Black littleman portfolio optimization

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Black littleman portfolio optimization

  1. 1. JOHN VON NEUMANN INSTITUTE VIETNAM NATIONAL UNIVERSITY HOCHIMINH CITY BLACK-LITTERMAN PORTFOLIO OPTIMIZATION Hoang Hai Nguyen nguyen.hoang@jvn.edu.vn HCM City, July- 2012 1
  2. 2. Abstract: In practice, mean-variance optimization results in non-intuitive and extreme portfolio allocations, which are highly sensitive to variations in the inputs. Generally, efficient frontiers based on historical data lead to highly concentrated portfolios. The Black-Litterman approach overcomes, or at least mitigates, these problems to a large extent. The highlight of this approach is that it enables us to incorporate investment views (which are subjective in nature). These aspects make the Black-Litterman model a strong quantitative tool that provides an ideal framework for strategic/tactical asset allocation. In this project, we will apply the Black-Litterman model for the context of VietNam equity markets. To represent the VietNam equity markets, we select top K = 10 most market capitalization stocks of the Ho Chi Minh city stock exchange with historical data of at least 1 year for tactical asset allocation. As extension part, we empirically compare the performance of the two approaches. The study found that the BlackLitterman efficient portfolios achieve a significantly better return-to-risk performance than the mean-variance optimal approach/strategy. 2
  3. 3. 1. Introduction Since publication in 1990, the Black-Litterman asset allocation model has gained wide application in many financial institutions. As developed in the original paper, the BlackLitterman model provides the flexibility of combining the market equilibrium with additional market views of the investor. The Black-Litterman approach may be contrasted with the standard mean-variance optimization in which the user inputs a complete set of expected returns 1 and the portfolio optimizer generates the optimal portfolio weights. Because there is a complex mapping between expected returns and the portfolio weights, and because there is no natural starting point for the expected return assumptions, users of the standard portfolio optimizers often find their specification of expected returns produces output portfolio weights which do not seem to make sense. In the BlackLitterman model the user inputs any number of views, which are statements about the expected returns of arbitrary portfolios, and the model combines the views with equilibrium, producing both the set of expected returns of assets as well as the optimal portfolio weights. Although Black and Litterman concluded in their 1992 article [Black and Litterman, 1992]: “. . . our approach allows us to generate optimal portfolios that start at a set of neutral weights and then tilt in the direction of the investor’s views.” they did not discuss the precise nature of that phenomenon. As we demonstrate here, the optimal portfolio for an unconstrained investor is proportional to the market equilibrium portfolio plus a weighted sum of portfolios reflecting the investor’s views. Now the economic intuition becomes very clear. The investor starts by holding the scaled market equilibrium portfolio, reflecting her uncertainty on the equilibrium, then invests in portfolios representing her views. The Black-Litterman model computes the weight to put on the portfolio representing each view according to the strength of the view, the covariance between the view and the equilibrium, and the covariances among the views. We show the conditions for the weight on a view portfolio to be positive, negative, or zero. We also show that the weight on a view increases when the investor becomes more bullish on the view, and the magnitude of the weight increases when the investor becomes less uncertain about the view. The rest of the article is organized as follows. In Section 2, we review the basics of the Black-Litterman asset allocation model. In Section 3, we present our empirical findings of the study and data description. Then we present our main results in Section 4 3
