Markowitz model
All portfolios that lie below the Efficient Frontier are not good enough because the return would be lower for the given risk.
Portfolios that lie to the right of the Efficient Frontier would not be good enough, as there is higher risk for a given rate of return.
An investor who is highly risk averse will hold a portfolio on the lower left hand of the frontier, and an investor who isn’t too risk averse will choose a portfolio on the upper portion of the frontier.
The investor's optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curve.
A portfolio with risk-free securities will enable an investor to achieve a higher level of satisfaction.
Any point on the line R1PX shows a combination of different proportions of risk-free securities and efficient portfolios.
In the market for portfolios that consists of risky and risk-free securities, the CML represents the equilibrium condition.
The Capital Market Line says that the return from a portfolio is the risk-free rate plus risk premium.
Figure 5 shows that an investor will choose a portfolio on the efficient frontier, in the absence of risk-free investments.
The portion from IRF to P, is investment in risk-free assets and is called Lending Portfolio.
Unless positivity constraints are assigned, the Markowitz solution can easily find highly leveraged portfolios (large long positions in a subset of investable assets financed by large short positions in another subset of assets) [citation needed], but given their leveraged nature the returns from such a portfolio are extremely sensitive to small changes in the returns of the constituent assets and can therefore be extremely 'dangerous'.
Positivity constraints are easy to enforce and fix this problem, but if the user wants to 'believe' in the robustness of the Markowitz approach, it would be nice if better-behaved solutions (at the very least, positive weights) were obtained in an unconstrained manner when the set of investment assets is close to the available investment opportunities (the market portfolio) – but this is often not the case.
Practically more vexing, small changes in inputs can give rise to large changes in the portfolio.
[3] In the real world, this degree of instability will lead, to begin with, to large transaction costs, but it is also likely to shake the confidence of the portfolio manager in the model.
Furthermore, the information dependency and the need to calculate a covariance matrix introduces some, albeit manageable, computational complexity and constraint to model scalability for portfolios with sufficiently large asset universes.
The expected returns are uncertain, and when we make this assumption, the optimization problem yields solutions different from those of the Markowitz Model.
