The Efficient Surface
The Efficient Frontier
This post builds on the material in The Efficient Frontier post. Recap: In order for a portfolio to be considered optimal, it must be located on the efficient branch of the efficient frontier (highlighted in orange).
Figure 1: The Efficient Branch of The Efficient Frontier
Contract percent shares have non-negative lower bounds with finite upper bounds, thus our most basic loss model portfolio optimisation problem is constrained. Consider the maximum portfolio premium case, Pmax; the portfolio manager chooses to write the upper bound of every contract percent share. There is only one possible portfolio for Pmax and this is also the only efficient portfolio. The constrained efficient frontier (CEF) in this case is one single point.
By and large, as the total portfolio premium increases from Pmin ≥ 0 to Pmax; more contribution must be made from the more correlated/volatile/ loss making contracts in order to satisfy the necessary portfolio premium requirement.
The contract percent share upper bounds essentially interfere with the optimal portfolio construction as 𝑃 increases – leading to an increase in volatility and a reduction in the obtainable portfolio rate of return. The range of possible returns decreases and as 𝑃 approaches Pmax the associated CEF narrows to a single point.
Figure 2 shows how the whole solution space typically narrows as more premium is demanded from the same portfolio of contracts.
At Pmax the range of possible portfolio's has narrowed to one point.
The efficient branch of the efficient frontier is highlighted in orange in each case.
Figure 2: Narrowing Portfolio Space
The Efficient Surface
For loss modelling purposes the efficient frontier is incomplete. We must include the total portfolio premium as an extra dimension. This results in a 2-dimensional ‘efficient surface’ embedded in 3-dimensional (𝜇, 𝜎, 𝑃) space. (Note: the standard deviation is plotted instead of the variance as the units of 𝜎 conveniently match those of 𝜇)
The constrained efficient frontier associated with each 𝑃 stops at the minimum variance point. Hence, only the efficient branch of each CEF is shown.
Figure 3: The Efficient Surface
Figure 3 shows an example efficient surface:
The blue sphere represents the current (inefficient) portfolio co-ordinates.
The surface shows all of the efficient (𝜇, 𝜎, 𝑃) combinations given the bounds and constraints entered by the users.
The orientation of the associated blue pane shows the intersection of the surface for the current portfolio premium. The blue arrows show that for the same premium amount, the portfolio could attain either a higher return for the same level of risk, or a lower level of risk for the same return. A combination of higher return and lower risk can be achieved and this is best quantified via the Sharpe ratio.
The user chooses an efficient portfolio from the surface. More expected return => More Risk. More premium => More Risk
The efficient surface is the key result that RiskWave produces. From this, further useful analytics can be derived (such as a value at risk surface) which aid the PM in understanding all of the efficient possibilities of their portfolio.