The purpose of this essay is to generalize the Markowitz portfolio choice problem and derive solutions to the variance minimization and utility maximization problems considering both general linear constraints and a single quadratic constraint. The Markowitz portfolio choice problem (introduced in [1] ) has been studied from many different perspectives. The problem is essential to minimize the variance of a linearly combined set of correlated random variables (returns), conditioned on the mean of these variables, with a constraint on the scalers or weights. The set of all such points is called the efficient frontier, i.e. the minimum variance envelope. The modern expression of the optimization problem outlined by Markowitz is due to Merton [2] who derives the solutions analytically using linear algebra. His solution considers the problem as a quadratic programming problem with two predefined linear constraints, the weight constraint (scalar constraint) and the expected return constraint (the mean). The statement of the problem is as follows. Matrices and vectors are in boldface.

w - n × 1 weight vector of the portfolio.

Ω - n × n covariance matrix (symmetric and positive definite).

r - n × 1 vector of expected returns.

1 - n × 1 vector of 1’s.

min w w ′ Ω w     s .t .     w ′ 1 = 1     and     w ′ r = r . (1.1)

However, the problem can also be solved with a general linear constraint as demonstrated by Best and Grauer [3] . The general linear constraint opens up possibilities to make considerations for and constraints on portfolio beta, sector or industry weights, multi-stage optimization, etc. I derive the solution to the general linear constraint problem in part 2 of this essay to familiarize the reader with my notation and so I can use it in my derivation of the maximum Sharpe Ratio problem.

However, since the original introduction of the model, modern thinking on portfolio construction (and financial theory in general) has evolved to consider the differences in incentives between the individual making the portfolio choices, the agent, and the individual hiring the delegated manager, the principal. This spawned a theory of active portfolio management, or delegated portfolio management, which centers on a time series of excess returns, essentially the returns of a security or portfolio in excess of a benchmark. This is a space transformation that causes agents to optimize in excess return space rather than raw return space (or even another excess return space with a different benchmark). Principals are then potentially motivated to constrain agent behavior in this space by putting limits on the variance of the delegated manager’s excess return, i.e. their tracking error variance.

There are many modern examples of papers analyzing delegated or relative problems. Bisburgen, Brandt, and Koijen [4] revisit the problem of multiple external agents relaxing the assumption about the certainty of the agents’ risk appetites and conclude that serious inefficiency exists without this assumption. Blake et al. [5] use the BBK framework and apply it to the delegated relationships in the pension industry and, among many other things, show that the delegated incentive is pervasive in professional investment management. Cuoco and Kaniel [6] analyze the linkage between compensation contracts and excess performance in delegated portfolio management. However, the delegated contracting problem can probably best be summed up by analyzing the tracking error constraint.

The first to analyze a constraint to tracking error in the context of a Markowitz portfolio choice model was Roll [7] . In this paper, Roll demonstrates the differences in portfolio choice for a decision-maker optimizing over variance versus one optimizing over tracking error. In a further refinement, Jorion [8] , applies a tracking error constraint to a mean/variance optimization problem, thus introducing a quadratic constraint into the problem. Whereas the Markowitz problem, under linear constraints, forms a minimum variance set shaped like a parabola in mean and variance, the tracking error constraint causes the efficient set to be shaped like an ellipse, with the feasible portfolios inside. See Figure 1 for a visual example of these curves.

The closest application of a utility problem directly to Roll’s TEV frontier by Bertrand [9] . In this paper, he considers the problem of a fixed risk aversion constraint in mean/variance and its ability to generate preferred portfolios.

The variance minimization problem is one approach to analyzing portfolio choice. However, it is also reasonable to consider the problem of maximizing utility. The popular utility function considered in this problem is a “quadratic-style” utility where the investor’s utility increases in the expected return, w ′ r , and decreases in the expected risk or variance, w ′ Ω w . And, the parameter, a, measures the sensitivity or the tradeoff: the level of risk aversion. The relationship is expressed as follows:

max w w ′ r − a w ′ Ω w (1.2)

Figure 2 shows a few of the curves depicted in Figure 1 with iso-utility curves corresponding to the same level of risk aversion, a.

Modern extensions of the Markowitz portfolio choice problem tend to focus more on altering something fundamental about the model or analyzing a specific constraint. For example, Chen et al. [10] optimize over semivariance but the model is adjusted to account for uncertainty in variable estimation based on historical data. Or authors may use modern techniques to analyze traditional problems like Kar et al. [11] that optimizes over the Sharpe Ratio and the Value at Risk simultaneously. These papers are very interesting and practical considerations for real world applications of this theory.

