# 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.

1. Using the data analysis results compute the excess return $\hat{\mu}$ and the covariance matrix $V$ for the two assets.

2. 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.

3. 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?

4. 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.

• Have you tried googling for Black-Litterman. There are tons of examples out there. – John Jan 13 '15 at 14:19
• @John I have tried googling Black-Litterman. I only really find the same one or two examples over and over. They are also not very clear. Perhaps you could recommend one? I thought that by getting a good answer to the question above we might put a good example on the internet. – Wintermute Jan 13 '15 at 14:58
• Work through Idzorek's paper (the second thing that comes up when googling) corporate.morningstar.com/ib/documents/MethodologyDocuments/… it is pretty clear how to implement your example. If you can point to a specific thing that you're having trouble setting up, then I can help with that. – John Jan 13 '15 at 15:46
• @John I'm having trouble figuring out how to incorporate the view that asset 1 will under perform the market by 60%. – Wintermute Jan 13 '15 at 16:33
• Two options. The first (and easiest if you generalize it) approach is to expand your mean and covariance to include the market. Then you can take the view directly. The second is to create a view on asset 1 that mimics this behavior. In particular, you'd take a view that asset 1 has a return of 40% (solving -60%=x-100%). – John Jan 13 '15 at 17:43