Using Black-Litterman model on example of russian stock exchange market

Galiev D.R., Isavnin A.G.

Kazan University, Russian Federation

Abstract

The problem of constructing optimal portfolios using the complex expert judgements and Bayesian methods is investigated. The Black-Litterman model was selected as a base model. A distinctive feature of this model is ability to combine the theory of market equilibrium with the expert judgements specified by confidence level. We consider the complete computational scheme for the model. The decision of the problem with constraints on the structure of the final portfolio is presented. It is suggested to use not only the subjective opinions of experts, but also indicators of sub-models such as the multifactor, neural networks, fuzzy-set, etc. We compare the results of our approach with the results of classical and traditional approaches.

Until recently, the modern portfolio theory was based on a bunch of models of Markowitz-Sharpe-Tobin. It was repeatedly criticized both by theorists and by practitioners [1,2]. The Black-Litterman model, for the most part, is able to cope with a number of shortcomings of the «classical» theory [3,4].

The Black-Litterman model was designed like a practice-oriented model. To do this, Black and Litterman proposed a theory, which they called «equilibrium» approach. They have established a market equilibrium as a starting point [1]. In this case, the equilibrium is understood as an idealized state in which the demand is equivalent to the proposal. Such a situation hardly occurs in the financial markets, but this idea has a number of attractive features. According to Litterman, there exist «natural forces» in economic system in the form of arbitrageurs whose operation eliminates the deviation from equilibrium. Even if the markets display disturbances, such as «noise» of speculators, uncertainty of information, lack of liquidity, eventually there will be a tendency to «adjust toward equilibrium». Firstly the model was applied to produce optimal and efficient portfolios of government bonds on international markets. Litterman claimed that the world CAPM (Capital Asset Pricing Model) is a starting point for the global equilibrium model. Black promoted the theory of global equilibrium, that formed the basis for the first modification of the Black-Litterman model. However, it can be used not only in global markets, but also in local national markets, to make up a portfolio of stocks and portfolios of fixed income instruments. Equilibrium returns are calculated with the method of «reverse optimization» :

(1)

where

is vector of the equilibrium return;

is risk aversion;

is covariation matrix;

is vector of asset’s market capitalization.

Coefficient of risk aversion () characterizes the investor’s willingness to sacrifice the value of the expected portfolio return for reducing the risk expressed by the dispersion of expected returns and plays the role of the scaling parameter. Higher values of yield per unit of risk (i.e. large ) lead to an increase in estimates of returns of assets. In the presence of assessing, the future profitability of the market or benchmark portfolio rate risk appetite can be calculated with the formula:

(2)

where is expected return of the benchmark, is risk-free rate, is benchmark’s variation.

To estimate equation (1), the next optimization problem was solved:

.

(3)

Let . U is a concave function and, therefore, has a single global maximum. In our case, without additional restrictions, it will be sufficient to find the derivative and equate it to 0:

(4)

In practice, due to possible statistical errors, the value of the equilibrium return has the following estimation:

,

(5)

Where is residual component, possible statistical error, that has the normal distribution .

After computing the aposterior value of the vector of return (see below), it is necessary to calculate the final weights of assets in the portfolio with new equilibrium returns, which take into account all the given expert estimates and confidence levels of them:

(6)

Consider the formula of Black and Litterman for aposterior return vector. That is the key for the calculation of the final portfolio. К is used to determine the number of subjective opinions, N indicates the number of assets. The equation for the new return vector :

(7)

where

is new aposterior return vector ();

is scalar value;

is matrix of covariation of historical returns ();

P is matrix identifying the assets, on which the investor has a subjective opinion ();

is diagonal covariance matrix with the levels of trust for each of the subjective views ();

is vector of equilibrium return ();

Q is vector of subjective views ().

Litterman’s paper [1] presented the equation of a new mixed return’s vector that is completely identical to the previous one:

.

(8)

The advantage of this way of writing the Black-Litterman’s equation is to simplify computations for software implementation.

Consider in detail the elements of the Black-Litterman equation. Some investors have their own position about the future profitability of an asset in the portfolio and this opinion may differ from the value of the vector equilibrium returns. For example, sometimes too strong fundamental factors (news, results of important meetings, political events, etc.) arise and affect asset prices and that can not be formalized. The Black-Litterman model allows us to consider these assumptions more accurately and with a certain confidence level. There exists the notion of confidence level of the provided subjective opinion (expert evaluation). Roughly speaking, if the investor has no assumptions about the behavior of an asset, the model suggests to adhere to market equilibrium portfolio (benchmark portfolio). The Black-Litterman model has several ways to specify the confidence levels of subjective opinion. There are examples below of the formation of such views in the framework of the Black-Litterman model.

o ▪ Stocks of Sberbank in this period will yield a 10% (confidence level = 25%).

o ▪ Stocks of Surgutneftegas will be more efficient than Rosneft shares by 2.5% (confidence level = 50%).

In terms of the Black-Litterman model, View 1 is an example of an absolute view, View 2 is a relative viewpoint. One of the difficult moments in the model is moving the formed opinions in the input parameters used in the Black-Litterman equation. Actually, it is not necessary for an investor to have unique views on each asset. The uncertainty of the subjective views is reflected in the error vector (), whose elements are normally distributed with zero mean value, and the matrix . Thus, the final values of subjective opinion is given as .

