Portfolio Allocation

The decision to modify the capital allocation of a portfolio generally follows two types of events. First, when new capital is made available, it needs to be invested. The allocation of new capital can be achieved either by acquiring new instruments or increasing existing positions. The second scenario for adjusting portfolio allocation is when there is a shift in expectations. Investors may seek a change in the characteristics of a portfolio to, for example, maintain its level of risk during turbulent market conditions or improve its expected return in the face of heightened inflation.

Capital allocation almost always involves making trade-offs between investment objectives and a number of constraints. A typical example of such a trade-off is striking a balance between reducing volatility (an objective) and maintaining the portfolio's level of diversification (a constraint). While these trade-offs are relatively easily to express in plain language, constructing portfolios that maximize an investment objective while complying with a number of constraints is generally quite involved. For portfolios with more than a few instruments, even simple optimization tasks are sufficiently involved in terms of data and calculations that require the help of a computer. In most practical cases involving different regions, currencies, and dozens of assets, the use of a computer is in fact unavoidable.

Computer programs used for portfolio optimizations first translate investment objectives and constraints into mathematical language and then execute an algorithm that returns a so-called "optimal" portfolio. The Scientific Portfolio (SP) platform performs this translation so that users can focus on defining their financial objectives and portfolio characteristics. As we detail below, the SP platform aims to offer a straightforward, user-friendly portfolio optimization experience by proactively shielding users from its most common pitfalls. The platform is equipped with the capability to optimize an existing portfolio, with metrics that are designed to be actionable.

Currently, the Scientific Portfolio platform allows the optimization of the following objectives:

• Risk reduction

• Increased factor intensity

• Enhanced factor quality

Reshuffling and completing

There are essentially two methods for adjusting a portfolio's capital allocation. One approach involves modifying the weights of existing instruments, a process that we refer to as reshuffling. The other approach is to acquire new instruments, which is referred to as completing. On the SP platform, these two operations are distinct and must be conducted sequentially. For instance, the integration of new instruments will often come before a reshuffling.

Reshuffling

When reshuffling a portfolio's weights, adjustments are applied to existing positions. It is important to note that depending on the user-defined constraints, some weights may be reduced to zero, effectively resulting in divestment. The constraints available to control the impact of the optimization process on the portfolio's characteristics will be detailed for each investment objective.

Completing

Portfolio completion involves finding new instruments that, when added to the portfolio, will result in an improvement of a specific investment objective, such as reducing risk or improving factor quality. Once the investment objective has been selected, the optimization process proceeds in two steps:

• The user is presented with a list of instruments that have the potential to improve the investment objective of the current portfolio.

• Once the user selects the instruments to be added to the portfolio, an optimization process is performed to determine the optimal weights for the newly added instruments.

During portfolio completion, the user is required to specify an investment universe from which the new instruments will be selected and the amount of new capital dedicated to the acquisition of these new instruments. An important point to note is that a completion task is always performed while keeping the weights of the current portfolio fixed.

Constraints

Portfolio optimization always adheres to a set of constraints. These constraints serve to control the impact of the optimization process on weight changes. There are essentially two types of constraints that offer control over either the weights themselves, or the portfolio's risk profile:

• Trading constraints control the maximum amount of capital that can be taken in and out of any particular instrument.

• Risk constraints control the portfolio's characteristics in terms of relative risk or factor exposure.

The following table provides a summary of all available constraints. As detailed below, for a certain objective, a number of constraints may become unavailable.

Compatibility of Objectives and Constraints

In many platforms and systems offering portfolio optimization tools, the user has the freedom to combine all of the available optimization objectives and constraints. However, this flexibility can make portfolio optimization a very delicate task for many users. First, the interactions between objectives and constraints may have unintended side effects. For instance, reducing the volatility may drastically reduce certain factor exposures. The avoidance of these side effects demands careful management of a number of technicalities, which requires a specific skill set. Second, and more importantly, portfolio optimization often fails as constraints are set in a way that makes the optimization program unfeasible. It is often difficult to understand why an optimization fails, and methods to pinpoint the origin of failures, such as relaxing each constraint separately, can prove to be very tedious.

The SP platform has developed and uses the concept of optimization atoms, or bricks. Each optimization task performed on the platform is indeed designed to guarantee success. This is achieved by proactively preventing constraints from making the optimization problem unfeasible. In such cases, the worst outcome for an optimization task is a return to the current portfolio, indicating that an improvement aligned with the investment objective was not identified. Consequently, not all constraints are universally available for every objective, and there may be instances where the scope or use of constraints is limited.

For instance, the platform does not allow to set a constraint on the factor intensity that is a higher value than that of the current portfolio. This is because setting such a constraint to a higher value could make the optimization fail if the portfolio already has the maximum intensity. This example illustrates how the platform may limit the range of constraints to prevent failures. When constraints are made available, their parameters cannot be set to unachievable values. Thus, the use of constraints is at times restricted so as to guarantee the success of every optimization task.

The fundamental concept behind optimization bricks is to avoid users setting unfeasible constraints, a situation often encountered without a clear understanding of the origin of the optimization failure. This feature makes the SP portfolio allocation tool a reliable partner for investors seeking to implement adjustments aligned with their investment objectives. Furthermore, bricks allows portfolios to be optimized for multiple investment objectives by simply running sequences of optimizations. Multi-objective portfolios are often sought by users, but many optimization tools are not built to deal with multiple objectives explicitly. Instead, many users integrate multiple objectives by bootstrapping a manual version of the so-called "epsilon-constraint method". This method involves progressively increasing the level of certain constraints until the optimization program eventually becomes unfeasible. For instance, with a standard optimizer, a user aiming to reduce risk while simultaneously increasing the factor intensity would need to designate risk reduction as an objective and carefully increase the factor intensity constraint until the optimizer starts to fail, thus indicating that the maximum value has been exceeded. This process of trial and error would need to be repeated each month, as changes in the underlying data would modify the value of the maximum feasible intensity.

