Portfolio Management puzzles, solved step by step

About Rs 6.3 crore over 10 days and Rs 9.2 crore over 21 days. With independent daily returns, variance adds across days, so volatility and value at risk scale with the square root of time: 2 x the root of 10 and 2 x the root of 21. The rule fails when returns trend or mean revert, when tails are fat, and when the book changes during the period.

Why the square root of time and not time itself?

Think of a person taking random steps left or right. After a hundred steps they are rarely a hundred steps from the start, because the left steps cancel the right ones; the typical distance is about ten, the square root of a hundred. Daily returns behave the same way when each day is independent. Variances add across independent days, so the spread of a ten-day return is the daily spread times the square root of ten, not times ten. Value at risk at a fixed confidence level is a multiple of that spread, so it scales the same way: Rs 2 crore becomes Rs 6.32 crore over ten days and Rs 9.17 crore over 21.

When does the rule give the wrong answer, and in which direction?

The rule rests on three assumptions, and each one breaks in real markets. If returns trend, so a bad day tends to follow a bad day, the true ten-day loss is larger than Rs 6.3 crore; if they mean revert, it is smaller. Fat tails make the 99% point further out than a normal curve suggests, and the ratio between the tail and the spread need not hold across horizons. And a book is not frozen: over a month a desk cuts losing positions, which the scaling ignores. Say which way each one pushes the number and the interviewer knows you understand the rule rather than having memorised it.

One more thing worth saying: the scaling also assumes the expected daily return is zero. Over a day that is harmless. Over a year, drift matters and a simple square root rule starts to overstate the loss for a portfolio with a positive expected return.

Sell Rs 35 crore of index futures notional. At a beta of 1.2 the portfolio moves like Rs 60 crore of the index. At the target of 0.5 it should move like Rs 25 crore. The difference, (1.2 minus 0.5) x Rs 50 crore, is Rs 35 crore, assuming the futures move one for one with the index. At an assumed Rs 10 lakh a contract, that is about 350 contracts.

Why is the portfolio's market exposure not simply Rs 50 crore?

Think of a car that goes 1.2 km for every km a reference car goes. Holding Rs 50 crore of it is like holding Rs 60 crore of the reference. Beta converts a portfolio's value into index-equivalent exposure, so a Rs 50 crore book at beta 1.2 carries Rs 60 crore of market risk. Once you see the exposure in index rupees, the hedge is a subtraction: you want Rs 25 crore left, so you take away Rs 35 crore by selling futures, which carry a beta of one to the index.

What does the hedge not do?

It removes market risk, not stock risk. After the hedge the portfolio still carries every stock-specific bet it had; only its sensitivity to the index has been cut. That is often the point: a manager who likes the stocks but not the market can keep the stock picks and trim the market bet. Say the limitations: beta is estimated from history and drifts, so the hedge is right only on average; futures need margin and must be rolled at expiry, and the futures price can move slightly differently from the index, which is called basis risk.

Expected return 12%, volatility 18% and the same Sharpe ratio of 0.33. With 1.5 in the asset and minus 0.5 in cash, the return is 6% plus 1.5 times the 4% excess, which is 12%, and the volatility is 1.5 x 12%, which is 18%. The Sharpe ratio is (12 minus 6) / 18, unchanged at 0.33. Leverage at the risk-free rate moves you along the line without changing its slope.

Why does leverage leave the Sharpe ratio alone?

Think of a recipe scaled up by half: every ingredient rises by 1.5 times, so the taste, which depends on the proportions, is the same. Borrowing at the risk-free rate multiplies both the excess return and the volatility by the leverage, so their ratio, the Sharpe ratio, does not change. The asset earns 4 points over cash with 12 points of volatility, 0.33 per unit of risk. At 1.5 times it earns 6 points over cash with 18 points of volatility: still 0.33. The borrowed cash itself has no volatility, so it adds none.

When does leverage lower the Sharpe ratio?

When borrowing costs more than the risk-free rate, which is the normal case for anyone but a government. At a borrowing rate of 8%, the borrowed half costs 2 points more than cash earns, so the levered return falls to 11% and the Sharpe ratio to 0.28. The line bends down to the right of the asset. Say the other limitations: volatility is not the only risk that scales, because a levered portfolio can be forced to sell after a large loss, and the higher volatility drags down compound growth even when the Sharpe ratio is unchanged. This is why a manager with a high Sharpe ratio, low-volatility strategy can lever it, while the same move on a volatile asset is far more dangerous.

About 25% for one year and about 1.75% for the ten-year average. One year: zero is 10 over 15, or 0.67 standard deviations below the mean, and the normal table gives 25.2%. The ten-year average has a standard deviation of 15 over the square root of 10, which is 4.74, so zero is 2.11 standard deviations away and the chance falls to 1.75%.

Why is a losing year so common when the portfolio expects 10%?

Think of a bus that is due every 10 minutes but can be 15 minutes early or late on a normal day. Being late is not rare; it happens about one day in four. A 10% expected return with 15% volatility means zero sits only two thirds of a standard deviation below the average, and about a quarter of any normal distribution lies further out than that. So a client holding this portfolio should expect a losing year roughly one year in four, even if nothing is wrong.

What does the square root of ten do, and what does it not do?

Averaging independent years cancels part of the noise: good and bad years offset. The standard deviation of an average of n years is the one-year figure divided by the square root of n, so ten years takes 15 down to 4.74. The chance of a losing average falls sharply with time, but the spread of the total amount you end up with keeps growing. The total ten-year return has a standard deviation of 15 times the square root of 10, about 47 points, against 15 for one year. A longer horizon makes a loss less likely, not smaller when it comes.

