3、 Methodology
3.2 Black-Litterman Methodology
The Black-Litterman model makes two significant contributions to the problem of asset allocation. First, it provides an intuitive prior, the CAPM equilibrium market portfolio, as a starting point for the application of Bayesian techniques to estimate returns. The idea that one could use „reverse optimization‟ to generate a stable distribution of returns from the CAPM market portfolio as a starting point is a significant improvement to the process of return estimation.
Second, it provides a clear way to specify investors‟ views and to blend the investors‟ views with prior information using Bayesian techniques. This process estimates expected return and covariance which can be used as input to an optimizer. Before their paper, nothing similar had been published. The mixing process had been studied, but nobody had applied it to the problem of estimating returns. No research linked the process of specifying views to the blending of the prior and the investors‟ views. The Black-Litterman model provides a quantitative framework for specifying the investors‟ views, and a clear way to combine those investor‟s views with an intuitive prior to arrive at a new combined distribution. This process is presented as Figure 1 and it can be divided into three parts: Reverse optimization, specifying the views, and the Black-Litterman formula. Besides, we consider use shrinkage techniques to shrunk the sample covariance matrix, modify the drawbacks that Black-Litterman model only consider the historical sample covariance matrix.
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Figure 1 : The process of deriving the New Combined Return Vector
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3.2.1 Computing the CAPM Equilibrium Excess Returns
The process of computing the CAPM equilibrium excess returns is straight forward. These returns will provide the prior distribution for the Black-Litterman model.
CAPM is based on the concept that there is a linear relationship between risk and return.
Further, it requires returns to be normally distribution. This model is of the form
m
f r
r r
E( ) (10)
where
rf The risk free rate
rm The excess return of the market portfolio
A regression coefficient computed as
m p
The residual or asset specific excess return
Under the CAPM theory the investor is reward for the systemic risk measured by, but is not rewarded for taking non-systemic risk associated with . This is because within a diversified portfolio the total should tend to 0 in the limit.
The CAPM theory states that all investors should hold the market portfolio as their risky asset.
They may hold arbitrary fractions of their wealth in the risky asset and the remainder in the risk-free asset depending on their degrees of risk aversion. The market portfolio is on the efficient frontier, and has the maximum Sharpe Ratio of any portfolio on the efficient frontier.
Because all investors hold only this portfolio of risky assets, at equilibrium the market capitalizations of the various assets will determine their weights in the market portfolio. Now we start with the „reverse optimization‟ for compute the equilibrium excess returns.
3.2.2 Computing Via Reverse Optimization
This section, we derive the equations for „reverse optimization‟ starting from the quadratic utility function. Throughout this paper, K is used to represent the number of views and N is used to represent the number of funds in the portfolio.
w w w
U T ) T
(2 (11) where
U : The investor‟s utility; the objective function during portfolio optimization w : The vector of weights invested in each asset (N x 1 column vector)
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: The vector of equilibrium excess returns for each asset (N x 1 column vector)
: The risk aversion parameter of the market
: The covariance matrix of excess returns (N x N matrix)
We will constrain the problem by asserting that the covariance matrix of the return is known3. The investor‟s utility U is a concave function, so it will have a single global maximum. If we maximize the utility with no constraints there is a closed form solution. We find the exact solution by taking the first derivative of (11) with respect to the weights and setting it to 0. Solving this for yields (12)
wmkt
(12)
where
wmkt : The market capitalization weight of the assets (N x 1 column vector)
The Black-Litterman model uses „equilibrium‟ returns as a neutral starting point. Equilibrium returns are the set of returns that clear the market. The equilibrium returns are derived using a reverse optimization method in which the vector of implied excess equilibrium returns is extracted from known information using formula (12).
In order to use formula (12) we need to have a value for, the risk aversion coefficient of the market. We can find by multiplying both sides of (12) by wmktT and replacing vector terms with scalar terms.
2
) (
m f
m r
r E
(13)
where
E(r) : the total return on the market portfolio (E(r)rf) rf : the risk free rate
2
m : the variance of the market portfolio (m2 wmktT wmkt)
As part of our analysis we must arrive at the terms on the right hand side of formula (13);
E(r),rf andm2 in order to calculate a value for. When we have a value for , then we plug w, and into formula (12) and generate the equilibrium asset returns. Formula (12) is the closed form solution to the reverse optimization problem for computing asset returns given an optimal mean-variance portfolio in the absence of constraints. Furthermore if we feed, and back into the formula (12), we also can solve for the weights (w). If we use
3 We use the historical covariance matrix as the .
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historical excess returns rather than equilibrium excess returns, the results will be very sensitive to changes in .with the Black-Litterman model, the weight vector is less sensitive to the reverse optimized vector. This stability of the optimization process is one of the strengths of the Black-Litterman model.
