33

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-...


24

This question goes to whether the historical returns to factors represent: Spurious results, overfitting, data mining... Mispricing Unexploitable effects Compensation for risk Case 1: Spurious results etc... If someone constructs a "stock tickers that begin with AAP or GOO" factor, the highly above average returns would almost certainly reflect a fishing ...


15

The following papers may help. A New Look at Minimum Variance Investing by Bernd Scherer Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that ...


14

+1 for asking an excellent question. I agree with the answers of @Owen and @chrisaycock - I'm late to the party but perhaps this will shed some light. How practitioners or academics answer this question will tell you a lot about their view on the nature and sources of returns and risk. For example, the Fama-French "equilibrium" school of thought would argue ...


12

I'll answer by way of example. Suppose I want to buy a stock that is relatively cheap. Firstly, I need to define what is meant by cheap, so I might choose to look at the price-to-earnings ratio. Then I need to define what is meant by relative, so I might compare stocks only within a given sector. This may work well at first, but then I notice that as I try ...


12

Unlike the tangency portfolio on the efficient frontier (which represents the most efficient portfolio in terms of max expected sharp ratio), min var portfolios have no ex ante theory that suggests it should outperform a cap weighted market portfolio. The same can be said about other risk-weighted portfolio construction schemes, including equal risk ...


11

In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have ...


11

After having done a lot of research on the topic I found the following excellent research piece on ETF.com: Wealthfront modifies historic asset-class returns with current market implied expected returns (Black-Litterman) as well as with the in-house views of Chief Investment Officer Burton Malkiel’s team. In addition, Wealthfront sets minimum and ...


10

This is indeed an interesting question. According to this website, a paper by Goldman Sachs [Tierens and Anadu (2004)] proposes three alternative methods for estimating average stock correlations: Calculate a full correlation matrix, weighting its elements in line with the weight of the corresponding stocks in the portfolio/index, and excluding ...


9

The minimum variance optimization framework does not guarantee positive return whatsoever. As a matter of fact what you are trying to do is something close to the following: $$\underset{w}{\arg \min} \quad w' Q w \quad \text{s.t} \quad Aw \leq b,\quad \sum_i w_i=1$$ The fact that you get positive return is a nice result that you get from your backtest (i....


9

Perform a returns analysis by regressing the returns of your composite strategy on the returns of the component strategies. Constrain the beta coefficients to sum to 100% and bound them from 0 to 1. You will then have the % explained by each component.


9

Have a look at this classic paper: Honey, I Shrunk the Sample Covariance Matrix by O. Ledoit and M. Wolf The abstract answers your question already: The central message of this article is that no one should use the sample covariance matrix for portfolio optimization. It is subject to estimation error of the kind most likely to perturb a mean-...


9

If you measure risk by the standard deviation of the portfolio return $$ \sigma = \sqrt{w^T \Sigma w}, $$ then it is usual to define risk contributions for each asset by $$ \sigma_i = w_i (\Sigma w)_i/\sigma, $$ then diversified could mean that these $\sigma_i$ are evenly spread over the assets in the portfolio. You find this approach and more in this paper ...


9

Define excess return $r^x_{it} = r_{it} - r^f_{t}$ as the return $i$ minus the risk free rate, and $f_{jt}$ similarly denotes the excess return of factor $j$ at time $t$. Let's say we have some factor model of returns where: $$ r^x_{it} = \alpha_i + \sum_j \beta_{i,j} f_{jt} + \epsilon_{it}$$ F-test / GRS Test If we assume the error terms $\epsilon_{it}$ ...


8

There's a strong theoretical argument that makes the case for active management that is also supported by empirical research. First, check out Jonathan Berk's paper "Five Myths of Active Management". The paper reads like a clever Gedankenexperiment. Starting with a theoretical approach is better than starting with an empirical approach because as Berk ...


8

The VaR constraint is convex and quadratic and can be handled with any solver supports quadratic constraints, like Guribi, cplex (from IBM) or xpress (from FICO). The CVaR can be formulated as a linear program if you are able to perform monte-carlo simulations on the returns. Briefly, the LP model is \begin{eqnarray*} c &\ge& \alpha + {1 \over (...


8

Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well. I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot ...


8

It appears that you are re-running the regression with each new data point. Instead, you should use an update/online formula (see an excellent answer by the famous Dr. Huber at stats.se). You can find an implementation in the R package biglm. If it doesn't have all the features you need (no windowing out of old data) you can at least adapt it and use it ...


8

The estimation of a covariance matrix is unstable unless the number of historical observations $T$ is greater than the number of securities $N$ (5000 in your example). Consider that 10 years of data represents only 120 monthly observations and about 2500 daily observations. Depending on the application, using data dating farther back than 10 years may be ...


8

Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas) would be zero. Discussion A carefully written version of a standard time-series regression of returns in excess of the risk free rate on market excess ...


8

Generally speaking, let us consider a problem where you have a series of simple payoffs $f_{K_i}(S_T)$ of strike $K_i$, $i \in I$, that depend on the value of $S_T$ at time $T$, as well as a more complex, laddered payoff $P_L(T)$ which pays a quantity $g_i(S_T)$ on regions of the form $\{K_i \leq S_T < K_{i+1}\}$ $-$ regions are delimited by the strikes ...


8

The underlying problem: your ACTR constraints aren't convex The $i$th constraint on your risk contribution can be written: $$ w_i \sum_j \sigma_{ij} w_j \leq c_i s$$ And this isn't a convex constraint because of the $w_j w_i$ terms (a function $g(x,y)=xy$ isn't convex in $x$ and $y$). They're not convex constraints, so you won't be able to write them as ...


7

Step 1: Get your data from SQL into R -> http://www.r-bloggers.com/?s=SQL Step 2: Run your analysis/optimizations like -> http://www.r-bloggers.com/portfolio-optimization-in-r-part-1/ or http://blog.streeteye.com/blog/2012/01/portfolio-optimization-and-efficient-frontiers-in-r/ or via RMetrics: http://www.statistik.wiso.uni-erlangen.de/lehre/bachelor/...


7

The term in sample and out of sample are commonly used in any kind of optimization or fitting methods (MVO is just a particular case). When you make the optimization, you compute optimal parameters (usually the weights of the optimal portfolio in asset allocation) over a given data sample, for example, the returns of the securities of the portfolio for the ...


7

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 ...


7

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 ...


7

Transaction costs - even for banks, funds etc, every trade has an associated cost, so if you would be buying a small number of shares, it's probably cheaper to carry the risk and not make those small trades. The source data is imperfect, and contains noise. A lot of the smaller components are simply artefacts of that noise so it would be both an unnecessary ...


7

A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few more recent articles about portfolio theory, in no particular order (all accessible online): Jorion: Bayes-Stein Estimation for Portfolio Analysis, JFQA, 1986 ...


6

"Skewness" quantifies how asymetric a distribution is about the mean. "Kurtosis" quantifies how peaked or flat the distribution is. Skewness is defined as: $E[ (X - mean)^3 ] = \frac{(\sum (x_i - x_{mean})^3 )}{N}$ and Kurtosis as: $E[ (X - mean)^4 ] = \frac{(\sum (x_i - x_{mean})^4 )}{N}$ where X is your distro values (x_1, x_2, ... x_N), mean is the ...


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