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 minimize 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).
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 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).
That's the first thing I thought of. I read the opening of this article and thought "oh this could be applied to a load balancing problem" but it immediately becomes obvious that you can't assume the variance is going to be uniform over time
This is why Markowitz isn't used much in the industry, at least not in a plug-and-play fashion. Empirical volatility, and the variance
-covariance matrix more generally speaking, is a useful descriptive statistic, but the matrix has high sampling variance, which means Markowitz is garbage in garbage out. Unlike in other fields, you can't just make/collect more data to reduce the sampling variance of the inputs. So you want to regularize the inputs or have some kind of hybrid approach that has a discretionary overlay.
Doesn't it make more sense to measure and minimize the variance of the underlying cash flows of the companies one is investing in, rather than the prices?
Price variance is a noisy statistic not based on any underlying data about a company, especially if we believe that stock prices are truly random.
What a weird way to write the harmonic average.
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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.Please will the mods implement maths rendering?? If the source were made available we could do it ourselves.
It’s much clearer when you write these problems in terms of matrix math. The minimum variance portfolio is very important in finance.
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 minimize 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).
It's hard enough as is to get a reliable variance-covariance estimate. I am immediately sceptical of anything more esoteric, it's a smell.
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 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).
There are lots of good books on writing mathematics:
1. How to Write Mathematics — Paul Halmos
2. Mathematical Writing — Donald Knuth, Tracy Larrabee, and Paul Roberts
3. Handbook of Writing for the Mathematical Sciences — Nicholas J. Higham
4. Writing Mathematics Well — Steven Gill Williamson
This is just the observed variance. Which means that you assume that this will be the variance in the future.
Don’t make decisions for evolving systems based on statistics.
Insider info on the other hand works much better.
That's the first thing I thought of. I read the opening of this article and thought "oh this could be applied to a load balancing problem" but it immediately becomes obvious that you can't assume the variance is going to be uniform over time
This is why Markowitz isn't used much in the industry, at least not in a plug-and-play fashion. Empirical volatility, and the variance -covariance matrix more generally speaking, is a useful descriptive statistic, but the matrix has high sampling variance, which means Markowitz is garbage in garbage out. Unlike in other fields, you can't just make/collect more data to reduce the sampling variance of the inputs. So you want to regularize the inputs or have some kind of hybrid approach that has a discretionary overlay.
Upvoting b/c this comment is true, obviously I disapprove of insider trading.
Volatility is fairly predictable. Or at least much more predictable than returns
Doesn't it make more sense to measure and minimize the variance of the underlying cash flows of the companies one is investing in, rather than the prices?
Price variance is a noisy statistic not based on any underlying data about a company, especially if we believe that stock prices are truly random.