Portfolio Management puzzles, solved step by step

Correlation 0.9, beta 1.0 and residual volatility of about 8.7%. Correlation is the square root of R squared, so 0.9. Beta is correlation times the ratio of volatilities, 0.9 x 20 / 18, which is exactly 1.0. The unexplained 19% of the fund's variance of 400 is 76, and the square root of 76 is 8.7%: the fund's own risk, on top of what the index explains.

Why does 81% explained still leave so much unexplained?

Think of a household's monthly spending. If rent explains most of how the bill moves, the groceries, travel and surprises that make up the rest can still swing it by a lot. The fund is the same. R squared splits variance, and variance is volatility squared, so a small share of variance becomes a much larger share once you take the square root back. The fund's variance is 20 squared, 400. The index explains 81% of it, which is 324. The other 76 belongs to the fund alone, and the square root of 76 is 8.72%.

How do you get the correlation and the beta from R squared?

In a regression on a single index, R squared is simply the correlation squared, so the correlation is the square root of 0.81, which is 0.9. Beta is the correlation scaled by how volatile the fund is relative to the index: 0.9 times 20 over 18 is exactly 1.0. So the fund moves one for one with the index on average, and carries about 8.7 points of volatility the index does not explain. Mention the sign: the root could be minus 0.9, but a long-only equity fund with a positive slope takes the positive root.

What does the residual number tell a portfolio manager?

With a beta of 1.0, the residual volatility is the fund's tracking errorThe volatility of the difference between a fund's return and its benchmark's return. against the index. An R squared of 0.81 sounds index-like, but 8.7 points of tracking error is a genuinely active book: almost half as volatile as the market itself. Say the limitation too. The split assumes the relationship is linear and stable over the sample; a fund whose beta drifted during the period shows a lower R squared for reasons that have nothing to do with stock picking.

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.

The single six in four rolls, which wins 51.8% of the time against 49.1% for the double six in 24. Work each through its complement. No six in four rolls has chance (5/6) to the 4th, 48.2%. No double six in 24 rolls has chance (35/36) to the 24th, 50.9%. The naive count, trials times chance, gives two thirds for both and is wrong for both.

Why does multiplying the trials by six not keep the odds the same?

A double six is six times rarer than a six, so 24 rolls look like a fair swap for four. The intuition treats the chance of success as growing in a straight line with the number of tries. It does not. What compounds is the chance of failing every time, and a rare event's failure chance, raised to a high power, falls more slowly than the straight line suggests. Think of looking for a friend in a crowd: glancing four times at a small crowd and twenty-four times at a crowd six times the size are not the same search, because each extra glance adds less as the misses pile up.

How many rolls would the double six bet need to be favourable?

Solve for the number of rolls where the chance of no double six drops below a half. (35/36) to the 24th is 0.5086, still above a half; (35/36) to the 25th is 0.4945, just below. So 25 rolls, not 24, is where the double six bet turns favourable, a gap of a single roll between a losing and a winning bet. That is the second way to show you understand the question: the naive rule is off by a small amount, and in a repeated game a small edge is everything.

On a desk the lesson is the difference between a rough count and an exact one: a strategy that looks equivalent on a back-of-envelope scaling can sit on the wrong side of break-even once you do the compounding properly.

Yes, it is accretive, by about 4.4%. Take Rs 50 crore of earnings on 10 crore shares at Rs 100, so EPS is Rs 5.00. The buyback spends Rs 100 crore, which was earning Rs 3 crore, so earnings fall to Rs 47 crore while the share count falls to 9 crore. EPS becomes Rs 5.22. It is accretive because the 5% earnings yield beats the 3% return on cash.

What is the company actually swapping?

Suppose you hold a fixed deposit paying 3% after tax and use it to buy out a partner's share of a shop that earns 5% on its price. Your income goes up, because you replaced a 3% asset with a 5% one. A buyback is the same trade. The company gives up the after-tax return on its cash and gets back a slice of its own earnings, priced at the earnings yield, which is one over the P/E. At 20 times earnings that yield is 5%, so the swap raises earnings per share. At 33.3 times earnings the yield would be 3% and the buyback would leave EPS unchanged.

