Portfolio Management puzzles, solved step by step

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All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
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  1. 004A portfolio has a 25% chance of a down year, and each year is independent of the others. What is the chance of at least one down year over ten years?Behavioural and decision trapsWarm upWealth managementRetirement and pensions

    Try it first

    Quick instinct: how likely is at least one down year in ten?

    Show the worked solution

    About 94%. The chance of avoiding a down year every single year is 0.75 multiplied by itself ten times, which is 5.6%. At least one down year is everything else: 1 minus 0.056, or 94.4%. Over a decade a falling year is close to certain, so a plan that treats one as a surprise is a plan built on the wrong base case.

    Why work through the chance of it never happening?

    Ask a commuter how likely they are to miss at least one train in a year of mornings, and the honest answer is: nearly certain, even if they miss one in a hundred. "At least one" questions have many routes to yes and a single route to no. Counting the one way it never happens and subtracting from 1 is always faster than counting every way it can happen. Here the only route to no down year is ten good years in a row, each with chance 0.75, so 5.6% of decades are clean and 94.4% are not.

    Chance of at least one down year, when each year has a 1 in 4 chance of falling50%100%25%144%258%368%476%582%687%790%892%994.4%1096%1197%12Years heldNo down year in ten:0.75 to the 10th = 5.6%
    With a one in four chance of a down year, the chance of at least one down year passes 50% by the third year and reaches 94.4% by the tenth, because the chance of dodging every one shrinks to 5.6%.
    The relationship
    P(at least one)=1−(1−p)n=1−0.7510≈0.944P(\text{at least one})=1-(1-p)^n=1-0.75^{10}\approx 0.944
    pthe chance of a down year, 25%
    nthe number of years, 10
    What it says in wordsThe chance of at least one down year is one minus the chance that every year is up.

    Why does a wealth desk ask a probability question like this?

    Because clients judge a portfolio year by year and plan over decades. A 25% chance in any one year sounds like a risk you might avoid; over ten years a down year is close to certain. The point of the number is to change the conversation from whether a down year comes to what the plan does when it does. Say the limitation as well: independence is an assumption. Real market years are not coin flips, and a regime of bad years clusters, which changes the count of down years without changing the lesson.

    Where candidates lose it

    The trap is adding: 25% times ten is 250%, which is obviously wrong, and candidates who spot that often retreat to 25%, which is just as wrong. The interviewer wants the complement said out loud.

    The quieter trap is getting 94% and stopping. On a wealth or pensions desk, the follow-up is what that means for a client, so have one sentence ready on setting expectations before the first bad year arrives.

    What the interviewer asks next

    • What is the chance of at least two down years in ten?
    • How many years before a down year is more likely than not?
    • If down years cluster, does the chance of at least one go up or down?
  2. 078You bought a stock at Rs 800. It now trades at Rs 500, and your updated estimate of fair value is Rs 450. Do you hold it until it gets back to Rs 800?Behavioural and decision trapsWarm upAsset managementWealth management

    Try it first

    Which numbers belong in the decision?

    Show the worked solution

    No. The Rs 800 you paid does not enter the decision; on your own numbers the stock is worth less than it trades for. Getting back to Rs 800 needs a 60% rise. Your own value of Rs 450 sits 10% below the Rs 500 price. Holding it is a fresh decision to own an overpriced stock.

    Why does the purchase price feel like it matters?

    Someone who paid Rs 2,000 for a concert ticket will go out in a storm with a fever rather than waste it, although the money is gone either way. The only question left is whether the evening is worth it now. Money already spent is sunk: it is the same whatever you do next, so it cannot help choose what to do next. The purchase price of a stock is exactly that kind of number.

    The purchase price is not on the decision; only today's two numbers are400450500550600650700750800850Rs 800: what you paidsunk, not a live inputPrice today Rs 500Your value Rs 450+60% needed just to get back: irrelevant to the choiceThe live gap: value is 10% below today's priceTest: would you buy this stock today at Rs 500 if you had never owned it? If not, holding it is the same bet.
    The Rs 800 purchase price sits outside the decision, and the 60% rise needed to reach it is irrelevant; the live comparison is today's price of Rs 500 against your updated value of Rs 450, which says the stock is overpriced.

    How do you say it so it sounds like judgement rather than a slogan?

    Turn the question round. If you had Rs 500 in cash today and no history with this stock, would you buy it at a price above your own value? If the answer is no, holding it is the same bet in disguise. The anchoringLeaning on a reference number, here the purchase price, when judging something that does not depend on it. pull comes from the Rs 800, and the disposition effect, holding losers to avoid booking the loss, is a well documented habit among professional managers as well as individuals.

    Then give the honest limits. Your Rs 450 is an estimate, so ask how confident you are and whether anything has changed that the price already reflects. A booked loss can also offset gains for tax, which is a reason to act, not to wait. Replacement matters too: the money should go to whatever has the best expected return per unit of risk, which may or may not be this stock.

    Where candidates lose it

    The trap is answering the question as asked, with a view on how long the stock might take to get back to Rs 800. That accepts the anchor, and the interviewer is testing whether you reject it.

    The opposite slip is a flat sell with no reasoning. Say the sunk cost point, give the would-I-buy-it-today test, then note that your Rs 450 value is an estimate that deserves a second look.

    What the interviewer asks next

    • Your value was Rs 900 instead of Rs 450. What changes?
    • Why do managers hold losers longer than winners, and how would you guard against it in your own process?
    • How would you explain this to a client who refuses to sell below cost?