B costs Rs 4,000 today. Selling lot A costs nothing and books a Rs 20,000 loss; if he has other gains this year, that loss can reduce their tax by up to Rs 4,000. The swing between the two choices is up to Rs 8,000 for the same stock and the same cash. Part of it is timing rather than a permanent saving: the gain inside lot B is still there, and will be taxed if he sells it later.
Frame it for the client without judging the instinct, because nearly everyone has it. The current tax rates, the holding periods that decide short or long term, and the rules on which losses can be set off against which gains all change over time, so confirm the current ones before acting on the numbers.
Where candidates lose it
Candidates say sell the winner and bank the profit, which is the exact bias the interviewer is probing. Others say it makes no difference at all, which misses the one thing that genuinely differs.
Give both halves: the purchase price is irrelevant to the stock's future, and relevant only through tax. Then quantify the tax, and say that part of the saving is a deferral, not a gift.
What the interviewer asks next
- Would your answer change if the lot bought at Rs 60 had been held long enough to qualify as long term?
- The client says he cannot bear to realise a loss. How do you handle that conversation?
- What if he has no other capital gains this year to use the loss against?
061A wealthy client refuses a fair coin flip that wins Rs 1.5 lakh or loses Rs 1 lakh, even though the expected value is plus Rs 25,000. If he weighs each rupee lost more heavily than each rupee gained, what weight on losses makes him exactly indifferent?Wealth management
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What loss weight makes the flip feel worth exactly nothing?
Show the worked solution
A loss weight of 1.5. He is indifferent when half the gain equals half the weighted loss: 0.5 x 1.5 = 0.5 x weight x 1.0, so the weight is 1.5. Refusing the flip tells you losses hurt him at least one and a half times as much as equal gains please him. For a client with crores invested, a Rs 1 lakh swing cannot threaten his wealth, so the refusal is loss aversion, not prudence.
How do you turn a refusal into a number?
Think of a child who is offered a toffee if a coin lands heads and must give back one if it lands tails, and still says no. A refusal of a positive bet reveals a weight on losses larger than the ratio of gain to loss, here 1.5 to 1. Write the felt value as half the gain minus half the weight times the loss and set it to zero. The halves cancel, and the weight is simply Rs 1.5 lakh over Rs 1 lakh. The idea has a name, loss aversionThe tendency to feel a loss more strongly than a gain of the same size. Described by Daniel Kahneman and Amos Tversky in prospect theory, 1979., from Kahneman and Tversky's prospect theory.
The flip wins Rs 1.5 lakh or loses Rs 1 lakh, worth plus Rs 25,000 on average, but a client who weighs losses 1.5 times as heavily feels the Rs 1 lakh loss as Rs 1.5 lakh, exactly cancelling the gain, so he is indifferent. The relationship1.5 the gain if heads, Rs lakh 1.0 the loss if tails, Rs lakh lambda the weight the client puts on each rupee lost What it says in wordsThe loss weight that makes a fair coin feel worthless is the gain divided by the loss.Why is this a trap for the client and not just a preference?
Because he judges each bet alone. Paul Samuelson described a colleague who refused one such bet but said he would take a hundred of them. Over 100 independent flips the expected gain is Rs 25 lakh, and the chance of ending behind is about 1.8%, because he loses only if fewer than 40 of the 100 flips land heads. Loss aversion applied one decision at a time rejects a set of choices that, taken together, almost never loses. A client who checks his portfolio daily and feels every red day is making the same mistake with his own money.
In the room, the good answer gives 1.5, then the reframing: show the client the portfolio of decisions, not the single flip. That is not persuading him to gamble; it is making sure he rejects bets for reasons he would still accept after seeing the whole picture.
Where candidates lose it
The arithmetic trap is setting the weight on the gain rather than the loss, or adding the stakes and answering 2.5. Write the indifference equation before touching the numbers.
The judgement trap is calling the refusal rational risk aversion. For a client with crores, Rs 1 lakh is too small to matter to his wealth; the refusal is about how the loss feels, and the interviewer wants to hear that distinction.
What the interviewer asks next
- The client also refuses win Rs 2.5 lakh, lose Rs 1 lakh. What does that tell you?
- Why might checking a portfolio monthly instead of daily reduce the pain a loss-averse client feels?
- How would you present a volatile but sound investment to a client with a high loss weight?
073A distributor shows a client that some of this year's top-quartile funds were also top quartile last year. If fund returns were pure luck, what share of this year's top-quartile funds would you expect to be top quartile again next year?Mutual fund distributionIndian wealth management
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Under pure luck, what share of top-quartile funds repeat?
