# Questions tagged [modern-portfolio-theory]

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

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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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### 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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### 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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### 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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### 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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### 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 You hear left and right that MVA (Mean Variance Analysis of Markowitz) is out of date, creates suboptimal portfolios in practice and so on ...
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### What does a portfolio risk of 20% mean?

From the book Active Portfolio Management there is a use of lingo I don't understand. Take this quote from pg. 100 "Why are institutional money managers willing to accept the benchmark portfolio ...
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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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### 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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### 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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### 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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### 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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### Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance

The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample. ...
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### Show that portfolio's percentage contribution to loss (PCL) equals PCR (risk)

I came across this question during self study on a quantitative book (Question 3.6 on Page 75 of Quantitative Equity Portfolio Management: Modern Techniques and Applications By Edward E. Qian, Ronald ...
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### Relation between CAPM and Portfolio Theory

can any of you explain to me in simple terms how CAPM and portfolio theory are related to each other? To my understanding: Portfolio theory helps to select the "right" stocks under risk/...
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### 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 ...
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### How do I maximize my expected utility of wealth?

Suppose I have a utility function say $U(p)=p^{1/2}$ and I bet on a basketball game. I have my initial investment, payouts and probabilities of winning, how can I determine the maximum I need to bet ...
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### Mean Semivariance Optimization VS PMPT

Mean Semivariance optimization defines semivariance, variance only below the benchmark/required rate of return, as: $(1/T).\sum_{t=1}^{T} [Min(R_{it}-B,0)]^2$ where $B$ is the benchmark rate, $R_{i}$...
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### Continuous formula for the price of an asset paying one terminal dividend?

I have been trying to come up with ways to come up with an answer for a question we got in my class of "Asset Pricing Theory". The question is as follows: "Write the price of the asset at time t in ...
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### How to find the tangency portfolio using quadprog in R with different risk free rates

I am trying to find the optimal tangency portfolio for the efficient frontier (calculated using qp.solver in quadprog) but subject to different risk-free rates. Demos for quadprog in R show that to ...
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### Some definitions in the BARRA Predicted Beta model

I'm studying the BARRA Predicted Beta model, and the common factor covariance between portfolio $p$ and the return on the market $m$ is defined as the product of the transposed vector of the factor ...
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### Is the Market Portfolio on the Markowitz Efficient Frontier?

I have seen "market portfolio" defined online (Wikipedia/Investopedia) as the bundle of all available investments where the assets are each weighted in proportion to their existence in the market. I ...
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### Problems with Black-Litterman: negative portfolio weights, and very poor returns

I am trying to implement the Black-Litterman model using own-defined views matrix (from consensus analysts). However, I have encountered the problems of negative portfolio weights in some periods, and ...
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### Mean Variance Optimization of 2000 pairs of securities (Python)

I would like to take the opportunity to ask for your help on an assignment I'm trying to complete. For this 'Modern Robo Advisory' course we are asked to solve a (target) goal-based investment ...
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### Portfolio risk decomposition using historical data: which weights to use for assets?

I am trying to decompose portfolio risk given historical returns of each asset in the portfolio. For a basic 2 asset portfolio, the portfolio risk is given as σ_p^2 = w_x^2 \cdot σ_x^2+ w_y^2 \...
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Given a portfolio of shares $I = \sum{w_iS_i}$ for some fixed weights $w_i$ where each stok $S_i$ has a log-normal distribution, what is the process / distribution followed by the portfolio? That is, ...