Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

Showing 21–30 of 100

021Analyst A has an information coefficient of 0.10 on 25 independent bets a year. Analyst B has an information coefficient of 0.03 on 400 independent bets a year. Whose information ratio is higher?Performance measurementHardQuantitative asset managementSystematic investing
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Which analyst has the higher information ratio?
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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.
Analyst A's skill of 0.10 on a root breadth of 5 gives an information ratio of 0.5, while analyst B's skill of 0.03 on a root breadth of 20 gives 0.6, unless B's bets are correlated enough to count as only 100, which cuts B to 0.3.
The relationship
$IR\approx IC\times\sqrt{BR} \qquad A: 0.10\sqrt{25}=0.5 \qquad B: 0.03\sqrt{400}=0.6$
IC the information coefficient: the correlation between forecasts and outcomes
BR breadth: the number of independent bets a year
IR the information ratio: active return per unit of tracking error
What it says in wordsSkill times the square root of independent bets gives the information ratio.
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.
Where candidates lose it
The instinctive answer is A, because a coefficient of 0.10 sounds far better than 0.03. The interviewer is testing whether you know breadth enters the formula at all.
The second trap is taking B's 400 at face value. The follow-up is almost always what if the bets are correlated, and the answer is that breadth shrinks and B's advantage can vanish.
What the interviewer asks next
How many independent bets would A need to match B?
How would you estimate the effective number of independent bets in a portfolio?
Why do trading costs hit analyst B harder than analyst A?
022A private fund calls Rs 100 crore from an investor today and returns Rs 200 crore in five years, an IRR of about 14.9%. If the fund instead uses a credit line to delay the call by one year, at a borrowing cost of Rs 8 crore paid out of the final distribution, what happens to the IRR and to the multiple of money?Private and real asset mathsHardPrivate markets
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With the credit line, what happens?
Show the worked solution
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.
Without the credit line the investor pays 100 at year 0 and gets 200 at year 5, an IRR of 14.9% and 2.00x; with it the investor pays 100 at year 1 and gets 192 at year 5, an IRR of 17.7% but only 1.92x.
The relationship
$\left(\frac{200}{100}\right)^{1/5}-1\approx 14.9\% \qquad \left(\frac{192}{100}\right)^{1/4}-1\approx 17.7\%$
200, 192 the distribution at year 5 without and with the line, Rs crore
1/5, 1/4 one over the years the investor's money is at work
What it says in wordsWith a single call and a single distribution, the IRR is the multiple raised to one over the years, minus one.
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.
Where candidates lose it
Candidates say both numbers fall, because the line costs money. They miss that IRR is time-weighted and rewards a later call.
The deeper trap is stopping at the arithmetic. The interviewer on a private markets desk wants to hear that a higher IRR here does not mean a better result for the investor, and that the multiple exposes it.
What the interviewer asks next
What if the line delays the call by two years at a cost of Rs 16 crore?
What return must the investor earn on the idle Rs 100 crore to break even?
Why do some investors prefer to see a fund's multiple before its IRR?
023A portfolio manager decides to buy 10,000 shares when the price is Rs 100. The trading desk fills 60% of the order at an average of Rs 101.2, and the rest goes unfilled as the price closes the day at Rs 104. What is the implementation shortfall?Funds, ETFs and implementationHardPortfolio implementationInstitutional asset management
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Which part of the shortfall is larger?
Show the worked solution
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.
Against a decision price of Rs 100, the 6,000 shares filled at Rs 101.2 cost Rs 7,200 and the 4,000 unfilled shares cost Rs 16,000 as the price rose to Rs 104, a total shortfall of Rs 23,200, or 2.32% of the intended trade.
Component Shares Per share, Rs Cost, Rs Share of Rs 10 lakh
Execution cost 6,000 1.2 7,200 0.72%
Opportunity cost 4,000 4.0 16,000 1.60%
Total 10,000 23,200 2.32%
The execution and opportunity costs add to a shortfall of Rs 23,200, 2.32% of the intended Rs 10 lakh purchase.
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.
Where candidates lose it
Most candidates measure only the execution cost, 1.2% on the filled shares, and forget that shares never bought still cost the fund. The unfilled 40% is the bigger number here.
