Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
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Covariance Matrix by Multi-Factor Model
I have been trying to find literature for the derivation of the covariance matrix, following a multi-factor model. I have had no luck at all, every single article I have found on the web already ...
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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 ...
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Tangency portfolio negative maximum Sharpe ratio
Suppose I have three assets: the market, factor A and factor B. The market is in excess returns of the risk free rate. The other two factors are long-short portfolios. I have net returns for these ...
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N asset covariance matrix vs N-1 asset covariance matrix
so I have been using a M-V framework to form M-V efficient portfolios. I have noticed that every time I make my investment universe smaller the minimum variance frontier moves to the right. This ...
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Why do we use half of the risk in objective function of markowitz portfolio theory
In some documents I have seen objective function of markowitz portfolio theory is as follows.
minimize 1/2 * w'Σw
where w is weights
Σ is covariance matrix
I could ...
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Risk Factors, Portfolio Optimization
I really need help with a project that I am working on, for my university.I study in Ecuador and the research material here is very limited. Nonetheless I have tried my best to start with the basics ...
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Unexpected Inflation and Asset Allocation
If asset allocation decisions were made prior to the news of unanticipated inflation, how should asset allocators incorporate the fact the inflation is now 5% higher than the 2% inflation target?
It ...
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Correlation Matrix to Variance Covariance Matrix Portfolio STDEV
I have a correlation matrix that I wanted to convert into a variance covariance matrix. I also have the weights in a column in excel along with each assets standard deviation. What excel function can ...
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Mean-variance optimization and hedging
I've read--and have been able to replicate empirically--that mean-variance optimization will trade two positions against each other if assets are highly correlated. For example, if stocks $A$ and $B$ ...
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Portfolio Theory - Maximizing Expected Utility Function
I am trying to implement a portfolio selection tool based on utility functions. So, I should maximize the expected utility of a given utility function:
$$
\begin{align}
&\max_{w}\ E[u(W_0(1+w^TR))]...
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Can I invest in the market portfolio of modern portfolio theory? [closed]
According to the theory, the market portfolio is composed of all assets weighted by their market capitalization, and this is the portfolio one should own. Is there a way to build a portfolio close to ...
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Why the market portfolio is the tangency portfolio in the Mean-Variance Optimization model?
I read in an explanation that the tangency portfolio has all securities with weights proportional to their market value because supply equal’s demand. But I can't understand why supply equals demand ...
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Utility maximization given tangent portfolio
I am currently working on a following problem:
Given risk free asset with return 1.1 and two normally distributed risky assets with returns (1.2, 1.3) and variances (4, 9):
1. Find weights that ...
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Is finding the efficient frontier a max or min problem?
I'm trying to understand where the efficient frontier comes from.
What I understand about the efficient frontier
I understand the efficient frontier is essentially a subset of the boundary of the ...
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Kelly Criterion for Multiple Simultaneous Correlated Bets [closed]
I am looking for an equation for the optimal fractional bet sizing for N number of simultaneous correlated bets.
I am looking specifically for an equation for binary bets, but an equation for bets ...
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Why additivity assumption holds in CAPM and factor models? (Screenshot of a textbook included) [closed]
All the excerpts are from the book investment, written by Bodie. At the bottom of this post, I attached pages of the the book that show a related part of my question.
Question
1. Why the variance of ...
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How to derive this mathematical equation from the perspective of the mean-variance portfolio optimization?
Question
I found a simplified inequation to decide whether the new asset A should be added to my current portfolio B. If the following inequation is satisfied, the new asset A should be added to my ...
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Best books on portfolio construction?
I am a master of finance student and although I understand the basics and the theory of portfolio construction I am still struggling when it comes to the practical side of things, i.e. building a real-...
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Factor Models vs. Risk-Adjusted Performance Measures
I am building a study focusing on portfolio returns relative to a number of portfolios constructed using various ESG screening techniques. My intention is to measure and compare the performance of '...
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Carhart 4-Factor Model intercept interpretation
I've been following studies such as Kempf & Osthoff (2007) and Statman & Glushkov (2009) in building a methodology measuring ESG portfolio performance centred around the Carhart 4-Factor Model....
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Portofolio selection by aggregating online expert advice
My question is about the following 2020 article : Universal portfolio selection strategy by aggregating
online expert advice.
I am at the moment trying to understand the underlying logic of the ...
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Consensus expected excess return from Active Portfolio Management
In the book Active Portfolio Management, when discussing components of expected return (page 92 in edition 2), the authors mention that the consensus expected excess return $\beta_n\mu_B$ is the ...
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Optimal portfolio with only n assets (with n less than total assets)
Given a time series of a set of N assets (let's say 100), how can I find the optimal portfolio, with the constraint that only n<N assets (let's say 10) can be in the portfolio? With 'optimal ...
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CAPM derivation - active portfolio management
In the book Active Portfolio Management by Grinold and Kahn, the author presents the derivation of CAPM. I am rather confused by the notation of $\beta$ in this derivation and hope to seek some help. ...
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Active portfolio management - characteristic portfolios derivation
In the book Active Portfolio Management by Grinold and Kahn, on page 30, when it derives the characteristic portfolio $h_a$ for some characteristic vector $a$, the problem is set up as
$$\min h^TVh$$
...
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Problem from Stochastic Portfolio Theory Textbook
I'm trying to do the following problem from Robert Fernholz's textbook Stochastic Portfolio Theory:
The assumptions mentioned are:
and my attempt is as follows:
I have no idea how to deduce that ...