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25%. If returns were pure luck, next year's quartile would be independent of this year's, so a top-quartile fund would have a one-in-four chance of landing in each quartile, top included. A quarter of the top funds repeat by chance alone, 6.25% of all funds. Persistence only counts as evidence of skill when the repeat rate is clearly above 25%, and with few funds that bar is higher than it looks.
Why is the luck baseline 25% and not zero?
Roll a die twice. The chance the second roll is a six does not care whether the first one was, so one in six of the first-roll sixes will be followed by another six. Luck does not avoid repeats, it ignores history, so under pure chance each quartile of this year's funds scatters evenly across next year's four quartiles. A quarter of the top quartile stays top, which is 6.25% of all funds.
If fund returns were pure luck, every cell of the four by four grid holds 6.25% of funds, so a quarter of this year's top-quartile funds land top again next year and the rest scatter evenly across the other three quartiles. How far above 25% is enough to mean something?
It depends on how many funds you are counting. With 100 funds there are 25 in the top quartile, and luck alone predicts about 6.25 repeats with a spread of about 2.2. Ten or more repeats, a 40% rate, still happens by luck about 7.1% of the time, so a single year's persistence among a small set of funds is weak evidence. Several years of repeats, or a very large sample, is what separates skill from a good draw.
Two more checks before believing a persistence table. Funds that closed or merged after bad years quietly vanish from later counts, which makes survivors look more persistent than they were. And funds in the same category share market conditions, so a style that is in favour can keep a whole group of funds on top together. Neither of those is manager skill.
Where candidates lose it
The common wrong answer is zero or near zero: candidates assume luck would never repeat, so any repeat looks like skill. That is exactly the error a persistence chart invites the client to make.
The second trap is 6.25%, the share of all funds in the top-top cell, quoted as if it were the share of top funds that repeat. Say which base you are using.
What the interviewer asks next
- What share of this year's top-quartile funds would be top in each of the next three years by luck?
- How does survivorship bias change a published persistence table?
- What evidence would make you believe a fund's top-quartile record reflects skill?
098A client keeps Rs 20 lakh in a 7% fixed deposit as his emergency fund, and at the same time carries a Rs 15 lakh personal loan at 14%. What does keeping the two apart cost him each year?Indian wealth management
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What does the habit cost him each year, before tax?
Show the worked solution
About Rs 1.05 lakh a year before tax, and more after it. On the Rs 15 lakh that overlaps, the deposit earns 7%, Rs 1.05 lakh, while the loan charges 14%, Rs 2.1 lakh. Repaying the loan from the deposit saves the 7 point gap. Deposit interest is taxed and personal-loan interest usually is not deductible, so at an illustrative 30% slab the cost is about Rs 1.37 lakh.
Why does it feel sensible to keep both?
Many families keep a jar labelled "emergencies" and never touch it, even while paying interest on a credit card. The label makes the money feel spoken for. Mental accountingTreating money differently depending on the label or pocket it sits in, even though a rupee is worth the same everywhere. makes a client treat the deposit and the loan as separate stories, but money is fungible: a rupee in the deposit and a rupee owed on the loan cancel. Every rupee kept in the deposit while the loan is open is effectively borrowed at 14% to earn 7%.
The same Rs 15 lakh earns Rs 1.05 lakh a year in the deposit pocket and costs Rs 2.10 lakh a year in the loan pocket, so keeping the pockets apart costs the client Rs 1.05 lakh a year before tax. The relationship15 the overlapping amount, Rs lakh 14% - 7% the gap between the loan rate and the deposit rate 0.30 an illustrative tax slab on deposit interest What it says in wordsThe cost is the overlap times the rate gap, and larger once tax on the deposit interest is counted.But does he not need the emergency fund?
He needs access to money in an emergency, which is a different thing from holding Rs 20 lakh in a deposit. Repaying the loan from the deposit and keeping Rs 5 lakh as a buffer leaves him with a smaller emergency pot but no 14% debt, and the saving of about Rs 1 lakh a year rebuilds the buffer. If a real emergency outruns Rs 5 lakh, he could borrow again, which is what he is already doing today.
Check the practical frictions before suggesting it: a prepayment charge on the loan, a penalty for breaking the deposit early, and whether the client's income is steady enough that a smaller buffer is safe. Those can shrink the saving, but they rarely close a 7 point gap. Say it with respect: the habit is common, and it comes from caution, not carelessness.
Where candidates lose it
The common answer is that nothing is lost because one is savings and the other is debt. That is the mental account talking, and the interviewer is testing whether you see through it.
The second loss is recommending he empty the deposit without a word about the buffer, prepayment charges or tax. Give the Rs 1.05 lakh, then the practical checks.
What the interviewer asks next
- The loan is a home loan at 8.5% with a tax deduction on interest. Does the answer change?
- How big should the emergency buffer be, and how would you decide?
- Name another everyday habit that comes from mental accounting.