28 votes
Accepted

What are reasons not to do factor investing in equity markets?

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 ...
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  • 6,334
13 votes
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Why shrink the covariance matrix?

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 ...
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13 votes
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cvxpy portfolio optimization with risk budgeting

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 ...
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  • 6,334
12 votes
Accepted

Which algorithms do robo-advisors use?

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 ...
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12 votes

Choosing the right statistical test for Mutual Fund Performance Evaluation

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 ...
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11 votes
Accepted

What is the total correlation between assets in a portfolio?

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: ...
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11 votes
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Calculating alpha and its meaning

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)...
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10 votes

Hedging Covid-19 and other low probability high loss risks

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 ...
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9 votes
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What to use as portfolio diversification measure?

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/...
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9 votes
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Why techniques for portfolio optimization do not take into account the non-fractionability of stock prices?

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 ...
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  • 7,652
8 votes

Calculate correlation between two sub portfolios and the combined portfolio

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}...
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8 votes

Why shrink the covariance matrix?

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 ...
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  • 3,250
8 votes

Creating a Beta-Neutral Portfolio

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'...
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  • 7,652
8 votes

What is the total correlation between assets in a portfolio?

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 ...
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  • 741
8 votes
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Finding arbitrage opportunity

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 ...
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8 votes
Accepted

Portfolio Risk Decomposition - different methodologies

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 ...
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8 votes
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How are modern portfolio theory (MPT) and CAPM related?

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 ...
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  • 496
8 votes
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Suppose that we are wrong about the relevant class of distributions for financial economics and econometrics. Now what?

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 ...
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  • 4,092
7 votes

Why shrink the covariance matrix?

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 ...
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  • 3,589
7 votes
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On learning the bayesian approach to portfolio optimization

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 ...
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  • 866
7 votes

non-subadditivity of VaR

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 = \...
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7 votes
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How to check if a portfolio has momentum bias

It kind of depends what your objective is. First, momentum 'bias' isn't well-defined. Are you looking to eliminate momentum exposure for some reason? Momentum itself isn't even well-defined really: ...
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  • 1,598
7 votes

Most significant research articles for practical investors with research perspectives

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 ...
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  • 9,077
6 votes

On the interface between Quant finance and actuarial-science/insurance-math

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 ...
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  • 10.6k
6 votes

Historical Financial Statement to Backtest in R

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 ...
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  • 404
6 votes
Accepted

Can I add the greeks of individual postions to obtain greeks for the portfolio

most models in financial maths are linear so prices and Greeks just add. This is in particular true of Black--Scholes so Yes. However, once one starts taking into account value adjustments non-...
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  • 6,743
6 votes
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Mean Variance Portfolio theory and real-world problem?

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 ...
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6 votes
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Maximum Certainty Equivalent Portfolio with Transaction Costs

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\...
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  • 116
6 votes

What are reasons not to do factor investing in equity markets?

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 ...
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  • 26.9k
6 votes
Accepted

How can I use a more efficient volatility estimator to improve the co-variance matrix?

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}...
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