Portfolio variance

[dinu]

Hi David

I have couple of doubt in Investment research. If we estimate a well diversified portfolio with limited number of stocks, i.e., the correlation between the stock is less and the stock have high expected returns. Given this portfolio, when we maximise the sharpe ratio from the historical returns, variance and covariance of these stocks, then whether the optimised weights would lie in the efficient forntier as a well diversified portfolio (market portfolio) weights would lie in the efficient frontier.

My second doubt is that whether withought increasing the number of stocks in a well-diversified portfolio, can we arrive at efficient weights for a portfolio.

 

[David Harper CFA FRM]

Hi dinu,

I think we may have 4 meanings of efficient:

1. efficient frontier (EF) of n-risky assets (plot return against risk)
2. efficient frontier of all risky assets (before intro the riskfree asset); the upper curvy line
3. CML - which becomes efficient after we intro the riskfree asset
4. market portfolio: most efficient portfolio of risky assets "highest sharpe ratio"

Your optimized portfolio would definitely lie on the efficient frontier of the plot that included only your portfolio assets (#1 above).

However I *think* under the onerous CAPM assumptions, if your portfolio does not include all risky assets, it will be sub-optimal vis a vis the "market portfolio;" I do not think it lies, strictly speaking, on the EF of #2 above, which includes the "market portfolio." The thing about the market portfolio is that it's based on the set of incredibly stringent assumptions including "all investors have the same information and views" and so, no asset can be omitted from the market portfolio. So, in regard to your 2nd and 3rd questions, I think the answer is: your portfolio will be efficient in regard to its own risk/reward, but can always be improved by adding asset(s) that make the portfolio resemble the market portfolio (the portfolio with the highest sharpe ratio). B/C under the unrealistic assumption, all investors will hold the market, their only decision is allocation between the market portfolio and the riskfree rate; i.e, they are allocating along the efficient (#3 above) CML, which include market portfolio.

as i think this out, i think we can conclude that, since the CML is tangent to the market portfolio, any risky portfolio different than the market portfolio must be "less efficient" than the any point on the CML...hope that helps..David

 

[ashm07]

Hi David,

Since market portfolio does not exist in the real world. Even the riskfree rate is not truly exist. So how do people calculate CML and beta, or they dont? Is it all just academic concept? In real world, would people use just any discount rate for the portfolio of risky assets and find an optimal mix and then use the beta thus calculated to compare different investment alternatives? This optimal mix will be called just an optimal portfilio and not a market portfolio, right?

Thanks.

 

[David Harper CFA FRM]

Hi seaTurtle - Big topic, the CFA Level 3 is a lot about this (optimization, asset allocation). You've got some professional optimizers out there. In my (limited) experience, this CML is academic, I never saw it for real, but maybe it's used ... For beta, many use a index proxy (S&P 500, Rusell) for the market; you can use beta (estimated) without going all the way to Markowitz (this is really what the FRM should study, right? actual practices not Markowitz...) but when i was consulting (I am not current this way) I sort of saw two sorts of practice: simple (old school, would laugh at CAPM/beta) or sophisticated (and therefore beyond single factor models, and would also laugh at CAPM/beta). To tell the truth, the only folks i saw talk about CAPM/beta were consultants

You don't need the entire Markowitz to solve for the efficient portfolio among a set of stocks, or the highest sharpe ratio (the subset equivalent to the market portfolio - I don't know what to call it...). As you suggest, it's a valid to simply solve for optimal portfolio within subset universe of available securities. The issue isn't getting market beta, to my knowledge, rather it's the undue sensitivity to input assumptions (where, as Jorion points out, with 100 assets, you need a covariance matrix = 100*101/2 covariance entries plus, worse, expected returns for all of them. Tweak a little and you can get big allocation differences. Epic model risk. The kicker, as has been proven now, is that your covariance matrix goes to mush during a crisis precisely when you need it). Black-Litterman was promoted by Goldman back when i was consulting, don't know how that's doing in practice...the Amenc text, beyond our assigned chapter, goes into some detail. But the best resource I know on this is CFA L3...David