How do you get the exact number quickly?

Pick round figures and the answer falls out: Rs 1,000 crore of market value, Rs 50 crore of earnings and 10 crore shares. Earnings drop by 10% of the market value times 3%, which is Rs 3 crore, and shares drop by 10%, so EPS moves by 0.94 divided by 0.90. That is 1.0444, an accretion of 4.44%. The shortcut works for any size of company because only the ratios matter.

Does accretive mean the buyback was a good idea?

No, and a portfolio manager is expected to say so. Accretion is arithmetic about earnings per share; value is created only if the company paid less for its shares than they are worth. A company on a low P/E almost always gets an accretive buyback, even if the shares are overpriced, and one on a high P/E can create value with a dilutive buyback if the shares are cheap relative to future growth. Accretion also ignores risk: swapping cash for shares makes the remaining equity more levered.

About 15 years. The t-statistic of a track record is the information ratio times the square root of the number of years. Set 0.5 x root T equal to 1.96: the root of T is 3.92, so T is about 15.4 years. A manager twice as good, with an information ratio of 1.0, would still need nearly four years, and most careers and fund lives are shorter than this test demands.

Why does proof take so long even for a good manager?

Think of a cricketer whose true average is a little above the team's. In one season, luck swamps that small edge; only after many seasons does the average clearly separate. Skill adds up in proportion to time, but noise adds up in proportion to the square root of time, so the ratio between them grows only with the square root. An information ratio of 0.5 means one year's excess return is half a standard deviation of noise. It takes four years to reach a t-statistic of 1.0 and about 15.4 years to reach 1.96.

What should a fund selector do with that number?

Accept that statistics alone will not settle the question within a useful time. A fund selector who waits for significance hires managers after their best years are behind them; one who does not wait must lean on evidence other than the return series. That means the process, the people, turnover, whether the returns came from the bets the manager says they make, and how much of the record is explained by factors that could be bought cheaply. State the limitation: the calculation assumes independent years and a constant information ratio, and real skill decays as assets grow.

About 8.7%. The IRR is the discount rate at which Rs 7 crore a year for five years plus Rs 110 crore at year five is worth exactly the Rs 100 crore paid. A quick estimate adds the 7% income yield to the annual growth in value, 100 to 110 over five years or 1.9% a year, which gives 8.9%. The exact answer is a little lower, 8.68%.

What is the IRR actually solving for?

Think of a savings account that pays you Rs 7 of interest each year on Rs 100 and then hands back Rs 110. You want the single interest rate that account must have been paying. The IRR is the one discount rate that makes the present value of everything you receive equal to what you paid. Try 9%: the rents are worth about 27.2 and the sale about 71.5, a total of 98.7, below 100, so 9% is too high. Try 8.5%: the total is about 100.7, so the answer sits between them, at 8.68%.

Why is the rough split a little too high?

The rule of thumb, IRR roughly equals income yield plus growth, is exact only when the rent grows at the same rate as the value, so the yield stays at 7%. Here the rent is flat while the value rises, so by year five the rent is only 6.4% of the property's worth, and the average yield across the hold is below 7%. That is why the answer is 8.68% rather than 8.92%. In the room, give the rough split first to show the structure, then the exact figure. Say the limitation too: an IRR assumes the rents can be reinvested at the IRR itself, and it says nothing about how much money was at work.

It creates new units and sells them, keeping about 1.2 per unit, or 1.2% of NAV. The authorised participant buys the underlying basket at the NAV of 100, delivers it to the fund, receives new ETF units and sells them on the exchange at 101.5. Revenue of 101.5 less 100 for the basket and 0.3 of creation costs leaves 1.2. Its selling pushes the ETF price back toward NAV until the gap no longer covers the cost.

