This study examines how effective the Black Litterman model is when used to optimize portfolios. The Black Litterman model (BL), when used to optimize portfolios, aims at improving the Markowitz portfolio optimization technique. The primary goal of this model is establish a portfolio allocation model that allows a natural and stable result, while at the same time overcoming the problems observed when modeling data using the Markowitz approach. The paper attempts to explain the model and test its effectiveness in the London Stock Exchange by constructing a benchmark portfolio from the FTSE 100 share index. The paper then integrates the different investor views into our portfolio and compares it to a portfolio weighted using the Markowitz’s Mean-Variance Optimization method and market capitalization.
Background of the Study
Portfolio optimization is a term that describes how investors choose various proportions of several assets, include the assets into their portfolio as a way of maximizing their returns and minimizing risks. Mostly, the assets belong to different asset classes. The attempt to apportion the limited resources of an investor to the asset in a portfolio prompted Harry Markowitz to develop the Mean-Variance Optimization (MVO) model in 1952.
This model was first established before the modern portfolio theory. This model assumes that the investors intend to maximize all the returns they get from their investments while incurring the least acceptable risk. This risk varies for various investors. A risk-averse investor accepts the minimum possible risk for any level of the returns. On the other hand, a risk-taker accepts the investment promising higher returns despite the level of risk associated with it. An efficient portfolio offers the lowest risk levels and highest returns level for any portfolio. The graphical representation of an efficient portfolio uses an efficient frontier, which is a line that shows the various combinations of assets returns and risks. The combinations falling below the efficient frontier lack enough return to match the levels of risk at the point.
Similarly, according to Markowitz, investors are faced with a tradeoff between risks and returns. The high returns are associated with high levels of risk, while low returns associate themselves with low levels of risk. The risk-return tradeoff shows that the money invested renders higher profit margins only when investors accept the possibility of losses. To balance this tradeoff, investors understand that the more risk-averse they are, the less compensation they will receive, and the higher the risk tolerance, the greater the compensation.
The developments made by Markowitz have various extensions, such as the arbitrage pricing, the capital asset pricing model (CAPM) and the single asset pricing model. Markowitz’s contribution does not apply in real financial markets since there are several problems associated with the discovery: Markowitz optimizers’ maximum errors, unintuitive portfolios in the mean-variance model, and the lack of robustness in the MV model.
These errors prompted Fischer Black and Robert’s model to come up with a model that would correct the errors. They invented the BL model as a result. The Black Litterman model is a combination of both the CAPM and the MVO models. The relevance of this model to investors lies in the investors’ ability to come up with varying combinations of unique views on the different asset performance in a market equilibrium and towards both an intuitive and a diversified portfolio. The BL model uses the investors’ insight in the creation of a stable mean-variance efficient portfolio. The unique features of such a portfolio is in their ability to overcome the input sensitivity error that results from the mean-variance model.
This model spreads the errors made by investors throughout the vector of expected returns and mitigates the estimation error problem. Black and Robert attempted to solve the classic problem of portfolio optimization using the BL model and applying financial qualitative and quantitative aspects. Qualitative aspects are investors view for any financial security while the quantitative aspect depends on the portfolio’s Mean-Variance analysis. This model uses the Bayesian approach to combine both the investors’ subjective views on the returns of an asset and the expected returns’ market equilibrium vector, which serves as the prior distribution to form new return estimates that from the posterior distribution.
This study considers stocks on the London Stock Exchange. The stocks include Unilever, Royal Dutch Shell, and HSBC Holdings. The study’s primary purpose is to assist investors and investment companies in modifying their current portfolio to optimize the portfolio. This research will also form a foundation for other researchers who would seek to come up with better models or enhance the Black Litterman model of portfolio optimization.
Statement of the Problem
This research seeks to construct intuitive portfolios in the London Stock Market. investors have previously used wrong models to determine their portfolio. This makes it necessary to come up with a model that can provide the right asset proportions for an optimal portfolio. The BL model calculates the different investors’ views as a way of solving the problem of unintuitive portfolios that was encountered in previous portfolio optimization models. The paper also seeks to create a stable mean-variance efficient model that relies on the investors’ unique insight into the efficient portfolios to mitigate the input sensitivity problem. This model spreads the errors in computing risk-return combinations by spreading them through the expected returns vectors.
