Portfolio Management puzzles, solved step by step

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%.

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.

About 0.54, close to 0.55. Add up 21 daily returns in each market. Each monthly variance is 21 times the daily variance. The monthly covariance collects 21 same-day terms of 0.30 and 20 lagged terms of 0.25, where A's day falls inside the same month as B's next day. So the correlation is (21 x 0.30 + 20 x 0.25) over 21, which is 0.538, tending to 0.55 for long windows.

Why would daily data understate how much two markets move together?

Two friends in different cities hear the same news; one hears it at lunch, the other the next morning. Compare their moods hour by hour on the same day and they look unrelated; compare their moods over the whole week and they look very similar. When two markets close at different times, news that arrives between the two closes shows up in one market's return today and the other's tomorrow, so same-day daily correlation misses part of the true co-movement. Longer windows add the delayed part back in.

How do you add up the covariance terms?

Write each month's return as the sum of its daily returns. The covariance of two sums is the sum of every pairwise covariance. With a unit daily variance, same-day pairs contribute 21 x 0.30, and the pairs of B's day t with A's day t minus 1 contribute 20 x 0.25, since only 20 such pairs sit inside a 21-day month. The variances are 21 each, with no autocorrelation to add. So the monthly correlation is (6.3 + 5.0) over 21, which is 0.538.

This matters for portfolio construction. A risk model built on daily same-day correlations between markets in different time zones will think they diversify each other far more than they do over the horizon an investor actually holds. The standard fixes are to use weekly or monthly returns, or to add lagged terms to the daily covariance, as this calculation does. The limitation: with only 12 monthly observations a year, the monthly estimate is itself noisy.

A has the lower error, 10 against 12.25, and an equal blend cuts it to about 5.6. Mean squared error is bias squared plus variance: A is 1 plus 9, B is 0 plus 12.25. Averaging the two halves the bias to 0.5 and, because the errors are independent, quarters each variance: 0.25 plus 5.31, or 5.56. The blend beats both forecasters by a wide margin.

How can a biased forecaster beat an unbiased one?

Think of two watches. One always runs a minute fast but is otherwise steady; the other is right on average but wanders a few minutes either way. If you need to catch a train, the steady fast watch may serve you better. Mean squared error charges for two things, the average miss and the scatter around it, so a small steady bias can cost less than extra scatter. A's bias of 1 adds 1 to its error. B's standard deviation of 3.5 instead of 3 adds 3.25. On this measure, A is the better forecaster.

Why does averaging the two help so much?

Because independent errors partly cancel. When A is too high, B is as likely to be too low as too high, so the average of the two misses by less than either. An equal blend halves the bias and, with independent errors, cuts the variance to a quarter of the sum of the two variances. Here that is a quarter of 9 plus 12.25, which is 5.31, plus a bias term of 0.5 squared, 0.25. The total, 5.56, is little more than half of A's 10.

You can do slightly better by leaning towards A. The weight on A that minimises the error is B's variance over the sum of A's mean squared error and B's variance, about 55%, which gives 5.51. The gain over a plain 50/50 is tiny, which is the practical lesson: a simple average of decent, independent forecasts captures almost all of the benefit. The limitation is the word independent. Two economists reading the same data make correlated errors, and then averaging helps far less.

Because the observations overlap, the honest t-statistic is close to 1, about 1.04. Ten years of monthly data give 109 rolling 12-month windows, but each shares 11 months with its neighbour. OLS assumes independent errors, so it treats 109 observations as 109 pieces of information. With the overlap, the variance of the estimate is understated about twelvefold, and 3.6 over the square root of 12 is about 1.0.

Which OLS assumption breaks?

Picture asking the same twelve people for their opinion, then swapping one person each month and asking again. You would not claim 109 independent surveys. OLS standard errors assume the regression errors are uncorrelated with each other, and overlapping windows make them strongly correlated, so the standard errors come out far too small. The alpha estimate itself is not biased; what is wrong is the claim about how precisely it is known, which is exactly what the t-statistic reports.

How do you get to about 1?

Add up the overlap. If monthly returns are independent, two 12-month windows k months apart have a correlation of 12 minus k over 12. The variance of an average of such overlapping sums is inflated by one plus twice the sum of those correlations, which here is 1 plus 2 times 5.5, or 12. The t-statistic shrinks by the square root of the inflation factor, 3.6 divided by the square root of 12. That gives about 1.04: nowhere near significant, and consistent with having only about ten independent years.

How would you fix it? Three answers, in order of simplicity. Run the regression on non-overlapping monthly returns, which uses all the data with independent errors. Or keep the overlap and use standard errors that allow for autocorrelation, such as Newey-West or Hansen-Hodrick with 11 lags. Or sample the 12-month returns once a year, which leaves only ten points but honest ones. The interviewer wants to hear that the fix is about the standard errors, not the coefficient.

The factors explain 14.7%, leaving alpha of 2.3%. Start from the 6% risk-free rate and add each loading times its factor return: 1.1 x 8 = 8.8 for the market, 0.4 x 2 = 0.8 for size, and minus 0.3 x 3 = minus 0.9 for value. The fund's 11 points over cash are mostly priced exposure; 2.3 points is left for skill.

Why not measure the fund against cash or the market alone?

A tutor whose students all score well may simply have been given the strongest students. To judge the teaching, you first adjust for who walked in. A fund with a beta above 1 and a small-cap tilt should earn more than the market in a year when the market and small caps did well, so its exposures must be priced before anything is called skill. The Fama and French three-factor model does exactly that with market, size and value.

What does the split tell you about the manager?

Against the plain market's 14%, the fund beat by 3 points. Of those 3 points, 0.7 came from priced tilts in net terms, more beta and small caps less the growth tilt, and 2.3 is left over. The negative value loading matters too: a growth-leaning fund lost 0.9 points in a year value did well, so part of the manager's alpha was earned while swimming against a factor.

Then state the limits. One year of alpha is mostly noise; you would want a regression over many years with a standard error before calling 2.3 points skill. And the answer depends on the model: add momentum or quality factors and some of the 2.3 may turn out to be another priced exposure.