3.2.3 Specifying the Views
We will describe the process of specifying the investors‟ views of estimated returns. We define the combination of the investors‟ views as the prior distribution. First, we will require each view to be unique and uncorrelated with the other views. This will give the prior distribution the property that the covariance matrix will be diagonal, with all off diagonal entries equal to 0. We constrain the problem this way in order to simplify the problem and improve the stability of the results. Second, we will require views to be fully invested; either the sum of weights in a view is zero (relative view) or is one (an absolute view).
We will represent the investors‟ K views on N assets used the following matrices.
1. P is a K x N matrix of the asset weights within each view, for a relative view the sum of weights will be 0, for an absolute view the sum of the weights will be 1.
Different authors compute the various weights within the view differently; Idzorek uses a capitalization weighed scheme, whereas others use an equal weighted scheme. Here we use the capitalization weighed scheme.
2. Q is a K x 1 matrix of the returns for each view.
3. Ω is a K x K matrix of the covariance of the views. Ω is diagonal as the views are required to be independent. 1 is known as the confidence in the investors‟ views. The i-th diagonal element of Ω is represented asi.
As an example of how these matrices be populated we have 3 assets and two views. First, a relative view in which the investor believes that asset 1 will outperform asset 2 by 4% with confidence1. Second, an absolute view in which the investor believes that asset 3 will return 3% with confidence2. These views are specified as follows:
Given these matrices we can formulate the prior distribution mean and variance in portfolio
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space as:
PE(r) ~ N (Q, Ω) (15)
There are four main ways to calculate Ω. First, we actually compute the variance of the view.
Second, we can just assume that the variance of the view will be proportional to the variance of the assets, just as the variance of the sampling distribution is. He and Litterman (1999) use this method, and we can use the variance of the view computed from the sampling confidence in terms of the percentage move of the weights from no views to total certainty in the view.
Beach and Orlov (2006) introduce the final method. They consider about asset returns are characterized by several stylized facts: volatility clustering, excess kurtosis, asymmetry, autocorrelation in risk, time-varying volatility. They use GARCH-derived views as an input into the Black-Litterman model. Here, we use GARCH-derived views as proxies for investor views in the Black-Litterman model. It is useful for expository purposes, since there is no model that effectively describes an investor‟s views.
3.2.4 Predicting of Covariance Matrix by GARCH Model
The GARCH model offers a more parsimonious model that reduces the computational burden.
It uses past variances and past variance forecasts to forecast future variances. The model is quite successful in predicting conditional variances. GARCH models allow not only forecasting conditional means of asset returns, but also conditional variances. We combine the ARCH-in-Mean model to relate the expected return on assets to the expected risk, and the Exponential GARCH model to allow for asymmetric shocks to volatility.
GARCH (p, q)
In general, a GARCH model can be represented by two equations － one for the conditional mean and the other for the conditional variance:
)
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where:
yt : The dependent variable (i.e. excess return) xt : Vector of exogenous variables
, and are the coefficients to be estimated. The one-period ahead forecast variance
2
t (conditional variance) depends on the mean (), news about volatility from the previous period (t2i, the ARCH term), and last period‟s forecast variance (t2 j, the GARCH term).
GARCH (p, q) refers to the presence of q-order GARCH term and p-order ARCH term.
EGARCH-M (p, q)
Exponential GARCH models are able to account for asymmetric shocks to volatility.
ARCH-in-Mean models introduce the conditional variance into the mean equation.
ˆ2 capture the empirical regularity that a negative shock leads to a relatively higher conditional variance than a positive shock of the same magnitude.
3.2.5 The Black-Litterman Formula
Applying Bayes theory to the problem of blending the sampling and prior distributions, we can create a new posterior distribution of the asset returns. We can derive the equation4 for the posterior distribution of asset returns.
)
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P : a matrix that identifies the assets involved in the views (K x N)
: a diagonal covariance matrix of error terms from the expressed views (K x K)
: the implied equilibrium return vector Q : the view vector
We can represent the same formula for the mean returns in an alternative way:
-1
Eq. (22) is the new combined return vector, and with all of the inputs and then entered into Eq.
(22) new combined return vector is derived. The new recommended weights (w ) can be * calculated by solving Eq. (9). The table 1 will show the detail about the Black-Litterman model‟s parameters.
Table 1: the parameter about Black-Litterman model
Parameter Estimate
The vector Q can be compute by the GARCH-derived mean equations and compute the error terms () from the view
Lee typically sets the scalar (τ ) between 0.01 and 0.05 Satchell and Scowcroft (2000) say the scalar (τ) is often set
to 1.
We set the scalar (τ ) be 0.01 , 0.005 , 0.001
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