P ORTFOLIO O PTIMIZATION M ODELS

The different models used in portfolio optimization are presented below. First portfolios are optimized based on capital marked predictions (CMP). Multiple CMP portfolios with different goals are optimized using Markowitz's standard mean-variance framework to satisfy various investors. Additionally, historical portfolios for two time periods are examined: July 2007 –

February 2021 and August 2009 – December 2019. As a result, we find comparable and complementary results for optimal portfolios for both crisis and non-crisis periods. The difference between CMP portfolios and portfolios constructed using historical data is mostly the returns. It is worth noting that CMP portfolios are based on the last ten years' historical risk and correlations; however, they are distinct from historical portfolios in that the estimated period varies.

In comparison to the CMP portfolios, historical portfolios are calculated using a variety of different methods. The first three historical models (MV, Mean-CVaR, Bayes-Stein) all use the same risk-minimizing and relative return maximizing principles when constructing the optimized portfolios, thus providing comparable results between them. The last two, on the other hand (GMM & MD), have different optimization objectives, which makes their results rather complementary than comparable. Based on the analysis of these results, we hope to understand whether and which liquid alternatives are rational additions for retail investor’s portfolios and based on which objectives.

6.1.1 Mean-Variance

The mean-variance (MV) model is the most classical approach to portfolio optimization. It assumes that investors prefer portfolios with high expected returns in relation to risk. The MV model is one of the building blocks of Markowitz’s modern portfolio theory. The main principles of MPT is explained in the theory chapter above. This model uses the arithmetic mean and standard deviation as return and risk estimates. The variance covariance matrix is then calculated based on these measures. Formulas for the arithmetic mean and standard

𝑥̅ = 𝑇ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛

Different portfolios are constructed here, first and foremost the minimum variance portfolio (MVP) and the tangency portfolio. The tangency portfolio is found by maximizing the Sharpe Ratio (1). The MVP is found by minimizing the portfolio variance (3). The weights that maximize and minimize these equations give the optimal allocations of each asset in a portfolio. Further using MV the efficient frontier and portfolios with risk and return constraints are constructed and analyzed. The optimization using the MV framework is done in R and Excel.

6.1.2 Mean-CVaR

A similar approach to the MV model is the Mean-CVaR model. The theoretical background of CVaR is presented in detail in the theoretical frameworks section above. In short, the biggest difference to MV is that CVaR includes tail risk and is more sensitive to the tail behavior of the distribution function. Regarding optimization, it is not limited to elliptical distributions as MV is.

Again, two portfolios are constructed using the Mean-CVaR model, the Minimum Risk Portfolio (MRP) and the Maximum Relative Performance Portfolio (MRPP). The optimization approach is identical to MV in minimizing risk and maximizing risk-adjusted returns, but instead of volatility as the measure of risk, this model uses CVaR. Because this model uses CVaR instead of volatility, the maximization portfolio is called MRPP instead of tangency,

although the ideology is the same. These portfolios are constructed using the “fPortfolio”

package in R (Wuertz, Chalabi, Chen & Ellis, 2010)

6.1.3 Bayes-Stein

While Mean-CVaR uses alternative estimates for risk, the Bayes-Stein model focuses on enhancing the return estimation. Equation (6) on page 53 shows the formal definition of the math on which this model is based.

The optimal portfolios using the BS model were built in R using the framework explained originally by Jorion (1986, 1991) and later modified by Avramov and Zhou (2010).

Hence, we find a new return vector for the assets using the BS shrinkage estimator and a new variance-covariance matrix based on these returns. The variance-covariance matrix is calculated using the formula below:

∑^𝐵𝑆 = (1 + 1

𝑇 + 𝜆̂) ∑̂ + 𝜆̂

𝑇(𝑇 + 1 + 𝜆̂)⋅ 1 ⋅ 1^′

1∑̂⁻¹1 (13)

where

𝜆̂ = (𝑁 + 2) [(𝐸(𝑟⁄ _𝑗) − 𝐸(𝑟_𝑀𝑉𝑃)1)^′ ∑̂⁻¹(𝐸(𝑟_𝑗) − 𝐸(𝑟_𝑀𝑉𝑃)1)],

∑̂ = 𝑇

𝑇 − 𝑁 − 2𝜎_𝑖𝑗

Using the new estimated returns and covariances, we construct two portfolios, the MRP and the tangency portfolio. The portfolio construction is done using the same principles as before, but in this case, using the BS estimators for inputs.

6.1.4 Geometric Mean Maximization

GMM was proposed as an alternative to mean-variance optimization. The main difference being that GMM considers wealth maximization as the main objective of investors rather than the optimization of risk-adjusted returns.

As shown by numerous researchers, the maximization of a portfolio’s geometric mean (GM) return can be done in numerous ways (Estrada, 2010). The method proposed by Estrada (2010), which is the one used in our analysis, is constructed using historical observations. The formal way to define this geometric mean maximization is presented below:

𝑀𝑎𝑥 𝐺𝑀_𝑝 = 𝑒𝑥𝑝 {𝑙𝑛(1 + 𝜇_𝑝) − 𝜎_𝑝² referred to in this paper as GMM. Equation (14) shows the role volatility in the GMM model compared to the MV model. In the MV, volatility is undesirable because it is synonymous with risk; in the GMM model, volatility is also undesirable, not because it means risk but because it lowers the geometric mean return. In other words, it lowers the rate of growth of the capital invested, ultimately lowering the expected terminal wealth which the model is designed to maximize.

6.1.5 Maximum Diversification

Maximum diversification portfolio (MD) was first introduced by Choueifaty and Coignard (2008) and gave the portfolio that maximizes the diversification ratio. Its true value is derived from informing investors about the degree of diversification available within that investment universe. This portfolio establishes a theoretical maximum level of diversification, giving insight into which assets provide the most diversification benefit when establishing a portfolio.

However, according to Choueifaty & Coignard (2008), the most diversified portfolio should not be considered an equilibrium model. It does not generally meet the objectives of most investors and thus should rather be seen as an idealized target. Additionally, Theoren and Vuuren (2017) investigated previous claims that the MD portfolio’s performance would be superior to MVP or tangency portfolios. They concluded that the MD portfolio outperforms the MVP portfolio in out-of-sample performance but falls short behind the tangency portfolio in total performance. This was especially the case in terms of cumulative returns.

This model is implemented using the same inputs as previous models: Holding period returns and standard deviations. A formal definition of the diversification ratio of a portfolio P is as follows:

Thus, as can be seen from (15), the diversification ratio is the ratio of the weighted average of volatilities divided by the portfolio volatility. By maximizing (15), we find the weights that provide the highest diversification benefit for an investor.