Portfolio Management puzzles, solved step by step

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  1. 017A coin-flip bet wins Rs 1.5 lakh or loses Rs 1 lakh. Many people refuse to play it once. Why, and what is the chance of ending with an overall loss if you play it ten times?Behavioural and decision trapsHardWealth managementAsset management

    Try it first

    Over ten independent plays, how often does the total end in a loss?

    Show the worked solution

    People refuse because a loss hurts more than an equal gain pleases, but ten plays lose money only about 17% of the time. Each play is worth plus Rs 25,000 on average. Over ten plays, four wins and six losses break even, so only three or fewer wins lose: 176 of 1,024 sequences, 17.2%. Judging each bet alone makes a good repeated bet feel bad.

    Why does a bet with a positive expected value get refused?

    Ask someone whether they would bet their weekend on a coin flip to win a second weekend, and most say no, because losing the one they have feels worse than gaining a new one feels good. This is loss aversionThe tendency, documented by Kahneman and Tversky in prospect theory, to feel a loss more strongly than a gain of the same size.. If a loss feels twice as bad as an equal gain feels good, this bet's felt value is 0.5 x 1.5 minus 0.5 x 2 x 1, which is minus 0.25 lakh, so refusing it once is consistent with how the person feels, even though its expected value is plus Rs 25,000. The factor of two is an assumption for the arithmetic, not a measured constant.

    How do you get the chance of a loss over ten plays?

    Count wins. With k wins and 10 minus k losses the total is 1.5k minus (10 minus k), which is 2.5k minus 10 lakh. That is negative only when k is three or less. Because each win outweighs each loss, you can lose six flips out of ten and still break even, which is why pooling the bets makes a loss so much rarer than a single flip suggests. The number of ways to get 0, 1, 2 or 3 wins is 1 plus 10 plus 45 plus 120, which is 176, out of 1,024 equally likely sequences: 17.2%.

    Ten plays of a +1.5 / -1 lakh coin flip: how often the total ends in a loss0.1%-10.01.0%-7.54.4%-5.011.7%-2.520.5%024.6%+2.520.5%+5.011.7%+7.54.4%+10.01.0%+12.50.1%+15.0Total after ten plays, Rs lakhLoss, 3 wins or fewer: 17.2%Break even, 4 wins: 20.5%Ahead, 5 or more wins: 62.3%Average: +2.5 lakh10 plays x Rs 25,000
    Over ten plays the total ends in a loss only with three or fewer wins, a chance of 17.2%, breaks even at four wins with 20.5%, and ends ahead 62.3% of the time, with an average gain of Rs 2.5 lakh.
    The relationship
    P(loss)=∑k=03(10k)(12)10=1+10+45+1201024≈17.2%P(\text{loss})=\sum_{k=0}^{3}\binom{10}{k}\left(\tfrac12\right)^{10}=\frac{1+10+45+120}{1024}\approx 17.2\%
    kthe number of winning flips out of ten
    \binom{10}{k}the number of sequences with exactly k wins
    What it says in wordsAdd up the sequences with three or fewer wins and divide by all 1,024 possible sequences.

    What does this mean for how clients see a portfolio?

    A client who checks a portfolio every day sees each day as a separate bet and feels every loss. Looking at the same holdings less often pools the bets, and the pooled result loses far less often than any single period does. The economist Paul Samuelson told the story of a colleague who refused one such bet but would take a hundred; the lesson for a wealth desk is to frame decisions at the horizon the money actually has. State the limitation: pooling helps only when the bets are independent and the investor can survive the bad runs, and ten plays still lose 17% of the time.

    Where candidates lose it

    Candidates call refusing the single bet irrational and move on. The interviewer wants the mechanism, loss aversion, and then the arithmetic that shows why the same person might accept the bet repeated.

    On the numbers, the slip is setting the loss threshold at fewer than five wins, as if wins and losses were the same size. Write the total as 2.5k minus 10 before you count anything.

