Formulating Views and Updating Returns in a Black–Litterman Portfolio
Summary
This question sets up a two risky-asset example for learning the Black–Litterman portfolio model. It starts with CAPM estimates for the assets, including a market return, asset betas, residual volatility inputs, and a risk-free rate, then asks how to derive expected excess returns and a covariance matrix. It also asks how to encode relative performance views and a view about performance versus the market as the model’s view matrix and view-return vector.
The proposed next steps are to combine those views with the CAPM prior using stated view uncertainty and a confidence parameter, then optimize a portfolio that includes the risk-free asset. The document provides the problem data and objective but no worked solution, so it does not establish numerical posterior returns or allocations. Readers would need to check notation and assumptions, including the covariance construction, view definitions, and how the risk-free asset enters the optimization.
Key ideas
- CAPM estimates can provide prior expected returns and covariance inputs for a Black–Litterman example.
- Relative views can be represented as linear combinations of asset returns.
- View uncertainty and the prior confidence parameter influence the posterior return estimate.
- The posterior estimates can serve as inputs to a constrained portfolio optimization.
- The document poses the calculation but does not provide its solution.
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# Black-Litterman with simple portfolio
# Black-Litterman with simple portfolio
In an attempt to learn Black-Litterman I have come across this "simple" example. Suppose that you analyze market data using CAPM $$r_i-r_f=\beta_i(r_m-r_f)+\epsilon_i$$
Suppose there are 2 assets in the market, $r_f=0.065$, and you find that $$\hat{u}_M=1,\; \sigma_M=1,\; \beta_1=0.5,\; \beta_2=-0.1,\; r_1=0.3,\; r_2=0.4$$ where $r_1$ and $r_2$ are the standard deviation of the regression residuals.
- Using the data analysis results compute the excess return $\hat{\mu}$ and the covariance matrix $V$ for the two assets.
- Suppose that your private information suggests that in the next period, asset 1 will out perform asset 2 by 20% and underperform that market by 60%. Formulate the private information into the investors views as defined in the Black-Litterman model.
- Given that the variance of your private information is 0.1 and your confidence parameter in the CAPM model is $\tau =0.01$, what is your best estimate on the expected excess return of the two assets in the next period?
- Based on your new estimate on the excess return of the two assets, solve that following portfolio selection problem with 3 assets (assets 1, 2, and the risk free asset): $$Max\; \mu_{bl}^Tx-\frac{\lambda}x^TVx$$ $$s.t. \;\; x^T\mathbb{1}=1$$
1 is straight forward using CAPM and 4 is easily computed using Lagrange multipliers if you have the results from 2 and 3. It's 2 and 3 I'm lost on. Can someone provide a detailed solution to this. I think it would provide a great simple example of BL in action, which I can't seem to locate anywhere on or off line.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.