The second slip is the base. Express the shortfall against the whole intended trade, Rs 10 lakh, not against the Rs 6 lakh filled, or the number cannot be compared across orders.
What the interviewer asks next
What if the price had closed at Rs 99 instead?
How would you decide how fast to trade the order?
Why might a VWAP benchmark make the same desk look good?
024A fund's measured alpha is 4% a year, with a standard error of 3%. Across all funds, true alphas average zero with a spread (standard deviation) of 1.5%. What is your best estimate of this fund's true alpha?Statistics and forecastingHardFund selectionQuantitative asset management
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Best estimate of the true alpha?
Show the worked solution
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%.
The prior for all funds is centred on zero with a spread of 1.5 and the fund's measurement on 4% with a spread of 3, so the combined estimate lands at 0.8%, with a spread of 1.34, pulled 80% of the way back to zero.
The relationship
$\hat\alpha=\frac{\tau^2}{\tau^2+s^2}\,\alpha_{obs}=\frac{2.25}{2.25+9}\times 4\%=0.8\%$
\tau the spread of true alphas across funds, 1.5%
s the standard error of this fund's measured alpha, 3%
\alpha_{obs} the measured alpha, 4%
What it says in wordsThe estimate is the measured alpha scaled by how much of the total variance comes from real differences between funds.
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.
Where candidates lose it
Most candidates either take the 4% as measured, ignoring the noise, or answer zero, ignoring the evidence. The interviewer wants the weighted middle and the reason for the weights.
The arithmetic slip is weighting by standard deviations, 1.5 and 3, instead of variances, which gives one third and 1.3%. Precision is one over the variance, so square before you weight.
What the interviewer asks next
What if the fund had 20 years of data and a standard error of 1%?
How would you estimate the spread of true alphas across funds?
Why does this argument make top-quartile rankings unstable from one year to the next?
025An asset has an expected return of 10% and volatility of 12%, and the risk-free rate is 6%. If you lever it 1.5 times, borrowing at the risk-free rate, what are the expected return, the volatility and the Sharpe ratio?Portfolio risk mathsCoreMulti-assetHedge funds
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What happens to the Sharpe ratio when you lever at the risk-free rate?
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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.
The asset at 12% volatility and 10% return and the 1.5 times levered portfolio at 18% and 12% sit on the same line from the 6% risk-free rate, so both have a Sharpe ratio of 0.33, while borrowing at 8% instead drops the levered point to 11% and a Sharpe ratio of 0.28.
The relationship
$E[r_L]=r_f+L(E[r]-r_f)=6+1.5(4)=12\% \qquad \sigma_L=L\sigma=18\% \qquad S=\frac{6}{18}=\frac{4}{12}\approx 0.33$
L the leverage, 1.5 times
r_f the risk-free rate, 6%, also the borrowing rate here
S the Sharpe ratio, excess return over volatility
What it says in wordsLeverage scales the excess return and the volatility by the same factor, so the Sharpe ratio is unchanged.
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.
Where candidates lose it
The common answer is that leverage raises the Sharpe ratio because it raises return, or lowers it because it raises risk. Both miss that the two rise together.
The arithmetic slip is levering the whole 10% return, answering 15%, instead of levering the 4% excess and paying 6% on the borrowed half. Write return as the risk-free rate plus leverage times the excess.
What the interviewer asks next
What leverage gives an expected return of 14%, and what is its volatility?
Why do investors who cannot borrow tend to hold riskier assets instead?
How does the answer change if the borrowing rate is 8%?
026A portfolio has an expected return of 10% a year and a volatility of 15%, and yearly returns are roughly normal and independent. What is the chance of losing money in any one year, and the chance that its average annual return over ten years is below zero?Portfolio risk mathsCoreWealth managementMulti-asset
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Before you calculate: roughly how likely is a losing ten-year average?
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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.
Both curves are centred on 10%, but the one-year curve is wide and 25.2% of it lies below zero, while the ten-year average curve is narrow and only 1.75% of it lies below zero. The total ten-year return still spreads out, with a standard deviation of about 47 points.
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.
The relationship
$z = \frac{0 - \mu}{\sigma / \sqrt{n}} = \frac{-10}{15/\sqrt{10}} = -2.11 \qquad P(\bar{R} < 0) = \Phi(-2.11) \approx 1.75\%$
\mu the expected annual return, 10%
\sigma the annual volatility, 15%
n the number of independent years averaged, 10
