Portfolio Management puzzles, solved step by step

About 0.62. Think of each asset's returns as a vector and correlation as the cosine of the angle between two of them. A correlation of 0.9 is an angle of about 25.8 degrees. B and C are each within 25.8 degrees of A, so they are at most 51.7 degrees apart, and cos 51.7 degrees is 0.62. The formula gives the same: 0.81 minus the square root of 0.19 x 0.19.

Why can the third correlation not be anything you like?

If your office is 10 km from your home and the gym is also 10 km from your home, the office and the gym cannot be 50 km apart. Distances have to fit on a map. Correlations work the same way: they are cosines of angles between return vectors, and angles have to fit together in space, so two strong correlations force a third. A correlation matrix that breaks this rule does not describe any real set of assets, and it can make a risk model report a negative variance.

How do you get 0.62 without drawing angles?

Split B and C into a part driven by A and a part independent of A. Each has 0.9 of A in it, and the independent parts carry the remaining variance, 1 minus 0.81, or 0.19. The correlation of B and C is 0.81 from the shared A part plus up to 0.19 either way from their independent parts, depending on whether those parts move together or against each other. So the range is 0.81 minus 0.19 to 0.81 plus 0.19: 0.62 to 1.00. The lowest value comes when the independent parts are perfectly opposed.

Why this matters on a risk desk: when analysts override individual correlations in a model, for a stress test or a view, they can create a matrix that no real market could produce. The fix is to check the matrix is positive semi-definite, and the lesson generalises. The bound is only strong when the given correlations are high; with two links of 0.7, the minimum for the third is -0.02, which forces almost nothing.

A proposes 98 for himself, 0 for B, 1 for C, 0 for D and 1 for E. Solve from the end. With two pirates, D keeps all 100 because his own vote is half. With three, C buys E for 1 coin. With four, B buys D for 1. With five, A needs two votes and buys the two pirates who get nothing under B's plan, C and E, for one coin each, keeping 98.

Where do you start?

At the end, where there is no choice left. Think of planning a train journey with connections: you start from the time you must arrive and work back to when you must leave. A sequential game is solved backwards, because each pirate's vote depends only on what he would get if the current proposal failed. With two pirates left, D proposes 100 for himself; his own vote is half, which is enough. So E gets nothing if it ever comes to that, and E knows it.

How does each step follow from the one below?

With three pirates, C needs two votes, his own and one more. E gets 0 if C dies, so one coin buys E: C proposes [99, 0, 1]. With four, B needs two votes; under C's plan D gets 0, so B buys D for one coin: [99, 0, 1, 0]. A vote is worth exactly one coin more than that pirate's fallback, so a proposer always buys the cheapest voters, the ones left with nothing in the next round. With five, A needs three votes. Under B's plan C and E get nothing, so one coin each buys them, and A proposes [98, 0, 1, 0, 1].

State the assumptions, because the answer rests on them. If an indifferent pirate voted yes, A could buy votes for zero coins. If the rule needed a strict majority, the counts change. Interviewers often change one rule as a follow-up to see whether you rebuild the chain or reach for a memorised answer. The buy-side lesson is about incentives: what someone will accept depends on their alternative, not on fairness.

Roughly 5 lakh people, within a range of about 3 to 10 lakh. Start from about 18 crore demat accounts, an assumption to confirm, and divide by 1.5 accounts per person: 12 crore individuals. Perhaps 40% hold a meaningful portfolio, 4.8 crore. Wealth is heavily skewed, so assume about 1 in 100 of those hold more than Rs 50 lakh: about 4.8 lakh. A Pareto-tail check gives the same order of magnitude.

Why is the headline account count a trap?

Think of a cricket academy with thousands of registered players. Asking how many could play first-class cricket is not a question about registrations; it is about the very top of the talent curve. Portfolio sizes are highly skewed, so the count above a high threshold is a small slice of the total, and the size of that slice, not the headline, decides the answer. Crores of accounts become a few lakh people above Rs 50 lakh.

How do you defend the 1 in 100?

With a second route. Suppose 20% of the 4.8 crore holders have more than Rs 5 lakh. Wealth tails often thin out like a Pareto distributionA skewed distribution in which the share above any level falls by a fixed power as the level rises, often used for wealth and city sizes.: raise the threshold tenfold and the share falls by a factor of 10 to about 20. Two independent routes landing in the same range is what turns a guess into an estimate. From Rs 5 lakh to Rs 50 lakh is a tenfold rise, so between 4.8 and 9.6 lakh people sit above Rs 50 lakh, which brackets the first answer.

Then say what the number is for, and what it misses. A portfolio management service can also be funded from bank deposits, mutual fund units or property sales, so the true addressable pool is wider than direct equity holders. The Rs 50 lakh minimum and every count used here are figures to confirm against current SEBI rules and depository data; the structure of the estimate is the part to trust.

About 40 basis points per rupee, and 8 times the total cost. Under a square-root law, cost per rupee rises with the square root of size: four times the size is twice the cost per rupee, 40 basis points. Total cost is size times cost per rupee, so it rises four times two, or eight times. On Rs 50 crore of daily volume, Rs 1 lakh becomes Rs 8 lakh.

Why does a bigger order cost more per rupee?

Selling one flat in a building is easy at the going price; selling ten at once means working down to buyers who would only pay less. A large order uses up the willing counterparties near the current price and has to reach further, so each extra rupee traded moves the price more than the one before. Empirical studies of trading costs across markets find that the cost per rupee grows roughly with the square root of the order's share of volume, a rule practitioners use widely as a first estimate.

What does the portfolio manager do with this?

