Portfolio Management puzzles, solved step by step

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.

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.

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%.

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.

About 47%, close to a coin toss. One manager beats the benchmark five years running with probability one half to the fifth, or 1 in 32. The chance that none of the 20 does it is 31/32 to the 20th power, about 53%. So the chance that at least one does is about 47%. In a crowd of unskilled managers, a five-year streak is almost as likely as not.

Why is a five-year streak weak evidence of skill?

Put 20 people in a room and ask each to call five coin tosses. Somebody calling all five correctly would look gifted, but with that many people, somebody usually does. The chance of a streak for one named person is small, but the chance that someone in a crowd produces one is large, and fund selection looks at the crowd. An investor who screens a category for managers with five straight winning years is sampling exactly the lucky tail of this distribution.

How do you compute at least one without adding up cases?

Use the complement. The chance that at least one of many independent trials succeeds is one minus the chance that every trial fails. Each manager fails the streak with probability 31/32. All 20 fail with probability (31/32) to the 20th, which is about 0.53. One minus that is 0.47. The expected number of streaks is 20 over 32, or 0.625, which is a separate and also useful number: less than one streak on average, yet close to even odds of seeing at least one.

The limitation runs in two directions. Real managers are not independent, since many hold similar stocks, which makes streaks cluster and the calculation rougher. And some managers do have skill; the point is only that a streak on its own cannot tell the skilled from the lucky. That takes longer records, consistent process and a view on why the edge should persist.

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.

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.

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.

About Rs 87,000 a year, and more after tax on the deposit interest. The card costs 36% of Rs 3 lakh, Rs 1,08,000. Using Rs 3 lakh of the deposit to clear it gives up 7% on that money, Rs 21,000. The difference, Rs 87,000, is a certain loss every year the household keeps both, paid for the comfort of a bigger deposit balance.

Why do people keep both?

Because money gets labels. The deposit is the emergency fund or the daughter's education money, and touching it feels like failing; the card is day-to-day spending and feels temporary. A rupee is a rupee whatever label it carries, so money earning 7% while debt costs 36% is simply borrowing at 36% to lend at 7%. The habit has a name, mental accountingTreating money differently depending on which mental category it sits in, a term from the economist Richard Thaler., and it is one of the most common and costly patterns a wealth adviser meets.

How do you get the number right?

Compare two whole choices. Keep both: the household earns Rs 35,000 and pays Rs 1,08,000, a net of minus Rs 73,000. Clear the card: it earns 7% on the remaining Rs 2 lakh, Rs 14,000, and pays nothing. The cost of a choice is the difference between the two outcomes, not the interest bill on its own. That difference is Rs 87,000, the same as 29 points of rate gap on Rs 3 lakh.

Tax widens the gap. Deposit interest is taxed at the household's slab while card interest is paid from income already taxed. At an illustrative 30% slab, confirm the current rates, the cost rises to about Rs 93,300. The honest limit is liquidity: a family with no buffer at all may need some cash on hand. Here the Rs 2 lakh left in the deposit is that buffer, so the case for keeping the debt is gone.

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.

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.

Each cycle wins Rs 1,000 with probability 63 in 64 and loses Rs 63,000 with probability 1 in 64, so the expected value is exactly zero. The capital covers six stakes, Rs 1,000 up to Rs 32,000, so six straight losses, a 1 in 64 chance, wipe it out. Doubling down turns many small wins into one rare, large loss without creating any edge.

Why does it feel like a sure thing?

A driver who never buys insurance saves the premium every month and feels clever for years, until the one accident. Doubling after each loss produces a long run of small wins because the one outcome that loses, six tails in a row, is rare, but when it comes it is sixty three times the usual win. Short track records of such strategies look excellent, which is exactly the danger.

What does this teach about judging a strategy?

Run it 50 times and the chance of at least one wipe-out is about 54%, yet most individual traders running it for a few weeks will show a smooth profit. A win rate says nothing on its own; what matters is the size of the losses relative to the wins, and whether the worst case can end the game. Selling far out-of-the-money options has the same shape: steady premium income, and occasionally a loss that erases years of it.

Say the limitation of the fix people suggest. More capital only pushes the wipe-out further out and makes it bigger: with infinite money and no stake limit the strategy would work, and nobody has either.