\Phi the standard normal cumulative probability
What it says in wordsMeasure how many standard deviations of the average zero sits below the mean, then read the tail off the normal table.
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.
Where candidates lose it
The common slip is to say the risk disappears with time, or to divide the 25% by ten. Neither is how averages behave: the spread of the average shrinks with the square root of the number of years, not with the number itself.
The subtler loss is stopping at 1.75% and calling long horizons safe. Say the second half: the total outcome still spreads out with time, so a patient investor faces fewer losing decades but not smaller losses when one arrives.
What the interviewer asks next
How many years until the chance of a losing average drops below 1%?
What happens to both answers if yearly returns have fat tails?
Is the chance of losing money over ten years the same as the chance of a negative average annual return?
027A trader is right on 70% of trades. Each winning trade makes 1% and each losing trade loses 3%. What does the average trade return, and what hit rate would the trader need just to break even?Probability and expected valueWarm upHedge fundsAsset management
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Gut call first: is this trader making money?
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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.
Over ten trades, seven winners of 1% add 7 points and three losers of 3% remove 9 points, so the trader nets minus 2 points, or minus 0.2% a trade. Expected return crosses zero only at a 75% hit rate, five points above what the trader achieves.
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.
The relationship
$E = pW - (1-p)L \qquad p^{*} = \frac{L}{W+L} = \frac{3}{1+3} = 75\%$
p the hit rate, 70%
W the average win, 1%
L the average loss, 3%
p^{*} the hit rate at which expected return is zero
What it says in wordsA strategy breaks even when the chance of losing, times the loss, equals the chance of winning, times the win.
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.
Where candidates lose it
Candidates hear 70% and say the trader is good, or multiply 70% by 1% and forget the losers entirely. The interviewer built the question so the hit rate looks impressive and the payoff ratio quietly wins.
The second loss is getting minus 0.2% and stopping. The follow-up is always the break-even hit rate or the break-even payoff ratio, so have the one-line formula ready.
What the interviewer asks next
Keep the 70% hit rate. How big must the average win be to break even?
A second trader is right 40% of the time, wins 3% and loses 1%. Who would you rather back?
How do trading costs of 0.05% a round trip change the break-even hit rate?
028Rs 10 a year forever, starting next year, is worth Rs 100 at a 10% discount rate. What is the same stream worth today if the first payment arrives only in year 4?Valuation riddlesWarm upAsset managementMutual funds
Try it first
Pick the answer before you work it.
Show the worked solution
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.
The stream of Rs 10 a year from year 4 is worth Rs 100 at year 3, and three years of discounting at 10% brings that to Rs 75.1 today. Subtracting the three missing payments, worth Rs 24.87 today, from an immediate perpetuity of Rs 100 gives the same Rs 75.1.
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.
The relationship
$PV = \frac{C/r}{(1+r)^{3}} = \frac{100}{1.1^{3}} = \frac{100}{1.331} \approx 75.13$
C the yearly payment, Rs 10
r the discount rate, 10%
3 the years between today and one period before the first payment
What it says in wordsValue the perpetuity where it begins, then discount that single sum back to today.
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.
Where candidates lose it
The two wrong answers come from the timeline. Discounting by 1.1 to the fourth assumes the formula lands at the first payment date rather than one year before it, which gives about Rs 68.3. Subtracting Rs 30 at face value ignores that early money is worth more than later money.
Draw the timeline, even in the air with your finger. Say where C over r lands before you discount, and offer the subtraction check.
What the interviewer asks next
What is the stream worth if the payments grow at 3% a year from year 4?
How much is the delay costing, as a share of the undelayed value?
Where does the same off-by-one error show up in a DCF terminal value?
029A fund returned 14% with 18% volatility in a year when cash paid 6%. Its benchmark returned 12%, and its tracking error against that benchmark was 4%. What are its Sharpe ratio and its information ratio, and what does each one tell you?Performance measurementWarm upPerformance analysisAsset management
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Which number goes in the denominator of the information ratio?
Show the worked solution