In contrast, the same optimization with the SP platform would simply involve executing two bricks. The first optimization would reduce the volatility, while the second would maximize the factor intensity while maintaining the current level of risk. These two optimizations implement the same investment objective, yet their implementation is guaranteed to succeed, regardless of the underlying data and without the need for further intervention. Notice that the sequence of bricks matters, as the user must decide which objective to optimize first. This forces users to explicitly define hierarchies among objectives, a step that is typically only implicitly done, and generally not in a controlled fashion, when balancing objective and constraints in a single optimization.

Convexity and Tractability

Before proceeding to the more technical part of this document, it is useful to understand which conditions are required for a portfolio optimization to be successful.

In the world of optimization, the word difficult often does not mean that a problem has many variables or that the objective of the optimization is very complex. In fact, the concept of difficulty is closely related to three important qualities: convexity, tractability, and the absence of combinatorics.

In general, the optimization of a portfolio with convex objectives and constraints can be solved efficiently. This is because convex functions have a unique minimum and there is, so to speak, no obstacle between the optimizer and the solution. In other words, the optimization process can freely navigate between the initial value and the solution. Standard optimizers can quickly and efficiently solve convex problems for sometimes millions of variables. Therefore, difficulty here is not related to size or complexity but rather to convexity. Many common investment objectives, such as those involving some form of equalization of risk contributions, lack convexity and are therefore computationally challenging. However, excellent approximate solutions are available for this class of problems, and therefore although they are not strictly convex, they are (or will become) available on the platform.

The absence of combinatorics is another often-required condition for a problem's solvability in principle. Combinatorics refers to the need to test numerous combinations of potential values in a problem. For instance, consider the seemingly straightforward problem of finding the smallest non-zero integer that satisfies

. In this case, the constraint that the solution must be a non-zero integer means that the optimizer will need to test all possible combinations of integers multiplied by +1 and -1. This becomes difficult when the number of possible combinations grows, which is often the case as the number of variables increase. Unfortunately, a common problem in finance involves limiting the number of positions to a fixed quantity. This constraint involves combinatorics since it requires all combinations of positions to be tested until the one that minimizes the investment objective is identified. There are many methods that facilitate the integration of such constraints, but their execution time is often unreliable. This is why such constraints are not available on the SP platform.

Even if a problem is both convex and devoid of combinatorics, it does not guarantee that it can be solved easily by a computer. In fact, an additional quality that a function must have to be optimized is called tractability. By tractable, we mean that the function must be expressible in a form that can be solved easily with existing optimization algorithms. Tractability is hardly ever a problem for commonly used investment objectives, although some objectives involving ratios may not be solved at once and could therefore require a (simple) iterative approach. This is typically the case for the optimization of, for example, factor quality or risk contribution objectives.

Standard form

Let us now introduce some notation. In this paper, all optimization programs will be written in the following form:

where the function

is referred to as the objective, the functions

and

as the constraints and the sign ≼≼ corresponds to the element-wise inequality if

is a vector. In human language, the above equation reads as follows:

Find values for each element of

so that the function F(x) is minimized subject to the constraints

and

.

This way of specifying optimization programs is known as the standard form. It is a convention whose purpose is to bring some standardization into the definition of optimization problems, both in academic papers and software documentation. The direction of the inequalities also means that for the optimization problem to be convex, both the objective and constraint functions must be convex.

In the sequel, the weights of the portfolios will be denoted by

with

representing the number of instruments in the portfolio. However, in many portfolio optimization programs, we will consider many more variables than just the weights of the portfolio. These variables will be added to the optimization in order to implement the different objectives and constraints. As a result, we will always write the variables to optimize as

where the additional variables will be denoted by

​.

Linear constraints

Linear constraints are the simplest form of constraints, yet they play an incredibly important role in portfolio optimization. As we will see below, these constraints make it possible to set budget constraints, limit turnover between two portfolios or simply limit the capital invested in a given region or instrument. Indeed, many investment constraints are implementable with linear constraints.

Linear equality and inequality constraints are defined as follows:

where the matrix

contains

rows. In the sequel, we will denote by

​ the

row of the matrix

. Each row defines an inequality which effectively limits the range of a variable. For instance, if we wish to limit the first weight of the portfolio to a positive value, one would have to add a

line to the matrix

and a

element to the vector

such that:

​In this way, all portfolios produced by the optimizer will have the constraint that

, which is otherwise identical to

. All linear constraints are obtained in a similar fashion, that is, by adding rows and elements to

and

, respectively.

Positive weights constraint

On the SP platform, all portfolio optimizations implement the positive weight constraint, which means that:

This constraint is linear and requires the addition of

linear constraints that are similar to the one shown above.

Budget constraint

On the SP platform, all optimizations also have a budget constraint defined as:

​where

​ is the initial portfolio's sum of weights. Therefore, the total weight of the final portfolio is equal to the total weight of the initial portfolio. In general, this weight should be equal to one.

Maximum weight difference and total turnover

The maximum weight difference limits the difference between the weight of an instrument in the initial portfolio and its weight in the final portfolio. This is a linear constraint whose technical implementation requires the introduction of two intermediary variables

and

.

By setting the following linear constraints

​we create the following equality:

Consequently, setting a limit on the the total turnover of the portfolio or the maximum weight difference is obtained by simply adding the linear constraints:

Factor Intensity constraint