What a weird way to write the harmonic average.

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Write v_i = Var[X_i]. John writes

    t_i = \frac{\prod_{j\ne i} v_j}{\sum_{k=1}^n \prod_{j\ne k} v_j}.

   t_i = \frac{1/v_i}{\sum_{k=1}^n 1/v_k}.

If you plug those optimal (t_i) back into the variance, you get

    \min Var[\sum t_i X_i] = 1/(\sum_{k=1}^n 1/v_k) = H/n,

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).

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.