Portfolio Management puzzles, solved step by step

About 19.9% at plus 0.3 and 15.6% at minus 0.3. The unhedged return is roughly the local return plus the currency return, so the variances add with a correlation term: 16 squared plus 8 squared, plus or minus 2 x 0.3 x 16 x 8. That is 396.8 or 243.2, with square roots of 19.9% and 15.6%. With negative correlation the unhedged position is less volatile than the hedged one at 16%.

How can adding a second risk reduce the total?

Think of a shop that sells umbrellas and sunglasses. Each product's sales swing a lot with the weather, but in opposite directions, so the till is steadier than either product alone. When two sources of return tend to move against each other, combining them lowers risk even though each one is volatile on its own. For an Indian investor holding foreign shares, the currency is the second source. If the foreign currency tends to strengthen against the rupee when that market falls, it cushions the loss, and the unhedged position is steadier: 15.6% instead of 16%.

So should a global portfolio hedge its currency?

The puzzle gives the risk side of the answer, not the whole decision. A hedge removes the currency's volatility, but it also removes the currency's correlation with the market, and when that correlation is negative the hedge adds risk rather than cutting it. The rest of the decision is cost, which depends on the interest rate gap between the two currencies, and the investor's own liabilities in rupees. Say the limitation: correlations are measured on history and tend to shift in a crisis, so the minus 0.3 that makes the unhedged position look safer is the number least likely to hold when it matters. The formula also ignores the small cross term from multiplying the two returns.

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.

Interview and pass on the first 3, then hire the first manager better than all of them. You get the best one about 40% of the time. The first three set the bar at no cost but the chance the best is among them. Passing on 4 gives almost the same, 39.8%. As the number of candidates grows, the rule becomes: pass on about 37%, one over e, and the chance of success tends to 37% too.

Why pass on anyone at all?

Think of house hunting in a fast market where every flat is gone the moment you walk away. If you sign the first one, you have no idea whether it was good. If you look at every flat before deciding, the best has already been taken. A short look-only phase buys you a benchmark; after that, the first candidate who beats the benchmark is likely to be the best overall. The cost is that the best might be inside the look-only phase, which happens with chance r in 10 if you pass on r. Hiring the first manager blindly succeeds only 10% of the time.

How do you compute the chance for a given rule?

Suppose you pass on r and the best manager sits at position i, after r. You hire them only if nobody between r and i beat the first r, which happens when the best of the first i minus 1 candidates is among the first r: a chance of r over i minus 1. Averaging over where the best one sits gives r over 10 times the sum of 1 over (i minus 1), for i from r plus 1 to 10. For r equals 3 that is 0.3 times (1/3 + 1/4 + ... + 1/9), which is 0.3987. The curve is flat near the top: 3 and 4 differ by less than half a point.

Say the limitation, because allocators hear this puzzle and then ask about real selection. The rule maximises the chance of the single best and scores a second-best hire as a total failure. Real allocators care about hiring someone good, can often revisit a manager, and rarely see candidates in random order, and each of those changes the rule.

About 25% a year for ten years. With no dividend, all of the 12% return must come from price, so the price in ten years must be 60 x 1.12 to the tenth, about 186 times today's earnings. If the stock then trades at 20 times, earnings must be 186 divided by 20, or 9.32 times today's. That is growth of 25.0% a year. At 15% growth the investor would earn only about 3% a year.

How do you turn a multiple into a growth forecast?

Think of paying 60 years of a shop's current profit for the shop. That only makes sense if the profit is going to be far bigger soon. A high multiple is a forecast you can read: work forward from the return investors want, and back from the multiple the stock should end at, and the growth in between is what the price assumes. Here the investor wants 12% a year with no dividend, so the price must grow 3.11 times in ten years, to about 186 times today's earnings. At a mature 20 times, earnings must reach 9.32 times today's level.

What do you do with the number once you have it?

You ask how often a company sustains that growth for a decade, which is rarely, and you say so. The question the interviewer wants answered is not whether the company is good but whether the price has already paid for more than the company is likely to deliver. At a still strong 15% a year, earnings reach 4.05 times today's, the price at 20 times is about 81, and the investor earns about 3.0% a year rather than 12%. Say the limitations: the exit multiple of 20 is an assumption, and buybacks, dividends or a higher exit multiple would lower the growth needed.

Analyst B, with 0.6 against 0.5. By the fundamental law of active management, the information ratio is roughly the information coefficient times the square root of the number of independent bets. A gets 0.10 times the root of 25, which is 5, for 0.5. B gets 0.03 times the root of 400, which is 20, for 0.6. Breadth beats accuracy here, but only if B's bets really are independent.

How can a less accurate analyst add more value?

Think of two shopkeepers. One makes a large margin on a few sales a day; the other makes a thin margin on hundreds. The second can earn more, and with less swing day to day, because many small edges average into a steadier total. Active return grows with the number of bets while its noise grows only with the square root, so the ratio of the two rises with the square root of breadth. This is Grinold's fundamental lawRichard Grinold's result that a strategy's information ratio is approximately its information coefficient times the square root of its breadth. of active management. B's skill is under a third of A's, but B's root breadth is four times A's.

Where does the argument break?

At the word independent. Four hundred bets that all lean on the same factor, say cheap stocks, behave like far fewer independent bets, and breadth has to be counted in independent bets, not trades. If B's 400 positions carry the information of only 100 independent ones, B's ratio is 0.03 times 10, which is 0.3, and A is ahead again. The law also ignores costs: 400 bets a year means more turnover, and a thin edge of 0.03 is the first thing trading costs eat. A portfolio manager hiring between the two would ask B how correlated the signals are before believing the 0.6.