  4. 4. 2. The Black-Litterman model The Black and Litterman (1990, 1991, 1992) asset allocation model is a sophisticated asset allocation and portfolio construction method that overcomes the drawbacks of traditional mean-variance optimization. The Black-Litterman model uses a Bayesian approach to combine the subjective views of investors about the expected return of assets. The practical implementation of the Black-Litterman model was discussed in detail in the context of global asset allocation (Bevan and Winkelmann, 1998), sector allocation (Wolfgang, 2001) and portfolio optimization (He and Litterman, 1999). In order to incorporate the subjective views of investors, the Black-Litterman model combines the CAPM (Sharpe, 1964), reverse optimization (Sharpe, 1974), mixed estimation (Theil, 1971, 1978), the universal hedge ratio/Black‟s global CAPM (Black and Litterman 1990, 1991, 1992; Litterman, 2003), and mean-variance optimization (Markowitz, 1952). The Black-Litterman model creates stable and intuitively appealing mean-variance efficient portfolios based on investors‟ subjective views and also eliminates the input sensitivity of the mean-variance optimization. The most important input in mean-variance optimization is the vector of expected returns. The model starts with the CAPM equilibrium market portfolio returns starting point for estimation of asset returns, unlike previous similar models started with the uninformative uniform prior distributions. The CAPM equilibrium market portfolio returns are more intuitively connected to market and reverse optimization of the same will generate a stable distribution of return estimations. The Black-Litterman model converts these CAPM equilibrium market portfolio returns to implied return vector as a function of risk-free return, market capitalization, and covariance with other assets. Implied returns are also known as CAPM returns, market returns, consensus returns, and reverse optimized returns. Equilibrium returns are the set of returns that clear the market if all investors have identical views. The following is the Black-Litterman formula (Equation 1) along with detailed description of each of its components. In this project, K represents the number of views and N represents the number of assets in the model. [ ] = ( ∑) + ′Ω ( ∑) ∏ + ′Ω (1) where, E[R] is the new (posterior) combined return vector (N × 1 column vector); τ, a scalar; Σ, the covariance matrix of excess returns (N × N matrix); P, a matrix that identifies the assets involved in the views (K × N matrix or 1 × N 4
  5. 5. row vector in the special case of 1 view); Ω, a diagonal covariance matrix of error t erms from the expressed views representing the uncertainty in each view (K × K matrix); ∏, the implied equilibrium return vector (N × 1 column vector); Q, the View Vector (K x 1 column vector) The Black-Litterman model uses the equilibrium returns as a starting point and the equilibrium returns of the assets are derived using a reverse optimization method using Equation 2 ∏ = ∑ (2) where, ∏, is the implied equilibrium excess return vector; , a risk aversion coefficient; ∑, the covariance matrix, and , is the market capitalization weight of the assets. The risk aversion coefficient characterizes the expected risk-return tradeoff and it acts as a scaling factor for the reverse optimization. The risk aversion coefficient can be calculated using equation 3 = (3) The implied equilibrium return vector is nothing but the market capitalization-weighted portfolio. In the absence of views, investors should hold the market portfolio. However, Black-Litterman model allows investors to incorporate their subjective views on the expected return of some of the assets in a portfolio, which may differ from the implied equilibrium returns. The subjective views of investors can be expressed in either absolute or relative terms. where, Q, the view vector, which is k × 1 dimension; k, the number of views, either absolute or relative. The uncertainty of views results in a random, unknown, independently, normally distributes error term vector ( ) with mean 0 and covariance matrix Ω. Thus a view has the form Q+ 5
