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Hot answers tagged modern-portfolio-theory

6

As a practitioner, I have worked on the following Maximize Yield/OAS for a Fixed Income Portfolio keeping the Rates Duration (Key Rate Durations) and Spread duration in a constrained range . There are other constraints such as No short selling Max amount you can buy is X% of Max outstanding amount in market Maximum exposure to a perticular country , ...

5

if you take the variance of a single asset it scales as a quadratic, $$var(\lambda X) = \lambda^2 var(X)$$ so it's not surprising that the general case gives a quadratic form.

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Lots of wealth management firms still use MPT; in my experience regulators like it because they understand it. If asset returns are normally distributed, the standard deviation of the portfolio is a coherent risk measure (this can be seen by noting that the normal distribution's CVaR, which is a coherent risk measure, can be written as $$\mu+c \sigma$$ ...

4

If you can add linear constriants (as you can do in quadprog) then you can formulate $w \mu = c_1$ as linear constraint, no matter what $\mu$ is (and first delete it from the objective by setting the parameter to zero. The only problem is the one norm. Let my clarify, this is: $$\sum_{i=1}^n |w_i| < c_2$$ Thus you allow for short sales but you want to ...

3

An introductory presentation by Michael Brandt from a seminar of Inquire Europe is Bayesian Portfolio Construction. His review Portfolio Choice Problems has a section on decision theory which could also be useful to you. Another good choice is Attilio Meucci's Risk and Asset Allocation book which contains a whole chapter (ch 9) on Bayesian techniques in ...

3

Initial capital is not a real constraint in theoretical analysis, but might be a practical constraint in reality. The objective function you gave defines the efficient frontier corresponding to a given risk tolerance $q \in [0, \infty]$: $$\min\{w^T\Sigma w-qR^Tw\}$$ This criterion is among the other popular optimization criteria, such as minimum variance, ...

3

This problem is not interesting enough, because putting your money in the bank guarantees you zero volatility (and a zero return on investment). In practice, whatever set of assets you chose you would get a very extreme solution (e.g. 100% weight on one asset with very low volatility.) With a minor tweak, you can get a very interesting problem. You can ...

2

Pretty good explanation is in Schweser CFA Study Notes for CFA level III. Books 3 and 5, at least from 2009, if I remember right. See also Tsay R.S. Analysis of Financial Time Series (Wiley Series in Probability and Statistics). // 2010. - good example with implementation in R.

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I am engineer studying Finance, therefore Im not an expert in Math/Stat, but not noob. I disagree with the previous answer. In fact, I know portfolio managers and hedge fund assesors that usses MPT. It must be said that you need to know what that represents, and also not only focus your investment in MPT, but consider other methods. Like in every other ...

2

You are asking two different questions: what would be the model result, and what would be the actual performance of an actual portfolio. The optimal model results with the S&P 1500 will be at least as good as the model results with the S&P 500. The S&P is a proper subset of the S&P 1500, so you can get the results of the S&P 500 model by ...

2

Well, you are asking something very subjective. In addition it should be mentioned that S&P500 are the companies with higher capitalization of S&P1500. Therefore a huge weight of S&P1500 is set by S&P500. In fact, as it can be seen in 2008 both went down a 37%, in the other hand S&P500 has 80% of the total of the US equity Market. After ...

2

The coefficients assuming they are statistically significant can be interpreted whether or not the underlying portfolio is efficient. The CAPM or FF4 simply tries to decompose a portfolio into a series of linear exposures + an intercept (alpha) which can be viewed as constant added value. In mathematical terms the regression is explaining how much of ...

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Let $s_1 = r_1 -r_f$ and $s_2 =r_2-r_f$. Then, this is the maximization problem: \begin{align*} & \ \max_{w_1, w_2} SR = \frac{\mu_p}{\sigma_p}, \, \mbox{ subject to}\\ \mu_p = & \ w_1 s_1 + w_2 s_2,\\ \sigma_p^2 = & \ \sigma^2\big(w_1^2 + w_2^2 + 2 w_1 w_2 \rho\big),\\ 1 = & \ w_1+w_2. \end{align*} By certain substitution, we convert the ...

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Sure a lot of traditional (mutual) buy side funds use MPT. They also mostly subscribe to the efficient market hypotheses. And they also do not hide the fact that they have no interest to lobby many retirement investment and savings schemes to allow for long/short investments but hold on to long-only. And finally, most of them underperform simple benchmark ...

2

I assume you're talking about this formula: $$U(w) = w'\mu - \frac{1}{2} \lambda w' \Sigma w = w'\mu - \frac{1}{2} \lambda \sigma_\omega^2$$ where $\sigma_\omega^2$ denotes the portfolio variance for a portfolio with weights $\omega$. Dividing by two is purely done for convenience, optimizing this formula requires taking the derivative with respect to ...

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Use PortfolioAnalytics See my previous response here: http://quant.stackexchange.com/a/16002/2154 , you will find links to the documentation there. You can use the constraint function to add a factor exposure constraint of 0. use add.constraint(your_portfolio_name,type='factor_exposure',B = your_vector_of_betas,lower=0,upper=0)

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Here is a guide by morningstar: " A step by step guide to the black litterman model" https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/BlackLitterman.pdf

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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 ...

2

If you give a covariance matrix an inverse Wishart prior, then it simplifies a lot of math in the calculations. This is called a conjugate prior. If you don't understand conjugate priors, you might want to work through the math on the univariate normal case with an inverse gamma or chi square prior for the variance. The Wishart distribution is just a ...

2

You do note require a sum up constraint that gives you that the weights sum up to 1? Then the problem is equivalent to a maximization without constraints: $$Z(\omega)=w'\mu - \frac{\gamma}{2}w'Vw$$ then it holds that $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = ... 2 since you've assumed that all returns are independent, the covariance matrix, C, is diagonal. In the comments, you are assuming that the investor is a mean-variance investor. It's a general result that every portfolio that maximizes return for a given variance is a tangent portfolio for some risk-free rate, R. Let e=(1,1,...,1). and let \mu be the ... 2 PerformanceAnalytics in R and PortfolioAnalytics in R Here is a tutorial from UW http://faculty.washington.edu/ezivot/econ424/portfolioFunctionsPowerPoint.pdf 2 If you take the a sample of historical asset returns as model for the risk then you can do two things: You calculate r_j = \sum_{i=1}^n w_i r_i^j thus for each scenario j you aggregate the individual asset returns to get a scenario for the portfolio. Then you can calculate Var(r_j) the variance of the sample of portfolio returns. This is the same as ... 2 I wonder if it's possible to use solve.QP from quadprog by using dummy variables. One dummy variable y_i would be used for each w_i, each y_i would be constrained to be greater than zero, and the leverage constraint would be applied to the sum of the y_i. Problem formulation would look like$$ \text{min } w^tΣw  subject to the ...

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I think generally there are two approaches: "calendar rebalancing" (such as monthly as you mention) and "optimal corridor width". For the first option, the danger is the portfolio could stray considerably from your benchmark between rebalancing dates. For the second option, track tactical deviation on a continuous basis. When you are outside the corridor, ...

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As you are especially interested in applications in Finance I'll recommend this book of Rachev which focus on Bayesian Methods in Finance

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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 ...

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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.

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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 ...

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Why not just use PortfolioAnalytics, as if your matrix is non positive definite you will have problems using non optimization approaches. Here is an example taken from my blog: retmat is a matrix of returns library(PortfolioAnalytics) moms_portfolio = portfolio.spec(assets=colnames(retmat)) moms_portfolio = ...

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