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Interaction between raw position signals and portfolio optimisation methodologies [closed]
I'm trying to get my head around how the various aspects of constructing a final position generally interact and wonder whether anyone could expand on my (tentative) understanding currently.
As I see ...
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How to deal with securities that has short historical data when performing mean-variance portfolio analysis?
I am trying calculate expected return and risk (stdev) based on historical data using Mean Variance Analysis framework. Let's say the portfolio has 10 stocks, 9 of them have more than 10 years history ...
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Strange efficient frontier, when I try to calculate BTC & ETH ratios using MPT(Modern Portfolio Theory) [closed]
The 10k Monte-carlo simulations all fall on the same line, instead of a proper scatter plot..
Not sure what I'm doing incorrect. It all works fine, if I include Monero in the mix.
Any pointers ?
I'm ...
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How to prove that the return criteria for adding an investment A to an existing portfolio can be represented using Sharpe Ratio Approach
How can I prove that the return criteria for adding an investment A to an existing portfolio can be represented as the below inequality using the Sharpe Ratio Approach for risk adjusted returns as ...
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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
...
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Sub-portfolio correlation
I am trying to reduce correlation matrices into sub portfolios.
For example, I have a covariance matrix $\Sigma$ and weight-vector $w$ of two line items which I blend together into a sub-portfolio $\...
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Equivalence of Standard Deviation and Variance as a risk measure - WRONG?
In Modern Portfolio Theory, I often see that people seem to view Standard Deviation and Variance as equivalent. Example from Markowitz himself:
"Thus far I have used the standard deviation ...
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RIsk-retun of 2-asset portfolio with perfect negative correlation
Risk-retun of 2-asset portfolio with perfect negative correlation $(\rho=-1)$ is a straight line with slope of $\frac{|\mu_2 - \mu_1|}{\sigma_2+\sigma_1}$ since $\sigma_P=|\omega_1\sigma_1 -\omega_2\...
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global minimum variance portfolio vs all-bond portfolio
I'm leaning portfilio theory and have got some questions. global minimum variance portfolio is defined as the leftmost point on the efficient frontier which suggest it is a all-bond portfolio if risk ...
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Optimal Portfolios with Skewed and Heavy-Tailed Distributions
I am learning about portfolio theory and been using Markowitz. I wondered, however, if I can use distributional and asymmetric information of the returns to solve the problem. For instance, I have a ...
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Should a stock with high return autocorrelation be weighted more heavily in a portfolio?
Some say the presence of autocorrelation (aka serial correlation) in a stock's financial return time series helps with forecasting its next-day movements, unlike a stock that has low serial ...
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Portfolio rebalancing to optimal weights including transaction costs and without cash component
Consider a portfolio with 4 assets (A, B, C, D) with prices, quantities, current weights, and target weights as follows:
I want to rebalance the portfolio from the current weights to the target ...
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Portfolio variance $<=$ weighted average of individual variances [closed]
In portfolio theory, I often (with some justifications but the message is the same) come across the following statement:
"The most important quality of portfolio variance is that its value is a ...
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Why is a smaller portfolio norm better?
If the norm of the portfolio weight vector, $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ for $p=1,2$, of portfolio A is 0.6, and the norm of portfolio B is 0.4, then portfolio B is considered more attractive ...
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Prove norm $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ of min-variance portfolio $\leq$ max-Sharpe portfolio
The minimum-variance portfolio weight vector is
$$\boldsymbol{w}_{MV} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$
whereas the maximum ...
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For portfolio variance, why doesn't $Var(X w) = w^\top \Sigma w$? [closed]
From multivariate asset returns $X$, we can calculate the sample covariance matrix $\Sigma$.
The definition of (any) portfolio variance is $w^\top \Sigma w$, where $w$ are portfolio weights.
If $X w$ ...
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Higher risk = high reward?
Some theory (in my understanding) suggests that price is the expectation of future cash flows discounted by expected return:
$$p_t=\frac{\mathbb{E}^m_t[c_{t+1}+p_{t+1}]}{1+\mathbb{E}_t^m[r_t]}$$
where ...
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Criteria for excluding an Asset Class from a Strategic Asset Allocation
While historically the return, volatility and correlation characteristics justified the inclusion of Sovereign Bonds (US Treasuries, European Central Bank Debt, etc) in Strategic Asset Allocation ...
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Maximum skewness portfolio solution derived from its Lagrangean formulation
$$\arg \min_w \quad w^\top \Sigma w$$
\begin{align}\text{s.t.} \quad \mathbf{1}^\top w = 1 \end{align}
is the optimization problem for the minimum-variance portfolio weights, whose analytical solution,...
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Why isn't the asset with minimum variance given a 100% portfolio weight? [closed]
The maximum expected return portfolio is the one that assigns a 100% weight to the asset with the highest expected return amongst all assets under consideration.
Shouldn't then the asset with the ...
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How to compare mean-variance-skewness-kurtosis portfolios obtained by expected utility maximization?
Suppose I have some portfolios which are the result of maximizing the expected utility of different approximations of a utility function, how do you test these portfolio's out-of-sample and how do you ...
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Correlation between mean-variance efficient portfolios
If the covariance solution between the returns series of the minimum-variance portfolio ($A$) and any other portfolio along the efficient frontier ($B$) is
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
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Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution
\begin{align}
\arg \min_w \enspace & -w^\top \mu \\
\mathrm{s.t.} \enspace & 1_N^\top w = 1 \\
& w_i \geq 0 \enspace \forall i=1,\dots, N
\end{align}
is the optimization problem for ...
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Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$
How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to
$$w_A^\top\Sigma w_B$$
where $w_i$ is a unique portfolio weight vector, and $\...
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