The primary contribution of this essay is in solving the problem proposed by Jorion [8] , optimizing with a tracking error constraint, under the conditions of the general linear constraint of Best and Grauer [3] . This expands and unifies two important developments in the mathematical analysis of the original Markowitz portfolio optimization model. I solve this problem with both a variance optimization approach and a utility maximization approach. The remainder of this paper precedes as follows. In section 2, I revisit the Best and Grauer [3] analysis with only a general linear constraint. In section 3, I derive the solution to the problem with the addition of a quadratic constraint (the tracking error constraint). This section contains this paper’s primary contribution and is a significant development in the ability to apply the general linear constraint to analyze modern delegated contracting with constraints and considerations to relative risk. In section 4, I consider an alternative utility function, the maximization of the Sharpe Ratio, Sharpe [12] , given these constraints. Lastly, section 5 concludes and suggests potential paths of future research.

The general linear constraint problem analyzed in Best and Grauer [3] comes in two forms, variance minimization given linear constraints and utility maximization given linear constraints. Although these problems are already solved, I repeat this calculation to familiarize the reader with my notation and so I can refer to my solutions throughout the paper. Additionally, I leave the expressions in the purest form possible, without making extraneous substitution. As before, the notation used throughout this paper is as follows:

w - n × 1 weight vector of the portfolio.

Ω - n × n covariance matrix (symmetric and positive definite).

E - n × k matrix of constraint conditions.

c - 1 × k vector of constraint constants.

The variance optimization problem given a tracking error constraint was originally solved in the context of portfolio choice in Jorion [8] . In this framework, he used the standard Markowitz constraints, the length of the weight vector and the weighted average expected return, to derive an ellipse in mean/variance space inside of which contains all portfolios with tracking error less than the constraint. In this section, which is the primary contribution of this essay, I derive “Jorion’s ellipse” with the general linear constraint of Best and Grauer [3] . This contribution unifies the analysis of linear constraints on the portfolio optimization model and the significance of relative portfolio optimization to modern delegated contracting. The closed-form nature of these solutions gives analysts a conceptual, theoretical framework to perform comparative analysis without having to rely on numerical simulation.

With the introduction of the Capital Asset Pricing Model, finance theory went the direction of assuming a risk free asset when analyzing linear asset pricing problems. The Sharpe Ratio, introduced in Sharpe [12] and defined below:

Sharpe   Ratio = w ′ r − r f ( w ′ Ω w ) 1 / 2 ,

is popular even today and assumes a risk free asset, which has expected return, r f . This ratio is simply the slope of a line connecting the risk free asset to any risky point in the Markowitz feasible set. See Figure 4 for representations of this line.

Maximizing the slope of this line in mean/variance space generates what is known as the Capital Market Line (CML) and its intersection with the minimum variance envelope is called the market portfolio.

The problem with the risk free asset in the context of a general solution to the variance minimization problem is that the covariance matrix, Ω , is no longer invertible because the risk free asset creates a row and column of zeros. Thus, Equation (2.1.1) is not a valid solution in this scenario. Finding the CML under the standard linear constraints of portfolio optimization is rather straight forward but with a general linear constraint, it becomes slightly more complicated.

Below is the solution to the Sharpe Ratio maximization problem. It is important to note that the solution to the following problem gives the weight vector for what is called the Market Portfolio if the constraints are the standard weight and return constraints. The statement of the problem is below:

max w S = w ′ r − r f ( w ′ Ω w ) 1 / 2     s .t .     w ′ E = c .

This formulation doesn’t lend itself well to quadratic programming, so in order to solve it we will use a well-known result called the mutual fund theorem, from Merton [2] , or the two fund theorem. Essentially, a linear combination of any two portfolios on the efficient envelope also resides on the efficient envelope. Thus, the solution to the maximization problem above is a linear combination of any other two portfolios on the efficient set. Therefore, I will use two of the results I already derived, w ∗ the variance minimization solution, Equation (2.1.1), and w U the utility maximization solution, Equation (2.2.1).

w ∗ = Ω − 1 E ( E ′ Ω − 1 E ) − 1 c ′ .

w U = Ω − 1 E ( E ′ Ω − 1 E ) − 1 c ′ + Ω − 1 ( r − E ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r ) .

And, their linear combination is below with weight x:

w = ( 1 − x ) w ∗ + x w U .

For simplicity, I define a new weight vector w H ,

w H = w U − w ∗ = Ω − 1 ( r − E ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r ) .

And, the weight vector, w, becomes:

w = w ∗ + x w H .

