Hedge Funds puzzles, solved step by step
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022Two independent, unbiased forecasts of next quarter's GDP growth have error standard deviations of 2% and 3%. How should you combine them, and how accurate is the blend?Quant and systematic funds
Try it first
What weight should the 2% forecast get?
Show the worked solution
Weight them 9/13 and 4/13, about 69% and 31%, and the blend's error falls to about 1.66%. For independent unbiased forecasts the best weights are proportional to one over each error variance: 1/4 for the 2% forecast and 1/9 for the 3% one. The blended error variance is 1 over (1/4 + 1/9), which is 36/13, so its standard deviation is 1.66%, better than either forecast alone.
Why does blending two forecasts beat the better one?
Ask two people to guess the weight of a pumpkin at a village fair. One is usually closer, but their mistakes are unrelated, so averaging tends to cancel part of each. Independent errors partly cancel when you average, so even a weaker forecast adds information, provided it gets a smaller weight. Throwing the 3% forecast away leaves you at 2%; blending it in well gets you to 1.66%.
Blending with inverse-variance weights of 9/13 and 4/13 gives an error of 1.66%, lower than the better forecast's 2.00%, while equal weights give 1.80% and weights of 3/5 and 2/5 give 1.70%. The relationship\sigma_A, \sigma_B the two forecasts' error standard deviations, 2% and 3% w_A the weight on the sharper forecast What it says in wordsEach forecast is weighted by its precision, one over its variance, and the blend's precision is the sum of the two.Why inverse variance and not inverse error?
The blend's error variance is w squared times 4 plus (1 minus w) squared times 9. Setting its slope to zero gives w = 9/(4 + 9), so the weights follow one over the variance, which penalises the noisier forecast harder than one over the standard deviation would. Check the alternatives: equal weights give an error of 1.80%, weights of 3/5 and 2/5 give 1.70%, and the inverse-variance weights give the minimum, 1.66%.
What would you check before trusting the blend?
Two assumptions carry the answer. The forecasts must be unbiased and their errors independent; if both forecasters lean on the same survey, their errors are correlated and the gain from blending shrinks. With an error correlation of 0.5, the best blend gives the sharper forecast 6/7 of the weight and improves the error only from 2.00% to 1.96%. Ask where each forecast comes from before you average them.
Where candidates lose it
Candidates either average equally, which overweights the noisier forecast, or keep only the better one, which throws information away. Both miss that the right weights come from the variances.
The subtler slip is weighting by one over the standard deviation, 3/5 and 2/5. It is close but not optimal; state the inverse-variance rule and show that the blended error beats 2%.
What the interviewer asks next
- What if the two forecast errors have a correlation of 0.5?
- How would you estimate each forecaster's error variance in practice?
- One forecast is biased upwards by 0.5%. What do you do?
073A thousand fund managers have no skill at all: each has a 50% chance of beating the market in any year, independently. How many will beat it five years running, and what does that say about track records?Quant and systematic funds
Try it first
How many of the 1,000 unskilled managers beat the market five years in a row?
Show the worked solution
About 31 managers, 1,000 halved five times. Each year roughly half the unbeaten managers beat the market by luck, so 500 survive year one, 250 year two, then 125, 62.5 and 31.25. A perfect five-year record is something luck hands to about 3 managers in every 100, so in a large crowd it cannot on its own separate skill from chance.
Why does a crowd produce streaks even without skill?
Fill a stadium with a thousand people and ask each to toss a coin five times. Someone will throw five heads, and about 31 will. A result that is rare for one person is almost certain somewhere in a large group, so the question is never whether a flawless record exists but how many you would expect by chance. Each manager's chance is 1 in 32; across 1,000 managers that is 31.25 expected perfect records.
Starting from 1,000 unskilled managers, half fall away each year, leaving 500, 250, 125, 62.5 and finally about 31 with a flawless five-year record produced by chance alone. The relationship1,000 the number of managers 1/2 each manager's chance of beating the market in a year 5 the number of years What it says in wordsMultiply the crowd by the chance that one member gets the streak.If some managers really are skilled, how much does a perfect record tell you?
