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Questions tagged [portfolio-optimization]

Questions related to mathematical methods used for searching of optimal portfolio structures. Also related to questions on optimal structure of portfolios from both strategic and tactical point of view

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What do the existence and parameters of an efficient investment tell you about the value of a risk-free return?

I'm working on an unassessed course problem, Consider the following risky investments \begin{matrix} \text{name} & \text{expected return} & \text{standard deviation of return} \\ A & 9\% &...
mjc's user avatar
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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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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 ...
user1120695's user avatar
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Combining many trading strategies in an efficient

I have a lot (>50) of back tested (and naively "validated") trading strategies. They trade different ETFs, mostly equities, but also others (like GLD, USO, ...). These are all strategies ...
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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 ...
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Portfolio construction: Over/underweighting assets with a given active risk budget

I am trying to refresh my knowledge of portfolio risk calculation but would like to get a second opinion on the best approach. I have a set of 10 assets that together make up the benchmark and I have ...
K. Leblora's user avatar
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260 views

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 ...
soulsbornefan's user avatar
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Interpretation of optimal weights in portfolio for risk-adjusted return maximization

To start, I'm not an expert in portfolio management. My research involves examining the effects that one financial asset has on another, specifically looking at the spillovers between cryptocurrency ...
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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 ...
helloimgeorgia's user avatar
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1 answer
337 views

PCA for portfolio optimization (Markowitz)

Suppose that I've used the spectral theorem of linear algebra to completely decompose the covariance matrix. I now know the largest and smallest eigenvalue, which corresponds to the largest and ...
Marlon Brando's user avatar
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Maximizing the expected log utility

Let's assume that we have a self-financing portfolio made by $\delta_t$ shares and $M_t$ cash, so that its infinitesimal variation is: $$ dW_t = rM_t \, dt + \delta_t \, dS_t $$ We define $\alpha_t$ ...
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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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What do we know about "overconfidence" w.r.t asset characteristics and how can the overconfidence bias be implemented in a Markowitz portfolio modell?

Disclaimer: I have a simnilar question already in quora with some answers which are good but not really satisfying (they were taken from Wikipedia or ChatGDP). If I break down overconfidence into ...
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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 ...
Zac's user avatar
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How to find the expression for the SDF and solve this exercise?

I'm struggling to solve point a and b of this exercise, while in point c I got a very close result to the reciprocal of the relative risk aversion. If you can help me and explain how to do it, it ...
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Maximising skewness for a portfolio

I am trying to recreate the Mean-Variance-Skewness-Kurtosis-based Portfolio Optimization work done by Lai et Al. (2006) (link). I reached the part where in order to run the PGP model, you need to feed ...
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Static Multiperiod Optimal portfolio

I am interested in optimal portfolios in a multi-period setting To be more precise, say I have an investment horzion over $T$ periods and the market consists of $N$ assets. I simulated future asset ...
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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 ...
T123's user avatar
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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 ...
T123's user avatar
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1 vote
2 answers
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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 ...
deblue's user avatar
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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 ...
KIM Kyuhyong's user avatar
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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 ...
user14894283's user avatar
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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 ...
Jack's user avatar
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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 ...
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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. ...
herminat0r's user avatar
6 votes
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146 views

How can one quantify the incremental value of better covariance matrix modeling in portfolio optimization?

Let's say we have two estimators of the covariance matrix, $\hat{C}_1$ and $\hat{C}_2$, and the latter is an improvement on the former. Is there any measure of the improvement that can be sensibly ...
Slow Learner's user avatar
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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}\...
T123's user avatar
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Terminology: "global" in "global minimum-variance portfolio"

I am confused about the meaning of "global" in "global minimum-variance portfolio". The sources that I have encountered so far do not explicitly state what "global" means....
Richard Hardy's user avatar
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"fix" a sample covariance matrix which is not positive semidefinite by using daily returns instead of monthly

In the portfolio optimization problem at hand, one of the constraints is that the tracking error should not be greater than $\gamma$. The constraint is therefore: $(\textbf{x}-\textbf{w})^\mathrm{T}\...
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How to derive the optimal option structure given investor views, i.e. is it optimal to buy a call option, a risk reversal or a butterfly

Optimizing a position typically requires two things: An assumption about how prices will behave in the future An objective function to maximize/minimize For certain cases in finance, we have closed-...
user1590123's user avatar
6 votes
1 answer
856 views

Markowitz Eigenvalues & PCA

I came across this passage in a book about PCA and denoising of Markowitz: But eigenvalues that are important from risk perspective are least important ones from portfolio optimization perspective. ...
Markowitz's user avatar
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Portfolio factorization for portfolio optimization

I am looking to do some basic portfolio constructions as an experiment to learn more about it. I have been researching a bit and what I have found is that one of the purposes of factors models (Fama-...
deblue's user avatar
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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 ...
we_are_all_in_this_together's user avatar
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Optimal portfolio as combination of target and minimum tracking error portfolios?

