Minimum-variance frontier


Does the minimum-variance frontier and the global minimum-variance portfolio only include risky assets?

Yes. It is the capital market line that would include a riskless asset.

Note that the point of tangency of the efficient frontier and the capital market line means that your exposure to the riskless asset is zero and the portfolio consists of the risky assets at their market weights.

Moving the the left of the point of tangency (market portfolio) indicates lending at the risk free rate (buying bonds), and anything to the right of the market portfolio indicates borrowing at the risk free rate.

Academics like to say that the tangency portfolio has to have all risky assets at their market weights. I know of no mathematical proof that this must be the case, and no intuition that suggests that it must be so.

Thank you guys!

Just another quick question, can you confirm the below is correct please?

Under the assumption of Homogenous Expectations : All investors face the same efficient frontier of risky portfolios and will all have the same optimal risky portfolio ( Market portfolio of risky assets ) and CAL.

So the only difference would be each investors indifference curves? Risk averse investors would choose a higher weight of the risk free asset and a lower weight of the market portfolio? and therefore, would receive a lower return than an investor who is less risk averse?

You got it!

It’s called Tobin’s Separation Theorem: you separate:

  • Which risky portfolio to choose – everyone chooses the same one (the market portfolio) – from
  • How much risky portfolio/risk-free asset you have in your portfolio (determined by your indifference curves: your level of risk aversion.

What I have learned is that assuming the (efficient) market is in equilibrium there will be no excess demand for any given asset. Because of this and homogeneous expectations, we can say that investors all hold the same tangency portfolio, which consists of all risky assets in their market weights. The idea is that the market equilibrium requires the supply of all assets to be equal to the demand for these assets, meaning no asset is held outside of its market weighting. This information (and a lot of other stuff) is also useful in deriving CAPM. Not sure if this helps with the intuition, though.

Just remember, CAPM assumptions are CHORD FU:

_ C _ompetive markets

H omogenous Expectations

_ O _ne-period time horizon

_ R _isk aversion

_ D _ivisible assets - Infinitely

_ F _rictionless market

U tility maximizing investors

Excellent! thank you guys :slight_smile:

My pleasure.

Also need to add the assumption that investors can borrow and lend unlimitedly at the risk-free rate (frictionless markets allow for the borrowing/lending rate to be the same). Additionally, no imperfections such as taxes (might be another one or two depending how finely you want to dice these up).

Thanks, glad we can fill the gaps in our studies.