Analytical Overview and Applications of Modified Black-Litterman Model for Portfolio Optimization

Abstract

<jats:title>Abstract</jats:title><jats:p>The paper makes analytical overviews of the Markowitz portfolio and the Capital Asset Pricing models and motivates the advances of the Black-Litterman (BL) one. This overview implies that for a small set of assets the BL model needs the characteristics of a specific market point, which cannot be taken from a global market index. The paper derives analytic relations for the new specific market point with analytical approximation of the efficient frontier. The BL model insists also expert views, which influence the portfolio solution. The paper derives formalization of the expert views from the difference between the evaluated implied returns and historical mean assets returns. Such form of expert views makes modifications of the BL model. This allows comparisons between Markowitz (MV) and BL portfolio performance. Benefits of this research are demonstrated with market data and comparison of the MV and BL portfolio results.</jats:p>

The Probabilistic Risk Measure VaR as Constraint in Portfolio Optimization Problem

“…The modern portfolio theory starts its flourish with quantitative estimation and evaluation of assets characteristics. The initial parameters, which have been mainly evaluated were the mean assets returns, the risk of each asset and the mutual influence between the assets returns, numerically assessed by values of the components of the covariation matrix [6,17]. Having these initially estimated asset characteristics the portfolio optimization has been formalized as a static optimization problem defined by [12] in the well-known forms where: w T = (w1,…,wN) is the vector of weights, which gives the relative value of the investment, allocated to the asset i, i=1, N is the number of assets in the portfolio; relation 𝐰 T |𝟏| = 1 gives limitations about the investment amount to be totally allocated to the portfolio; w T ≥ 0 means that the assets must be bought for the portfolio; Risk(w) and Return(w) are analytical relations, describing these portfolio characteristics as functions of the arguments w;…”

“…The components of the correlation matrix 𝚺 are evaluated according to (4) 𝚺 = | 0.0455 0.0182 0.0182 0.0360 |. These initial data about 𝐸 𝑖 , 𝑖 = 1, 2, and 𝚺 are used for the definition of two portfolio problems: problem (5) without constraint for VaR and problem (5) with VaR constraint of form ( 16) (17) P1:…”

“…The analytical form of the VaR constraint in problem ( 17) becomes 𝐄 T 𝐰 − 1.645√𝐰 T 𝚺𝐰 ≥ −0.3. Solving problems (17) and (18) for different values of the parameter 𝜆 ∈ (0, ∞), points of the corresponding efficient frontier are evaluated. From these sets of portfolio, for comparison reasons, it has been chosen the portfolios from ( 17) and (18), which have maximal Sharpe ratio = Portfolio return/Portfolio risk.…”

“…4. The "Efficient frontier" is the same for problems (17) and (18) To find the reason why problems (17) and (18) have equal solutions here it is graphically presented the influence of the VaR constraint. The form of constraint (16) is quadratic one and it makes and elliptic curve in the space of the assets weights w2(w1).…”

“…6 it has been illustrated the case for the equal solutions of problems ( 17) and ( 18) even for the value γ=0.3. Graphically the solution of ( 17) is presented with the tangent point between the objective function of (17) and the line 𝐰 T |𝟏| = 1. The objective function also is an elliptic curve, which makes a tangent point with 𝐰 T |𝟏| =1.…”