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

Filter by
Sorted by
Tagged with
3 votes
2 answers
2k views

Replicate a Portfolio with Given Payoff

Looking for a convincing general strategy [not trial and error] to solve these kind of questions: Any help will be super helpful! Thanks a bunch! Replicate a portfolio on an underlying asset $S$ ...
user avatar
2 votes
2 answers
243 views

How to add the effect of skewness in the portfolio optimisation objective function?

I have the following risk adjusted portfolio which I optimise, where gamma is the risk return trade off, $r$ are the returns and $C$ is the covariance matrix which considers scenarios, so it is not ...
user avatar
  • 314
8 votes
3 answers
5k views

Maximum Sharpe portfolio (no short selling restrictions)

Suppose we have $n$ assets whose expected return vector is $r$ and is positive, and whose covariance matrix is $\Sigma$. Is there a closed form or quasi closed form (like the eigenvector of a matrix ...
user avatar
  • 921
4 votes
1 answer
803 views

Monte Carlo (resampling) in m.v. portfolio optimization

The instability and high sensitivity of optimisation results can be augmented by adding another layer of quantitative methodology in the form of Monte Carlo Simulation. The name Monte Carlo alludes to ...
user avatar
  • 349
4 votes
2 answers
1k views

Closed-form analytical solution for the variance of the minimum-variance portfolio?

The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution, $$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
user avatar
  • 2,835
4 votes
1 answer
2k views

CVAR alternatives for optimization

Are there some alternatives to the CVaR measure for portfolio optimization, which are easier to implement for ex. with a linear program? They can be just approximations of CVaR or measures ...
user avatar
7 votes
2 answers
638 views

Maximum Certainty Equivalent Portfolio with Transaction Costs

Out of curiosity I tried to compute the portfolio weights of a maximum certainty equivalent allocation, however, by incorporating (quadratic) transaction costs. However, my result is not as intuitive ...
user avatar
  • 1,436
3 votes
2 answers
4k views

maximize Sharpe ratio in portfolio optimization

I am trying to understand how to maximize Sharpe ratio in portfolio optimization. $\boxed{\begin{align}\max\>&\frac{r^Tx-r_f}{\sqrt{x^TQx}}\\ & \sum_i x_i = 1\\ & x_i\ge 0\end{align}}$ ...
user avatar
  • 413
10 votes
3 answers
413 views

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 $R_{t+1, i}$ ...
user avatar
6 votes
1 answer
760 views

Rockafellar-Uryasev mean-CVaR optimiztion

In Rockafellar-Uryasev 2001 paper the mean-CVaR optimization can be written as a linear programming optimization problem as: $$P_{\text{CVaR}} = \arg \min_w \text{VaR}_\alpha+\frac{1}{(1-\beta)S}\...
user avatar
5 votes
1 answer
402 views

Quasi Random Monte Carlo in m.v. portfolio optimization

Not specifying a correlation matrix for the Monte Carlo Simulation's random returns is equivalent to assuming no correlation or a correlation coefficient of zero, which will seriously and adversely ...
user avatar
  • 349
2 votes
2 answers
270 views

name of this portfolio optimization strategy

I have come across a portfolio selection strategy that buys in equal amounts the top decile of expected earners, and simultaneously short sells the lowest decile in a similar fashion. What is this ...
user avatar
  • 534
23 votes
5 answers
15k views

Portfolio optimisation with VaR or CVaR constraints using linear programming

I would like to optimize a portfolio allocation (maximizing the exposure or the expected return), but with VaR or CVaR contraints. (some parts of my portfolio cannot exceed a certain VaR) How can I ...
user avatar
  • 1,993
10 votes
1 answer
2k views

Minimum Variance and Minimum Tracking Error portfolio as second order cone program

The quadratic optimization (min variance) $$ w^{T} \Sigma w \rightarrow \text{min}, $$ where $w$ is the vector of portfolio weights and $\Sigma$ is the covariance matrix of asset returns, is a well ...
user avatar
  • 13.3k
20 votes
4 answers
14k views

Why shrink the covariance matrix?