  6. 6. Q+ = : : : + : (4) Investor views on the market and their confidence level on the views form the basis for arriving at new combined expected return vector. With respect to investor views, we need to consider the following aspects while developing the Black-Litterman model: 1. Each view should be unique and uncorrelated with the other. 2. While constructing the views, we need to ensure that the sum of views is either 0 or 1, which ensures that all the views are fully invested. The investor view matrix (P) was constructed differently by various authors. He and Litterman (1999) and Izorek (2005) used a market capitalization weighted scheme. However, market capitalization weighted scheme is applicable only in relative views.The expected return on the views is organized as a column vector (Q) expressed as Kx1 vector. Omega, the covariance matrix of views, is a symmetric matrix with non-diagonal elements as 0s. For calculating it, we have assumed that the variance of the views will be proportional to the variance of the asset returns, just as the variance of the prior distribution is. This method has been used by He and Litterman (1999) and Meucci (2006). Using these expected return, risk aversion coefficient (λ) and covariance matrix (∑), new asset weights can be allocated using equation 5. = ( ∑) * E[R] (5) Before we attempt to detail the empirical examination of the Black-Litterman model, it might be useful to give an intuitive description of the major steps, which are presented in Figure 1 6
  7. 7. Figure 1: Major steps behind the Black-Litterman model. 3. Empirical findings of the study and Data description Data description The current study is based on various stocks constructed and maintained by the Ho chi minh city stock exchange (HSE), VietNam. We select top K = 10 most market capitalization of Ho chi minh city stock exchange with historical and data of at least 1 year and use daily closing prices from January 1st, 2011 to January 31st, 2012. List of 10 stocks are selected such as: No. Stocks Code 1 Baoviet Holdings PetroVietnam Fertilizer and Chemicals Company Vietnam export import Bank FPT Corporation Hoang Anh Gia Lai JSC Masan Group Corporation Saigon Securities Inc Sai Gon Thuong Tin Bank Vingroup Vinamilk Corp BVH Market capitalization (billion VND) 46,272 DPM 12,160 4.61% EIB FPT HAG MSN SSI STB VIC VNM 15,523 10,610 13,645 64,409 7,549 14,040 35,595 47,095 5.89% 4.02% 5.17% 24.39% 1.79% 5.31% 13.48% 17.84% 2 3 4 5 6 7 8 9 10 Proportion 17.51% 7
  8. 8. Empirical findings of the study As VietNam is an emerging economy that could withstand the after-effects of global financial meltdown, several foreign institutional investors are keen on parking their investments in the country. Each of them has different long-term and short-term views on different sectors of the VietNam equity market. This has motivated to empirically examine the tactical asset allocation across different sectors of VietNam equity market through Black-Litterman approach. The study has considered the monthly closing price of ten stocks of HSE from January 1st, 2011 to January 31st, 2012. The daily closing price of stocks has been taken to compute the continuous compounded return of daily these stocks by taking the natural logarithmic of price difference. This is represented as follows: = ln( ) − ln( ) where, is the return at time t , price at time t, and , price at time t-1 A risk-return profile of 10 stocks over a one years, from 1st, 2011 to January 31st, 2012, is presented in the Table 1 and Figure 2. Table 1 and Figure 2 indicate the risk-return profiles of ten stocks of HSE. Table 1.Historical risk-return profile of different sectors (1st, 2011 to January 31st, 2012) No. 1 2 3 4 5 6 7 8 9 10 Stocks Risk (%) BVH DPM EIB FPT HAG MSN SSI STB VIC VNM 52.61% 35.56% 18.52% 29.26% 41.33% 48.04% 40.62% 25.00% 42.67% 27.32% Return (%) 10% 12% 15% 15% 16% 15% 18% 20% 15% 25% 8