The study’s Objectives
The primary goal of the research is to use the BL Model for portfolio optimization.
The following specific objectives of the study will help to achieve the main goal of the study.
- To compute the market weights and market capitalization values.
- To find the vector of the excess equilibrium’s return(implied).
- To form a vector of new returns.
- To calculate Black- Litterman weights, historical weights, and CAPM weights.
- To use the Sharpe’s ratio in showing that the portfolio obtained from Black-Litterman weights is optimal.
Investors using the MVO framework use the BL model when producing various sets of expected returns, which allows these investors to overcome the unintuitive portfolio, input sensitivity, error maximization problem and highly concentrated portfolio that is introduced by the MVO model. This research will form a foundation for other researchers who would seek to enhance the Black Litterman model or come up with better models of portfolio optimization. Investors and investment companies can use the model to modify their current portfolio to optimize the portfolio.
Chapter 2: Literature Review
Many researchers have researched on the subject of portfolio optimization, and several optimization models have been developed. Markowitz was the pioneer of this researches. His development of the Modern Portfolio Theory (MPT) served as a mathematical framework for investors to optimize their returns and risks (Markowitz, 2009). This theory’s primary goal was to diversify returns of investment by forming a portfolio that reduces the risks associated with asset returns. In this model, portfolio returns are a function of the portfolio’s expected returns. Markowitz used this invention to develop the mean-variance analysis model in 1952. This model helped investors choose the best portfolio from different asset ranges.
Later on, David Wilkinson, William J. Bernstein, and Markowitz improved the mean-variance model used for portfolio optimization by forming the Mean-Variance Optimization (MVO) technique. This technique is quantitative. It helps investors to determine the existing tradeoff when allocating assets in a portfolio. They developed a single period MVO model. This model helps in the portfolio allocation of a model for the next single period. The goal of the model is to help the investors determine the asset combination that will give the highest returns of a portfolio while incurring the lowest possible levels of risk (Markowitz, 2009). Markowitz argues that if an investor considers the risk of a portfolio represented by the covariance of assets, they can construct a portfolio that generates two scenarios of returns. First, they can generate high returns under the same level of risk, and second, they can generate the same level of expected return at a lower risk level.
On the other hand, Bernstein and Wilkinson modified the model developed by Markowitz by coming up with the multi-period Mean-Variance Optimization, which focused on how investors used portfolio rebalancing strategies for a specified allocation of the assets at the end of each period (Bernstein and Wilkinson, 1997). They named the strategy Constant Ratio or Constant Proportion of Asset Allocation. The two scholars aimed at maximizing the geometric mean (true period mean return) for a specified level of fluctuation. Wilkinson and Bernstein showed that the formation of a multi-period Mean-Variance Optimization Model was a contributing factor to the acceptance of the MV model in the industry since it gave the most preferred portfolio optimization models.
Contrary to the researchers that said that Mean-variance Model was the best portfolio optimization model, Michaud 1998, discovered that this model had some limitations. Some of the limitations included, the model assumed that data is normally distributed and used historical data rather than expected return. These two assumptions led to error maximization since the return data is not normally distributed, as Markowitz assumed it to be. Furthermore, the historical return does not give an accurate indication of expected returns since the return series is not autocorrected (Michaud, 1998). The investors had to provide return vector inputs. The model also required that these investors estimated their inputs for the variance-covariance matrix. The model limited the optimization models using less data, which posed a challenge in the computations since they depend on the values obtained in the variance-covariance matrix.
Michaud also noted that the model optimizes error since there is no correct and exact mean and variance. The errors, at times, gave a false impression of the performance of a particular asset. Sharpe, 1967 agrees with Michaud when he says that although the mean-variance model may seem appealing, several problems arise on putting the model in practice. He further points out that Markowitz doesn’t explain how people should allocate their wealth in the portfolio. Still, he only explained how investors select the assets to include in their portfolio.
In 2015, Walter argued that the BL Model’s use was necessary for overcoming the challenges introduced by the mean-variance model. Black and Goldman Sachs had developed this model, which according to Walter, overcame the Capital Asset Pricing Model (CAPM) problems and the mean-variance approaches through the use of both the qualitative and quantitative aspects of finance (Walters, 2014). Like the previously used models, the quantitative aspect is the portfolio assets’ historical data. On the other hand, the qualitative aspects are the investors’ views for any particular security.
