Mutual Fund Mastery puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 29
- Topics
- 13
- Hard
- 30
020Ten per cent of fund managers are skilled and beat their index in 70% of years; the rest have no skill and beat it in 50% of years. A manager has beaten the index three years running. What is the probability that this manager is skilled?Fund research and ratingsIndian AMCs
Try it first
After a three-year streak, how likely is the manager to be skilled?
Show the worked solution
About 23%. Picture 1,000 managers. The 100 skilled ones produce 100 times 0.7 cubed, 34.3 three-year streaks. The 900 unskilled produce 900 times 0.5 cubed, 112.5. Of 146.8 streak holders, 34.3 are skilled, 23.4%. The streak more than doubles the odds of skill, from 10%, yet most streak holders are still lucky.
Why does a three-year streak prove so little?
Think of a test for a rare condition that catches most real cases but also flags plenty of healthy people. If the condition is rare, most positive results are false alarms. When the thing you are looking for is rare, even good evidence leaves most positives coming from the larger group, so the starting proportion matters as much as the evidence. Here skill is the rare condition, at 10%, and a three-year streak is a test that unskilled managers pass one time in eight.
Of 1,000 managers, the 100 skilled produce 34.3 three-year streaks and the 900 unskilled produce 112.5, so only about 23% of managers with a streak are actually skilled. How do you set it up fast in the room?
Use counts, not formulas. Pick 1,000 managers, split them by skill, then split each group by whether it produced the streak, and the answer is one box over the sum of two boxes. Skilled: 100 times 0.343 is 34.3. Unskilled: 900 times 0.125 is 112.5. The answer is 34.3 over 146.8, 23.4%. The same result in odds form: prior odds of 1 to 9, times a likelihood ratiohow many times more likely the evidence is if the manager is skilled than if not of 0.343 over 0.125, about 2.74, gives 2.74 to 9.
The relationshipS the manager is skilled 0.1 and 0.9 the shares of skilled and unskilled managers 0.7^3 and 0.5^3 each group's chance of three wins in a row What it says in wordsThe chance of skill given a streak is the skilled managers' share of all the streaks.Push it once to show judgement. A ten-year streak changes the picture: 100 times 0.7 to the tenth against 900 times 0.5 to the tenth gives about 76% skilled, because the unskilled pass a ten-year test only about once in a thousand. The limitation is that the 10% and 70% are assumptions, and real years are not independent coin tosses; managers with a style that suits a market phase can string wins together without skill.
Where candidates lose it
The common slip is answering 70%, mixing up the chance a skilled manager wins with the chance a winner is skilled. It is the same inversion that makes people overrate a positive result on a test for a rare condition.
The other slip is ignoring the 10% starting share and answering from the hit rates alone, 0.343 over 0.343 plus 0.125, about 73%. That treats skilled and unskilled managers as equally common, which is the one thing the question told you they are not.
What the interviewer asks next
- How many consecutive winning years would push the probability of skill above 50%?
- How would the answer change if skilled managers made up 30% of the population?
- Why does survivorship in fund databases make this problem worse in practice?
058A fund's yearly returns average 12% with 18% volatility, and are roughly normal and independent from year to year. What is the chance of losing money in any one year, and what is the chance that its average return over ten years is negative?Fund research and ratingsIndian AMCs
Try it first
Which pair is closest?
Show the worked solution
About 25% in any one year, and about 2% over ten years. One year: zero sits 12/18 = 0.67 standard deviations below the mean, which leaves 25.2% of outcomes below it. The ten-year average keeps the 12% mean but its spread falls to 18 divided by the square root of 10, 5.7%, so zero is 2.11 standard deviations away and the chance is 1.8%.
How likely is a losing year?
Picture the daily commute. Any one day might be twenty minutes late because of rain or a breakdown; your average over a month is almost never more than a few minutes off. Single outcomes are noisy; averages of many independent outcomes are much less noisy, because the bad days and the good days partly cancel. For one year, the question is how far zero sits below a 12% mean when the spread is 18%. That distance is 12/18 = 0.67 standard deviations, and the normal table puts 25.2% of the curve below it. One year in four is a loss, which matches what investors in equity funds actually live through.
The one-year return curve, centred on 12% with an 18% spread, has 25% of its area below zero, while the ten-year average curve has the same centre but a 5.7% spread and only 1.8% of its area below zero. Why does the ten-year chance collapse to about two percent?
