Mutual Fund Mastery puzzles, solved step by step
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009A thousand fund managers have no skill at all: each has a 50% chance of beating the index in any year, independently. How many beat it five years in a row by luck, and what is the chance that at least one beats it ten years running?Fund research and ratingsIndian AMCs
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Guess the chance that at least one of the thousand posts a ten-year streak.
Show the worked solution
About 31 managers beat the index five years running by luck alone, and there is about a 62% chance that at least one beats it ten years running. Each year halves the survivors: 1,000, 500, 250, 125, 62.5, 31.25. For ten years, one manager's chance is 1 in 1,024, so the chance none of the thousand does it is (1023/1024) to the 1,000th, about 38%.
Why do perfect records appear even when nobody is skilled?
Ask a stadium of a thousand people to toss a coin and sit down on tails. After five rounds about 31 are still standing, and each of them has a perfect record. A streak that is rare for one person is expected somewhere in a large enough crowd, so the size of the starting group matters as much as the length of the streak. The five-year count is just 1,000 halved five times.
Starting from 1,000 managers with no skill, halving each year leaves about 31 with a perfect five-year record, and the chance that at least one of the 1,000 posts a ten-year streak is about 62%. How do you get the ten-year figure without a calculator?
Go through the complement: work out the chance that nobody does it. Each manager fails with probability 1023 over 1024. For many small independent chances, (1 minus 1/n) to the power n is close to 1 over e, about 0.37, and here the power is 1,000 against 1,024, so the chance nobody does it is about 0.38. That leaves about 62% for at least one ten-year streak.
The relationship1/1024 one manager's chance of ten wins in a row, one half to the tenth 1000 the number of managers trying What it says in wordsTake the chance that every manager fails, and one minus that is the chance at least one succeeds.Say what it means for fund selection. A long record of beating the index is evidence, but weaker evidence than it looks when it is picked from a large universe after the fact. The limitation of the model is that real returns are not coin tosses and some managers do have skill; the puzzle only shows how much luck alone can produce.
Where candidates lose it
The trap in the second part is answering 1 in 1,024 or 0.1%, the chance for one named manager, when the question asks about anyone in the group. It is the same slip as being amazed that someone at a party shares your birthday.
The other slip is adding the chances, 1,000 times 1 in 1,024, to get 98%. That double counts the cases where two or more managers succeed; the complement avoids it.
What the interviewer asks next
- How many managers would you expect with exactly four wins out of five years?
- If one manager in the group truly beats the index 60% of years, how likely is a ten-year streak for them?
- Why does survivorship, funds closing after bad years, make published track records look better still?
032A corporate bond pays 11% for the year, with a 4% chance of default and 40% recovery if it defaults. A AAA bond pays 7.5% for certain. Which has the higher expected return, and why might a debt fund still choose the AAA bond?Fixed income desksIndian AMCs
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What is the risky bond's expected return for the year?
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The risky bond, 8.16% against 7.5%. Ninety six times in a hundred it pays 11%; four times it returns 40 of 100, a 60% loss. Weighted, 0.96 x 11 plus 0.04 x (minus 60) is 8.16%. A debt fund may still prefer the AAA because its investors treat it like a deposit: a single default is a sudden loss, triggers redemptions, and in a fund of twenty such bonds, one default a year is more likely than not.
How do you set up the expected return?
Think of lending Rs 100 to twenty-five friends at a good rate, knowing that one of them, on average, will repay only Rs 40. You have to count that friend before you celebrate the rate. Expected return weights every outcome by its chance, and the default outcome is a large loss, not a zero. Recovery of 40% means you lose 60 of your 100, and for simplicity the coupon is also lost in default.
