Pillai 'Portfolio Optimization'

How to distribute a fixed amount of capital among various stocks/commodities so as to generate a "good" portfolio is addressed here along the classic ideas of Markowitz, where the volatility of the portfolio as measured by its overall variance is minimized by adjusting the capital allocation percentage on each item invested. This approach is further explored here, by simultaneously maximizing the overall portfolio gain while minimizing the portfolio variance. Towards this, their ratio is maximized by borrowing ideas from "matched filter" design in signal processing.

We also introduce the notion of "tight stocks", whereby it is meant that the inverse of their covariance matrix of the returns of such investments has all nonnegative entries. They are said to be "loosely tight stocks" if each row sum or a generalized sum of the inverse of their covariance matrix is nonnegative. Such sets of stocks/investments as a group are shown to be natural candidates to form Portfolios.

[probabilitystochasticproce]

At T=21.55 min, the inverse matrix R^-1 having nonnegative entries might be a too severe constraint, but all we need is that each row sum of R^-1 be positive (nonnegative).
Same comments apply at T=42 min. All we need is that each row of R^-1 scaled by the \mu vector and summed be positive.

[probabilitystochasticproce]

List of minor Errors:
At T= 16.53 min, the constraint (second) term on the right side has a negative sign in front of it, and to be consistent, the positive sign at T=16.33 min should be changed to a negative sign in front of the constraint term there.


Also, a T=17.58 min, the secnd entry on the last rowof the matrix should be C_n, 2 (and not C_2, 2 as shown there).

[probabilitystochasticproce]

List of minor Errors:
At T=30.50 min, the vector "a^T" should be just "a" at the end of the variance expression.

At T=31.16 min, the second term [r(t) - \mu] inside the expectation should be transposed as in the line above. Also the next vector "a^T" should be just "a" at the end.