Plugging this vector into our function for S yields:

S = w ′ ∗ r + x w ′ H r − r f ( w ′ ∗ Ω w ∗ − 2 x w ′ ∗ Ω w H + x 2 w ′ H Ω w H ) 1 / 2 .

w H is particularly convenient because it can be shown that

w ′ ∗ Ω w H = c ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r − c ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r = 0.

Thus, S becomes:

S = w ′ ∗ r + x w ′ H r − r f ( w ′ ∗ Ω w ∗ + x 2 w ′ H Ω w H ) 1 / 2 .

Now basic calculus can be applied to find the maximum with respect to x.

d S d x = − x w ′ H Ω w H ⋅ ( w ′ ∗ r − r f ) − w ′ H r ⋅ w ′ ∗ Ω w ∗ ( w ′ ∗ Ω w ∗ + x 2 w ′ H Ω w H ) 3 / 2 = 0.

And the solution follows easily:

x = w ′ H r ⋅ w ′ ∗ Ω w ∗ w ′ H Ω w H ⋅ ( w ′ ∗ r − r f ) .

Next, we need to calculate the expected returns and variances to reparametrize the expression into the proper variables and find the constant:

x = r ′ Ω r − r ′ Ω E ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r r ′ Ω r − r ′ Ω E ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r ⋅ c ( E ′ Ω − 1 E ) − 1 c ′ c ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r − r f = c ( E ′ Ω − 1 E ) − 1 c ′ c ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r − r f .

And lastly, plug this constant back into the function for w = w ∗ + x w H :

w = Ω − 1 E ( E ′ Ω − 1 E ) − 1 c ′ + ( c ( E ′ Ω − 1 E ) − 1 c ′ c ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r − r f )     ⋅ Ω − 1 ( r − E ( E ′ Ω − 1 E ) − 1 E ′ Ω − 1 r ) . (4.1.1)

Thus, this is the weight vector that maximizes the Sharpe Ratio given a general linear constraint, or equivalently, if the constraints are the standard Markowitz constraints, this is the weight vector of the market portfolio. It is expressed in the familiar and convenient form of a shift to the minimum variance weight vector.

In this paper, I derive the solutions to portfolio choice problems with a general linear constraint when they are also subject to a tracking error constraint. The primary method used to derive the solutions is quadratic programming. This essay connects the research of some of the most fundamental papers in the field of portfolio choice and generalizes a solution to the Tracking Error Volatility constraint problem of Jorion [8] . I set up the objective function in both variance minimization and utility maximization. Although Jorion solves the variance minimization problem for specific constraints, my method solves the problem with a general constraint. Moreover, Jorion does not derive the utility maximization problem with a tracking error constraint, which I have done in the essay with a general linear constraint.

Additionally, I have revisited the problems of Best and Grauer [3] and reformulated them so the similarity is more apparent between their solutions and mine. I have discussed how the general constraint matrices need to be set in order to replicate the problems of Markowitz [1] and Roll [7] . I use my derivations of the general linear constrained problems to tackle the problem of maximizing the Sharpe Ratio and derive the optimal solution to this problem. Maximizing the Sharpe ratio can be considered an alternative form of utility and is a very important concept in the foundation of linear asset pricing, particularly when a risk free rate is assumed to exist.

There are limitations to empirical research given these derivations. Real world data is oftentimes too volatile to apply the methods of modern portfolio theory directly. These derivations are best used as a conceptual framework to make theoretical assertions about the behavior of investment managers. The implications from these theories can, however, be creatively tested and existing empirical data can be used to discover nuances of the decision-making or real world investment managers. One example in particular that this model exposes is that managers that are incentivized by relative risk and return metrics will usually build portfolios that are riskier than a principal would desire. Although these equations likely couldn’t test that. This result could be tested in numerous other ways.

There are many other examples of constrained portfolio choice problems that fall into the framework outlined in this paper. Although my final expressions are cumbersome, they are also left in their purist form so the reader can see the form unobstructed by substitution. The general linear constraint is very useful when considering real world constraints to asset management, and the tracking error constraint is a pervasive, explicit and implicit constraint to which almost all asset managers abide. This paper will hopefully allow researchers to compare the effects of differing constraints on portfolios in a more approachable manner, with the solutions readily available.

The author declares no conflicts of interest regarding the publication of this paper.

Stowe, D.L. (2019) Portfolio Mathematics with General Linear and Quadratic Constraints. Journal of Mathematical Finance, 9, 675-690. https://doi.org/10.4236/jmf.2019.94034