  9. 9. 30% VNM 25% STB Return 20% HAG FPT EIB 15% SSI VIC 10% MSN DPM BVH 5% 0% 0% 10% 20% 30% 40% 50% 60% Risk Figure 1. Scatter plot of risk-return profile of different sector (1st, 2011 to January 31st, 2012) Traditional mean variance optimization often leads to highly concentrated, undiversified asset allocations. When developing an opportunity set, one should select non-overlapping mutually exclusive asset classes that reflect the investors‟ investable universe. In this project, we have presented two types of graphs – efficient frontier graphs and efficient frontier asset allocation area graphs. Efficient frontier displays returns on the vertical axis and the risk (standard deviation) of returns on the horizontal axis. Efficient frontier is the locus of points, which represents the different combination of risk and return on an efficient asset allocation, where an efficient asset allocation is one that maximizes return per unit of risk. This is presented in Figure 3. 35% Assets Implied_EF 30% Return 25% 20% 15% 10% 5% 0% 10% 20% 30% 40% 50% 60% Risk Figure 2: Efficient frontier, historical return versus risk. 9
  10. 10. Efficient frontier asset allocation area graphs complement the efficient frontier graphs. They display the asset allocations of the efficient frontier across the entire risk spectrum. Efficient frontier area graphs display risk on the horizontal axis. The efficient frontier area graph displays all the asset allocation on the efficient frontier. This is helpful to visualize the efficient frontier graphs and the efficient frontier asset allocation area graphs together because one can simultaneously see the asset allocations associated with the respective risk-return point on the efficient frontier, and vice versa. To avoid the limitation of efficient frontiers based on historical data leads to highly concentrated portfolios in the mean variance approach of Markowitz‟ s theory, the BlackLitterman model (1992) proposed a better solution. This was further researched and emphasized by Von Neumann, Morgenstern and James Tobin. A rich literature on this was well documented by Sharpe (1964, 1974), respectively. The pivotal point of BlackLitterman model is implied returns. Implied returns (otherwise known as equilibrium returns) are the set of sectoral indices returns that clear the market if all investors have identical views. This means the market follows the strong form efficiency of the efficient market hypothesis or leads to a perfect competitive market. To compute the equilibrium returns of the sectoral indices, we need an input parameter, that is, risk aversion coefficient. The risk aversion coefficient characterizes the risk-return trade off. Risk aversion coefficient is the ratio of risk-return and variance of the benchmark portfolio. The mathematical representation of risk aversion coefficient (denoted by λ) is as follows: = − where, is the return on benchmark; , the risk free rate, and , is the variance of the benchmark. This project considered HSE as the benchmark index to compute the risk aversion coefficient. We have considered the risk free rate to be 8%. By computing the ratio of risk premium and variance of HSE, we have calculated the risk aversion coefficient (λ) at 4.2%. The risk aversion coefficient characterizes the risk return trade off. From the daily return series of stocks, we have generated the covariance matrix. This is represented in Table 2 10
