# Questions tagged [modern-portfolio-theory]

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

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### Derivation of the Sharpe ratio [closed]

How to derive the Sharpe ratio of a portfolio? Is it different or similar to the CAPM derivation of the Sharpe ratio? How to reconcile the two derivations Line by line, plus official sources would be ...
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### Is quadratic programming used to maximize portfolio skewness and kurtosis?

Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$ because the objective function coincides ...
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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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### SML Interpretation

I follow this paper and estimated two different asset pricing models via systems of deep neural networks. Both models have the exact same input: firm-specific features for 10'000 (unique) US stocks ...
56 views

### Ridge and Quadratic Programming for Portfolio Norm Optimization

Much like this post: https://stats.stackexchange.com/questions/119795/quadratic-programming-and-lasso, I'm trying to integrate RIDGE Penalty in a dedicated quadratic solver. In my case, I am working ...
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### 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 ...
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### Why does the likelihood of corner solutions in portfolios increase as the number of assets grows?

A three- asset portfolio doesn't seem prone to generating corner solutions, which are very high allocations to one of the assets and $0$ to the others. Instead, when the number of assets is low, these ...
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### Non-linear correlation (co-dependence) and the efficient frontier

The graph below shows how the efficient frontier for 2 assets bends into a sharp bisection as correlation decreases from $1$ to $-1$, with $\rho=-1$ being the most diversified, and highly unattainable ...
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### Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?

The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio ...
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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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### Portfolio return distribution as a mixture distribution

For a returns data set with $K$ stocks that are each normally distributed, can I represent the portfolio return distribution to be a weighted sum of the $K$ asset distributions, aka a mixture ...
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### Asset Allocation with near zero rates

With central banks pegging interest rates to near zero rates, an argument could be made that the future distribution of interest rates and bond returns are not normally distributed. How has modern ...
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### Is the portfolio weight vector an eigenvector?

If $A$ is a symmetric matrix, then $b$ is its eigenvector if $Ab =\lambda b$, where $\lambda$ is a scaling constant, which could even equal 1, leaving $Ab = b$. In portfolio theory, the problem is to ...
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### Interpretation and units of a covariance element in portfolio risk

Given portfolio risk is $\mathbf{w}\boldsymbol{\Sigma}\mathbf{w}$ where $\boldsymbol{\Sigma}$ is the covariance matrix whose diagonal elements $\sigma^2_{n}$ are individual asset return variances and ...
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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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### Meaning of an identity matrix for the covariance in portfolio optimization

Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $I$. This ...
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### What is the formula for the global minimum variance portfolio with positive weights?

I know how to algebraically solve for the weights when short selling is allowed but I can’t seem to find the formula for when it’s strictly positive an the weights sum to 1 anywhere online.
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### Is there a performance measure for the entire efficient frontier?

The Sharpe ratio is an example of a performance measure for individual mean-variance efficient portfolios, regardless if they maximize the Sharpe ratio or not. The efficient frontier, however, ...
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### Why do only portfolios of indices show elliptical dependence?

Elliptical distributions imply an asymmetric relationship between variables such as financial returns of different assets. I'm guessing this is mainly due to skewness, although I might be wrong and ...
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### Portfolio Optimization sum of weights constraint with short selling

For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint. In most resources I've seen, such as https://www.coursera.org/learn/...
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### Are mean-variance efficient portfolio weights random variables with probability distributions?

The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
103 views

### Why is diversifiable risk unrewarded?

I am currently looking through some actuarial study materials (CM2, formerly CT8) in which models of asset returns are being discussed. One such model is the market model (A.K.A The single-index model)...
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### Are heuristic portfolios efficient portfolios?

Markowitz's definition of an efficient portfolio is one that minimizes portfolio risk for a given level of expected return. He therefore calls portfolios along the efficient frontier "frontier ...
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### James-Stein estimator for superior estimates of returns in m.v. portfolio optimization

I am currently learning about statistical techniques to enhance the estimation of input parameters in a m.v. optimization. Specifically I have some doubts about the James-Stein estimator applied as an ...
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### Mathematical techniques for Trading signals

I'm trying to come up with a reasonable and mostly mathematical way to trade signals between two people with interests in collaboration but still wary and skeptical. The idea being that you start ...
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### SDF as an affine transformation of the tangency portfolio

I'm studying this paper. In the formulation of the theoretical setup they state: Our goal is to explain the differences in the cross-section of returns $R$ for individual stocks. Let $R_{t+1, i}$ ...
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### Is it possible to make a portfolio with higher expected return and lower standard deviation than constituent securities?

Assume we are working in the framework of modern portfolio theory. Now, let's say we have two securities (they could also be portfolios themselves) A and B. Portfolio A has expected return 10% and ...
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### Questions about Sharpe Ratio calculation

Let's say I have daily returns. Don't they depend on the risk per trade I am using? Obviously, if I'm risking 2% of equity per trade returns will be drastically different than when I'm using 10%? So ...
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### Is this methodology for finding the minimum variance portfolio with no short-selling sound?

I have below here an excerpt from a book on (among other things) mean-variance analysis showing how to find the minimum variance portfolio (Risk and Portfolio Analysis: Principles and Methods, by Hult,...
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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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### Time varying weights in a portfolio

As I have seen in my portfolio theory class, we define the weights of some assets and quantify the risk and return of the whole portfolio. In this setup, the weights do not change in time. What if the ...
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### portfolio return, sharpe ratio and value at risk

Can you please help me to confirm if my calculations are correct or need improvement, or (too simplistic...) : - portfolio return, - portfolio standard deviation, - portfolio sharpe ratio - ...
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### How can we quantify time varying portfolios?

In portfolio management, it is assumed that the assets and the weights in the portfolio are static and do not change in time. By the help of this static structure of the portfolio, we can talk about ...
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### Transform Hierarchical Correlation structure to Standard Form

In the standard portfolio risk setup, we have $\sigma_{\Pi} = \sqrt{(w' B (VFV) B' w) + w'Dw}$ where $w$ is our weight vector for N assets $B$ is the Nxm factor beta matrix $V$ is the factor ...
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### Finding the Efficient Frontier in Tensorflow

I have two assets A and B. Asset A has an expected return of 0.92% with a standard deviation of 2.10. Asset B has an expected return of 1.39% and a standard deviation of 2.20. I want to find the set ...
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### ESG Style Analysis

Hi all and thank you in advance. Do you think that implementing a style analysis on ESG equity portfolios is feasible? When I mean style analysis I refer to the seminal paper of Sharpe (1992) but I ...
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### Portfolio construction in reality

I have a very basic knowledge of portfolio construction and optimization aside from the mean-variance efficient portfolio theory. I am looking for a series of resources to dig more into the topic ...
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### Do equities have spread duration?

I was reading a research article today that suggested that equities exhibit significant effective spread duration. I'm looking at this on an index level, not firm specific. I've tried several ways of ...
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### Using Implied Volatility for Portfolio Optimization

Hello I am interested in portfolio optimization . Previously I when I have done portfolio optimization I would take the historical returns of a stock and use them to perform a mean variance ...
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### Benchmark of a Dollar Neutral Strategy

A dollar neutral strategy invests the same amount of money long and short without accounting for the volatility (risk) of either side. Depending on volatility you either end up positively or ...
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### 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 ...
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