Why can anyone profit from the gap at all?

Picture a shop selling gift hampers for Rs 101.5 when the items inside cost Rs 100 at the market next door, and assembling a hamper costs Rs 0.3. Someone will buy the items, pack hampers and sell them until the hamper price falls. An ETF unit is a claim on a basket, and the creation and redemption window lets a large dealer convert one into the other, so the unit's price cannot drift far from the basket's value. The authorised participantA large dealer appointed by the fund to create and redeem ETF units in bulk directly with the fund. is that hamper maker.

Where does the arbitrage stop?

Every new unit sold adds supply on the exchange and every basket bought adds demand for the underlying shares, so the gap closes from both sides. The trade keeps paying until the premium falls to the creation cost of 0.3, so in a calm market the ETF price sits inside a band of roughly 99.7 to 100.3. Below 99.7 the trade reverses: buy cheap units, redeem them for the basket and sell the shares. The band is only as tight as the cost: an ETF holding illiquid bonds or foreign shares that trade in another time zone can show a wider gap for days, and that gap is a cost to whoever trades against it.

The portfolio management point: this machinery is why an ETF can trade close to its holdings without the fund itself selling anything. Mutual fund investors transact at NAV once a day; ETF investors transact at a market price that stays near NAV only because this arbitrage is open.

About 45%. One rupee at 15% grows to 2.66 after seven years and 4.05 after ten. The total gain is 3.05, and the last three years add 4.05 minus 2.66, which is 1.39. That is 45.5% of the decade's gain in 30% of the time, because each year's 15% is earned on a larger base than the year before.

Why is the gain not spread evenly across the years?

Think of a snowball rolled down a long slope. In the first few metres it picks up a little snow; near the bottom each turn picks up much more, because the ball itself is bigger. Compounding earns the same rate on a growing base, so every year's rupee gain is 15% larger than the year before. Year 1 adds 0.15 on each rupee. Year 10 adds 0.53, three and a half times as much. The first three years together add only 0.52; the last three add 1.39.

What does this mean for an investor who leaves early?

Someone who exits after seven years has sat through 70% of the time but collected only 55% of the decade's gain. Because compounding back-loads the reward, leaving a long plan early costs far more than the fraction of time given up. The same arithmetic runs against the investor with fees: a charge taken every year compounds too, and its cost is also concentrated at the end. Say the limitation plainly: 15% is an assumed rate for the arithmetic, not a forecast, and real returns arrive unevenly, so the actual last three years could be the worst three.

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.

About 64%, and a per-test threshold of about 0.25%. Each worthless strategy passes by luck 5% of the time, so all 20 fail with chance 0.95 to the 20th, 35.8%, and at least one passes 64.2% of the time. To hold the overall rate at 5%, test each at 5% divided by 20, which is 0.25%; the exact version, 1 minus 0.95 to the power of one twentieth, is 0.256%.

Why does testing more ideas throw up a false winner?

Ask a room of 20 people to each flip a coin five times, and there is a fair chance someone gets five heads. Nobody in the room has a lucky hand; there were simply enough tries. A 5% test lets one worthless idea in twenty through by chance, so a researcher who tests twenty ideas should expect about one false winner, not be impressed by it. The expected number of false positives here is 20 x 0.05, exactly 1, and the chance of at least one is 64.2%.

What does a quant desk actually do about it?

Dividing the threshold by the number of tests is called the Bonferroni correctionA rule that divides the significance level by the number of tests run, so the chance of any false positive across all of them stays near the original level.. The real discipline is counting every test you ran, including the ones you dropped quietly, because the correction is only as honest as that count. A researcher who tried 200 variants and reports the best 20 has a far bigger multiple testing problem than the 20 suggest. Desks also hold out data the research never touched and demand a reason for the edge before the backtest. Say the limitation: the correction assumes independent tests, and for correlated strategies it is too strict, which costs real ideas.