Questions tagged [modern-portfolio-theory]

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

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2 votes
0 answers
103 views

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 ...
0 votes
1 answer
121 views

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 ...
1 vote
0 answers
32 views

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 ...
2 votes
1 answer
152 views

Assumptions of the CAPM

As to my understanding, the CAPM assumes that all investors behave as described in the portfolio theory. Consequently, all investors hold a combination of the risk-free investment and the efficient ...
3 votes
1 answer
414 views

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 ...
1 vote
1 answer
85 views

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 ...
0 votes
1 answer
86 views

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? ...
0 votes
0 answers
31 views

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 ...
1 vote
0 answers
102 views

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 ...
0 votes
2 answers
469 views

Cover's universal portfolio vs. Markowitz's mean-variance model

Cover's universal portfolio maximizes the wealth growth rate Markowitz's mean-variance model minimizes portfolio variance Both allocate assets based on historical returns. How do these two models ...
5 votes
3 answers
3k views

How can I find the portfolio with maximum Sharpe Ratio - Using Lagrange Multipliers

In Markowitz' portfolio theory we can construct portfolios with the minimum variance for a given expected return (or vice versa). Across expected risks, this traces out the well-known efficient ...
1 vote
2 answers
978 views

Do linear combinations of two efficient portfolios cover the entire efficient frontier?

Note : We are considering the case of N risky assets. I think the answer is 'Yes', although I am not sure as I am unable to prove it. The reasons for me thinking that the answer is 'Yes' are - 1) ...
2 votes
1 answer
235 views

Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions

Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions? During my engineer student days I saw the Markowitz theory for portfolio selection ...
1 vote
0 answers
22 views

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 ...
1 vote
1 answer
126 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 ...
2 votes
0 answers
39 views

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 ...
7 votes
2 answers
1k 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 ...
1 vote
0 answers
82 views

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 ...
1 vote
0 answers
42 views

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 ...
1 vote
1 answer
582 views

Use of AI in portfolio optimization [closed]

A new Danish hedge fund has received quite a large amount of attention in the last month or so. They are launching a new investment fund that uses proprietary AI to select stocks to invest in. They ...
0 votes
0 answers
38 views

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 ...
2 votes
0 answers
31 views

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 ...
0 votes
0 answers
53 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 ...
4 votes
0 answers
101 views

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
0 answers
65 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 ...
3 votes
1 answer
344 views

Proof that Sharpe ratio of the benchmark is related to the maximal information ratio and Sharpe ratio

I understand the economic logic behind it, that the active portfolio with the highest information ratio will also have the highest Sharpe ratio, but I can't see how $SR_B^2 = SR_P^2 - IR^2 $
0 votes
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58 views

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 ...
0 votes
0 answers
75 views

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 ...
1 vote
2 answers
137 views

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 ...
0 votes
0 answers
79 views

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 ...
0 votes
0 answers
126 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 ...
5 votes
2 answers
606 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 ...
2 votes
1 answer
325 views

annualized vs annual returns

For the purposes of MPT, to compute return of an asset, one typically uses the daily log return of the assets and then anualizes it and the same goes for stddev ...
0 votes
0 answers
53 views

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. ...
3 votes
2 answers
615 views

Portfolio Systematic Risk, Breaking it down into factor % contributions

I have a portfolio (p) of N equities, with let's say weights vector (m) at the start of the calculation period. Each equity has its own set of factors (like corresponding country, industry index, etc.)...
1 vote
0 answers
77 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 ...
0 votes
0 answers
67 views

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 ...
2 votes
1 answer
343 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}\...
4 votes
1 answer
952 views

How to derive the CAPM from maximizing the Sharpe ratio?

I know how to derive at the CAPM from a microeconomic foundation. In a recent University course I stumbled over a slide that derived the CAPM solely from the Sharpe ratio: I cant come up with that ...
0 votes
1 answer
59 views

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 ...
0 votes
1 answer
213 views

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 ...
19 votes
2 answers
9k views

Why is the first principal component a proxy for the market portfolio, and what other proxies exist?

Let's say that I have a universe of stocks from a certain sector. I want to compute the market portfolio of this sector. Beta is the covariance between each stock and the market. But how do you ...
2 votes
2 answers
279 views

Portfolio construction for almost identical assets

The problem I am looking at concerns the treatment of almost identical assets in portfolio construction. Let us assume that we have two assets, both with a standard deviation $\sigma=0.2$ and a ...
14 votes
2 answers
2k views

Models crumbling down due to negative (nominal) interest rates

Given that the negative interest rates on a lot of sovereign bonds with maturity under 10 years are trading in the negative (nominal) interest rate territory (recently also the short term EURIBOR has ...
0 votes
3 answers
375 views

Why is the performance of a portfolio based on geometric means boosted by positive correlation?

https://qoppac.blogspot.com/2017/02/can-you-eat-geometric-returns.html The blog post above by Rob Carver discusses the use of geometric means to evaluate investments. The section "The ...
3 votes
1 answer
213 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 ...
0 votes
0 answers
108 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 ...
9 votes
2 answers
1k views

Why do low standard deviation stocks tend to have superior future returns?

I've recently stumbled on something that really surprised me. These papers (1, 2) find that past standard deviation of returns is inversely related to future returns. That is, portfolio of low ...
1 vote
1 answer
372 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 ...
6 votes
2 answers
7k views

Calculating alpha and its meaning

According to wikipedia, CAPM model is described by: $E(R_{i})=R_{f}+\beta _{{i}}(E(R_{m})-R_{f})$ And according to website such as http://investexcel.net/jensens-alpha-excel/, $\alpha = E(R_{i}) - ...

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