Questions tagged [mean-variance]

Mean-variance is the starting point of most portfolio optimisation techniques.

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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 ...
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How to prove that the feasible set of a two-asset portfolio is a hyperbola?

The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book does not give a complete proof, and I did not find a ...
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Portfolio optimization with Scipy in Python

I performed Scipy portfolio optimization in two scenarios: 1) when I cannot lend or borrow at the risk-free rate; 2) when I can lend and borrow at rf=1.5%. Now, optimal risky portfolio weights anyway ...
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How to prove the inequality for the standard deviation of a linear combination of two random variables

The variance of the linear combination V of random variables X₁ and X₂ is given by the following formula: $$ \sigma_{V}^{2} = s^{2} \sigma_{1}^{2}+(1-s)^2 \sigma_{2}^{2}+2 s(1-s) c_{12} $$ where s and ...
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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 ...
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Calculate minimum variance hedge ratio for foreign-denominated asset hedged to domestic currency

The formula for minimum variance hedge ratio (MVHR) is conceptually the correlation multiplied by the ratios of volatilities. correl (Y,X) * (STDEV Y / STDEV X) ...
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What is the mathematical difference between Mean-Variance Optimization and CAPM?

I have spent some time going through the maths of both Mean-Variance Optimization and CAPM, and I'm trying to pin down the mathematical differences between them. For both, let $p$ be a portfolio ...
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Find variance of Asset with lesser return to make a pure portfolio of it the min-variance portfolio [duplicate]

I need to solve the question mentioned above. For an asset with a worse payoff than another, I need to determine a variance for which the minimum-variance portfolio only consists of this asset. There ...
gerscorpion's user avatar
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Closed form solution for Mean-Variance optimization without short-selling

So I am writing my bachelor thesis about the naive portfolio vs mean-variance portfolio and I am currently a bit stuck at the part about describing the mean-variance portfolio. I know that if there ...
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Robust estimates of variance covariance matrix

I am looking for help from other people with experience creating variance covariance matrix that have enough predictive power to actually lower portfolio volatility out of sample. Using real world ...
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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 (...
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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 ...
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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 ...
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If Kelly and tangent portfolios have the same weights, do they differ only empirically?

I studied Kelly portfolio and tangent portfolio and found that they have the same weights. But the empirical studies that I have seen so far show that Kelly portfolio has a smaller number of stocks ...
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Markowitz Optimization with 2 assets

Suppose there are only two risky assets and we want to optimize our portfolio. Constraints are that we have a minimum return $\overline{r}$ and we can only invest $w_1 + w_2 = 1$. Is it possible that ...
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Beyond the mean-variance framework, can expected returns be HIGHER for an individual due to a HIGHER risk aversion?

In the mean-variance framework, the only way to get a higher expected return is to be exposed to a higher beta, and the more risk-averse an agent, the lower the beta of their portfolio (lending ...
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Naive Diversification under mean variance

I'm looking for a way to introduce naive diversification bias in a mean variance framework and had the idea to model it as some sort of "aversion to extreme portfolio weights" of the ...
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Alternative form of mean-variance optimization that uses standard deviation

I'm curious about an exercise found in Optimization Methods in Finance. Exercise 8.2 (pg 143) explores a variant of the more commonly used form of MVO. When I refer to the more common variant I'm ...
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Maximizing Mean+Variance in a Portfolio

Mean-Variance optimization trades off expected returns with portfolio variance. The idea is that excess variance is not desirable. But what if you weren't averse to high variance and you wanted to ...
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Mean-variance optimization - objective function formation with factor models

Tradition mean-variance optimization uses the following objective function in optimization: $$ \mu w^T - \lambda w^T \Sigma w $$ Which I'm trying to adapt to a factor model. I've come up with: $$ f \...
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Is this equation correct for portfolio optimization for CARA normal with N risky and one riskless asset?

Suppose the consumer Solves $\max -e^{-\gamma W}$ where $W=X^T D -X^Tp R_f$ where $X$ is the vector invested in a risky asset and $D\sim N(E[D],\Sigma^2_D)$ and $R=\sim N(E[R],\Sigma^2_R)$. Then ${ X=(...
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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?...
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Why is the dynamic mean-variance problem time-inconsistent?