The IRR rises to about 17.7% while the multiple falls from 2.00x to 1.92x. With the line, the investor pays Rs 100 crore at year 1 instead of year 0 and receives Rs 192 crore at year 5 after the borrowing cost. That is 1.92 times the money over four years, 17.7% a year, against 2.00 times over five years, 14.9%. The investor ends with Rs 8 crore less, and the reported IRR looks better.

How can the return rise when the investor gets less money?

Imagine lending a friend money: getting back Rs 192 after four years can be a better annual rate than Rs 200 after five, even though Rs 200 is more money. IRR measures speed, not size, so anything that shortens the time the investor's money is out raises it, even at a cost. A subscription lineA short-term loan to a private fund, secured on investors' commitments, used to delay capital calls. does exactly that. The fund's deal is identical; only the investor's clock starts a year later.

Is the investor better or worse off?

It depends on what the investor does with the Rs 100 crore during the extra year. If the idle money earns less than the Rs 8 crore the line costs, the investor is worse off even though the fund reports a higher IRR. That is why allocators look at the multiple and the IRR together, and increasingly ask for IRRs calculated both with and without the effect of credit lines. Say the limitation: this example uses one call and one distribution; real funds call and return money in many pieces, and the line's effect on IRR is largest in the early years of a fund.

Rs 23,200, or 2.32% of the Rs 10 lakh intended trade. Against a paper portfolio that bought all 10,000 shares at the decision price of Rs 100, the 6,000 filled at Rs 101.2 cost Rs 7,200 in execution. The 4,000 unfilled shares missed a rise to Rs 104, an opportunity cost of Rs 16,000. The order you did not fill cost more than the one you did.

What is the shortfall measured against?

Think of deciding to buy a train ticket at the counter price, then finding the queue slow: you pay a bit more for some of the group's tickets, and the rest miss the train and take a dearer taxi. The cost of the trip is measured against the plan, not against what you ended up doing. Implementation shortfall compares the real portfolio with a paper portfolio that traded everything instantly at the decision price, so both the extra paid and the move missed count as cost. The idea is due to Andre Perold, and the benchmark here is Rs 100, the price when the decision was made.

What should the manager take from the split?

That trading patiently is not free. A desk that works an order slowly to keep execution cost down can lose more to the price running away, so the right trading speed depends on how fast the manager's idea is likely to be priced in. Here the price moved 4% in a day, a sign the idea was urgent, or that the buying itself pushed the price. Say the limitations: explicit costs such as commission and taxes should be added on top, and the unfilled shares are charged at the closing price by convention; had the order been cancelled deliberately, a different end point might be fairer.

About 0.8% a year. Combine the fund's noisy measurement with what you know about funds in general, weighting each by its precision. The measurement's variance is 3 squared, 9; the spread of true alphas has variance 1.5 squared, 2.25. The measurement gets 2.25 / 11.25, a weight of 0.2, so the estimate is 0.2 x 4% + 0.8 x 0%, which is 0.8%. A noisy 4% deserves heavy shrinkage toward zero.

Why not take the 4% at face value?

A new restaurant with two five-star reviews is probably good, but you would not bet it beats every restaurant in town with thousands of reviews. When a measurement is noisy compared with how much things really differ, most of an extreme reading is luck, so the best estimate sits much closer to the average. Here true alphas across funds rarely stray far from zero, a spread of 1.5, while one fund's measured alpha can miss the truth by 3. A reading of 4% is far more likely to be a modest fund that got lucky than a fund with a true 4%.

How does a fund selector use this?

By refusing to rank funds on raw past alpha. Shrinkage keeps the ranking but compresses it, so a fund with a long, steady record keeps more of its measured alpha than one with a short, volatile record showing the same number. A fund whose standard error was 1% instead of 3% would keep 2.25 / 3.25, about 69% of its 4%. Say the limitations: the result depends on the assumed spread of true alphas, which itself is estimated, and on the average being zero; if the fund belongs to a peer group with a known positive or negative average, shrink toward that instead.

Volatility is about 11.3%, and equities contribute about 93% of the risk. The variance is 0.36 x 324 plus 0.16 x 36 plus 2 x 0.6 x 0.4 x 0.1 x 18 x 6, which is 116.64 plus 5.76 plus 5.18, or 127.58. The square root is 11.3%. Equities' own term plus half the cross term is 119.2, which is 93% of the total.

Why does 60% of the money carry more than 90% of the risk?

Picture a household with two earners, one a salaried clerk and one a commission-only salesperson. The salesperson may bring in 60% of the money but almost all of the month-to-month swings. Risk contribution depends on weight times volatility, and variance squares it, so a sleeve three times as volatile with more capital swamps the other. Equities' variance term is 0.36 times 324, or 116.6; bonds' is 0.16 times 36, or 5.76. Twenty to one before the cross term.

How do you split the risk between the two sleeves?

Give each asset its own variance term plus half of the cross term. An asset's risk contribution is its weight times its covariance with the whole portfolio, and the contributions add to the total variance. For equities that is 0.6 x (0.6 x 324 + 0.4 x 10.8), which is 0.6 x 198.72, or 119.23. Divided by 127.58, that is 93.5%. Bonds take the remaining 6.5%.

This is the arithmetic behind risk parity, which sizes each sleeve so that the risk contributions are equal rather than the capital. To give bonds half the risk here, the portfolio would hold roughly three times as much bond capital as equity, and often borrow to lift the return. The limitation: correlation is not stable. In some years stocks and bonds fall together, and the 7% can grow quickly.

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.