Efficient Frontier

Some say the world will end in fire,
Some say in ice.
From what I’ve tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To say that for destruction ice
Is also great
And would suffice.
- Robert Frost

Definition

In finance, the efficient frontier is the set of portfolios maximizing return (benefit) for a bounded risk (cost). The resource allocation is therefore “efficient” - for the same or less cost, no better benefit can be achieved. Given time, any position not along the efficient frontier will come to be dominated by a position that is.

Concretely, for a $\text{Portfolio } p$:

$$Benefit(p) = B(p) = ExpectedReturn(p)\text{, and}$$ $$Cost(p) = C(p) = Risk(p)$$

Given cost $C(p)$ and benefit $B(p)$ functions, the $EfficientFrontier$ function $EF$ finds the set of porfolios $p$ maximizing benefit, for a certain risk budget.

$$EfficientFrontier(B, C) = \lbrace p \text{ s.t. } B(p) \ge B(p\prime), \text{ } \forall p\prime \text{ where } C(p \prime) \le C(p) \rbrace$$

Generalizing the idea beyond portfolios, consider the functions $C: \vec X \to Cost$ and $B: \vec X \to Benefit$. For brevity, since $C$ and $B$ share a domain $\vec X$, let $F_c = C, F_b = B$, and denote the pair simply as $F$. The generalized $EfficientFrontier$ could then be written as:

$$EfficientFrontier(F) = \lbrace \vec x \text{ s.t. } F_b(\vec x) \ge F_b(\vec x \prime), \text{ } \forall \vec x\prime \text{ where } F_c(\vec x\prime) \le F_c(\vec x) \rbrace$$

In a resource-bound optimization, the cost could simply be usage of the limiting resource: $F_c(\vec x) = R(\vec x)$.

The efficient frontier is a powerful tool, and can generalize well.

Example

To provide a concrete example benefitting from the efficient frontier, consider optimizing for (perceived) beauty, described by Taylor Swift:

“There’s always some standard of beauty that you’re not meeting. Because if you’re thin enough, then you don’t have that ass that everybody wants, but if you have enough weight on you to have an ass, then your stomach isn’t flat enough…”

The benefit to optimize, $F_b(\vec x) = Beauty(\lbrace x_1 = Ass, x_2 = FlatStomach \rbrace)$. Attempts to globally maximize $Beauty$ lead to a sense of hopelessness:

“..It’s all just fucking impossible.”

The key idea, therefore, is to introduce efficiency with a defined cost $F_c$, and constrain the search to $EfficientFrontier(F)$. In this case, one suggested cost constraint is: $F_c(\vec x) = -Health(\vec x)$, to ensuring a minimum, baseline health level. Under this constraint, globally maximal beauty may be unattainable, but sanity will remain.

Assumptions

The efficient frontier has several assumptions. Completeness of a search, for a given cost, assumes the search can both:

1. Fully cover all sufficiently relevant dimensions (no surprise dimensions outside of $\vec X$ affect $F_c$ or $F_b$, too much).
2. Fully expand each dimension $X_i$ (no surprise outliers in a given dimension $X_i$ affect $F_c$ or $F_b$, too much).

How grounded are these assumptions? The question of the missing dimension (1) is surprisingly difficult, and should wait for another post ¹.

For (2), however, consider the tool of the $\text{utopian extreme}$²: for each dimension $X_i$ under consideration, find a solution $\hat x_i$ so purely and singly optimized along that axis, that the solution is fatal ($F_b(\hat x_i) = 0$). In Frost’s terms, form each dimension $X_i$ in death by fire, or death by ice. Often, such a point $\hat x_i$ will have either infeasible, or trivial cost.

The $\text{efficient frontier}$ lies between $\text{utopian extremes}$; of course, one could find balance between the pure, fatal extremes. Further, constraints on $F_c$ can help to achieve a more reasonable solution.

Building on the previous example, Swift finds herself between two $\text{utopian extremes}$:

• $Ass \to \infty$: Death by obesity, and $Beauty(Ass \to \infty) \to 0$
• $FlatStomach \to \infty$: Death by malnutrition, and $Beauty(FlatStomach \to \infty) \to 0$

The utopian extremes serve two main purposes:

1. Compare benefit more absolutely: obsession with relative changes to benefit, $F_b$, can cause one to lose sight of how effectively the current state $\vec x_t$ currently optimises benefit $F_b$. Taylor Swift would be considered beautiful realive to a skeleton and a morbidly obese person.
2. Directionally, deterimine if the cost-restriction imposed on $F_c(\vec x_t)$ could be relaxed or tightened. Sometimes, thinking in extremes allows one to break out of an artificial limitiation. Taylor Swift could allow herself to be healthier, at a slight cost to some perceived benefit.

Ultimately, Taylor Swift realizes both purposes of the $\text{utopian extreme}$, and constrains her search to save her health:

I worked hard to retrain my brain that a little extra weight means curves, shinier hair and more energy.

Philosophical Frontiers

The concept of the efficient frontier becomes especially interesting when applied philosophically.

Below are some more abstract examples of how one could use utopian extremes to determine a good philosophical balance, for a given effort level.

As an example, consider optimizing the benefit of $Truth$ (Row 1), i.e. $F_b = Truth(\lbrace x_1 = Idealism, x_2 = Realism \rbrace)$, with cost $F_c = Action$. How much $Truth$ one can know, depends on a balance between $Idealism$ (to see what could be) and $Realism$ (to see what is). Action is the constraining resource in increasing both of these dimensions; it takes action to see what is, consider what could be, and change what is into what could be. More examples are provided in a table below.

Next, consider how each dimension could derange, at a utopian extreme. $Truth$ approaches zero, and action is not required, when idealism wholeheartedly ignores reality, or realism wholeheartedly accepts it.

Finally, establish the efficient frontier, lying between the two extremes, and constrained by how much of each resource one may invest towards each benefit. $EfficientFrontier(Truth, Action)$ could be Peace.

To guide action, one could simulate first how to act unreasonably, in $\text{utopian extremes}$ outside of an $EfficientFrontier$, and think of the consequences of these actions. Then, after calibrating the full range of possible outcomes, one may work to balance the good and bad of the extreme points, along an $EfficientFrontier$ with some bounded, feasible cost.