# Tag Info

6

This is indeed a subtle point. What is generally meant with this statement is that correlation is going up in bear markets, so it is not so much the "turmoil" part (i.e. volatility per se) but the "trend" (i.e. negative in this case) part. Putting it another way is that when you control for volatility not the correlation but the covariance (which is the part ...

5

This optimization is trivial $$w^{T,J}_i = \begin{cases} 1 \quad \text{if } i=\arg \max_i R^{T,J}(S_i) \\0 \quad \text{otherwise} \end{cases}$$ That is to say, when you optimize only one weight will be nonzero. That's because these ratios incorporate no notion of distributional width, and therefore do not reward diversification. With no concentration ...

5

The weak EMH states that it is impossible to earn an excess return given publicly known information such as past prices. Clearly, different securities have different expected returns. For example: the bond and the stock of one company or a security that generates twice the return of another one. This difference in expected return is explained by a ...

5

To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1$. Further, you have Estimators for the true Variance $\operatorname{Var}\left[\mathbf{a}\right]$ resp. $\operatorname{Var}\left[\mathbf{b}\right]$ and the ...

5

Accurately stated: Diversification helps during turmoil, but helps less as what would be expected by using $w^T \Omega w$ as the portfolio variance where the off-diagonal covariances are estimated during tranquil periods. This is because correlations and covariances change during turmoil, typically increasing. This reduces the benefit of diversification ...

4

With respect to issue one, it can be simpler to consider the case where the constraint on the expected return is an equality. In that case, the first problem can be transformed to Minimize with respect to $\left\{ x,\lambda_{1},\lambda_{2}\right\}$: $x'\Sigma x + \lambda_{1} (\mu'x - r) + \lambda_{2} (1'x - 1)$ by the technique of Lagrangian multipliers, ...

4

The initial investment is the capital in the account used to support the portfolio, not the cost of the assets in the portfolio. For example, when you sell a stock or bond short, your account doesn't actually accrue any cash. Instead you start receiving a regular cash flow. There isn't necessarily a difference between these quantities in a long-only ...

4

There are plenty of books on portfolio issues built according to formula "some theory + some R code (or Matlab, or S - which is very similar to R)". See for example Pfaff B. Financial Risk Modelling and Portfolio Optimization with R.// 2013. Best M.J. Portfolio Optimization. Chapman & Hall, 2010. Würtz D. et al. Portfolio Optimization with ...

4

The Efficient Market Hypothesis (EMH) states that you cannot beat the market on a risk-adjusted basis by looking at past prices. You can certainly earn higher returns than the market if you take on more risk (by leveraging, for example). Modern Portfolio Theory allows you to construct portfolios that are efficient. According to this theory, you still cannot ...

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

You can obtain the covariance between 2 portfolios by multiplying the row vector, containing the weights of portfolio A with the variance-covariance matrix of the assets and then multiplying with the column vector, containing the weights of assets in portfolio B. Equally you can set up a new portfolio A+B by creating a new column vector that contains the ...

3

There are many papers on this subject (try googling portfolio optimization skewness kurtosis) that can describe the assumptions of including skewness and kurtosis in a utility function (if that's what you're interested in). I would highlight two main points. Mean-variance optimization does not make an assumption of normality. Assume returns are ...

3

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 $$\mu+c \sigma$$ ...

2

Check out following link. In page 23 you'll find the derivation. http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf

2

The Lyxor white paper Regularization of Portfolio Allocation contains a lot on this topic. The head of quant research there, Thierry Roncalli, also held a talk about this recently.

2

The concrete (general) answer to part (ii) of my question seems to be contained in Equation 8 of the following link: http://www.columbia.edu/~ks20/FE-Notes/4700-07-Notes-portfolio-I.pdf In particular, interpreting $\sigma$ as volatility, take for example $E_A=0.10,\sigma_A=0.15,E_B=0.25,\sigma_B=0.40$ and $\rho =−0.2$. I get that about 83 percent of the ...