An article by Winkelmann and Bevan described how they had adopted and implemented the BL when allocating assets in a company, Goldman Sachs. They had successfully calibrated the process, which was a major discovery of the unique features of the BL model. They proved that when investors can make views, they should take a risk, which should be proportional to these views. A higher risk taken is associated with the most influential views, while a lower risk is associated with weaker views. They successfully engaged Litterman in the discussion in 1999 (Bevan and Winkelman, 1998). This discussion was followed by Satchell and Scowcroft’s demystification of the BL model, who claimed that setting the parameter τ (“Tao”) to 1 yield a more stable return. However, their study was not conclusive. It did not explain the validity of the argument. The two also came up with a step by step derivation of the BL model.
A study on the Bayesian inferences and the application of Bayesian Theory when giving the model assumptions developed the basic formulas for the portfolio’s posterior returns. Mishra and his colleagues introduced a recession factor to the BL model, which led to the formation of the two-factor BL Model. The market was not linearly related to this recession factor (Mishra et al., 2011). On the same note, another scholar, Meucci, furthered the research into the concept, making it easy for investment companies to comprehend and implement this new model in their daily modeling of returns. This invention was an incorporation of the non-normal investor views in the BLmodel. He extended these views in his paper, which allowed the inclusion of scenario analysis for various parameters when modeling. Barga and Natale also extended the research using uncertainty in views to track the volatility of errors in returns.
There is a vast amount of literature concerning the problem experienced when estimating the investors’ views using the BL model. A paper written by Indore in 2005 calibrated the model to allow its effectiveness when used by non-quantitative investors. In this model, he allowed the investors to specify their views on a confidence interval ranging from 0-100%. This interval measured the percentage change in the posterior weights compared to the prior weights (Indore, 2005). He related this confidence level to 100% which was the conditional estimate. Similarly, Becker and Gutler used the analysts’ forecasts in estimating the confidence in views using Monte Carlo simulation and the dividend discount model (Becker & Gutler, 2009). The project showed that the use of the BL model outperformed other models when implemented using several analysts’ forecasts.
In another paper, the authors expressed the qualitative views as linear inequalities quickly incorporated into the portfolio MVO model. Using the inequality expressions, the authors computed the portfolios’ expected alpha (Chiarawongse et al., 2012). In the paper, alpha represented a risk-adjusted measure of an investment’s active return. The expectation conditioned the qualitative views, which had been combined at a pre-determined degree of confidence.
In 2006, Mankert used behavioral finance concepts with the Black-Litterman model. In her article, she successfully introduced the home bias factor. This condition does not only influence the portfolio weights at given confidence levels but also increased the riskiness of foreign investment assets (Mankert, 2006). From her discussion, a biased home investor has less confidence in the views regarding foreign investments when compared to domestic investments. The foreign-invested assets got closer to the benchmark weights, unlike the domestically invested assets.
According to Guangliang and Litterman, investor views are derived from the views of the various renowned financial analysts (Guangling & Litterman, 1999). The essence of using a large number of the analyst is to prevent the views from being subjective. Financial websites like The Financial Times Interview, The Financial Analyst include a comprehensive view of the securities’ expected performance in the London Stock Exchange and other exchanges across the whole world. Since the certainty of this view is not known, Black and Litterman suggested the creation of the uncertainty matrix to reduce the level of uncertainty. Guangliang and Litterman also say that most investors prefer investing in stocks based on the advice they get from a financial analyst (Guangliang & Litterman, 1999). Most investors are afraid to invest a large sum of money in new stocks they do not know. Thus, the Black Litterman model offers the best criteria to determine which security should be given the highest amount of capital based on the analyst’s fundamental analysis.
Walters (2014) explains that the BL model calculates weights that, when considering, leads to the highest portfolio return. How the weights are apportioned is thus important in overcoming the weight being concentrated to few assets in a portfolio, which does not necessarily result in high return. The validity of the above statement has not been ascertained yet despite it being important. This project will seek to illustrate how to calculate the Black-Litterman and how the weights will be used to come up with a portfolio with the highest return.
