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.

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.

The average trade loses 0.2%, and break-even needs a 75% hit rate. Expected value is 0.7 x 1% minus 0.3 x 3%, which is 0.7% minus 0.9%, or minus 0.2%. To break even the winners must pay for the losers: p x 1 = (1 minus p) x 3, so p is 3 over 4, or 75%. The trader is right more often than wrong and still loses money.

Why does a 70% hit rate not settle the question?

A shopkeeper who makes a small profit on seven sales out of ten but sells the other three at a big loss can still close the month in the red. Counting the happy sales tells you nothing until you know how big each one was. Expected return is each outcome's probability times its size, added up, so a hit rate only means something next to the payoff ratio. Here the losers are three times the size of the winners, and that ratio is doing all the damage.

How do you find the break-even hit rate in one line?

Set expected value to zero and solve. The break-even hit rate is the loss size divided by the sum of the win and loss sizes: 3 over 1 plus 3, which is 75%. Every point of hit rate is worth 0.04% a trade here, because moving one trade in a hundred from loser to winner swings 4 points in total. So the trader is 5 points of hit rate, or 0.2% a trade, short of break-even.

Then say what you would change. The trader can cut losers sooner, let winners run further, or be more selective; lifting the average win to 1.5% with the same losses moves break-even to 3 over 4.5, about 67%. Averages also hide transaction costs, which push break-even higher still.

About Rs 75.1. The formula C over r gives the value one year before the first payment. With the first payment in year 4, the stream is worth 10 over 0.10, or Rs 100, at year 3. Discount that back three years: 100 over 1.1 cubed is Rs 75.1. As a check, Rs 100 less the present value of the three missing payments, Rs 24.9, gives the same answer.

Where does the perpetuity formula put its answer in time?

Think of a pension that starts paying at retirement. Its value on the day before the first cheque is one number; its value to a 30-year-old is that number shrunk by three decades of waiting. C over r values a level perpetuity exactly one period before its first payment, so a delayed stream is the ordinary perpetuity discounted from that point. Here the first payment is in year 4, so C over r lands at year 3, where it reads Rs 100. One more discounting step gets it to today.

How do you check it without the formula?

Start from what you know: Rs 10 a year from year 1 is worth Rs 100. The delayed stream is that same stream with the first three payments cut out. Their present values are 9.09, 8.26 and 7.51, which add to 24.87, and Rs 100 less 24.87 is 75.13. Two methods agreeing to the paisa is what makes the answer safe to say out loud.

This is the same step as a terminal value in a discounted cash flow. The Gordon growth value at the end of year 5 is a year 5 number and must be discounted five years, not six. Counting the periods wrong by one is the most frequent error in valuation models, and this puzzle is the cleanest place to catch it.

The Sharpe ratio is about 0.44 and the information ratio is 0.5. Sharpe divides the excess over cash, 14 minus 6 or 8 points, by total volatility of 18: 0.44. The information ratio divides the excess over the benchmark, 14 minus 12 or 2 points, by the tracking error of 4: 0.5. Sharpe judges the whole portfolio's risk; the information ratio judges only the manager's active bet.

Why are there two ratios for one fund?

Picture judging a cook. One question is whether the whole meal was worth its price. Another is whether the chef's changes to the standard recipe made it better. The Sharpe ratio asks whether the fund's total return beat cash by enough to justify all its risk, and the information ratio asks whether the manager's departures from the benchmark earned enough to justify that active risk. An investor choosing an asset mix cares about the first; one who has already chosen the market and is picking a manager cares about the second.

How do you read 0.44 and 0.5 once you have them?

The tracking errorThe standard deviation of the difference between a fund's return and its benchmark's return, a measure of how far the fund strays. is small against total volatility because most of the fund's ups and downs are the market's, which the benchmark shares. A 0.5 information ratio from a single year is respectable on paper but statistically weak: one year of 2 points against a 4-point tracking error is half of one standard deviation. It would take many years at that rate before anyone could separate skill from luck with confidence. The Sharpe ratio is also best compared with the benchmark's own Sharpe over the same period, not read alone.

Say one limitation in the room: both ratios assume returns are roughly normal. A fund that sells insurance-like options can post a smooth, high Sharpe for years and then lose a large amount in one month, which neither ratio sees in advance.

Light rope 1 at both ends and rope 2 at one end, together. When rope 1 burns out, light the other end of rope 2; when rope 2 burns out, 45 minutes have passed. Rope 1 lasts 30 minutes because two flames share its 60 minutes of burning. At that moment rope 2 has 30 minutes left, and lighting its other end halves that to 15. 30 plus 15 is 45.

Why does uneven burning not spoil the halving?