General case: Example:

(9)

Excluding the case when there is complete confidence in the subjective view, the elements of the error vector () are non-zero values. Error vector () is not introduced directly into the Black-Litterman equation. Nevertheless, the variations () of each element of the error vector, which is absolutely different from the error vector (), are a set of input parameters of the formula. Variations of the elements of the error vector form matrix, where is a diagonal covariance matrix. The matrix is diagonal indeed, because, according to the prerequisites of the model, subjective opinions are independent of each other. Variations of the error vector () show a measure of uncertainty of subjective views. If the variation of the error vector () is larger, the uncertainty of the subjective view will be greater. Variation () of zero value characterizes the total confidence in View.

General case:

(10)

Estimation of individual variation of the error vector (), which defines a diagonal matrix , is the most difficult aspect of the computing part of the model. There are several methods for determining the elements of [3,4,6]:

o Proportional historical variation;

o Using confidence intervals;

o Using factor models (including AR-type models GARCH);

o Thomas Aydzorek’s Method.

In this work, with specific examples, we consider integrating in the Black-Litterman model different kinds of predictions:

o peer review of analytical departments

o forecasts for multifactor models;

o forecasts on Intelligent Techniques (studying neural networks with the architecture of multilayer perceptron);

o forecasts heuristic techniques, forecast models and technical analysis.

In the original papers of Fisher Black and Robert Litterman most of these methods (especially intellectual) are not discussed in detail. It should be noted that the Black-Litterman model takes into account expert assessments and indicators of sub-models with levels of confidence, calculated both numerically and semantically (on a scale of confidence levels). In the case of a quantitative method it is necessary for random residues of forecasts to have «Gaussian» distribution. Values of returns on subjective views, which are in the vector-column of Q, are introduced into the model by the matrix P. Meaning of the result of influence of each of the subjective views are in the vector-row dimension. Thus, for the K views, we obtain the matrix P, with dimension.

General case: Example (continue):

(11)

The first row of the matrix P reflects the View 1 (the absolute opinion, the continuation of the previous example). View 1 includes only one asset: Sberbank. Assume that we have considered four assets: Sberbank, Surgutneftegas, Rosneft, Rostelecom. In this example, Sberbank is the first serial number, hence the «1» is in the first column and the first row of P. View 2 reflects the second row. In the case of relative views, the sum of all row items must be 0. In the matrix P nominally superior assets receive positive weight, while nominally yielding assets - negative.

Consider the results of using the model in practice. Within this work several experiments were made: the Russian market (MICEX), foreign (NYSE), with the marked growing trend, and in its absence. One of the experiments was conducted on Russian market (MICEX), in post-crisis period. Let’s examine in more detail its results. To make up the portfolio stocks of six companies were selected: Aeroflot (AFLT), Gazprom (GAZP), MTS (MTSI), Rosneft (ROSN), Sberbank (SBER), Uralkaliy (URKA). The reasons to choose these assets are high liquidity, different profile of activity (which provides good diversification and risk reduction), positive predictions of experts and submodels. The beginning of the experiment is on 08/24/2009, the end is on 29/01/1910. This medium-term period (5-6 months), according to research [5], in the Russian market is the most stable period for the covariance matrix of returns. Using daily stock rates of these companies and trading volume for the period from 10/31/2004 to 20/08/2009, statistical analysis was performed to determine the model input parameters. Further, using public available information about the alleged behavior of the stock rates over the next 6 months, and using additional models described in this paper, we got input information for the Black-Litterman model. As expert’s views we used only information from analytical agencies with adequate predictive ability [8]. The optimization problem’s solution is to build a portfolio within framework of the Black-Litterman model. A portfolio of the Black-Litterman model (41.71%), taking into account expert assessments for the period 08.24.2009 - 29.01.1910, is ahead of the market portfolio return (28.41%). Market portfolio, in turn, is slightly ahead of the yield of the MICEX index (profitability of the MICEX index = (1419.42-1120.54) / 1120.54 * 100% = 26.67%) and a portfolio constructed with the classical theory of H. Markowitz (27.44%) (see Figure 1). The level of the risk is approximately the same.

Figure 1. Comparison of portfolio returns at approximately equal levels of risk

For all the results obtained, it can be generally concluded that the Black-Litterman model is an essential tool in modern financial management for the prompt, optimal and intelligent management of the investment portfolio. The resulting portfolio is assessed both ex post and ex ante. The model takes into account the complex nature of expert judgments: analytic solutions departments, factor analysis, technical analysis, neural network models, etc. The results of both the national and foreign markets, allow to make a conclusion about the practical suitability of the Black-Litterman model in combination with the proposed methods of accounting. Investment portfolio built with the Black-Litterman model takes into account the complex nature of expert’s review, has the best indicator of profitability compared with the whole market equilibrium portfolio and portfolio of classical theory. Several additional experiments carried out in a similar way, but in other times on the Russian and U.S. markets, confirmed this fact.

Bibliography

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