Chapter 3: Methodology
To compute the required objectives, data was obtained from yahoo finance on two well-performing stocks in the FTSE-100 share index, Royal Dutch Holdings and Unilever
The Black-Litterman Model
The BL portfolio allocation strategy is used on the London stock market. The results are compared with the results from an equally weighted portfolio and a MV portfolio. The study uses the same assets and includes them in another portfolio where the CAPM determines the weights of each return to test the results from the Black-Litterman model. The expected results will explain whether the BL model’s performance is significantly better than the CAPM. Data used for the model is the daily data from December 31, 2014, to December 31, 2018. The four-year historical data will be used to compute a BL portfolio with determined weights and investor views.
The Financial Times Stock Watch will provide the annual investor views from January 2015 to December 2018. A renowned individual analyst and major financial institutions present recommendations about the buy, sell, and neutral views on the stocks in the market. The website includes a target price that determines the size of the expected returns that correspond to the investor views. The paper sets the target prices concerning the spot prices in the market at the time of the recommendation. Setting these targeted prices makes it possible to obtain the expected future returns of the investor views.
The strategies, buy, neutral, and sell are represented by values 1,0 and -1 assigned to them. A daily basis returns of different funds will be necessary for achieving a significant number of observations. The paper will not include the view of an asset that lacks investor recommendation. Each new calculation of views and weights will lead to an update of the portfolio weights. The returns of the previous year will calculate the weights of the upcoming year.
Investors are differently subjective leading to subjective views. Also, different investors forecast the future returns differently. To reduce the biasness in the computations, the paper will compile all the information from various sources as provided by the Financial Times. For instance, an asset having ten investors who would want to buy it means that it is more likely to outperform the other assets in the market.
The equation gives the expected utility function to use in the computation.
Maximizing the utility function yields
Differentiating the expected utility function concerning w and setting the value equal to 0 gives the weights of the model using the formula below.
Σ = An N x N various covariance matrix for the assets
r = Risk-free rate (treasury bills interest rates)
s = the variance of the portfolio
ω = Weight of each asset as given by the BL Model
λ = the coefficient of Risk aversion
The BL model will perform the portfolio optimization technique following the steps explained below.
The data from financial times from December 2018, using the market capitalization of assets, gave the market weights. Market capitalization is the value obtained after multiplying the outstanding shares sold within a particular day and the stock price during that day.
Step 2: Implied Returns
The implied return is calculated since it is important in finding the BL weights. The value of the implied returns will, therefore, be replaced in the BL formula
Π = λ Σ ω
Where the Risk aversion coefficient is denoted by λ.
This coefficient of risk aversion shows the risk of portfolio construction. For this study, the coefficient risk aversion is the risk of the portfolio that has the stocks from the portfolio that shall be constructed.
E(r) = Expected market return (used the FTSE share index)
rf = Risk free rate (364 days-treasury bills interest rates)
ω = Weights of assets based on their market capitalization
Σ = An N x N variance covariance matrix of the assets
= Market variance (FTSE share index variance)
The values recorded in the Variance Covariance Matrix represents the deviations of the daily returns from the mean return. In excel, the daily excess returns can be used to construct a matrix.
Excess returns = (daily log returns – average log return)
The daily share prices from January 2014 to December 2016 will be used to calculate daily log returns. The reason for using log-returns is that they give the continuous result of how the returns of the seven stocks have been moving over the past three years.
The log-returns are given by
Where: p = share price of day t
Pt-1 =share price of day t-1
The average of the returns of each company gives the daily log returns
Step 3: computing the expected return and the view matrix
The view matrix in this paper will be a K*N matrix representing each of the assets included in the views. The Black-Litterman model’s view matrix is denoted by P. Each row in the matrix denotes a view while each column denotes a company under investigation. The sum of the rows will be either 0 or 1 for relative and absolute views. Absolute views are the views for which an investor believes that the given security returns will decrease or increase at a specified percentage. On the other hand, relative views are the views of an investor who believes that an asset’s returns will decrease or increase when compared to other assets in the portfolio. In this paper, the view matrix will be a 3×3 identity matrix.
The matrix Q (the expected return matrix) is a column vector with each view’s estimated returns. The view matrix P specifies how much the investor views the assets. The Q matrix shows the size of the views. The matrix P does not quantify the views of the investors about the securities.