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.

2

Let there be n stocks, 2 portfolio a and b. c is a combined portfolio of portfolio a and portfolio b. $\Sigma$ is variance-covariance matrix of the n assets. Weight vectors for portfolios a and b are $$w_{pa},w_{pb}\in\mathbb{R}^{n} ,$$ $$\left\|w_{pa}\right\|_{1}=\left\|w_{pb}\right\|_{1}=1$$ then $$Var(a)= w_{pa}' \Sigma w_{pa}$$ $$Var(b)= w_{pb}' ... 2 EMH says that one can not earn excess return using some information. This is known as joint-hypothesis problem: to test for market efficiency one have to determine first what is "normal" market return, i.e. what type of information is normally priced by the market. Usually to test for EMH they use CAPM or 3-factor Fama-French model (which is a kind of ... 2 I assume you're talking about this formula:$$U(w) = w'\mu - \frac{1}{2} \lambda w' \Sigma w = w'\mu - \frac{1}{2} \lambda \sigma_\omega^2$$where \sigma_\omega^2 denotes the portfolio variance for a portfolio with weights \omega. Dividing by two is purely done for convenience, optimizing this formula requires taking the derivative with respect to ... 1 MPT should be called Medieval Portfolio Theory, it is a theory from 50 years ago with huge theoretical flaws (mean-variance utility, use of Pearson's correlation that is not coherent, based on historical data). Come on, it is an error maximizer. The least one could do is Michoud resampling, but it is patented. Or a bayesian Black-Litterman would be more ... 1 a) The formula for Beta is:$$\beta_i=\frac{\sigma_{i,M}^2}{\sigma_M^2}=\frac{0.165^2}{0.11^2}=2.25$$b) So by the CAPM equation, the expected return for the asset is:$$E(R_i)=r_f+\beta(R_M-r_f)=0.04+2.25(0.12-0.04)=0.22=22\%$$c) If the variance of the stock is 0.22^2, since this variance was multiplied by \beta=2.25, we get: ... 1 MPT uses expected values for its parameters. How these expected (future) parameters are estimated, is another question. Usually one takes historic averages when its the only information available, but one could for example also use analysts forecasts or other advanced estimation methods. 1 Only certain aspects of the risks that you bear in power markets given exposure to variable quantity swaps can be hedged. To your point, you have to have some expectation of what the load will look like. Even if you immediately go out and buy power against this expected qty you are subject to the risk that the load will deviate from said qty. There is no ... 1 If you don't know the meaning of the other matrices, I'd look more at the docs and the definition of the quadratic program: http://cvxopt.org/userguide/coneprog.html#quadratic-programming This is also an example from the book: http://www.ee.ucla.edu/~vandenbe/publications/mlbook.pdf And there is a good deal of explanation there. Finally, if you don't ... 1 Others may have different views, but I've tried applying Kelly formula/fractional Kelly strategies to capital allocation, and find it rather unpractical and risky. I would honestly suggest a three-tier optimization framework that I am myself adopting: Assuming you have M number of models covering multiple instruments and strategies. Your goal is to pick ... 1 Bayesian Odds Ratios can be used to compare models and allocate wealth to various models based on the relative probability that each particular model is "best." You could begin to look into it more on the wiki site. 1 Have a look at my paper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2259133 I checked Kelly formula and found the answer from it is exactly as Markowitz's theory. >Thus, most issues on mean-variance theory (e.g. noise of estimation for mean and >variance) applies here. Kelly is not exactly as Markowitz's theory but they are indeed closely ... 1 Sure, the variance of the total wealth can be expressed in terms of the variances and covariances of the prices of the assets. If$$ W = \sum_{i} \pi_i P_i $$where \pi_i is the total dollar amount invested in asset i with price P_i. The variance of total wealth is then$$ Var(W) = \sum_i \pi_i Var(P_i) + \sum_i \sum_{j, j\neq i} \pi_i \pi_j Cov(P_i, ...

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