I'm trying to understand why it's useful to shrink the covariance matrix for portfolio construction or in fact general. Think I missing something. I know if you have 5,000 stocks it's a lot of ...
user avatar
  • 347
15 votes
1 answer
2k views

Optimization: Factor model versus asset-by-asset model

In portfolio management one often has to solve problems of the quadratic form $$ w^T \Sigma w + w^T c \rightarrow \min_{\omega} $$ with portfolio weights $w \in \mathbb{R}^N$ a constant $c \in \mathbb{...
user avatar
  • 13.3k
8 votes
2 answers
2k views

Random Portfolios vs Efficient Frontier

I understand the concept of the efficient frontier and am able to calculate it in Python. But even when generating 50'000 random 10 asset portfolios, the single portfolios are not even close to the ...
user avatar
3 votes
1 answer
988 views

Hamilton-Jacobi-Bellman equation in Merton Model

I'm trying to study the Merton Model for portfolio optimization and the document doesn't explain a quite important step : if $$V(t,x)=\sup\{E[U(X_T(\phi))~|~X_t=x]~~ |~~\phi~~\text{an admissible ...
user avatar
  • 287
5 votes
1 answer
303 views

Reference Request: Horse Race for Portfolio Allocation

Probably the most popular horse race study for portfolio strategies is Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?, with DeMiguel, L. Garlappi and R. Uppal. ...
user avatar
  • 1,436
3 votes
2 answers
526 views

Contribution of an asset's variance to portfolio variance

How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$? I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
user avatar
  • 2,835
7 votes
1 answer
2k views

Sharpe Maximization under Quadratic Constraints

When doing Sharpe optimization $$ \max_x \frac{\mu^T x}{\sqrt{x^T Q x}} $$ there is a common trick (section 5.2) used to put the problem in convex form. You add a variable $\kappa$ such that $x = ...
user avatar
  • 1,561
4 votes
3 answers
3k views

Compute tangency portfolio with asset allocation constraints

I am looking to compute the tangency portfolio of the efficient frontier, but taking into account min_allocations and ...
user avatar
  • 241
2 votes
2 answers
1k views

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-...
user avatar
2 votes
3 answers
2k views

Regularizers to compute Minimum Variance Portfolio weights

I need to compute the mimimum variance portfolio using different regularizers, to compare the results and use validation methods to find the optimal parameters. Currently my work has been performed ...
user avatar
  • 103
2 votes
1 answer
2k views

short-sale constraint with nonpositive-definite matrix in portfolio optimization

I need help about portfolio optimization in R. I have inverted matrix and I want to use it as an input in portfolio optimization. It was non-positive definite before I have handled it. In portfolio ...
user avatar
1 vote
2 answers
499 views

Portfolio Optimization - Equal Weighting Algorithm

I am trying to write an algorithm which can output the number of stocks to purchase so that it equal weights positions in a portfolio of stocks. Say we want to invest $1000 in 5 stocks with equal ...
user avatar
  • 21
8 votes
2 answers
4k views

Portfolio Optimization : Shrinkage of Covariance Matrix when data is available

It seems that shrinking the covariance matrix is especially useful if the number of individual stocks is greater than the number of data points. However is there any special gain if you're not ...
user avatar
4 votes
2 answers
3k views

How can I find the portfolio with maximum Sharpe Ratio - Using Lagrange Multipliers

In Markowitz' portfolio theory we can construct portfolios with the minimum variance for a given expected return (or vice versa). Across expected risks, this traces out the well-known efficient ...
user avatar
  • 143
4 votes
1 answer
2k views

Portfolio optimization with Portfolio CVaR Constraint

I wanted to optimize a portfolio based on a portfolio-wide CVaR constraint (i.e. $CVaR_p \leq 0.08$). Unfortunately, I only find solution that minimizes the entire CVaR of the Portfolio. Do you mind ...
user avatar
  • 141
4 votes
3 answers
493 views

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 ...
user avatar
  • 5,325
3 votes
1 answer
209 views

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 ...
user avatar
  • 2,835
2 votes
1 answer
3k views

Turnover as a soft constraint for portfolio optimization

I am using cvxpy to do a simple portfolio optimization. I implemented the following dummy code ...
user avatar
2 votes
1 answer
914 views

Minimize overall portfolio turnover under constraints

Assume I have M portfolios, each of them can be represented as a T by N matrix, where N represents number of stocks traded and T represents number of days. For each portfolio matrix, each row is under ...
user avatar
  • 145
2 votes
2 answers
7k views

Calculating the efficient frontier from expected returns and SD

I'm trying to calculate the efficient frontier (and the optimal portfolio at the Sharpe ratio) given two vectors for a portfolio: (1) expected returns and (2) historical standard deviations. I would ...
user avatar
2 votes
1 answer
200 views

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 ...
user avatar
  • 2,835
2 votes
3 answers
3k views

Portfolio Optimization - Zero beta portfolio

I am trying to solve a optimization portfolio in R in which I do the following constraints: Set weight sum to within a boundary Set return to a certain value Set portfolio beta to 0 The purpose is ...
user avatar
1 vote
1 answer
299 views

Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?