Name the assumptions when you give the number. Real yearly returns have fatter tails than a normal curve and are not fully independent; losing years tend to cluster. Both make the true ten-year figure higher than 1.75%, so treat it as a floor, not a promise.

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.

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.

35% for one stock, about 20% for ten, and a floor of 17.5% for infinitely many. Portfolio variance is the stock variance times one over n, plus the correlation times what is left. With ten stocks that is 0.1225 times 0.325, a volatility of 20.0%. As n grows the one-over-n part vanishes and only the correlation term remains: 35% times the square root of 0.25, or 17.5%.

Why does adding stocks lower risk at all?

Ten shops in ten different towns do not all have a bad week at once; ten shops in one mall often do. Some of what moves a stock is its own news and some is the market everyone shares. Stock-specific shocks cancel out as you add names, because one company's bad quarter is offset by another's good one, but the shared market shock hits every name together and does not cancel. Correlation measures how much of each stock's movement is shared.

How much of the benefit do the first ten stocks deliver?

Most of it. Going from one stock to ten cuts volatility from 35% to 20.0%, about 86% of the whole distance to the floor. Going from ten to thirty takes it only to 18.4%. Diversification pays off quickly and then almost stops, and the level where it stops is set by correlation, not by the number of holdings. That is why a manager worried about risk gains more from adding assets that are less correlated than from adding a fortieth stock of the same kind.

Say the limitation. Correlations are not fixed: in a sell-off they tend to rise together, which raises the floor exactly when diversification is needed. Real stocks also differ in volatility and correlation, so this uniform market is a teaching model; the shape of the curve survives, the exact numbers do not.

A's expected shortfall is Rs 6.0 crore; B's is about Rs 12.0 crore. With 100 days, the worst 5% are the five worst days, and expected shortfall is their average. A's five add to 30, an average of 6.0. B's add to 60.2, an average of 12.04. Both portfolios cross Rs 5 crore on the same number of days, but when B has a bad day it is, on average, twice as bad.

What does value at risk leave out?

Think of a river's flood mark. Saying the river tops the bank five days a year tells you nothing about whether it rises a hand's width or floods the town. Value at risk tells you the loss that is exceeded on the worst 5% of days, and nothing about how big those losses are. Both portfolios breach Rs 5 crore on five days in a hundred, so their VaR is identical. What happens beyond that line is where they differ, and it is exactly the part the measure does not describe.

Why do risk managers prefer expected shortfall?

Because it cannot be gamed as easily and it adds up sensibly. A trader can sell deep out-of-the-money options, which earn small premiums almost every day and lose heavily now and then. That book can show a low VaR while hiding a large tail. Expected shortfall looks inside the tail, so strategies that pile risk just past the cut-off show up in it. It also behaves well when books are combined: diversification never makes it worse, which VaR cannot promise.

Be honest about the limits. Five observations are a thin sample for estimating anything, and one Rs 25 crore day dominates B's figure. A longer history, or a model of the tail, would be needed before setting limits on it. The comparison is still the right first read: same VaR, and one portfolio's bad days are twice as painful.

About Rs 16.1 lakh, a diversification benefit of about Rs 3.9 lakh. Under normal returns value at risk is a fixed multiple of standard deviation, so it combines the same way: the square root of 10 squared plus 10 squared plus 2 x 0.3 x 10 x 10, which is the square root of 260. Adding the two desks' figures would overstate the risk by Rs 3.9 lakh.

Why can you not just add the two numbers?

Two friends each walk 10 minutes from a crossing, one north and one north-east. They do not end up 20 minutes apart; the angle between their paths matters. Under normal returns, a desk's value at risk is a fixed multiple of its standard deviation, and standard deviations combine like arrows: the angle between them is set by the correlation. Only at a correlation of 1 do the arrows point the same way and add to 20.

What assumption is doing the work, and when does it fail?

The square-root rule holds only when returns are jointly normal, or close to it, so that value at risk is a clean multiple of standard deviation. With fat tails or options in the book, value at risk need not be subadditive, and the combined figure can even exceed the sum. Correlations also rise in a crisis, so the Rs 3.9 lakh benefit is thinnest on exactly the days it is needed. Firms often report both the diversified total and the sum of the parts for that reason.

A tracking error of about 1.25% a year. Tracking error is the volatility of the active weights. A's bet contributes (5% x 30%) squared, 2.25; B's contributes (5% x 25%) squared, 1.5625; and because the bets are opposite on correlated stocks, the covariance term is minus 2.25. The sum is 1.5625, whose square root is 1.25%.

Why does adding a second bet reduce the risk?

Buying an umbrella and selling a raincoat leaves you with little net exposure to rain, because both move with the weather. Overweighting A and underweighting B, when the two stocks tend to move together, is partly a hedge: when both rise, the gain on A is partly offset by the shortfall on B. Tracking error measures the risk left after that offset.

What is the neat coincidence, and what does it hide?

Here the covariance term exactly cancels A's own variance, so the answer equals B's bet alone, 5% x 25% = 1.25%. That is a coincidence of these numbers, not a rule: the covariance term is 2 x 0.6 x 30 x 25, which happens to equal 30 squared. Change the correlation to 0.5 and the answer moves. The general lesson holds, though: tracking error depends on the size of the bets and on how much they cancel.

Say the limits. Correlations are estimated and unstable, so a pair that looks like a hedge in calm markets can decouple when a stock-specific event hits. Real tracking error also includes every other small active weight, and many small bets can add up to more than one large one.