The risky bond pays 11% with 96% probability and loses 60% with 4% probability, an expected 8.16%, above the AAA bond's certain 7.5%. The whole case against it sits in the loss branch, which a debt fund's investors are not expecting to see. The relationshipp probability of default in the year, 4% c the coupon, 11% R recovery, 40% of the money lent What it says in wordsExpected return is the coupon when paid, weighted by its chance, plus the default loss, weighted by its chance.If the risky bond pays more on average, why hold the AAA?
Because an average is not what a debt fund investor experiences. A debt fund is bought as a place for money that must not fall, so the question is not only the average but the chance and size of a loss. On one bond, the return has a standard deviation of about 13.9%, against zero for the AAA. Spread across twenty such bonds, the chance that at least one defaults in a year is 1 minus 0.96 to the twentieth, about 56%. Each default cuts the NAV overnight and can set off redemptions that force the fund to sell its better bonds.
There is also a margin-of-safety check: the risky bond only matches the AAA if the default chance rises to (11 minus 7.5) over 71, about 4.9%. A small error in the 4% estimate wipes out the advantage. Default probabilities are estimates, not facts, and they tend to rise together in a downturn, which is exactly when investors redeem.
Where candidates lose it
Candidates often forget that recovery is 40, not zero, and compute 0.96 x 11 = 10.56% or subtract only the coupon. Others treat default as a zero return and get 10.56% as well. The loss in default is the principal not recovered, 60%.
The bigger miss is stopping at 8.16% and declaring the risky bond better. The interviewer asked why a fund might still choose the AAA; the answer is about who holds the fund and what a loss does to them, not about the average.
What the interviewer asks next
- What default probability makes the two bonds equal on expected return?
- How does holding 50 such bonds instead of one change the picture?
- Why would the default probability of these bonds be correlated, and why does that matter?
072A toll-road InvIT unit pays Rs 12 in a good year (40% chance), Rs 8 in a normal year (45%) and Rs 2 in a bad year (15%). What is the expected payout, and what is the unit worth at a 10% required return if this pattern continues forever?NuveenChicago · 2023
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What is the unit worth at a 10% required return?
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The expected payout is Rs 8.70 and the unit is worth about Rs 87. Weight each year by its chance: 0.40 x 12 = 4.80, 0.45 x 8 = 3.60, 0.15 x 2 = 0.30, which sum to Rs 8.70. A payment expected every year forever is worth that payment divided by the required return, so 8.70 / 0.10 = Rs 87. Valuing the likeliest year alone would give Rs 80.
How do you value income that changes every year?
A farmer whose crop is good four years in ten, ordinary in about half, and poor in the rest does not plan around a normal year. He plans around what the land produces on average over many years. When income is uncertain, you value the expected valueThe probability-weighted average of all possible outcomes: each outcome times its chance, added up. of the income, not the likeliest outcome and not the plain average of the outcomes. For this unit: 0.40 x Rs 12 = 4.80, 0.45 x Rs 8 = 3.60, 0.15 x Rs 2 = 0.30, a total of Rs 8.70 a year.
Weighting Rs 12, Rs 8 and Rs 2 by their chances gives an expected payout of Rs 8.70, which at a 10% required return forever values the unit at Rs 87, while valuing only the likeliest year would give Rs 80. The relationshipE[D] the expected yearly distribution r the required return, 10% V the value of the unit, assuming the pattern repeats forever with no growth What it says in wordsAverage the payouts by their chances, then treat the average as a level payment forever.Where do the two common wrong answers come from?
The likeliest year pays Rs 8, so some candidates value the unit at Rs 80. That throws away the 40% chance of Rs 12, which is worth Rs 7 of value here. Others average 12, 8 and 2 to get Rs 7.33, as if each year were equally likely, which gives Rs 73 and punishes the unit for a bad year that happens only 15% of the time. The weighting is the whole answer.
Now the honest limits, because the follow-up is usually how you would assess such an asset in practice. The expected value hides the spread: the payout's standard deviation here is about Rs 3.36, large against Rs 8.70, and an investor needing steady income cares about that. Uncertainty usually shows up in a higher required return, not in a lower expected payout, so the 10% must be chosen with the spread in mind. A toll road's traffic also trends and its concession ends, so the forever assumption is a simplification to state openly.
