# Questions tagged [modern-portfolio-theory]

A theoretical framework for analyzing investment portfolios based on their expected return and risk.

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### Why minimum variance portfolio is used to construct factor models

I am reading Tsay's classic "Analysis of Financial Time Series" and I have seen him using minimum variance portfolio Relevant passage on the minimum variance portfolio here (Chapter 9, Page ...
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### Asset rate (elasticities ?) of substitution

I'm kind of a newbie in the finance research area. However, I'm working on cross-asset spillovers (transmission of shocks between assets) and my guess is that it comes from investors behaviors. ...
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### Risk Aversion Coefficient Literature Rationale and Sources

I'm running a Black-Litterman model and for the Risk Aversion Coefficient I have two potential formulas. The first is the standard formula which I believe is used in the original Black-Litterman ...
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### Risk free rate must be lower than expected return of global minimum variance portfolio

I heard a professor say: "We know the return of the risk free asset must be less than the expected return of the global minimum variance portfolio, otherwise there would be arbitrage ...
205 views

### "Risk Matters Hypothesis" - does it really?

Risk.net has recently run a story about the "risk matters hypothesis" which refers to Sharpe’s Arithmetic and the Risk Matters Hypothesis by Haghani, Ragulin and White (2023). If I ...
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### Excess Return Covariance Matrix is Singular - Cash return and risk free rate are the same [closed]

I've created a three asset excess return covariance matrix. The assets are; equity, bonds, and cash. However, my cash return is the same as my risk free rate ( i.e. 3 month Euribor). This is leaving ...
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1 vote
189 views

### Reverse optimization: How to generate the expected portfolio returns given the weights and a series of constraints on those weights?

I have the below function in Python. My objective is to back out the expected returns associated with certain portfolio weights given a series of assumptions. From this I want to generate the expected ...
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138 views

### How to construct a delta-neutral portfolio containing stocks using correlations?

I’m aware of the mean-variance framework where we construct a portfolio such that we attempt to minimise the variance and maximise returns. What if instead we’re in a scenario where the main goal is ...
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1 vote
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1 vote
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### What is Ei in paper "How to Combine a Billion Alphas" by Zura Kakushadze? [closed]

I am reading paper "How to Combine a Billion Alphas" by Zura Kakushadze. In the paper, it has Ei which are the expected returns for alphas. It also has Ri hat as follows. I wonder what the ...
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### Why not inequality constraint in mean-variance portfolio optimization?

Question 1: In Modern Portfolio Theory, the case where we minimize variance given a set return and that the weights sum to 1, why is the return set as an equality constraint, not an inequality? ...
1 vote
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### How can equilibrium weights be found for momentum factor in Black-Litterman model?

I have a momentum factor which consists of going long in three rising ETFs and going short in three falling ETFs. I want to use this factor as part of my portfolio for Black-Litterman model, however I ...
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1 vote
139 views

### What is the meaning of Beta of an individual asset in relation to a portfolio, not the market?

Assume I've got a portfolio "A" with an expected return of 14% and a volatility of 20% and my broker suggests to add a new share "H" to my portfolio which has an expected return of ...
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### Intuition behind portfolio weights with lower RMSE but higher variance

I have recently encountered a phenomena in portfolio optimization that has baffled me for days. I was experimenting with different ways of transforming a covariance matrix to get a stable minimum ...
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1 vote
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### How to adjust an assets position to target volatility in a long-short portfolio?

I have a portfolio of weights $\mathbf{x}$ where some positions in $\mathbf{x}$ are short s.t. $\Sigma_i x_i=0$ (dollar neutral). The standard way to estimate the volatility contribution per asset is ...
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1 vote
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### What is the meaning of the asset risk contribution in a long-short portfolio?

If I have a portfolio of weights $\mathbf{x}$ and the covariance matrix of asset returns $\Sigma$ then the volatility contribution per asset is given as standard $\mathbf{x}' \Sigma$. For a standard ...
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### Computation of tangency portfolio [duplicate]

Good morning, I would like to solve the maximization problem that you can find at pag 23 of this source (http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf) in order to find the ...
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### Are there known benchmark examples where Cover universal portfolio performs better than naive uniform CRP and Split-and-Forget?

I am investigating the performance of Cover universal portfolios cf. https://en.wikipedia.org/wiki/Universal_portfolio_algorithm (and references therein). I would like to know if there are any ...
63 views

### Calibration of Covariance Matrix for a Cumulative Period Return

I am trying to compute optimized weights (minimum-variance portfolio) for a cumulative return over a period (weekly or fortnightly). In a daily return setting, it is quite simple, I just compute a ...
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### Evaluating estimate of covariance matrix

I am testing out different methods / shrinkages to estimate a covariance matrix and I am wondering what is the best method of comparing the estimated covariance matrix to the true covariance matrix (...
1 vote
90 views

### Portfolio risk of correlated assets using Mahalanobis distance

I am trying to understand if there is an agreed methodology to measure the total risk in a portfolio of correlated assets. I am taking a simple model of stock prices following geometric Brownian ...
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### How to change the covariance matrix for a parallel-shift of the efficient frontier?

I'm trying to obtain a parallel shift in my efficient frontier based on the Merton 1972-parameters. As i think a picture tells you more than 1000 words here is what i tried: The setting of my problem ...
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### How to construct the behavioral efficient frontier

I just stumbled across an interesting chart in Meir Statman's book "Finance for Normal People" where he introduces his behavioral portfolio theory. There, he also provides the following ...
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1 vote
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### Linear programming and factor models vs M-V optimization?