Two things. Because total cost grows faster than size, spreading a large order over several days is usually cheaper: four days at 10% each costs about Rs 4 lakh against Rs 8 lakh in one go, provided the price impact of one day's trading has largely faded by the next and the price does not drift away meanwhile. And a strategy's capacity is limited: a fund that doubles in size pays about 2.8 times the rupee impact on each rebalance, which eats into returns quickly.

Say the limits. The square-root law is an empirical average, with a coefficient that differs by stock and market conditions, and splitting an order adds timing risk: if the information behind the trade leaks or the price moves, waiting costs more than impact would have.

About 73%, against 50% for a fair bet. This is the gambler's ruin problem. With the ratio of loss to win odds r = 0.45 / 0.55, the chance of reaching 10 units from 5 is (1 minus r to the 5) over (1 minus r to the 10), about 73.2%. Smaller bets make it higher still: with Rs 50,000 bets it is about 88%.

Why is the answer so much higher than 55%?

A slightly better tennis player might win 55% of points, yet win most matches, because a match needs many points and the small edge keeps adding up. Reaching Rs 10 lakh before zero needs a net five wins, which takes many bets, and every bet tilts the race a little in your favour, so the edge compounds into a much bigger survival advantage.

Why does bet size matter so much?

Keep the edge and the rupee distances fixed and change only the stake. With Rs 5 lakh bets the whole outcome rides on one toss, 55%; with Rs 1 lakh bets it is 73%; with Rs 50,000 bets it is 88%, because more bets give the law of large numbers room to work. That is the portfolio lesson: a real edge is only worth something if position sizes are small enough that an unlucky run cannot end the game first.

The price implies about 10.2% growth for ever, below the 12% that reinvestment could support today. Setting 25 equal to 0.4 x (1 + g) / (0.12 - g) gives g of 2.6 / 25.4. Retaining 60% at a 20% return supports 12%, but 12% for ever equals the cost of equity and would make the stock infinitely valuable, so the market is pricing in returns fading to about 17%.

What does it mean to work backwards from a multiple?

If a flat rents for Rs 30,000 a month and sells for Rs 1.2 crore, the price tells you what buyers assume about future rents, whether or not anyone says it. A P/E is the same kind of compressed forecast: fix the payout and the cost of equity, and the multiple pins down the one perpetual growth rate the price assumes. Reverse engineering the price like this is safer than forecasting growth and seeing what multiple falls out.

Is the stock cheap because it can grow faster than the price assumes?

Not so fast. Sustainable growth is return on equity times retention, 20% x 60% = 12%. But 12% growth for ever equals the cost of equity, which would make the stock worth an infinite multiple, so no price could reflect it; the gap tells you the market expects the 20% return on new investment to fade. At the same 60% retention, the price is consistent with a long-run return on equity of about 17.1%.

That turns the question into a sharper one: will this business keep earning well above 17% on new money for a long time? That is a question about competition and moats, and it is the right place to spend the next five minutes of the interview. Say also that one-stage models are crude; a two-stage model with high growth fading to a lower rate is the usual next step.

Allocation added 0.70 points, selection cost 0.50 and interaction cost 0.30, netting to the -0.1 shortfall. Allocation prices the weight bets at benchmark returns: +10% x 12 and -10% x 5. Selection prices the return gaps at benchmark weights: 50% x (10 - 12) and 50% x (6 - 5). Interaction is the weight bets times the return gaps.

Why split the result at all, if it only lagged by 0.1?

A team can lose a match by one run after a brilliant bowling spell and a dismal batting collapse; the scoreline hides both. Attribution separates the decision of how much to hold in each asset class from the decision of what to hold within it, so each can be judged on its own. Here the allocation call was right and the equity picking was poor, which is a very different story from a flat 0.1 point miss.

What is the interaction term, and why does it bite here?

Interaction is the weight bet times the return gap. The manager overweighted equities by 10 points and underperformed within them by 2 points, so the extra equity weight magnified the poor picking by 0.2; underweighting bonds, where the picking was good, gave up another 0.1. Some firms fold interaction into selection, since the weights are the manager's choice; say which convention you are using.

One more check an interviewer may ask for. The Brinson-Fachler version measures allocation against the total benchmark return, (weight bet) x (segment return minus 8.5). It gives the same total, 0.70, here, because the weight bets sum to zero, but it splits the credit between segments differently.

The barbell wins both ways on an immediate parallel move, by about 0.12 points if yields fall and 0.10 if they rise, but it yields about 0.15 points less a year. With duration matched, convexity is the only difference: 71 for the barbell against 49 for the bullet. Over a year the barbell needs a move of roughly 113 to 123 basis points to earn back its lower yield, and it loses if the curve steepens.

Why does spreading money to the two ends add convexity?

Balancing a plank on a see-saw with one heavy child in the middle is steady; balancing it with a child at each end makes it swing more for the same push. Price sensitivity grows faster than linearly with maturity, so a mix of short and long bonds with the same average duration as a middle bond has more curvature than the middle bond: here about 71 against 49. More convexity means gaining a little more when yields fall and losing a little less when they rise.

What does the barbell give up, and when does it lose?

First, yield. Weighting the 2-year at about 50% to match duration gives a blended yield near 6.85%, about 0.15 points below the bullet's 7.0%. That is the price of convexity: if yields sit still, the bullet simply earns more, and over a year the barbell's edge only pays for itself on a parallel move of more than about 1.2 points. Second, curve shape. If the curve steepens, with short yields falling and long yields rising, the barbell's long leg loses and the bullet wins; if the 7-year sector rallies against the two ends, the bullet wins too.

State the assumptions: zero-coupon bonds, a blended yield taken as a weighted average, which is an approximation, and an instantaneous shift. The conclusion survives all three: the barbell buys protection against large parallel moves with yield, and takes on curve-shape risk.