Chapter 6: Mean-Variance Optimisation Loss Function

[Anjum48]

Hi Cam,

Could you explain how to use (or reference to a source) this loss function:

I'm assuming that the loss function is attempting to minimise the portfolio weights for the 4 stocks (which can be done using scipy optimise etc.), but I'm not sure what the lambda parameter is.

Thanks!

[pieteradejong]

@Anjum48 which chapter is this?
@CamDavidsonPilon lambda seems like the regularization parameter?

[CamDavidsonPilon]

loss function is attempting to minimise the portfolio weights for the 4 stocks

This is trivially solved by setting them all to negative infinity, so I don't think you meant to say that. It's two counteracting forces: the larger the weights, the larger the first term (return), but also the larger the variance (second term).

The lambda parameter is a user specified parameter that is a tradeoff between maximizing the return (the first term) and the second term (the portfolio variance). It's not a regularizer in the traditional regression sense but it is analogous.

This is a pretty common formula in finance - it's the lagrangian of a quadratic problem: http://www.actuaries.org/AFIR/Colloquia/Rome2/Cesarone_Scozzari_Tardella.pdf

[yiyuezhuo]

@CamDavidsonPilon
I don't understand that the optimizer use min, I mean, you should want to maximize your return and
be penalized by variance. So you should use max or append a negative sign instead of using min.

[mdengler]

The lambda 𝛌 is often referred to (e.g. here) as the "risk aversion" parameter -- the tradeoff @CamDavidsonPilon mentions.

[rsvp]

If lambda 𝛌 = 1, then the objective function approximates the portfolio's
geometric mean return, and that's something one should maximize
with respect to the weights (which by construction should sum to 1).

[mdengler]

There definitely seems to be a minus sign missing.

[CamDavidsonPilon]

Yea it should be max 👍

[rsvp]

OK, fix done:

#347 for two notebooks.

For the record, another good reference, by William Sharpe:
https://web.stanford.edu/~wfsharpe/mia/opt/mia_opt3.htm --
and I'll be doing a tutorial at https://git.io/fecon235
in regards to Boltzmann portfolios.