# Black-Litterman

A Bayesian equilibrium expected return model

Black-Litterman returns combine the market equilibrium return expectation (CAPM) with the investor’s unique market views. The investor’s views for a given asset can be either absolute or relative to another asset. The Black-Litterman model is designed as a means of constructing a sensible set of expected returns that is consistent with the equilibrium framework.

## The Model

The model assumes that there are two sources of information on expected returns, the investors views $(Q,\Omega)$, and market equilibrium $(E[r]_{\text{equilibrium}},\Sigma)$ . The following figure describes how Black-Litterman combines the distribution of views and market equilibrium to form a posterior view, or its expected returns.

The prior equilibrium is assumed to be Gaussian with a mean of $E[r]_{\text{equilibrium}}$ and a (co)variance of $\tau \times \Sigma$ . In WPA 3.0, we assume that $\tau =1$ , this free-parameter is a measure of investor's confidence in the prior. The covariance matrix of the prior is specified in WPA's risk screen as the active risk and correlation model.

## Views and Confidence

The systematic framework of the Black-Litterman model allows investors to specify both relative views and absolute (idiosyncratic) views on an assets return.

To manage your views for the Black-Litterman model, click on the Views and Confidence button on the parameters panel.

To add a new view (absolute or relative), follow the on-screen instruction and right-click to add a view. This will create a new column for you to specify your new view.

### Absolute Views

An absolute view is an idiosyncratic view for a particular asset. For example, an investor may have a view that domestic bonds will return 7% over the next year with 99% confidence. You can specify when using the Black-Litterman return model in the Views and Confidence dialog

The qualifyer long and short are used to indicate +1 or -1 in an indicator matrix that corresponds to the mathematics as described by the model.

\begin{align} P &= [0, 0,0, 1, 0, 0, 0] \\ Q&=0.07 \end{align}

For absolute views, you will want to flag the asset that your value specification would apply to with the long marker; i.e. select long for the corresponding asset to apply your expected return view value and confidence, and none for other assets.

In the example above, we are only specifying one absolute view corresponding to U.S. bonds (4th asset) producing a non-zero element in $P$ .

### Relative Views

Relative views specifies comparative description of an investors outlook for an asset (or group of assets). For example, let's suppose that the investor considers that U.S. stocks will outperform Non-U.S. stocks over the next year by 20% with 75% confidence.

In the WPA, we would configure this by adding a new view (observe the second column below)

The long/short markers are use to indicate the direction of the relative view, i.e. U.S. stocks will outperform relative to Non U.S. stocks.

\begin{align} P &= [1, -1, 0, 0, 0, 0, 0] \\ Q&=0.20 \\ \end{align}

### Multiple Views

You can specify as many combination of views as there are number of assets in your case file. The Black-Litterman model will reconcile your combination and specification of views as described by its mathematical model.

### Confidence

The WPA solves for the implied covariance from the confidence specified for each view using the Gaussian probability density function.

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