Below is an efficient frontier, i.e. the navy-blue continuous curve that passes between points a b c and d. It illustrates the trade-off between return and risk. The efficient frontier is essentially a universe of assets, or a set of portfolios of assets that offer the minimum risk for a given level of return. The black line, that is a tangent to the efficient frontier, and is passing between points c and d is the CAPM line. It contains, at its intersection with the efficient frontier, all iterations of portfolios containing the risk free asset that rational market participants can select.
Therefore, given the objective stated in Part 1, we seek to find point c on the intersection of the CAPM and the efficient frontier.
Therefore, given the objective stated in Part 1, we seek to find point c on the intersection of the CAPM and the efficient frontier.
Click on Image for Better VisibilityAs I mentioned before, the optimization problem introduced in part 1 is unconstrained, and generally it is standard practice to trace-out the efficient frontier by introducing a weighting parameter λ (λ is greater than or equal to zero but less than or equal to one), and considering:
We'll call the expression above equation 5 and it is subject to equations 6 and 7 which follow respectively:
In equation five, the case λ = 1 represents minimizing the risk, and when we substitute λ for 1 in equation 5, we get equation 4, which can be expanded into the following expression (which we'll call equation 8), in the case of a two asset portfolio:
Where:- X1 and x2 represent wi and wj respectively
- σ1 and σ2 represent the standard deviation of asset i and the standard deviation of asset j respectively
- p12σ1σ2 is equivalent to σij and represents the covariance of asset i and asset j
Stay tuned for the third part of this installation, where I'll convert the optimization problem into a binary quadratic problem, and plug in variables to generate a problem matrix for DWave's Orion to solve.
