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Questions tagged [modern-portfolio-theory]

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

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Excess Return Covariance Matrix is Singular - Cash return and risk free rate are the same

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 ...
Farrep7's user avatar
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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 ...
Farrep7's user avatar
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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 ...
Xerium's user avatar
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Distribution of sample covariance times inverse covariance times sample covariance

I want to understand the distribution of the random variable: $$S_n = \frac{1}{n^2} 1'\hat \Sigma \Sigma ^{-1} \hat \Sigma 1$$. 1 is a vector of ones of size n, and the variance is of size nxn. $\hat \...
alejandroll10's user avatar
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Optimal weights in portfolio after rebalancing

I have a quite simple question but while looking for answers in research papers I couldn't find anything. The question can be summarized as : if you expect a shock on an asset, why don't you rebalance ...
krauuuus's user avatar
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Question about marginal risk contribution / portfolio volatility decomposition

I am trying to understand the rule where you add a new asset to a portfolio if its Sharpe ratio is greater than the product of the portfolio sharpe ratio and the correlation between the portfolio and ...
Steve R's user avatar
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Application of Leverage in Different Interest Rate Environments to an Efficient Portfolio

I have read that some Institutional Investors are utilizing leverage. According to Modern Portfolio Theory, to apply leverage one would: a) find the tangency portfolio on the efficient frontier from ...
AlRacoon's user avatar
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Calculating marginal risk contribution of FX for foreign asset portfolio

I am a European investor investing in US equities. My US equities portfolio returns in EUR can be broken down into (1) equities returns in USD terms, and (2) USDEUR spot currency returns. Using the ...
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Analytical solution to short-sale constrained portfolio

Say that we want to find the efficient mean-variance portfolio (i.e. minimize variance given that weights sum to 1 and given a set target return) and impose a short sale constraint such that $w_i \geq ...
Mr Entscheidung's user avatar
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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 ...
Li Mike's user avatar
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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? ...
Mr Entscheidung's user avatar
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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 ...
Karina's user avatar
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1 answer
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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 ...
j3141592653589793238's user avatar
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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 ...
Jay's user avatar
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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 ...
PyRsquared's user avatar
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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 ...
PyRsquared's user avatar
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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 ...
Absbert's user avatar
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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 ...
user1120695's user avatar
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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 ...
KaiSqDist's user avatar
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4 votes
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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 (...
xxanissrxx's user avatar
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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 ...
Zac's user avatar
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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 ...
T123's user avatar
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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 ...
T123's user avatar
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1 vote
2 answers
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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 ...
deblue's user avatar
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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 ...
user14894283's user avatar
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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 ...
Jack's user avatar
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5 votes
2 answers
628 views

Question about adding new investment A to portfolio B

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 ...
jauyjad's user avatar
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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. ...
herminat0r's user avatar
1 vote
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83 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 ...
Jess's user avatar
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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 ...
user3741124's user avatar
2 votes
1 answer
351 views

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}\...
T123's user avatar
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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 ...
we_are_all_in_this_together's user avatar
3 votes
1 answer
221 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 ...
FISR's user avatar
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3 votes
1 answer
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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 ...
LattePrincess's user avatar
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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 ...
TLS's user avatar
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7 votes
2 answers
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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 ...
deblue's user avatar
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1 vote
1 answer
392 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 ...
jam123's user avatar
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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 ...
Red's user avatar
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1 vote
2 answers
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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 ...
Simon Rydstedt's user avatar
1 vote
0 answers
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 ...
AlRacoon's user avatar
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0 votes
1 answer
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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 ...
Andrei's user avatar
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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 ...
Tom Boshoff's user avatar
2 votes
0 answers
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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 ...
deblue's user avatar
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2 votes
0 answers
255 views

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 ...
TrueWarrior09's user avatar
1 vote
1 answer
62 views

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 ...
user64858's user avatar
4 votes
0 answers
285 views

Two approaches to optimizing quadratic utility

My understanding of the traditional Markowitz portfolio optimization process is as follows: Let’s say I have data from year 1 to year 10. At the end of year 10 (having information about year 10), I ...
shenflow's user avatar
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3 votes
1 answer
300 views

Covariance Between Two Frontier Portfolios

Based on the definitions of A, B, C, and D in "An Analytic Derivation Of The Efficient Portfolio Frontier" by Robert Merton (1972), how can I prove the following in a line-by-line derivation?...
David 's user avatar
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3 votes
0 answers
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How to deal with benchmark timing in quantitative portfolio management?

In Grinold & Kahn (2000), the authors emphasized the separation of stock selection and benchmark timing in active portfolio management. So if we avoid benchmark timing, the optimal portfolio's ...
Gödel's user avatar
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Have more complex MVA-style models become obsolete?

Just reading a book about about portfolio optimisation. You hear left and right that MVA (Mean Variance Analysis of Markowitz) is out of date, creates suboptimal portfolios in practice and so on ...
not_sure95's user avatar
1 vote
1 answer
274 views

Why the portfolio return is defined as a weighted return?

After reading the modern portfolio theory, I am wondering why the portfolio return is defined that way. Suppose there are $n$ assets in a portfolio, the simple return of an individual asset $i$ at ...
chichi's user avatar
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