I wish there was a Strunk and White for mathematics.
While by no means logically incorrect, it feels inelegant to setup a problem using variables A and B in the first paragraph and solve for X and Y in the second (compounded with the implicit X==B, and Y==A).
Yeah and this is a much more intuitive way of generalising from the n = 2 case. Weights are proportional to inverse variance even for n > 2. Importantly this assumes independence so it doesn’t translate to portfolio optimisation very easily.
If A and B have different volatilities, it's rather counter-intuitive to allocate proportionally rather than just all to the one with the lower volatility... :-/
I agree, and I had to think about it for a second, but now it seems obvious. It works for the exact same reason that averaging multiple independent measurements can give a more accurate result. The key fact is that the different random variables are all independent, so it's unlikely that the various deviations from the means will line up in the same direction.
Yes, I think that's part of the point of the post. One intuition is that allocating only a little bit to a highly volatile asset creates a not-very volatile asset. Investing a little bit is the same as scaling the asset down until it's not very volatile.
I realize that this is meant as an exercise to demonstrate a property of variance. But most investors are risk-averse when it comes to their portfolio - for the example given, a more practical target to optimize would be worst-case or near-worst-case return (e.g. p99). For calculating that, a summary measure like variance or mean does not suffice - you need the full distribution of the RoR of assets A and B, and find the value of t that optimizes the p99 of At+B(1-t).
A handful of the comments are skeptical of the utility of this method. I can tell you as a physical scientist, it is common to make the same measurement with a number of measuring devices of differing precision. (e.g. developing a consensus standard using a round-robin.) The technique Cook suggests can be a reasonable way to combine the results to produce the optimal measured value.
I'm not a physical scientist, but I spend a lot of time assessing the performance of numerical algorithms, which is maybe not totally dissimilar to measuring a physical process with a device. I've gotten good results applying Simple and Stupid statistical methods. I haven't tried the method described in this article, but I'm definitely on the lookout for an application of it now.
I wonder if this minimum variance approach of averaging the measurements agrees with the estimate of the expected value we'd get from a Bayesian approach, at least in a simple scenario, say a uniform prior over the thing we're measuring and assume that our two measuring devices have unbiased errors described by normal distributions.
There exists a problem in real life that you can solve in the simple case, and invoke a theorem in the general case.
Sure, it's unintuitive that I shouldn't go all in on the smallest variance choice. That's a great start. But, learning the formula and a proof doesn't update that bad intuition. How can I get a generalizable feel for these types of problems? Is there a more satisfying "why" than "because the math works out"? Does anyone else find it much easier to criticize others than themselves and wants to proofread my next blog post?
Here's my intuition: you can reduce the variance of a measurement by averaging multiple independent measurements. That's because when they're independent, the worst-case scenario of the errors all lining up is pretty unlikely. This is a slightly different situation, because the random variable aren't necessarily measurements of a single quantity, but otherwise it's pretty similar, and the intuition about multiple independent errors being unlikely to all line up still applies.
Once you have that intuition, the math just tells you what the optimal mix is, if you want to minimize the variance.
This all hinges on the fact the variance is homogeneous to X^2, not X. If we look at the standard deviation instead, we have the expect homogeneity: stddev(tX) = abs(t) stddev(X). However, it is *not linear*, rather stddev(sum t_i X_i) = sqrt(sum t_i stddev(X_i)) assuming independent variables.
Quantitatively speaking, t^2 and (1-t)^2 are always < 1 iff |t| < 1 and t != 0. As such, the standard deviation of a convex combination of variables is *always strictly smaller* than the convex combination of the standard deviations of the variables. In other words, stddev(sum_i t_i X_i) < sum_i t_i stddev(X_i) for all t != 0, |t|<1.
What this means in practice is that the convex combination (that is, with positive coeffs < 1) of any number of random variables is always smaller than the standard deviation of any of those variables.
This is equivalent to inverse variance weighting. For independent random variable, this is the optimal method to combine multiple measurements. He just used a different way to write the formula and connect that to other kinds of functions.
He also frames it as a different goal too: normally when we (as a physicist) talks about the random variables to combine, we think of it as different measurements of the same thing. But he didn’t even assume that: he’s saying if you want to have a weighted sum of random variables, not necessarily expected to be a measurement of the same thing (eg share same mean), this is still the optimal solution if all care is minimal variance. His example is stock, where if all you care is your “index” being less volatile, inverse variance weighting is also optimal.
As I’m not a finance person, this is new to me (the math is exactly the same, just different conceptually in what you think the X_i s are).
I wish he mention inverse variance weighting just to draw the connection though. Many comments here would be unnecessary if he did.
The primary problem with this method is that while one correctly assumes one cannot forecast future returns, it incorrectly assumes one can correctly forecast future volatility of those returns. To be sure, Vol / variance of returns is more predictable than returns, but it's not perfectly predictable.
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[ 4.5 ms ] story [ 51.5 ms ] threadDon’t make decisions for evolving systems based on statistics.
Insider info on the other hand works much better.
While by no means logically incorrect, it feels inelegant to setup a problem using variables A and B in the first paragraph and solve for X and Y in the second (compounded with the implicit X==B, and Y==A).
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Write v_i = Var[X_i]. John writes
But if you multiply top and bottom by (1 / \prod_{m=1}^n v_m), you just get No need to compute elementary symmetric polynomials.If you plug those optimal (t_i) back into the variance, you get
where `H = n / (\sum_{k=1}^n 1/v_k)` is the Harmonic Mean of the variances.There exists a problem in real life that you can solve in the simple case, and invoke a theorem in the general case.
Sure, it's unintuitive that I shouldn't go all in on the smallest variance choice. That's a great start. But, learning the formula and a proof doesn't update that bad intuition. How can I get a generalizable feel for these types of problems? Is there a more satisfying "why" than "because the math works out"? Does anyone else find it much easier to criticize others than themselves and wants to proofread my next blog post?
Once you have that intuition, the math just tells you what the optimal mix is, if you want to minimize the variance.
Is it?
You have ten estimates of some distance with similar accuracy of the order of 10m : you take the average (and reduce the error by more than half).
If you increase the precision of one measure by 1% you will disregard all the others?
Quantitatively speaking, t^2 and (1-t)^2 are always < 1 iff |t| < 1 and t != 0. As such, the standard deviation of a convex combination of variables is *always strictly smaller* than the convex combination of the standard deviations of the variables. In other words, stddev(sum_i t_i X_i) < sum_i t_i stddev(X_i) for all t != 0, |t|<1.
What this means in practice is that the convex combination (that is, with positive coeffs < 1) of any number of random variables is always smaller than the standard deviation of any of those variables.
He also frames it as a different goal too: normally when we (as a physicist) talks about the random variables to combine, we think of it as different measurements of the same thing. But he didn’t even assume that: he’s saying if you want to have a weighted sum of random variables, not necessarily expected to be a measurement of the same thing (eg share same mean), this is still the optimal solution if all care is minimal variance. His example is stock, where if all you care is your “index” being less volatile, inverse variance weighting is also optimal.
As I’m not a finance person, this is new to me (the math is exactly the same, just different conceptually in what you think the X_i s are).
I wish he mention inverse variance weighting just to draw the connection though. Many comments here would be unnecessary if he did.