Dear Quant StackExchange I seek some intuition for how my portfolio behaves given constraints. In a universe of say 5 assets, I have a "target portfolio" with weights that are found from ...
fdp1996's user avatar
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3 votes
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Tail Risk Hedging for Public Pension Plan

Very simplistically, ERISA rules require corporate pension plans to use market rates to discount their liabilities. If interest rates go up, the value of their pension liabilities goes down. Since ...
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1 vote
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Is It necessary to make each strategy's volatility almost equal before solving weight when constructing a risk parity portfolio? Or use other model?

Say I want to construct a portfolio consisting of different trading strategies (but trades same pool of assets) with cross-sectional varying volatility. It makes me feel uncomfortable since the one ...
NewBieQuant's user avatar
1 vote
1 answer
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Faster Portfolio Optimization under rank 1 updates

I was studying Markowitz portfolio optimization and had a question on the practicality of this in the setting of high frequency trading. Optimization seems like a cumbersome process. But at each tick ...
user50123's user avatar
1 vote
1 answer
52 views

Is there economic/intuitive reason why the treynor-black model favour low delta instruments?

In the treynor-black model optimal instrument weights are proportional to: $w_i = \frac{\frac{\alpha_i}{\sigma_i^2}}{\sum_j \frac{\alpha_j}{\sigma_j^2} }$. Let Instrument 1 be a stock with $\alpha_1$ ...
mbison's user avatar
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1 vote
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Risk Budgeting with negative covariance

Let's say I want to optimise allocations between strategies in a multi-strategy fund. There are 3 strategies, and the CIO want me to solve the portfolio that has 50% of risk in 1st strategy, 40% in ...
MackMalc's user avatar
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How do I allocate between passive and active strategy using active risk budget?

Lets say I have 100 million dollars. My active risk budget is 5%. I have an active fund that has active risk of 10%. What will be my mix in dollar terms of this active port and passive etf (assume ...
Michael's user avatar
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Derivation Treynor-Black model

In the treynor-black model the assumption is that markets are not fully optimal and it is possible to achieve additional alpha on top of the market portfolio. After a mean-variance optimization ...
mbison's user avatar
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3 votes
1 answer
225 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 ...
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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 ...
LattePrincess's user avatar
1 vote
1 answer
418 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 ...
jam123's user avatar
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1 vote
1 answer
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Optimal leverage for strategy with normal returns

Given a strategy with normal returns with mean 5% and standard deviation 10% what is the optimal leverage (up to a maximum of 2x) to maximize the expected wealth? With the same setting, if trading is ...
Mattiatore's user avatar
1 vote
0 answers
199 views

Minimum transaction size for portfolio optimization with CVXPY

Long time reader, first time asker! I am working on a portfolio optimizer where I have a universe which is much larger than potential portfolio and where I want to exclude small transaction, i.e. a ...
herminat0r's user avatar
1 vote
1 answer
207 views

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 ...
ethor's user avatar
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1 vote
3 answers
264 views

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 ...
ethor's user avatar
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210 views

L1 norm equality constraints in portfolio optimization, pros and cons

Can I use L1 norm equality constraints $\sum_{i}{|w_i|}=c,~c>0$ (or $\|\mathbf{w}\|_1=c$) in portfolio optimization, instead of the $\sum_{i}{w_i}=c$ (or $\mathbf{1}^T\mathbf{w}=c$) constraint? Any ...
user29988's user avatar
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1 answer
269 views

Market portfolio and portfolio with three risky assets

I'm trying to solve a problem with portfolios, but I cannot get to the solution. There are three risky assets A, B and C whose betas are respectively 0.6 (A), 1.5 (B) and 1.1 (C). Besides, we know ...
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