Private Wealth Management puzzles, solved step by step
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032Suppose every fund manager is pure luck: each fund has a 50% chance of beating its benchmark in any year, independently. Out of 1,000 funds, how many will beat the benchmark five years in a row?Mutual fund distributionIndian wealth management
Try it first
Quick: how many of the 1,000 funds show a five-year streak by luck alone?
Show the worked solution
About 31 funds. The chance of beating the benchmark five years running by luck is one half multiplied by itself five times, 1 in 32. Out of 1,000 funds that is 1,000 divided by 32, or about 31.25. So a large field of funds throws up dozens of perfect five-year records even if nobody has any skill, and a streak on its own is weak evidence.
Why does a large field guarantee some long streaks?
Put 1,000 people in a hall and ask each to toss a coin five times. Nobody would be surprised that about 31 of them get five heads, and nobody would ask them for coin-tossing lessons. A rare event for one participant becomes a near certainty somewhere in a big enough crowd. Funds are the same crowd, and the ones with perfect records are the ones that get advertised.
If each fund beats its benchmark with a coin-flip chance each year, 1,000 funds shrink to 500, 250, 125, 62.5 and finally about 31 with perfect five-year records. Those 31 streaks appear with no skill anywhere in the field. The relationshipN the number of funds, 1,000 p the chance of beating the benchmark in one year, 0.5 k the length of the streak, 5 years What it says in wordsThe expected number of lucky streaks is the size of the field times the chance that any one fund strings the wins together.What should an adviser take from this when screening funds?
That a track record needs a comparison against luck, not just a count of good years. Even ten straight years of beating the index would be expected from about 0.98 fund out of 1,000 by chance, so one such fund in a large category is not proof of skill. What helps more: a consistent process, returns that survive a fair risk adjustment, and results after costs over several market conditions.
The limitation of the puzzle is the 50% and the independence. In reality, after costs the average fund beats its benchmark less than half the time, and good and bad years may be correlated with a manager's style. Both change the count, but not the lesson.
Where candidates lose it
The first trap is saying zero or a handful, because five in a row feels rare. It is rare for one fund; it is common in a field of 1,000. The interviewer is testing whether you think about the size of the crowd.
The second trap is getting 31 and stopping. Add the adviser's conclusion in one line: streaks are what luck produces in a large field, so screen on process and risk-adjusted results, not on the streak.
What the interviewer asks next
- If the true chance of beating the benchmark after costs is 40%, how many five-year streaks do you expect?
- How many funds would you need for one ten-year streak by luck?
- What evidence would make you believe a streak is skill?
058A fund has not had a single losing year in seven years, and the salesperson presents that as proof it is safe. If any given year had a 30% chance of a loss, how likely was a clean seven-year record anyway?Mutual fund distributionIndian wealth management
Try it first
Your estimate of the chance of seven clean years?
Show the worked solution
About 8%, which is unusual but not rare. A clean year has probability 0.7, and seven independent clean years is 0.7 to the 7th, 0.082. Among 100 funds carrying the same 30% yearly chance of a loss, about eight would show a clean seven-year record by luck. The record makes a safer fund more likely, but it does not prove the fund is safe.
How do you get to 8% in your head?
Square, then keep multiplying. 0.7 x 0.7 is 0.49, call it a half. Two more years take it to about a quarter, 0.24, two more to about an eighth, 0.12, and the seventh year times 0.7 gives about 0.08. Each extra clean year multiplies the chance by 0.7, so the probability of an unbroken run falls away quickly but never reaches zero. It is the same arithmetic as a batsman who survives each over with probability 0.7 lasting seven overs.
If each year carries a 30% chance of a loss, the chance of no losing year falls from 70% after one year to 49% after two and to 8.2% after seven, so about 8 funds in 100 would show a clean seven-year record by luck alone. So is the record evidence of anything?
Yes, but weigh it properly. Compare with a fund that truly has only a 10% chance of a losing year: it keeps a clean seven-year record 0.9 to the 7th of the time, 47.8%. A clean record is about 5.8 times more likely from the safer fund than from the riskier one, which shifts the odds but does not settle them. With hundreds of funds on sale, some risky ones will carry spotless records, and those are exactly the ones placed in front of clients.
There is a second problem the arithmetic hides. Years are not independent across funds: in a strong market decade most funds have no losing years at the same time, so seven clean years may say more about the decade than about the fund. Ask how the fund behaved in the worst year the market had, not how many years it avoided a loss.
Where candidates lose it
The trap is hearing seven out of seven and treating it as near-certain proof. Candidates rarely compute the chance of the record under the risky assumption, so they cannot say how surprising it really is.
The opposite miss is dismissing the record as pure luck. It is evidence; the good answer quantifies how much, then names what it cannot show.
What the interviewer asks next
- How many clean years would you need before the chance under a 30% loss rate falls below 1%?
- If 200 funds each have a 30% yearly loss chance, how many clean seven-year records do you expect?
- Why might a clean record in a rising market be weaker evidence than one that spans a crash?