Ω represents the uncertainty in the views, and it is a KXK matrix. The Black Litterman model has several assumptions, with the main assumption that investors have uncorrelated views. Ω is diagonal and contains variances. The covariance of this matrix is zero. A square matrix represents the variance. It has a number of rows and columns equal to the number of views that investor has used. This matrix is important because it gives the possibility that the views may not be right at times, and the Black Litterman returns may not be true. This value is used to regulate the view matrix and thus protect the investor from losing a large amount of money if the financial analyst prediction is not true.
The uncertainty of view is calculated as
Ω= τP ΣPT
τ = A scalar number that indicates the uncertainty in the view (τ =1) an assumption of the model
P = A matrix(KxN matrix) representing investor views; each row represents a unique view of the market, and each entry of the row represents the weights of each asset
Ω = the variance-covariance matrix
The inverse of Ω gives the confidence of the investors
Black Litterman formula
Having all the information required by the model, the next step is the computation of the model. The Black Litterman model is defined as
E(R) = [ + ΩQ] (3.6)
τ=1 is a scalar number that indicates the uncertainty
P is a matrix with investors views;
Π is the Implied excess returns
Q is the expected returns of the portfolios from the views described in matrix P (Kx1 vector)
Ω = A diagonal covariance matrix (KxK matrix) with entries of the uncertainty within each view
New Black Litterman weights that will maximize the utility function in equation
Calculating CAPM and historical weights using excess returns from CAPM formula and historical data.
Where E(ri)=expected returns of an investment
rf= risk-free rate of interest
Bi = Beta of the returns
(rm)= returns on the market
Chapter 4: Data Analysis and Discussion
Figure 1: the plot for Unilever’s’ Average Returns
Figure 2: the plot for the Royal Dutch Average returns
The graphs above are a representation of the share prices behaviors over the past three years. Each company’s stock is plotted independently. Figure 1 above shows that the share prices of Unilever have been increasing gradually over the past four years. This increase is due to the improved technological trends in the country that has led to the growth of companies in the telecommunication sector.
The share prices of the Royal Dutch Company have been highly volatile over the past three years. This volatility is due to the competition, the seasonal consumption of their products, and government taxes. On an analysis of the Expected excess returns, the following observations are made.
The Black-Litterman expected excess returns to vary by a huge amount from the historical returns and CAPM returns. The historical returns are based on the trend of stock prices over the past three years. Unilever exhibits positive excess returns since the Unilever shares have been consistently going up over the past four years, and the daily returns exceed the risk-free daily rate. Although Royal Dutch’s average return is positive, it is less than the risk-free rate, making excess returns negative. The other five companies have both negative average returns and negative excess returns.
Both stocks exhibit positive excess returns when calculated using CAPM because the betas’ value is all positive. Although these values are positive, they are still less than those calculated using the Black Litterman Model. This is due to the investors’ views that are incorporated while using this model. The excess returns obtained using the Black Litterman model are all positive. These returns are the best among the three indicating that the Black Litterman model optimizes the expected excess returns. Investors who will make their investment decisions based on the Black Litterman model are likely to enjoy high returns incase the weights are true .
On analyzing the weights, the following were the observations. The Black Litterman model allocates more weight to the stocks whose views on expected future performance are better to maximize the portfolio’s overall return.
The traditional approach in which we use historical data to get weights gives more weight to those securities with large expected excess returns and small variance while giving fewer weights to those that have low expected excess returns and high variances. The negative weights obtained from historical data indicates short selling. The traditional portfolio approach advocates buying low and selling high, that is, selling off the stock whose return has been increasing.
An investor stands to gain some profits if he or she sells a stock at a higher price than the price it was purchased—for instance, this approach advocates for selling of Unilever, which have been increasing gradually. CAPM apportions weight to the stocks according to the proportion of the excess returns of each stock. That is, the stock with the highest expected excess return will be assigned the highest possible weight. We can see from figures 1, and 2 Unilever and Royal Dutch have superior excess returns according to CAPM, which translates to more weights for the three stocks.
On the other hand, the Black Litterman model advocates holding stocks with the expectation that their share prices will rise in the future based on the fundamental analysis conducted by the financial analyst. This is under the assumption that the financial analyst views will actualize, and the investment will earn more profits.
The efficient frontier
This diagram shows the efficient frontier constructed using the weights obtained using the Black Litterman Model.
Figure 3: The efficient frontier for the Black Litterman Model for the data
The optimal portfolios located on in the efficient frontier maximize the return for a given level of risk. Therefore, the portfolios on the efficient frontier are a set of optimal portfolios that satisfies the condition that no other portfolio exists with a higher expected return but the same risk.