    What the interviewer asks next

    • How many plays until the chance of an overall loss falls below 5%?
    • What if the loss were Rs 1.4 lakh instead of Rs 1 lakh?
    • Why might a client rationally refuse even the ten-play version?
  2. 049Fund rankings persist from one year to the next with a correlation of 0.2. A fund was in the top decile of its category last year. Where do you expect it to rank this year?Behavioural and decision trapsHardFund selectionPerformance analysis

    Try it first

    Where should you expect last year's top-decile fund to land this year?

    Show the worked solution

    Around the 64th percentile, still above average but most of the way back to the middle. On a normal scale, the top decile averages about 1.75 standard deviations above the median. With a correlation of 0.2, the expected position this year is a fifth of that, 0.35, which is about the 64th percentile. The quick linear version, 50 plus 0.2 x 45, gives the 59th.

    Why should a top-decile fund be expected to fall back?

    Think of a student who topped one exam. Part of that was knowing the subject and part was that the questions happened to suit them. Next time, the knowledge carries over and the luck does not. An extreme result is usually a mix of skill and a large dose of luck, and only the skill is expected to repeat, so the best forecast moves back towards the average by however much luck there was. A correlation of 0.2 says four fifths of last year's spread in rankings was the non-repeating kind.

    With persistence of 0.2, next year's expected rank hugs the middle00252550507575100100Percentile rank last yearExpected rank this yearperfect persistencepure luck: 50top decile: 96thTop-decile fundaverage z last year 1.75x 0.2 = 0.35about 64th pctexpected this year(shortcut: 59th)
    With a year-to-year correlation of 0.2, the expected rank this year is a flat curve close to the 50th percentile. A top-decile fund, averaging about the 96th percentile last year, is expected around the 64th this year, far below the diagonal that perfect persistence would give.

    How do you turn a rank correlation into an expected rank?

    Work on a normal-score scale, where the regression is a straight line through the middle. The expected score this year is the correlation times last year's score, so a correlation of 0.2 keeps one fifth of the distance from average. The top decile's average score is the normal density at its cut-off, 1.28, divided by 0.1, which is about 1.75. A fifth of that is 0.35, and the normal table turns it into the 64th percentile. If you skip the normal scale and pull the 95th percentile a fifth of the way from 50, you get about the 59th, a fair first answer.

    The relationship
    E[zt+1∣zt]=ρ ztzˉtop 10%=φ(1.28)0.10≈1.75Φ(0.2×1.75)≈0.64E[z_{t+1} \mid z_t] = \rho\, z_t \qquad \bar{z}_{top\,10\%} = \frac{\varphi(1.28)}{0.10} \approx 1.75 \qquad \Phi(0.2 \times 1.75) \approx 0.64
    \rhothe year-to-year correlation of rankings, 0.2
    z_tlast year's normal score
    \varphi, \Phithe standard normal density and cumulative probability
    What it says in wordsNext year's expected score keeps only the correlation's share of this year's distance from the middle.

    Then say what a fund selector does with it. Chasing last year's top decile buys mostly luck at a high price, often just as new money floods in. The answer is not to ignore performance but to weight longer records, look for a process that explains the result, and set expectations that even a genuinely good fund will usually rank well below where it did in its best year.

    Where candidates lose it

    The two extreme answers both lose the point. Expecting the fund to stay in the top decile ignores that a correlation of 0.2 is weak. Expecting it to fall below average overshoots: regression pulls results back towards the middle, not past it.

    Say one fifth of the distance, give the 64th percentile on the normal scale or the 59th with the quick version, and explain that the difference comes from how extreme the top decile really is. Then draw the selection lesson.

    What the interviewer asks next

    • What persistence correlation would keep a top-decile fund in the top quartile on average?
    • How does the answer change if you look at five-year rankings instead of one?
    • Why do flows into last year's best funds tend to make this problem worse?