Two people eating one plate of food from opposite sides finish it in half the time one person would take, whether the food is piled high on one side or spread evenly. They simply meet off-centre. A rope holds 60 minutes of burning in total, and two flames consume it twice as fast, so it is gone in 30 minutes wherever the flames happen to meet. The unevenness decides the place, never the time. That is the only fact the puzzle needs.

How do you know rope 2 has exactly 30 minutes left at the half-hour?

Rope 2 has been burning from one end for 30 minutes, so it has used 30 of its 60 minutes of material, whatever length that turned out to be. You never measure length; you only ever track time used and time left. Lighting the far end of what remains halves the 30 minutes left, and the rope finishes 15 minutes later. Say the timeline in that order and the answer is audible.

Interviewers often extend it: with the same two ropes you can also time 15, 30, 60 and 90 minutes, each by deciding which ends are burning at which moment. Answering one extension shows you own the principle rather than a memorised trick.

Why would a real asset desk ask this? It is a clean test of whether you separate the thing you can measure from the thing that is noisy. A rent roll is lumpy month to month; the annual total is what the valuation rests on. Saying that link in one sentence costs nothing.

Minus 35.4%, from the peak of 130 to the trough of 84. Track the running peak: 100, then 130, which holds until 140 in the final year. The deepest fall below that peak is 84, and 84 over 130 minus 1 is -35.4%. The earlier dip to 91 was only minus 30%, and the fall from the starting 100 looks like just 16%.

Measured from where?

Picture a climber on a ridge. How far she has fallen is measured from the highest point she reached, not from the car park and not from the step she took a moment ago. A drawdown is the fall from the running peak, the highest NAV seen so far, and the peak only resets when the NAV climbs above it. Here the peak jumps to 130 in year 1 and then sits there through the dip to 91, the recovery to 120 and the slide to 84. Every one of those years is inside the same drawdown.

Why does the recovery to 120 not end the drawdown?

Because 120 is still below 130. Anyone who invested at the peak was still under water at 120, and the slide to 84 took them further down. A partial recovery does not reset the peak, so two separate-looking dips can be one long drawdown. Treating 120 as a new start gives a fall of 30% to 84, which is wrong on the question asked, though it is a correct answer to how bad the single year was.

Say what the number misses as well. Maximum drawdown is one path and one worst point, so it tells you nothing about how long the fund stayed under water, here four years from year 1 to year 5, and a longer history almost always shows a deeper one. Allocators read it next to the recovery time for that reason.

Two weighings. Put three bags on each side and leave three aside. Whichever side rises holds the light bag; if the pans balance, it is among the three on the table. Take that group of three and weigh one bag against another: the one that rises is light, and if they balance, it is the third. A balance gives three outcomes per weighing, and three times three covers nine bags.

Why is halving the pile the wrong instinct?

Halving is what you would do with a question that answers yes or no, like guessing a number between 1 and 100. A balance tells you more than that. A balance scale has three outcomes, left pan rises, right pan rises, or level, so each weighing should split the suspects into three equal groups, not two. The group left on the table is not wasted: a level balance is information too. With four and four plus one aside you would learn little from a level result except that the odd bag is the spare one, and an unlucky run needs three weighings.

How do you prove two is the minimum?

Count outcomes. One weighing produces at most three different results, and three results cannot point to nine different bags. With k weighings you can separate at most 3 to the power k bags, so nine bags need at least two weighings, and the method above shows two is enough. The same count answers the natural follow-up: three weighings handle up to 27 bags, four handle 81.

Why would an asset manager ask this? It is a test of whether you use all the information a measurement gives you. A risk report, a performance attribution or a data screen can be read in more ways than pass and fail, and the candidate who spots the third outcome here is the one who reads the level result as a finding.

DV01 is Rs 5 lakh, so a 25 basis point rise costs about Rs 1.25 crore. Modified duration of 5 means the value moves about 5% for each percentage point of yield, or 0.05% per basis point. 0.05% of Rs 100 crore is Rs 5 lakh, and 25 basis points is 25 times that.

Why turn duration into rupees at all?

A shopkeeper who says the rent went up 4% has told you something; one who says it went up Rs 2,000 a month has told you what to do about it. Duration is the percentage version. DV01, the rupee change for a one basis point move, is the version a desk can add across positions, compare with a limit and put in a risk report. Two portfolios with the same duration but different sizes carry very different rupee risk, and only DV01 shows it.

Where does the straight-line answer stop being right?

The modified durationThe percentage change in a bond price for a one percentage point change in its yield, taken at the current yield. estimate is a tangent: it treats the price and yield relationship as a straight line. For 25 basis points the straight line is close enough, but for a 200 basis point shock the curve bends away from it and the true loss is smaller than DV01 times 200. That bend is convexity. Mention it in one sentence; the interviewer will often ask for it next.

Say also that DV01 assumes every yield in the portfolio moves by the same amount. If short rates rise and long rates stay still, a single DV01 number misses it, which is why desks also keep DV01 by maturity bucket.