The Black-Litterman Model Explained.

Active portfolio managementMean variance optimization has a few problems. For example, in Markowitzs paradigm optimal portfolios are highly concentrated and very sensitive to provided inputs. Empirical evidence also indicates that small change in expected returns can cause a drastic change in portfolio composition. So we get some wired portfolios sometimes. The Black-Litterman model is represented as an asset allocation model but is essentially a model to forecast expected returns and once we know expected returns we can use standard optimization techniques to arrive at the optimal portfolio. What BL do is what we can call reversed optimization in order to arrive to an estimate of implied equilibrium excess returns. It also allows us to incorporate our views about various stocks or assets that comprise our portfolio and also our confidence about our views to generate the expected returns vector.First, let us consider the notion of implied equilibrium excess returns. Let us start by writing the investors utility function:U= wR 0.5ww(1)So, U stands for utility. The utility function is equal to the expected return which is w transposed times the return. Now, the risk is going to be penalized by subtracting from our expected returns half times the risk aversion parameter or the price of risk (denoted by lambda here), times w transposed times the variance-covariance matrix () times w. This last term captures the variance of the portfolio.

We want to maximize the utility function subject to the constraint that the sum of the weights always should be equal to one. U= wR 0.5ww, s.t. w1=1(2)To maximize this function we are going to take a partial derivative of U with respect to w which is going to give us from the first term R and from the second term we are going to get one half times 2 because w transposed times w is equivalent to writing w square, so 2 times the risk aversion parameter times the variance covariance matrix times w. The two twos are going to cancel out. Thus we are going to have R minus the risk aversion parameter times the variance covariance matrix times w. =R- .2w=R- w(3)In order to maximize the utility, we set the derivative equal to zero.=R- .2w=R- w=0(4)Now, what Black-Litterman did was rather than solving for optimal weights, they argued that weights are already observed in the market and therefore we can compute them using market capitalization. What they did for that was to reverse the problem. This solves for R from equation (4). So we are going to see is that R is simply equal to the risk aversion parameter times the variance covariance matrix times the market weights. R= w(5)The risk aversion parameter or the price of risk can also be written as the excess return on the market divided by the variance of the market.=(6)Now, we are in agreement with the excess returns thus generated we should hold the market portfolio because in the absence of any other view these returns takes back to the market weights. Let us call this implied equilibrium excess returns as a vector and let us call .Now, let us introduce views about the stocks or the assets that comprise the portfolio. We can have absolute views. For example, the return on Googles stock will be 2% in the near future. We can also have relative views, for example, the return on Googles stock will be greater than the return on Apples stock by 1%. In practice, relative views are more common.Let us consider a three asset portfolio. We believe that the return in asset A is going to exceed the return on asset B by 1%: > by 1%We can also say; the return on asset C is going to exceed the return on asset A by 0.5%: > by 0.5%So we have two views, which can be expressed as a vector. Let us see how this vector looks like. Let us name this vector Q and call it views vector. And we can write this vector as simply as. In this case a 2X1 vector. In general terms, Q will be an NX1 vector, where N would be the number of views. The problem is that by looking at vector Q, we do not what the views are about. So what we need to have is a matrix which is going to establish a link between our views and what the views are about. Let us call this matrix, matrix P. In this matrix P, each row represents a view. So we have two views. We also have three assets that are going to be written in columns.AB C

The question now is what is going to be the elements of this matrix. For each positive view we can write a one, and for each negative view we can write a -1, but do keep in mind that the sum of relative views should be zero. What are our views? We believe that the return on asset A is going to be greater than the return on asset B, so we are positive about asset A. So, in the row one, the first view, and we are negative about asset B so we can write a minus one below B. The first view does not talk about asset C at all, so we can write a zero below C. The sum of these views is going to be zero. Likewise we can fill the row for the second view. We are positive about C and relative speaking negative about asset A. Second view does not talk anything about asset B.AB C P=(7)

Let us now focus on risk. It is pretty obvious that regarding expected returns there will be some uncertainty about them. If we want to have a measure of confidence about expected implied equilibrium returns, we can express this confidence by simply writing variance covariance matrix inverse. Because the variance covariance matrix represents uncertainty and the inverse is going to represent the confidence that we wave about estimated implied equilibrium excess returns. Likewise, the views that we have are just views or opinions, they are not facts. They are not set in stone. Therefore, there is an uncertainty about the views also. So if views are represented by vector Q, there is also going to be a vector for the error term. Let us write down this as follows. Because we are not 100% sure about views, there is going to be an error component: .(8)We are going to assume that the errors terms are normally distributed, which we can write them as follows: (9)The uncertainty about our views has a zero mean and some variance. The uncertainty matrix which in this case is a 2X2 matrix we will call it omega, .There is not one way to compute the elements of omega matrix. Black Litterman suggest that the elements can be calculated as a scalar times the link matrix times the variance covariance matrix times the transposed of the link matrix. (10)Tau is just a scalar and it is just as abstract as omega. Black-Litterman has used the value of 0.025 in their paper. Other people or researchers have used the value of one. For the sake of convenience, let us use 1 for tau. In any case, if omega represents the uncertainty and we want to have a measure of confidence about our views, we simply write the measure of confidence about our views as the inverse of the omega. Confidence on views: We have introduced a lot of concepts. Let us collect of all them in one place. The list of inputs for our three assets portfolio is the following:

(Implied equilibrium excess returns) (variance covariance matrix) (Views) (Link matrix) (Uncertainty about views)What Black Litterman formula does is to give us an estimate of excess returns by calculating a weighted average of two items. What are those two items? A weighted average of implied equilibrium excess returns, , and our views captured by a Q vector. We will have a weighted average of these two items. The question is what the weights are. The first weight is the confidence about And it is represented by the scalar times the variance covariance matrix and because we are measuring the confidence we take the inverse.

The second weight is going to be related about our views. How confident are we about our views. The confidence is to be represented by the omega inverse and we said that just by looking at the vector Q we would have it an idea of what the views are about. So, we will have a link matrix P transposed times the uncertainty matrix.

What we have is just a weighted average of about implied equilibrium returns and views. (12)

Weighted equilibrium returnsWeighted Views

The last term is actually the second term in the B-L formula. We know that the sum of the weights should be equal to one. This is assured by multiplying the last term by the first term in the B-L formula: (13)

Let us conclude by writing the complete B-L formula. What we have is that the excess returns are simply equal to the first term multiplied by the second term. (14)

Making sure the sum of weights will be equal to oneWeighted average of equilibrium returns and views

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