Where candidates lose it
The trap is valuing the most likely year, Rs 8, which gives Rs 80. It feels prudent but ignores that good years happen 40% of the time.
The second miss is double counting risk: weighting the payouts down for the bad year and then also using a high required return for the same risk. Say that probabilities go in the cash flow and the price of uncertainty goes in the rate, once each.
What the interviewer asks next
- The bad-year chance rises to 30%, taken from the good years. What is the unit worth now?
- Why might two investors pay different prices for this unit with the same expected payout?
- The concession ends after 20 years with nothing left. Roughly what is the unit worth then?
Asked at Nuveen, Multifamily, Chicago, 2023 (Wall Street Oasis):
Walk me through how you would assess the value of a property if the income stream is unpredictable?
098You roll a fair die and are paid the face value in thousands of rupees. After seeing the first roll you may reroll once, but then you must take the second roll. When should you reroll, and what is the game worth?Indian AMCsGlobal asset managers
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Which first rolls should you reroll?
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Reroll on 1, 2 or 3; keep 4, 5 or 6. The game is worth 4.25 thousand, Rs 4,250. A reroll is a new die worth 3.5 on average, so keep any face above 3.5. Half the time you reroll and expect 3.5; the other half you keep 4, 5 or 6, worth 5 on average. So the game is 0.5 x 3.5 + 0.5 x 5, or 4.25. The right to reroll adds Rs 750 to a plain roll's Rs 3,500.
How do you decide whether to keep a roll?
A friend offers you a sealed envelope known to hold Rs 350 on average, in exchange for the Rs 300 note in your hand. You swap; if the note were Rs 500, you would keep it. Keep any outcome that beats what the alternative is expected to give, and the alternative here is a fresh roll worth 3.5 on average. So 4, 5 and 6 are kept and 1, 2 and 3 are thrown back. The decision depends only on comparing the face in front of you with 3.5.
Each first roll below the reroll's expected 3.5 is thrown back and each roll above it is kept, which makes the game worth 4.25, or Rs 4,250, against Rs 3,500 for a single roll with no option. The relationship3/6 the chance the first roll is 1, 2 or 3 and you reroll 3.5 the expected value of the reroll (4+5+6)/6 the expected value from the faces you keep What it says in wordsWeight each branch by its chance: rerolled faces are worth 3.5, kept faces are worth themselves.Why is the option worth Rs 750, and what if there are more rerolls?
Without the reroll you get 3.5 on average. The option lets you throw away the low outcomes and replace them with an average one, so it lifts the value to 4.25: an option is worth something because you choose after seeing the outcome, and its value comes entirely from the bad outcomes it lets you escape. With two rerolls, work backwards: the last reroll is worth 3.5, so the middle roll is kept on 4 or more and is worth 4.25; the first roll is then kept only on 5 or 6, because 4 is below 4.25. The game rises to 4.67.
The fund-desk parallel is any decision with a later choice built in: a redemption right, a switch option, a stop-loss. Each one is worth what it lets you avoid, and each has a threshold set by what the alternative is expected to give. Saying the general rule, keep what beats the continuation value, is what lifts the answer above a piece of arithmetic.
Where candidates lose it
The common slip is setting the threshold by feel, keeping 3 because it is close to average or rerolling 4 because it feels low. The cutoff is exactly the reroll's expected value, 3.5, and a whole number on a die is either above it or below it.
The second loss is computing the value as the simple average of the best choices without weighting by probability. Say the threshold, then weight each branch: half the time 3.5, half the time 5.
What the interviewer asks next
- With two rerolls allowed, what is the game worth? (4.67)
- What would you pay to play if each reroll cost Rs 500?
- If the payout were the face value squared, would the threshold change?