I have been recently researching about portfolio optimization problems and it is unclear to me what is currently the state of art modeling choices when it comes to this topic. On one hand, I've ...
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### negative portfolio variance? Creating a positive semi definite matrix in excel

I am attempting a portfolio optimization model and ended up generating negative portfolio variance using 2WaWbσaσbcorrel(a,b) or 2WaWb*Cov(a,b) From reading the linked article where other users had an ...
187 views

### Information Ratio Confusion in Grinold's Signal Weighting Paper

In the procedure Grinold outlines in his 2010 paper "Signal Weighting" for optimally combinining $J$ raw alphas, $\mathbf{a}_j$, he first assumes each $\mathbf{a}_j$ has been scaled so its ...
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I've found a ton of sources that mention the classic rule of "If the Sharpe ratio of the new asset is greater than the Sharpe ratio of the existing portfolio times the correlation of the existing ...
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### Selection of Risk aversion in portfolio optimization

I have a portfolio of equities with a cross-sectional score as expected return (mean=0) and am using mean-variance optimization. However, the question is how one selects the risk aversion parameter. ...
1 vote
98 views

### How to calculate returns of a portfolio with rebalancing? [closed]

I would like to compare the performance between a portfolio with the 30% of firms in S&P500 that have the highest ESG score to a portfolio with the 30% with the lowest ESG score. Then I would like ...
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### Can anyone help me to understand why the GMV point is not on the efficient frontier?

I am following a course about portfolio construction with Python. I am able to successfully draw the efficient frontier and capital market line (CML), and the global minimum variance (GMV) point using ...
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### How to solve for the optimal portfolio weight with target variance?

I'm confused a bit with the following problem: As far as i understand, the following problem where $$\min_{w} \omega^{T}\Sigma\omega$$ $$\textrm{s.t.}\hspace{0.5cm} \omega^{T}\mu=E$$  \omega^{T}\...
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### Wider VaR for portfolio risk?

Is there a way to widen the 95% VaR by changing the distribution of a portfolio of stocks? When calculating 95% VaR of my portfolio using the holdings based approach (which requires the covariance ...
228 views

### Derivation of optimal portfolio weights using Risk Budgeting approach

In Thierry Roncalli's book Introduction to Risk Parity and Budgeting (2013), he gives an example of particular solutions to the Risk Budgeting portfolio such as for the $n=2$ asset case. The risk ...
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### How to Maximize Portfolio Sharpe Ratio using Lagrange Multipliers in a Factor Model

I've come across the notes of the 2003 lecture "Advanced Lecture on Mathematical Science and Information Science I: Optimization in Finance" by Reha H. Tutuncu. It describes on page 62 in ...
155 views

### How to calculate the ex-ante beta of a portfolio between several rebalancing?

I have a portfolio composed of $N$ assets. I know the one-year beta of these assets, I also know the past (ex-post) beta ($\beta$) of my portfolio. My portfolio changes allocation every month. So I ...
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2k views

### Fama-French factor model: why mimicking portfolios?

I am trying to understand the Fama-French factor model, or any kind of CAPM extensions really. What is really puzzling me is the use of mimicking portfolios. Fama and French create mimicking ...
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440 views

### Portfolio optimization on a subset of assets

My objective is a portfolio optimization of the type: given $N$ assets with expected returns $r_i$ and a fixed portfolio size $M$, with $M < N$, find weights $w_i$ (positive or negative) maximizing ...
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### Finding latest market price of market portfolio according to No Arbitrage

In Excel, I have the monthly stock price data for the past few years for Asset A and Asset B. I have calculated the monthly returns, mean returns, variances, and standard deviations for both stocks as ...
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1 vote
177 views

### Can there be different Sharpe Ratios for the same index? [closed]

I am reviewing a fellow students paper, and it is argued in this paper that the Sharpe Ratio can differ based on which model is used to analyze the portfolio returns. Here a model based on the ...
1 vote
56 views

### Does it make sense to have an allocation to short term fixed income and a leveraged or unfunded position?

This may sound like a basic question but I have seen many large institutional investors have this as part of their asset allocation and am wondering why they do this? Does it make sense to have a ...
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### Two specific questions about CAPM's assumptions and implications

I have two questions about the CAPM model: the first is theoretical while the second is related to observed market data. First question: let's say we have company A and company B and we want to ...
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### Kelly Criterion for cash game poker (normally distributed returns)

I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using outdated bankroll management techniques for decades, and I want to change that. My goal is to graph the log growth ...
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### Is CAPM + additions actually used in industry-level portfolio optimization? [closed]

I've been wondering how do institutional investors actually perform portfolio optimization, i.e., do the traditional methods work when combined together with superior data collections and quality, and ...
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### Does a portfolio on efficient frontier also lie on CML(capital market line)?

I am trying to solve this question: Assume that CAPM is true. The risk-free rate is 3%, the expected return on the market portfolio is 10% and the standard deviation of the return on the market ...
1 vote
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### Can someone briefly explain me what's the difference between Exact Factor Model (EFM) and Approximate Factor Model (AFM)? [closed]

I'm reading De Nard et al. paper "Portfolio Selection in Large Dimensions" and the authors talk about these two models, but I don't have any background on these topics from university. Can ...