  11. 11. BVH DPM EIB FPT HAG MSN SSI STB VIC VNM BVH 0.2768 0.0858 0.0126 0.0367 0.0927 0.1449 0.0761 -0.0044 0.0527 0.0210 DPM 0.0858 0.1264 0.0213 0.0517 0.0892 0.0704 0.0948 0.0120 0.0281 0.0352 EIB 0.0126 0.0213 0.0343 0.0114 0.0177 0.0095 0.0245 0.0141 0.0125 0.0040 FPT 0.0367 0.0517 0.0114 0.0856 0.0468 0.0323 0.0586 0.0078 0.0151 0.0217 HAG 0.0927 0.0892 0.0177 0.0468 0.1708 0.0702 0.0935 0.0059 0.0613 0.0276 MSN 0.1449 0.0704 0.0095 0.0323 0.0702 0.2308 0.0527 0.0001 0.0336 0.0283 SSI 0.0761 0.0948 0.0245 0.0586 0.0935 0.0527 0.1650 0.0132 0.0313 0.0326 STB -0.0044 0.0120 0.0141 0.0078 0.0059 0.0001 0.0132 0.0625 0.0089 0.0080 VIC 0.0527 0.0281 0.0125 0.0151 0.0613 0.0336 0.0313 0.0089 0.1821 0.0135 VNM 0.0210 0.0352 0.0040 0.0217 0.0276 0.0283 0.0326 0.0080 0.0135 0.0747 Table 2. The covariance matrix of our stocks. To compute the weighted market capitalization of each of stocks, we have considered the closing prices and shares outstanding of each member constituents on 1st November, 2011, as this is the end date of the study period. The market capitalization weights are represented in the following Table 3. Table 3. Implied returns, market capitalization weights ( No. Stocks Market capitalization* (billion VND) 46,272 12,160 15,523 10,610 13,645 64,409 7,549 14,040 35,595 47,095 BVH DPM EIB FPT HAG MSN SSI STB VIC VNM * Dated 1st November, 2011 1 2 3 4 5 6 7 8 9 10 ) weight (%) 17.51% 4.61% 5.89% 4.02% 5.17% 24.39% 1.79% 5.31% 13.48% 17.84% Finally, the implied excess returns of the sectoral indices have been computed by considering their corresponding risk aversion coefficient (λ) and market capitalization weights ( ). This is represented in Table 4. 11
  12. 12. No. 1 2 3 4 5 6 7 8 9 10 Stocks BVH DPM EIB FPT HAG MSN SSI STB VIC VNM Risk (%) 52.61% 35.56% 18.52% 29.26% 41.33% 48.04% 40.62% 25.00% 42.67% 27.32% Total implied return* (%) 44.84% 24.51% 5.25% 12.84% 27.04% 42.38% 22.22% 3.13% 21.51% 12.97% Table 4.Implied return (∏ = λ∑ ) of stocks - risk profile (January 1st, 2011 to January 31st, 2012). *Total implied return = implied excess return + risk free rate After generating the implied return and risk of the stocks, we have generated the optimized portfolio efficient frontier. Here, it is understood that implied returns are considered as the E[R] of the respective stocks. These implied returns are the starting point for the Black-Litterman model. However, it has been observed that most investors stop thinking beyond this point while selecting the optimal portfolio. If investors or market participants do not agree with implied returns, the Black-Litterman model provides an effective framework for combining the implied returns with the investor’s unique views or perception regarding the markets, which result in well diversified portfolios reflecting their views. To implement the Black-Litterman approach, an asset manager has to express his or her views in terms of probability distribution. Black-Litterman assumes that the investor has two kinds of views absolute and relative. For now, we assume that the investor has k different views on linear combinations of E[R] of the n assets. This is explained in details as an equation (Equation 1) in the methodology section. In this project, we have considered the combination of one absolute and one relative view on list of our stocks. These views are expressed as follows: Absolute view View 1 VNM will generate an absolute return of 10%. Relative view 12
  13. 13. Views 2 MSN outperform HAG by 8%. These two views are expressed as follows: µVNM = 0.1 strong view: = 0.0019 µMSN - µHAG= 0.08 weaker view: Thus P = 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 = 0.0065 1 0.1 0.0019 ,q= and Ω = 0 0.08 0 0 0.0065 Applying formula (1) to compute E[R], we get E[R] BVH DPM EIB 43.56% 23.97% 5.25% FPT HAG MSN SSI STB 12.55% 27.77% 39.43% 22.01% 3.04% VIC VNM 21.60% 11.46% Set up the quadratic problems for portfolion optimization: min ¸ μ x≥R Ax = 1 x≥0 where, x: weight vector of portfolio H: covariance matrix of our stocks μ: new combined return vector R: expected return contraint of portfolio A: unity vector 13