Averaging ten independent years keeps the centre at 12% but divides the spread by the square root of ten. The standard errorThe spread of an average. For independent draws it equals the spread of one draw divided by the square root of the number of draws. of the ten-year average is 18 / 3.16 = 5.7%, so zero is now 2.11 standard deviations below the mean instead of 0.67. The tail beyond 2.1 standard deviations is 1.8%. Five years sits in between: a spread of 8.0% and a chance of about 7%.
The relationshipr bar 10 the average yearly return over ten years 12 the mean yearly return, per cent 18 / sqrt(10) the spread of the ten-year average Phi the share of a normal curve below a given number of standard deviations What it says in wordsDivide the distance to zero by the spread of the average, not by the spread of one year, then read the tail.Now the limits, because a sharp interviewer will push. A negative arithmetic average is not quite the same as losing money: compounding knocks roughly half the variance off growth, so the fund compounds nearer 10.4% than 12%, and the chance of ending ten years below the starting amount is a little higher than 1.8%. Real returns also have fatter tails than a normal curve and are not fully independent, since bad years cluster. The direction survives all of that: time narrows the spread of the average, not the risk of a bad single year.
Where candidates lose it
The first trap is saying the risk of loss is the same at every horizon, or that it falls in proportion to time. It falls with the square root of time, which is why ten years cuts 25% to about 2%, not to zero and not to 2.5%.
The second is overselling the answer. Say what the two percent assumes: normal returns, independent years and a fixed mean, and that compounding and fat tails push the true figure somewhat higher.
What the interviewer asks next
- What volatility would make the one-year chance of loss exactly one in three?
- Why is the chance of ending below your starting value higher than the chance of a negative arithmetic average?
- Over how many years does the chance of a negative average fall below 1%?
084Two diversified funds each pick 50 stocks at random from the same universe of 100 stocks. How many stocks do you expect them to hold in common, and what does that suggest about a client who holds three large cap funds?Indian AMCsDistribution and sales
Try it first
How many stocks do you expect the two funds to share?
Show the worked solution
25 stocks, half of each fund, give or take about 2.5. Each of fund B's 50 picks has a 50 in 100 chance of being among fund A's, so the expected overlap is 50 x 0.5 = 25. Two funds fishing in the same pond share half their holdings by chance alone. With three such funds, about 12.5 stocks sit in all three, and together they hold only about 87.5 different names.
Why is the overlap so large with no coordination?
Two friends each order five dishes from a menu of ten without talking to each other. Each dish you pick has a 50% chance of being on your friend's list, so you expect to share about two and a half dishes. When each fund holds a large share of the same universe, overlap is not a coincidence to be explained; it is the default the arithmetic produces. The tool is linearity of expectation: go stock by stock, add up the chance that each one is in both funds, and the dependence between picks never needs to be modelled.
The relationshipN stocks in the universe, 100 k stocks each fund holds, 50 k/N the chance a given stock is in one fund What it says in wordsEach stock is in both funds with probability one half times one half, and there are 100 stocks, so expect 25 shared.In one random draw of two 50-stock funds from 100 stocks, 25 stocks land in both, matching the expected 50 x 50/100, and a third such fund would leave only about 87.5 distinct names across 150 holdings. How firm is 25, and what changes with three funds?
The count follows a hypergeometric distributionThe distribution of how many marked items you get when you draw a fixed number without replacement from a pool that holds a fixed number of marked items. with a standard deviation of about 2.5, so most random pairs share between 20 and 30 stocks. With three funds each holding half the universe, a stock is in all three with probability one in eight, about 12.5 stocks, and in none with probability one in eight too. Three such funds do not give three times the diversification: 150 holdings collapse to about 87.5 distinct stocks, and 50 of those are held by two or three of the funds at once.
Real funds are not random, and the difference cuts one way. A large cap mandate points every manager at the same biggest companies, and benchmark weights pull portfolios further together, so real overlap between large cap funds tends to sit above this random baseline rather than below it. Use the random case as the floor: if two funds overlap much more than chance would give, the second fund adds little except a second fee.
Where candidates lose it
The instinct is that two independent managers should have little in common, so candidates guess 5 or 10. They forget that each fund covers half the universe, which makes sharing the norm, not the exception.
The second loss is stopping at 25 without the portfolio point. The question is really about clients holding three or four funds from the same category who believe they are diversified; say what the overlap does to that belief, with the 87.5 distinct names as the number.
What the interviewer asks next
- If each fund picks only 20 of the 100 stocks, what overlap do you expect?
- How would you measure overlap by portfolio weight rather than by count?
- A client holds four large cap funds. What would you look at first?