Suppose 5% of the thousand are skilled and beat the market 60% of the time. They produce about 3.9 perfect records, while the 950 unskilled produce about 29.7, so a manager with five perfect years is skilled only about 12% of the time. A 60% manager has only a 8% chance of five perfect years, so most skilled managers do not have flawless records either. The record is weak evidence in both directions.
What should you look at instead?
Longer records, more decisions per year and a reason. Skill shows up more reliably in many independent decisions than in a handful of annual outcomes, and in a process that explains where the edge comes from. Allocators also check how many managers were in the starting pool, because the funds still reporting are the ones that survived; the ones that were closed after bad years have dropped out of the data. That is survivorship biasThe distortion that comes from studying only the survivors of a process, whose results look better than those of the whole starting group., and it makes every surviving record look stronger than it is.
Where candidates lose it
The first loss is saying none, or very few, because five in a row sounds impressive. The interviewer wants the crowd arithmetic: rare for one, expected for many.
The second is stopping at 31 without the conclusion. The number is only half the answer; say what it means for reading a track record, and name survivorship bias.
What the interviewer asks next
- How many of the 1,000 beat the market in at least four of the five years?
- How many years of beating the market would one unskilled manager in 1,000 be expected to reach?
- How would you design a test that separates a 60% manager from a 50% one?
088A stock's true model is: stock return = 0.5 x market return + 1.0 x sector return + noise. Regressing the sector's return on the market gives a slope of 0.4. If you regress the stock on the market alone, what slope do you get?Quant and systematic funds
Try it first
What does the market-only regression report?
Show the worked solution
About 0.9. The market reaches the stock by two paths: directly, with a coefficient of 0.5, and through the sector, which moves 0.4 for each unit of market and passes all of it on with a coefficient of 1.0. A regression on the market alone cannot separate the two and reports the total, 0.5 + 1.0 x 0.4 = 0.9. The extra 0.4 is omitted variable bias.
Why does leaving the sector out change the market slope?
Suppose you measure how much ice cream sales rise on hot days, but hot days also tend to be holidays, and holidays sell ice cream too. Leave holidays out and the heat gets the credit for both. A regression gives a left-out variable's effect to whichever included variable moves with it, in proportion to how strongly the two move together. Here the sector moves with the market, so the market's slope absorbs part of the sector's effect.
The market reaches the stock directly with a coefficient of 0.5 and through the sector with 0.4 x 1.0 = 0.4, so a regression of the stock on the market alone reports 0.9, of which 0.4 is the sector's effect credited to the market. How do you compute the bias?
Write the sector as 0.4 x market plus a part unrelated to the market, then substitute. Stock = 0.5 x market + 1.0 x (0.4 x market + other) + noise = 0.9 x market + (1.0 x other + noise). The bracket is unrelated to the market, so a regression on the market alone recovers 0.9. The bias is the omitted coefficient times the slope of the omitted variable on the included one, 1.0 x 0.4. A simulation of 20,000 days with these coefficients gives a slope of 0.897, matching the algebra.
The relationshipbeta_M the stock's true direct loading on the market, 0.5 beta_S the stock's loading on the sector that was left out, 1.0 delta the slope of the sector's return on the market's, 0.4 What it says in wordsThe short regression's slope is the true slope plus the left-out variable's effect times how much that variable moves with the one you kept.Is 0.9 wrong, or answering a different question?
It depends on what you use it for. If you want to hedge the stock with the market alone, 0.9 is the right hedge ratio, because it captures everything the market drags along with it. If you want the stock's exposure holding the sector fixed, say to build a sector-neutral book, 0.9 overstates it and 0.5 is the number you need. The bias can also run the other way: if the sector moved against the market, or the stock loaded negatively on the sector, the short slope would sit below 0.5. Naming both uses is what the interviewer is listening for.
Where candidates lose it
The fast wrong answer is 0.5: candidates assume a regression recovers the true coefficient whatever else is left out. It does so only when the omitted variable is unrelated to the included one.
The second loss is getting 0.9 and calling it simply wrong. It is the correct total effect of the market and the right number for a market-only hedge; it is wrong only as an estimate of the direct effect.
What the interviewer asks next
- What slope do you get if the sector's slope on the market is minus 0.4?
- You add the sector to the regression. What happens to the standard error of the market coefficient if the two are highly correlated?
- How does this bias show up when you estimate a stock's factor exposures with too few factors?