  14. 14. 4. The results Solving for R = 3.0% to R = 32% with increments of 1% we now get the optimal portfolios and the effcient frontier depicted in Table 5 and Figure 3 Table 5: Black-Litterman Efficient Portfolios Return Variance 3% 14.43% 4% 5% BVH Weights HAG MSN DPM EIB FPT SSI STB 1.05% 0.00% 44.77% 9.84% 0.00% 14.43% 1.05% 0.00% 44.77% 9.84% 14.43% 1.05% 0.00% 44.77% 9.84% 6% 14.43% 1.05% 0.00% 44.77% 7% 14.43% 1.05% 0.00% 8% 14.43% 1.05% VIC VNM 2.19% 0.00% 19.01% 4.52% 18.62% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62% 0.00% 44.77% 9.84% 0.00% 2.19% 0.00% 19.01% 4.52% 18.62% 9% 14.43% 1.53% 0.00% 43.91% 9.83% 0.00% 2.70% 0.00% 18.67% 4.79% 18.56% 10% 14.53% 2.72% 0.00% 41.80% 9.82% 0.00% 3.94% 0.00% 17.84% 5.48% 18.41% 11% 14.72% 3.91% 0.00% 39.70% 9.80% 0.00% 5.18% 0.00% 17.00% 6.16% 18.26% 12% 15.01% 5.10% 0.00% 37.59% 9.78% 0.00% 6.43% 0.00% 16.16% 6.84% 18.11% 13% 15.39% 6.11% 0.00% 35.45% 9.55% 0.68% 7.56% 0.00% 15.40% 7.36% 17.88% 14% 15.85% 7.10% 0.00% 33.32% 9.28% 1.47% 8.69% 0.00% 14.66% 7.84% 17.65% 15% 16.38% 8.09% 0.00% 31.18% 9.01% 2.25% 9.81% 0.00% 13.92% 8.33% 17.42% 16% 16.98% 9.07% 0.00% 29.05% 8.74% 3.04% 10.93% 0.00% 13.17% 8.81% 17.19% 17% 17.64% 10.06% 0.00% 26.91% 8.47% 3.83% 12.05% 0.00% 12.43% 9.30% 16.96% 18% 18.35% 11.05% 0.00% 24.78% 8.21% 4.61% 13.17% 0.00% 11.68% 9.79% 16.72% 19% 19.11% 12.03% 0.00% 22.64% 7.94% 5.40% 14.29% 0.00% 10.94% 10.27% 16.49% 20% 19.91% 12.95% 0.51% 20.46% 7.55% 6.01% 15.35% 0.00% 10.20% 10.79% 16.18% 21% 20.75% 13.82% 1.31% 18.25% 7.09% 6.52% 16.39% 0.00% 9.48% 11.32% 15.83% 22% 21.61% 14.68% 2.08% 16.04% 6.61% 7.01% 17.43% 0.08% 8.75% 11.85% 15.47% 23% 22.51% 15.52% 2.70% 13.80% 6.07% 7.43% 18.48% 0.47% 8.03% 12.38% 15.11% 24% 23.42% 16.37% 3.33% 11.56% 5.53% 7.85% 19.53% 0.85% 7.32% 12.91% 14.75% 25% 24.36% 17.22% 3.95% 9.32% 4.99% 8.27% 20.58% 1.24% 6.60% 13.45% 14.39% 26% 25.32% 18.06% 4.57% 7.08% 4.45% 8.69% 21.63% 1.62% 5.89% 13.98% 14.03% 27% 26.30% 18.91% 5.20% 4.83% 3.90% 9.11% 22.68% 2.01% 5.17% 14.52% 13.67% 28% 27.29% 19.75% 5.82% 2.59% 3.36% 9.53% 23.73% 2.39% 4.46% 15.05% 13.31% 29% 28.29% 20.60% 6.44% 0.35% 2.82% 9.95% 24.78% 2.78% 3.74% 15.59% 12.95% 30% 29.31% 21.52% 6.98% 0.00% 1.86% 10.41% 25.94% 3.07% 1.93% 15.99% 12.28% 31% 30.34% 22.48% 7.49% 0.00% 0.78% 10.89% 27.13% 3.35% 0.00% 16.36% 11.51% 32% 31.41% 23.98% 7.54% 0.00% 0.00% 11.60% 28.66% 3.26% 0.00% 16.23% 8.73% 33% 32.51% 25.58% 7.37% 0.00% 0.00% 12.32% 30.24% 2.97% 0.00% 16.03% 5.50% 34% 33.65% 27.18% 7.20% 0.00% 0.00% 13.03% 31.83% 2.68% 0.00% 15.82% 2.27% 35% 34.82% 29.23% 6.26% 0.00% 0.00% 13.87% 33.47% 2.02% 0.00% 15.16% 0.00% 36% 36.07% 32.36% 3.51% 0.00% 0.00% 15.01% 35.23% 0.48% 0.00% 13.41% 0.00% 14
  15. 15. Figure 3. Efficient Frontier and the Composition of Efficient Portfolios using the Black-Litterman approach Extension part: comparision of two approaches Figure 4 plots the efficient frontier generated by implied return and Black Litterman return. It can be concluded that Black-Litterman model provides the optimal portfolio with maximum return and minimum risk in comparison to implied return based and mean variance based portfolio optimization. 40% 35% 30% Return 25% 20% 15% 10% Implied_EF 5% Black-Litterman EF 0% 10% 15% 20% 25% 30% 35% 40% Risk Figure 4. Efficient Frontier: Black-Litterman versus implied return. 15
  16. 16. REFERENCES [1] The intuition behind Black-Litterman model portfolios - Guangliang He, [2] A step-by-step guide to the Black-Litterman model - Thomas M. Idzorek (2005), [3] Exercises in Advanced Risk and Portfolio Management – A. Meucci (with code) [4] Optimization Methods in Finance - Gerard Cornuejols (2005), [5] An equilibrium approach for tactical asset allocation: Assessing Black-Litterman model to Indian stock market - Alok Kumar Mishra (2011), 16

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