Portfolio optimization

TODO Business description

Context The fundamental premise of investing is that one expects to earn a return that compensates
for the risk taken. Investing in financial markets amounts to considering a vast number of different (such as stocks, bonds, etc.) that one could potentially include in their portfolio, and each securitiesof these securities carries its own level of risk and expected return. The goal is to maximize the expected return of a portfolio while minimizing the risk with the natural trade-off: higher potential returns come with higher risk.

Harry Markowitz developed in [1] the concept of mean-variance portfolio optimization (MVPO), also known and commonly reffered to as Modern Portfolio Theory (MPT), as a systematic approach to portfolio optimization. MPT revolves around the idea that by carefully selecting a combination of different assets, investors can maximize their expected return for a given (fixed) level of risk.

Problem Definition (High-Level) Let us break down the fundamental concepts of portfolio optimization by mean variance.

“Mean”, often denoted as μ, represents the expected return of a portfolio. A portfolio’s expected return is calculated as a weighted average of the expected returns of its individual assets.
“Variance”, often denoted as σ, represents the risk associated with a portfolio. Essentially, it measures how much the returns of a portfolio can deviate from their expected return μ. In other words, it measures the portfolio’s volatility. It takes into account not only the individual risk levels of the assets in the portfolio (standard deviation, σ, of the returns), but also how these assets are correlated with each other.
“Optimization” involves selecting the proportion of each asset in the portfolio such that the portfolio’s expected return is maximized for a given level of variance or, conversely, the variance is minimized for a given level of expected return.

What MPT tries to solve, is the problem

maximize Expected return of portfolio
subject to Risk willing to take

minimize Volatility of the portfolio
subject to Target expected returns

Difficulty (high-level) Due to the combinatorial nature of these problems, as the number of assets in the portfolio increases, the problem becomes exponentially difficult for classical computers. This is where quantum computers may be beneficial and could help to solve complex portfolio optimization problems faster and more efficiently, potentially opening up opportunities for real-time optimization or dealing with significantly larger asset sets.

Problem Definition (Detailed) In the context of MPT, a typical portfolio optimization problem for N assets can be formulated as follows (minimization version, Prob. (2) as it is naturally encoded onto
a quantum system). We represent by μi 45 the expected return of asset i, σij ≡ Cov(i, j) represents the covariance between the returns for assets i and j and by R the target return. The decision variables are the weights wi for each asset. At this point we make the assumption that the covariances are known to the portfolio managers and generally are computed as

Cov(i, j) = 1 n − 1 Xn t=1 (Ri,t − μi)(Rj,t − μj )

over a period of discrete time steps t = [n]. The problem to be solved is a linearly constrained quadratic optimization problem that reads

minimize wi , wj ∈ S X N i=1 X N j=1 wiwjCov(i, j)

subject to X N i=1 wi = 1,

N i=1 wiμi = R

Here, S denotes the set that the weights take values on, for example R in the simplest case or some subset of Z in the complex case. The latter, makes Prob. (3) intractable (see below).

Difficulty Implementing MPT can be a highly-complex task due to fluctuating asset interdependencies (correlations), and more often than not it requires advanced models and tools. Specifically, if the covariance matrix is positive definite and weights are assumed to be real numbers the problem can be solved efficiently using classical solvers. However, if the covariance matrix is not positive definite or if the weights are to be discrete the problem is significantly hard to solve at large instances and it has been shown to be a very hard problem.

TODO Technical description

Generic I/O The JSON schema is provided below:

{
	"title":"Markowitz Portfolio Optimization Input",
	"description":"Input data",
	"type":"object",
	"properties": {
		"N": {
			"description":"The number of assets in the portfolio",
			"type":"integer",
			"minimum": 1
			},
		"alpha": {
			"description":"The expected return of each asset in the portfolio",
			"type":"array",
			"minItems": 1,
			"items": {
				"type":"number"
			}
		},
		"beta": {
			"description":"The covariance between each pair of assets in the portfolio",
			"type":"array",
			"minItems": 1,
			"items": {
				"type":"array",
				"minItems": 1,
				"items": {
				"type":"number"
				}
			}
		}
	},
	"required": ["N", "alpha", "beta"]
}

Example Simple JSON example for a portfolio with N = 3 assets and known αi 140 , βij .

{
	"N": 3,
	"alpha": [0.1, 0.2, 0.15],
	"beta": [
		[0.05, 0.01, -0.02],
		[0.01, 0.06, 0],
		[-0.02, 0, 0.04]
	]
}

Quadratic Program As defined in Prob. (3).

Ising Model As defined in Prob. (8)

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