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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 ...

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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-...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $\epsilon_{it}$ ...

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There's no easy answer to your question, as noob2 pointed out. You can look online for info from Universa. That fund does exactly what you are asking: https://www.universa.net/riskmitigation.html Of course, post a crash, such as the one we just experienced, the cost of hedges is larger than it is prior to such events. Understand that you aren't going ...

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

There are a few related reasons: The optimization becomes a lot harder when only discrete values are considered. Mean variance has a closed form solution for the continuous case but the case with discrete holdings is quite hard; For small retail investors fixed trading costs will swamp the rounding in your example but also for larger amounts (at least ...

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 ...

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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 ...

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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 ...

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There are more ways to approach this but the method I propose should work reasonably well in practice, especially if you increase the number of assets you hold. Calculate the beta of the stocks you're holding with respect to an index Buy $N_f$ (sell when $N_f$ is negative) future contracts on that index $N_f$ can be calculated as $$N_f = \frac{\beta_T - \... 8 I just want to add to vonjd's answer some info on the comparison of the 3 methods. This is too big for a comment so I'm posting as a separate answer but please upvote his answer, not mine. Do the differences in methodologies matter in practice? To gauge the practical importance of the biases in methods 2 and 3, we calculate the weighted stock correlation ... 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 Different portfolio risk decompositions answer different questions. Before discussing what method to use, first ask why you want a decomposition and what definition of risk are you using. Is the point to examine how portfolio return volatility is affected by a tiny change in portfolio weights? On the other hand, if the point is to make a statement like, "30%... 8 CAPM states that the expected return of any given asset should equal ER_i=R_f+β_i (R_m-R_f), with α being the error term of the previous equation. Now, as α has an expected value of zero, then only way to achieve higher expected returns is taking on more β (given that E[(R_m-R_f )]>0). Every individual stock has some idiosyncratic risk in addition to ... 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 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 ... 7 I will be glad to help, but let me first advise you away from working on this topic until you have an academic position. This topic has been poison for me, but I am slogging on anyways. Before you use anything I do, get permission from your academic advisor. I have an unpublished article on options pricing, and I am proposing a new branch of stochastic ... 6 Actuarial science traditionally focuses on estimation of joint probabilities using real data where math finance is on valuation of contracts under an arbitrary distribution. It means the first one deals with methods of estimation of future distributions (the number of accidents of a given kind, the probability of someone with a given profile to have a ... 6 Both free and paid access to data sets conatianing company financial statement items is available from Quandl. The free data sets are sourced from the SEC based on compnay electronic filings and go back about five years. For example, you could obtain five years of MSFT's quarterly net income using the R call Quandl("RAYMOND/MSFT_NET_INCOME_Q") Lists of ... 6 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 ... 6 Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to "0" if the ... 6 Seems like a small mistake in the last equation. It should read \Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\right], which is not equivalent to your result. 6 To add another perspective see this current and very relevant article with many unique and original insights (Kritzman is one of my favorite authors anyway): Cocoma, Paula and Czasonis, Megan and Kritzman, Mark and Turkington, David, Facts About Factors (April 6, 2015). MIT Sloan Research Paper No. 5128-15. Available at SSRN: https://ssrn.com/abstract=... 6 Simple example where sub-additivity fails Let there be four possible outcomes i=1,2,3,4 that occur with equal probability \frac{1}{4}. Payoffs for X, Y, and X + Y are given by:$$ X = \begin{bmatrix}-1\\0\\1\\2 \end{bmatrix} \quad Y = \begin{bmatrix}0\\-1\\1\\2 \end{bmatrix} \quad X + Y = \begin{bmatrix}-1\\-1\\2\\4 \end{bmatrix}$$What's the ... 6 Let s be a N\times1 vector of standard deviations and C be an N\times N correlation matrix. The covariance matrix is equal to$$\Sigma=\text{diag}(s) \ C \ \text{diag}(s) where $\text{diag}(x)$ is a function that takes an $N\times1$ vector and puts it on the diagonal of a $N\times N$ matrix. If you get some better standard deviation estimates, ...

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