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Well there are two main things to consider here.
Many implementation of Black-Litterman use the market portfolio and the ex post volatility and correlation structure to back out implied returns to use as prior. As far as I know, there is no standard way to reverse-engineer the optimization problem in the presence of nonnormal markets. (the first guess is ...
5
I had a quick look at your code and it seems like you are applying the inverse function incorrectly in your calculations. For example, in the second equation I did the following changes:
sub1 = tau * S_cov_var[1:5 ,1:5]
Isub1 = solve(sub1)
IOmega = solve(Omega)
second <- (Isub1 %*% implied_equilib_excess_rets[,1] + (t(P) %*% IOmega %*% Q))
Which ...
4
It depends on your investment process: more specifically, on how you generate views. Here are three practical cases which lead to different choices for $\Omega$:
Let's assume you are an investor who acts on (more or less) arbitrary bits of opinion: e.g. you like Italian equities because you like Italy, and German equities because you find Angela Merkel's ...
4
Your statement about the properties of $\tau$ is correct. $\tau$ is a measure of uncertainty. I think the problem you are having is because in most practical situations nobody really knows what values should be used for $\tau$ and/or $\Omega$.
There is plenty of practical advice out there, and some of it is very confusing! For example, Jay Walters has ...
4
There are some technical problems with using your previous weights as priors (that is, they are point measures), but yes, the Black-Litterman framework is suitable for this. You can essentially include any view point you have on the market within the model and let it affect your position size. This also includes views on transaction costs (based on such ...
3
Do not use the capitalizations of the ETFs, they hold a small and variable proportion of the securities outstanding.
Instead base your weights on the papers by Swinkels et al. which have tried to estimate the actual amounts invested by all investors (not just ETFs!) in various assets. Their latest paper is Historical Returns of the Market Portfolio, 2019 ...
3
Any potential source of "alpha" would suffice, in fact. And your research would be research of how this "alpha" source is able to produce alpha. On my mind, candidates could be
(1) some well-documented predictions of somebody (like Prechter or Dow) - in that case you'll have 2-3 views for period, and the rest of assets or classes remain in equilibrium - see ...
3
Currency indices will require some base currency from which the other currencies weights will be determined. Also, the weights are determined as of some point in time and frequency, and are based on some economic statistic, such as the share of international trade, that reflects their relative importance in the global economy. Other possibilities could be ...
3
It's because the model assumes that the market will maximize its Sharpe ratio and your weights don't do that. Essentially, your example assumes investors are irrational in their allocation. If you solve for the weights that maximize the Sharpe ratio, the implied returns will equal the given returns.
In your example, the Sharpe Ratio reaches a maximum ...
3
In practice, $\Omega$ (the covariance of the investor views) often 'inherits' the market covariance $\Sigma$. A convenient choice is
$ \Omega = \left( 1/c -1 \right) P \Sigma P^T$
where $c$ is a confidence parameter: the case $c \rightarrow 1$ corresponds to a strongly peaked distribution of views (the investor views dominate the market), while $c \...
3
Of course you can choose the prior. As far as I understand the literature, the BL-model is characterized by using the equilibrium implied returns. Otherwise it would just be a Bayesian model.
If you estimate the returns in a different way (not taking implied returns from the market portfolio), you could lose the stabilizing inverse optimization step ...
3
Jay Walter's Paper on "The Factor Tau in the Black-Litterman Model"
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1701467 is also useful to review
3
In The Black-Litterman Model In Detail Jay Walters says the following on p. 13 (top paragraph):
First, by construction we will require each view to be unique and uncorrelated with the other views. This will give the conditional distribution the property that the covariance matrix will be diagonal, with all offdiagonal entries equal to 0. We constrain the ...
2
Yes, you can formulate such a view. A lot of ways to formulate views are described in the literature (one can start here). However, your view is based on the assumption that CAPM works precisely for single stocks. This assumption will be wrong in most cases.
EDIT after a comment by the OP:
I think now I understand. You want to replace expert's views (...
2
When I implemented a BL model, I chose to do the omega optimization using the technique Idzorek proposed here:
https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/BlackLitterman.pdf
It's a numerical procedure though.
2
I don't believe there is one generally accepted method and a number of papers are written on this issue.
The Black-Litterman Approach: Original Model and Extensions (2008) by Meucci has an overview and I believe is generally useful to learn more. It suggests using $\tau = \frac{1}{T}$ but notes more complicated approaches exist. A demystification of the ...
2
Under the logic of the CAPM, the equation $\operatorname{E}[R_i - R_f] = \beta_i \operatorname{E}[R_m - R_f]$ would hold for any return, whether it's a stock return, bond return, portfolio return, call option return, etc....
If the CAPM holds for a set of assets, it's easy to see that CAPM would hold for any portfolio over those assets. Let $\mathbf{w}$ be ...
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This issue is dealt in detail in Atillio Meucci's paper titled Fully Flexible Views: Theory and Practice. See Appendix A.4.
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The problem is not Black-Litterman.
B-L aims at finding an estimation of expected return based on a prior objective and some views.
The prior is based on market neutral portfolio composition (or your benchmark) and the views are for the returns, so no constraint on weights can be added in this phase.
The fact that you get negative weights is entirely due to ...
1
Yes all you have to do is estimate the Black Litterman covariance matrix that includes investor views using a shrinkage estimator. Covariance shrinkage like Ledoit Wolf is an old technique, however, that has been outperformed by the denoised or detoned covariance matrix estimated by random matrix theory, as well as the nested clustered optimization (NCO) ...
1
Since you add the bayes-theory tag here I'm gonna speak in bayesian interpretation; I'd say it's just because this is the simplest way to obtain the distribution of prior; A better way to do this is by finding a prior optimal (essentially finding best mean and variance that fits our assumption of return distribution based on the data you have; usually done ...
1
intuitively BL works as assuming return (can be factor return defined in APT model or return over some interval) follows normal distribution with mean $E$ and variance $V$, we want to infer such mean and variance based on noisy observations from view about portfolio, and then use mean variance optimization to get the optimal portfolio holding which is just $(...
1
You forgot to include constraints as a parameter to the call to Problem.
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Indeed I think you can, but it comes at a price. To be clear: Neither have I done this myself nor have I attempted it.
The main idea is typical Bayesian:
In computing the posterior return parameters for the random variable $\mu$, you can calculate the conditional expectation subject to your inequality constraints, say $A\mu \leq b$).
There are three, ...
1
There is also http://www.blacklitterman.org/ Where you can find an implementation under Excel and Matlab of the Model.
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