In this case, the optimal portfolio gives a return of 0.0036 and a risk of 0.0112, which is the return and standard deviation of the portfolio constructed using the Black Litteman weights. This portfolio is considered to be optimal since it forms a tangent with the capital market line. From this result, it can be shown that the Black Litterman model optimizes any portfolio.
The Sharpe’s Ratio
The Sharpe’s ratio is used to measure the performance of the portfolios obtained from the different models. Computing it using r gave the following values.
Table 1: The Sharpe Ratios for each model
The historical approach gives a negative Sharpe ratio to indicate that the return of the portfolio constructed using this method gives negative expected excess returns. The Black Litterman model gave the largest Sharpe’s ratio, meaning that the Black Litterman portfolio is the optimal portfolio. CAPM is a better model than the historical approach.
The main objective of this research was to use the BL model to optimize a portfolio. This model calculates the optimal weights to be assigned to the assets in the portfolio’s assets. We followed a stepwise approach and obtained the Black Litterman weights and later did a comparative analysis with the weights obtained from the historical and CAPM model. Afterward, we each of the portfolios measurement as constructed using the three models.
The BL model is usable in the asset allocation process. The model estimates the weights that lead to more diversified and more stable portfolios that the common estimates derived for the CAPM returns and the historical returns. This model’s nature makes it hard for investors to use CAPM or historical returns optimization to obtain a portfolio without extreme weights. The BL model’s disadvantage is the use of too much data, which may be hard to obtain.
The first step when using this model is identifying an investable universe and then computing each asset’s market capitalization. The next step involves estimating a variance-covariance matrix, which summarizes the excess returns of all the assets. The historical data for an appropriate time window helps in the formation of this matrix. The details on how to compute the variance-covariance matrix are elaborated by Bevan &Winkelmann (1998) and Litterman (2003). In their literature, they use daily historical excess returns to estimate the daily historical returns. An appropriate proxy, such as the FTSE-100 index, helps in the instances where the actual return cannot be used. The returns on treasury bills represent the investors’ risk-free rate.
Finding the market capitalization information is a challenge for an individual investor, especially when solving the asset allocation problem. However, institutional investors enjoy all the benefits of modeling with minimal obstacles since they have access to the index information provided by various providers. Investors have a hard time computing adequate marketing information, especially on illiquid assets. Inconsistencies, smoothing, and reporting delays also complicate the return data for the same asset classes.
An investor must quantify their view and apply them in the computation of new return estimates. The qualitative and quantitative processes can be used to derive these views, which have high possibilities of being incomplete or conflicting. The last step entails feeding the data into a portfolio selection model that generates the efficient frontier and the selected efficient portfolio. Bevan and Winkleman (1999) described how to use the Black Litterman model in their asset allocation process for international fixed income. They calibrated the information and computed covariance matrices.
Chapter 5: Conclusion
Implementing the Black Litterman model has complications, and the outcome of the model depends on how the components of the model interact and how accurate their relationship is determined. This model requires a thorough understanding of the components of the model on the investors’ end. This study shows that the Black Litterman Model performs well than the historical approach and the CAPM.
The Sharpe ratios for all the models show that the Black Litterman model has a better performance. The yields of stocks modeled using the model are higher than those modeled using the CAPM and the historical returns. However, the model does not assure that the yields are the most superior because the investor views var\y. It is worth noting that the underperformance of this model in this study results from inaccurate views. The problems in the CAPM can be curbed by using the Black Litterman Model. One advantage of this model is its robustness towards changing the confidence of views.
This model outperforms the Meucci (20100 model) since it makes the subjective parts more consistent, and it outperforms the model when imposing the short-selling restrictions. However, the model underperforms when solving analytical problems without imposing restrictions. There is a gap in determining which model is better. A sure conclusion is that the accuracy of the investor’s views determines the performance of the model. For an optimal allocation problem involving a wide range of assets, the Black Litterman model is better. Forecasting indices and asset classes are less subjective than when dealing with single stocks.
It is necessary to add additional asset classes with a high level of co-movement, use a wider range of asset classes, a more dispersed stock market, and a longer period to understand the inner working of this model. Incorporating the views directly into the covariance matrix will be important in determining the optimal weights and future covariances.