What I want to do is the following: Let's say I have two assets 1 and 2, and have a 2x2 covariance matrix. Then I have two portfolios A and B made of weights from assets 1 and 2. What I would like to ...
user avatar
5 votes
2 answers
934 views

Risk Parity / Equal Risk Contribution with Tail Risk Measures

Risk Parity or (synonymous) Equal Risk Contribution is an approach to portfolio construction which could work in theory with a broad class of risk measures. Yet, all references I have found so far ...
user avatar
  • 1,933
2 votes
3 answers
2k views

What are pros and cons of mean absolute deviation portfolio optimization?

In this question a paper about mean absolute deviation portfolio optimization is mentioned and in the answer a spreadsheet with an implementation is attached. What is the use of this procedure? Does ...
user avatar
  • 13.3k
1 vote
1 answer
1k views

How to implement Konno's Mean-Absolute Deviation Portfolio Optimization Model using LP methods in Excel

Konno proposed a LP method for portfolio optimization using the Mean Absolute Deviation (MAD)
user avatar
  • 77
1 vote
1 answer
164 views

On a source for a mean-variance portfolio optimization result

In the context of a mean_variance framework consider an optimizing investor who chooses at time $T$ portfolio weights $w$ so as to maximize the quadratic objective function: $$U(w) = E[R_p] - \frac{\...
user avatar
  • 357
1 vote
0 answers
119 views

How to transform a cubic optimisation problem into a quadratic for portfolio allocation

I have the following cost function for portfolio allocation: $$ w^T\mu-\frac{1}{2}\gamma w^T\Sigma w+\frac{1}{6}\gamma^2 w^TM_3(w\otimes w), $$ which considers also the co-skewness ($M_3$ tensor), $\...
user avatar
  • 314
3 votes
2 answers
261 views

Find k of n assets that "minimize" the correlation matrix

I'm trying to find an efficient way to select $k$ from $n$ risky assets that are the least correlated with each other. I know that I can perform a brute-force search of all $k$-sized combinations of ...
user avatar
3 votes
4 answers
534 views

Portfolio Optimization using S&P Universes

Assuming a set portfolio optimization problem, if all optimization inputs are kept constant, what would you expect, in terms of results, if you run the same optimization using the S&P500 as ...
user avatar
  • 652
3 votes
2 answers
1k views

How to perform portfolio optimization with user-defined expected return and variances using R?

I found some functions for Markowitz mean variance portfolio optimization in R such as portfolio.optim in tseries package. ...
user avatar
1 vote
1 answer
169 views

How to measure the practicality of a market portfolio for long-term investment?

Do you believe that the composition of the market portfolio that you have found is a desirable or practical one as an investment? Explain why or why not, based on the positions of your stocks. I ...
user avatar
  • 11
1 vote
2 answers
1k views

How to calculate optimal portfolio using sector constraints in python

I'm looking into CVXPY at the moment. Main goal would be to be able to calculate the optimal portfolio, which in my opinion would mean that we need to maximise (expected return - risk free) / ...
user avatar
1 vote
0 answers
444 views

Portfolio Optimization with equal weight for assets selected

I have a data frame of bets, with 1 being a win and 0 being a loss. These bets are correlated so I cannot just pick the highest winning percentage. Goal is to get 2 optimizations, 1 for max sharpe ...
user avatar
  • 41
1 vote
1 answer
641 views

Closed-form analytical solution for Markowitz efficient portfolio without short-selling

In a portfolio without risk-free assets I know that the efficient portfolio si given by: $\omega=\frac{1}{BC-A^2}[\mu(C\Sigma^{-1}R-A\Sigma^{-1}\mathbb{1})+B\Sigma^{-1}\mathbb{1}-A\Sigma^{-1}R]$, ...
user avatar
  • 23
0 votes
1 answer
259 views

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:...
user avatar
  • 123