Private Wealth Management puzzles, solved step by step
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- 30
009A client regrets missing two stocks on his 20-stock watchlist that doubled last year. Of the other 18, five halved and thirteen ended flat. What would an equal stake in all 20 have returned?Wealth management
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Before you add it up: what did the whole watchlist return?
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An equal stake in all 20 would have lost 2.5%. Put Rs 1 lakh in each, Rs 20 lakh in total. The two doubles end at Rs 4 lakh, the five halved end at Rs 2.5 lakh and the thirteen flat end at Rs 13 lakh, which is Rs 19.5 lakh. The winners he remembers were two picks out of a list that lost money as a whole.
Why does the regret feel bigger than the numbers?
Think of someone who says they almost bought a winning lottery ticket because the number was one digit off their birthday. Every number was one choice among many; only the winner gets remembered. Hindsight picks the winners after the result is known, which is a choice the client never actually had at the time. This is hindsight biasThe tendency to believe, after an outcome is known, that it was predictable and that you would have acted on it., and it makes the missed gains feel certain.
The client remembers two stocks that doubled, but an equal Rs 1 lakh in each of the 20 on his watchlist would have turned Rs 20 lakh into Rs 19.5 lakh, a loss of 2.5%. How do you use the number with the client?
Turn the regret into the decision he actually faced. At the start of the year he had twenty names and no way of knowing which two would double. The honest benchmark for a missed opportunity is the whole set of choices available at the time, not the best one in hindsight. On that benchmark his inaction cost nothing: holding cash lost nothing, while the list lost 2.5%.
The relationship2, 5, 13 the number of stocks that doubled, halved and stayed flat 20 the whole watchlist, equally weighted What it says in wordsThe equal-weight return is the simple average of the 20 returns.Note the arithmetic quirk it hides. A double and a halving look like mirror images, but the double adds Rs 1 lakh while the halving takes away only Rs 50,000. Five halvings still outweigh two doubles here, and the adviser who can show that in rupees wins the conversation.
Where candidates lose it
The trap is anchoring on the two doubles and guessing a positive return. Candidates who do this repeat the client's own bias back to him, which is the opposite of the job.
The second loss is getting minus 2.5% and not saying what it means for the conversation. The interviewer wants the reframing: judge a decision by the choices available at the time.
What the interviewer asks next
- If he had bought any two stocks from the list at random, what is the chance he would have picked both winners?
- How would you respond if he wants to buy only last year's winners now?
- What other bias sits next to hindsight in a client's review of his own record?
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
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Pick the average across all 16 schemes.
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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?
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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?
047The index has fallen five days in a row and a client says it is now due to rise. Suppose each day is independent and a down day has a 48% chance. How many losing runs of five or more days should you expect in a 250-day year, and what does the streak say about tomorrow?Wealth management
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After five down days in a row, what is the chance tomorrow is an up day, under these assumptions?
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About 3.3 losing runs of five or more days a year, and the streak says nothing about tomorrow. A run of five falls has probability 0.48 to the power 5, about 2.55%, and it can start on almost any of the 250 days, provided the day before was up. That gives roughly 3.3 runs a year, and a 97% chance of at least one. With independent days, tomorrow's chance of rising is still 52%.
How often should a five-day losing run turn up?
Toss a slightly unfair coin 250 times and look for five tails in a row. Any one stretch of five is unlikely, 2.55%, but there are about 245 stretches to look at. A run of five down days needs an up day, or the start of the year, followed by five falls, so the expected count is about 0.0255 x (1 + 245 x 0.52), roughly 3.3 runs a year. Across a year, the chance of seeing at least one is about 97%.
The relationshipp the chance of a down day, 48% n trading days in the year, 250 (1-p) the up day that must come just before a run for it to be a new run What it says in wordsCount every day a new five-day losing run could start, and multiply by the chance it does.In one simulated year of 250 independent days with a 48% chance of a fall, three losing runs of five or more days appear, of 7, 5, 5 days. Independence alone predicts about 3.3 such runs a year, and none of them tells you anything about the next day. Why does the client feel the market is due?
Because people expect short sequences to look like long-run averages, so a run of losses feels like a debt the market must repay. If the days are independent, the market keeps no ledger: the chance of a rise after five falls is the same 52% as after five rises. The adviser's job is to take the streak out of the decision and bring the conversation back to the client's plan and time horizon.
The limit is the independence assumption. Real markets show some short-term momentum and some mean reversion at different horizons, and volatility clusters, so streaks are a little more common than a coin predicts. None of that makes five falls a reliable signal to buy.
Where candidates lose it
One trap is agreeing with the client that the market is due, which is the gambler's fallacy with a market label. The other is calling a five-day run rare because 0.48 to the fifth is small, forgetting how many days it has to appear on.
Give the expected count, about three a year, and the 52% for tomorrow. Then say how you would steer the client back to his plan, because that is what the desk actually does with the maths.
What the interviewer asks next
- How many runs of ten or more down days would you expect in a year?
- What would you look for in the data before believing streaks carry information?
- The client wants to add money after every three-day fall. How do you respond?
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?
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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?
085A client bought a stock at Rs 500 and it now trades at Rs 300. He says he will sell only once it gets back to Rs 500. What return does he need just to get there, and what is wrong with the plan?Wealth management
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What rise takes Rs 300 back to Rs 500?
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He needs 66.7%, and the plan anchors on a price the market does not care about. Getting from Rs 300 to Rs 500 is a rise of 200 on 300. At 12% a year that is about 4.5 years. The purchase price is history; the question is whether he would buy the stock at Rs 300 today.
Why is 66.7% bigger than the 40% he lost?
A shirt marked down from Rs 500 to Rs 300 is 40% off. Marking it back up to Rs 500 is a 66.7% rise, because the markup is taken on the lower price. A percentage recovery is measured from the lower base, so the gain needed to recover always exceeds the loss suffered.
The relationshipg the rise needed to get back to the purchase price n years to get there at 12% a year What it says in wordsThe rise needed is the purchase price over today's price, less one; at 12% a year it takes about four and a half years.The Rs 500 purchase price sits above today's Rs 300 like an anchor, and reaching it needs a 66.7% rise, about 4.5 years at 12% a year, although the market sets no target from what the client paid. What is wrong with waiting to get back to Rs 500?
AnchoringLeaning on a reference number, here the purchase price, when judging a value that should not depend on it. on the purchase price makes a past number decide a future choice. The stock's next return is the same whether he bought at Rs 500 or Rs 200, so the only live question is whether Rs 300 of this stock is the best use of Rs 300 today. Holding a loser to avoid admitting a loss, while selling winners early, is a well documented pattern called the disposition effect.
Say it kindly. Nobody enjoys booking a loss, and the job is not to tell the client he was wrong. Ask him whether he would buy it today at 300; if the answer is no, he is holding it only because of a number from the past. A booked loss may also offset other gains for tax, which is a real reason to act rather than wait.
Where candidates lose it
The fast wrong answer is 40%, the size of the fall. It measures the recovery on the old base, and the interviewer hears that you would not spot the same error in a client's thinking.
The bigger loss is doing only the arithmetic. The question asks what is wrong with the plan: name anchoring, give the one question that breaks it, and say it without making the client feel foolish.
What the interviewer asks next
- The stock falls further to Rs 250. What rise is needed now?
- Why do investors tend to sell winners too early and hold losers too long?
- How would you raise this with a client who is emotionally attached to the stock?
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?
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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.