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.
The Sharpe ratio divides 8 points of return over cash by 18 points of total volatility and gives 0.44; the information ratio divides 2 points of return over the benchmark by 4 points of tracking error and gives 0.50. Each ratio pairs a reward with the risk that produced it.
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.
The relationship
$\text{Sharpe} = \frac{R_p - R_f}{\sigma_p} = \frac{8}{18} = 0.44 \qquad \text{IR} = \frac{R_p - R_b}{TE} = \frac{2}{4} = 0.5$
R_p the fund's return, 14%
R_f the cash rate, 6%
\sigma_p the fund's total volatility, 18%
R_b the benchmark return, 12%
TE the tracking error, 4%
What it says in wordsEach ratio is a reward divided by the risk taken to earn it; the two differ in which reward and which risk.
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.
Where candidates lose it
The frequent mix-up is putting total volatility under the information ratio, which gives 2 over 18, about 0.11, and makes a reasonable manager look poor. The other is measuring the Sharpe numerator against the benchmark instead of cash.
Say the pairing aloud before calculating: excess over cash with total risk, excess over benchmark with active risk. Then the numbers take ten seconds.
What the interviewer asks next
The benchmark had 16% volatility. What was its Sharpe ratio, and did the fund beat it on that measure?
How many years of a 0.5 information ratio before the excess return is statistically significant?
Why can a fund have a higher Sharpe ratio than its benchmark but a negative information ratio?
030You have two ropes and a lighter. Each rope takes exactly 60 minutes to burn from one end to the other, but it burns unevenly, so half the length does not mean half the time. How do you measure exactly 45 minutes?Logic brainteasersWarm upAsset managementReal assets
Try it first
What does lighting a rope at both ends give you, if it burns unevenly?
Show the worked solution
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.
Rope 1, lit at both ends, is gone after 30 minutes even though its flames meet away from the middle. Rope 2, lit at one end, has 30 minutes of burning left at that moment, and lighting its other end halves that to 15 minutes, ending at 45.
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.
Where candidates lose it
Candidates try to cut or fold the ropes, or to reason about lengths, which the uneven burning makes useless. The whole question is set up to see whether you let go of length and think only in minutes of burning.
The second slip is lighting rope 2's second end at the start. Rope 2 must be lit at one end at minute 0 so that exactly 30 minutes of it are used up when rope 1 finishes.
What the interviewer asks next
Using the same two ropes, how do you measure 15 minutes?
With three ropes, what is the longest time you can measure beyond 60 minutes?
Can you measure 20 minutes with two ropes? Why or why not?

IC	the information coefficient: the correlation between forecasts and outcomes
BR	breadth: the number of independent bets a year
IR	the information ratio: active return per unit of tracking error

200, 192	the distribution at year 5 without and with the line, Rs crore
1/5, 1/4	one over the years the investor's money is at work

Component	Shares	Per share, Rs	Cost, Rs	Share of Rs 10 lakh
Execution cost	6,000	1.2	7,200	0.72%
Opportunity cost	4,000	4.0	16,000	1.60%
Total	10,000		23,200	2.32%

\tau	the spread of true alphas across funds, 1.5%
s	the standard error of this fund's measured alpha, 3%
\alpha_{obs}	the measured alpha, 4%

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How can a less accurate analyst add more value?

Where does the argument break?

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

Is the investor better or worse off?

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What is the shortfall measured against?

What should the manager take from the split?

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Why not take the 4% at face value?

How does a fund selector use this?

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Why does leverage leave the Sharpe ratio alone?

When does leverage lower the Sharpe ratio?

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

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

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

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

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Where does the perpetuity formula put its answer in time?

How do you check it without the formula?

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Why are there two ratios for one fund?

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

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Why does uneven burning not spoil the halving?

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

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