A lot of the literature in dynamic mean-variance problem states that the dynamic mean-variance problem is time-inconsistent. Now I was not able to find an example of why the problem is time ...
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Mean-variance framework with endogenous correlations

In most mean-variance frameworks I have seen, once we clear markets in the model, it determines asset prices (and returns). However, all of these frameworks assume that the correlation matrix of the ...
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Why does the mean term have a higher effect than the covariance term in MV optimization? [closed]

I am trying to use the mean-variance (MV) optimization framework. When I change the mean term using future-ground-truth return (I am not supposed to do so), it has a higher effect on the MV ...
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Hedging with peer companies and optimize the weights

I am trying to long a security that is expected to outperform its peers after certain corporate actions, but want to hedge using the same group of peers (so short ~5 names). So the goal here is to ...
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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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Index Tracking Problem

I have set up a mean variance optimization problem, $$min:{W}^{\prime}{\Sigma_{\varepsilon}{W}}$$ $$s.t:{W}^{\prime}{\alpha}=R_B\;,\;\;W^{\prime}l={1},\;\;W'\beta=0,\;\;W'Z=\beta_p$$ where, $W$ is an (...
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Utility Theory and Mean Variance Analysis

I was wondering if it's pertinent to use this interpretation of the expected utility function given by the Taylor series expansion, $${E(U(W)}\approx{U[E(W)}]+\frac{U''[E(W)]\sigma^2_W}{2}\tag{1}$$ to ...
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Closed form solution for Mean-Variance optimization under constraint

Is there a closed form solution for the vector weight $w$ for the following mean-variance optimization problem? $\max_w w'\mu - \frac{\gamma}{2}w'\Sigma w $ s.t. $w'z\geq \bar{z}$ where $w, z$ are N ...
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Questions about Merton's derivation of the security market line

In Merton's "An Analytic Derivation of the Efficient Frontier" (PDF), he derives the security market line for the CAPM using the definition of the tangency portfolio. He writes: Here, $m$ ...
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Is this quadratic form the Sharpe ratio?

I'm reading Merton's An Analytic Derivation of the Efficient Portfolio Frontier. In section IV, he derives the efficient frontier with a riskless asset. Let $\mathbf{w}$ be a vector of portfolio ...
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mean-variance optimization === max sharpe ratio portfolio?

Noobie here. I just wanna ask a simple question: in the context of portfolio optimization, is Mean-Variance optimization the same as the max sharpe ratio portfolio?
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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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Monte Carlo vs. Block Bootstrapping vs. Bootstrapping

Because I can fit e.g. ~25 distributions via empirical cumulative distribution fitting to correlated data (including stable dist.), and then simulate the original data based on correlation (covariance)...
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Mean-Variance Portfolio Axis Description

I'm currently looking into the mean-variance approach to portfolio theory and I wonder, why the standard deviation $\sigma$ is graphed on the x-axis and not the variance $\sigma^2$ as a measure of ...
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Proof that mean-variance opportunity set is closed

In the book Financial Economics (2010) by Hens and Rieger, on page 101 we find the following Lemma 3.1: If we have finitely many assets, the minimum-variance opportunity set is closed and connected. ...
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Portfolio Optimization constrained to maximum N% of short selling portfolio weights

For mean-variance portfolio optimization with short-selling allowed, but restricted to a certain percentage of the portfolio weights (lets assume N), we can constrain it in the follwoing way: (from j=...
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Black-Litterman for quant portfolio

I have seen a lot of research around the Black-Litterman approach and I think theoretically, it is a nice framework. However, it appears that its main strength is from a practitioner's point of view, ...
Jim's user avatar
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CVXOPT quadratic programming mean variance example

Trying to learn how to use CVXOPT to do quant finance optimization. For the example given on page https://cvxopt.org/userguide/coneprog.html#quadratic-programming . I feel confused how this "S&...
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Mixed-integer programming approach for index tracking

Suppose you currently own a portfolio of eight stocks. Using the Markowitz model, you computed the optimal mean/variance portfolio. The weights of these two portfolios are shown in the following table:...
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Deriving the risk-aversion coefficient

By considering the parametrised formulation of the mean-variance criterion by Markowitz, the risk aversion coefficient $\lambda$ can be derived as follow. As suggested by Arrow and Pratt, given the ...
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Optimal Portfolio Formulation

I'm currently studying Luenberg's Article "Projection Pricing" (Jrl of Optimization Theory and Applications, Vol. 109, No. 1, pp. 1–25, April 2001) and there is a claim that I can't prove. ...
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Covariance Matrix for asset returns [closed]

Hey guys I'm pretty new here, not sure how to code my question so I'll include a picture reference instead. I'm a bit confused on how the standard deviation of F (commodity price) would affect the ...
Prisha Singh's user avatar
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Corner portfolios

This is more a theoretical problem rather than a technical one. I am looking for a clear and rigorous definition of corner portfolios and I like to understand more precisely their relation with the ...
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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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Calculate weight of an asset

Suppose there are three assets, and the first asset has volatility 18%, the second asset has volatility 16%, and the third asset has volatility 16%. Suppose also that the first two assets' returns ...
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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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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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