Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
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100 views
Does anyone know where I can find a free efficient frontier tool, or an informative and legitamate/academic graph of the efficient frontier?
I'd like to build a portfolio based upon modern portfolio theory and I'd like to find a tool I can use to calculate the proper mix of asset classes. Can anyone help with this? I think a good chart/...
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1answer
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)
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1answer
135 views
Risk minimization by investing in all assets with positive expected return
Suppose I have an amount $T$ to invest and $N$ available assets.
The stochastic return per invested unit of asset $i$ is $R_i$.
The variance and the expectation of $R_i$ are $\sigma^2_i$ and $\...
5
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3answers
1k views
How to apply the “Knapsack Problem” to minimise a portfolio's volatility?
Suppose I have a stock selection universe of 100 stocks.
I have estimated the covariance matrix of this 100 stocks.
I would like to create an equaly-weighted basket of 5 stocks which has the lowest ...
1
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1answer
153 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{\...
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1answer
700 views
Why model the variance-covariance matrix as an inverse-Wishart distribution in bayesian portfolio analysis?
I am following Risk and asset allocation (Attilio Meucci,2007). I must say I am enjoying this reading quite a lot so I hope nobody takes my question as a critique on the text.
When we are introduced ...
5
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2answers
2k views
On learning the bayesian approach to portfolio optimization
I am required by my course to write a small paper on the Bayesian approach to portfolio optimization, I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted ...
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2answers
788 views
Black-Litterman: Why should the views be independent of each other?
This question relates to this question.
In the Black-Litterman framework views of inverstors on the market are modelled.
These views have a covariance-matrix $\Omega$.
I always found it quite ...
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2answers
2k views
Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions from prior to posterior
In Black-Litterman we get a new vector of expected returns of the form:
\begin{align}
\Pi_{BL} = \Pi + \underbrace{\tau \Sigma P^T[P\tau\Sigma P^T+\Omega]^{-1}}_{\text{correction}}[Q-P\Pi]
\end{align}
...
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3answers
2k 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 ...
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0answers
798 views
Black-Litterman with simple portfolio
In an attempt to learn Black-Litterman I have come across this "simple" example. Suppose that you analyze market data using CAPM
$$r_i-r_f=\beta_i(r_m-r_f)+\epsilon_i$$
Suppose there are 2 assets in ...
4
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1answer
2k views
Portfolio Optimization with Monte Carlo Simulation - How to do it with Excel?
If I have three asset classes and their historical weekly returns for five years, how can I construct a minimum variance portfolio and an efficient frontier plot with Excel? To do that do I have to ...
4
votes
1answer
954 views
Black-Litterman example
I'm trying to learn Black-Litterman. I feel like I get the overall idea from books like Risk and Asset Allocation by Meucci as well as some of the early papers which develop the model. What I would ...
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1answer
188 views
Asset Liability Management Test Topic Interpretation
I will write a test based on Excel and one of the topics is "The Asset Liability related analysis: including the input assumptions generation, constraints, portfolio optimization analysis and results ...
2
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1answer
1k 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 ...
3
votes
1answer
117 views
How consequential are violations of the efficient diversification assumption of asset pricing models?
When using asset pricing models such as the CAPM or the Fama-French four factor model to determine the risk-adjusted return of a portfolio, does this strictly require efficient diversification of the ...
2
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3answers
1k views
Finding Expression for Optimal Markowitz Weights
So there are two assets with return rates $r_1$ and $r_2$ which have identical variances and a correlation coefficient $p$. The risk free rate is $r_f$.
I need to find an expression for the optimal ...
4
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2answers
301 views
Optimal Portfolios
In modern portfolio theory, one famous problem is the Markowitz mean variance optimal portfolio, defined by solving
$$\underset{\mathbf{w}}{\mbox{min}\,\,}\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}$$...
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0answers
137 views
Model-independent dynamic portfolio optimization techniques
For a problem where we need to optimize the portfolio based on the data, going for Markowitz MPT has the following advantage: we only have to estimate mean and covariance to find optimal weights. I'd ...
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4answers
391 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 ...
2
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2answers
5k 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 ...
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0answers
117 views
Transform MPT optimization problem
I am trying to teach myself about MPT and optimization.
I understand that MPT solutions can be found using three equivalent optimization problems:
Minimizing variance for given return limit
...
8
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4answers
2k views
Is there anyone still using Markowitz modern portfolio theory?
I was reading about the MPT (Use standard deviation as risk measure) on "Mathematics for Finance by Marek Capinski". I was just wondering is there anyone actually applying this theory to their ...
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1answer
751 views
CAPM (SML) Problem
I got 1.3636 for beta for the problem below(165/121).
But I became so unsure about the answer when I solved (c) because then the market risk becomes larger than the variance of Stock A.
Beta^2*Ļ(M)^2=...
2
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1answer
548 views
Efficient Frontier Derivation: why minimize half the portfolio variance instead of just the variance?
In Robert Merton's derivation of the efficient frontier of a portfolio, he minimizes
$\frac{1}{2}\sigma^2 $ over the investment weights in each asset, where $\sigma^2$ represents portfolio variance. ...
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1answer
61 views
MPT and the connection to asset prices / initial capital
I have some question about MPT. Suppose we want to build a portfolio given $N$ assets: $A_1,\dots,A_N$. At time $t$ we build the portfolio using MPT, which yields some weight vector $w_t=(\lambda_1,\...
4
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1answer
845 views
Recommended Literature for creating Factor Mimicking Portfolios
Is there a textbook that contains the basics for creating Factor Mimicking Portfolios? Although there is a lot of peer-reviewed literature on this, I cannot find textbooks on Asset Pricing that ...
2
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1answer
656 views
reference question about portfolio optimization
I know the "classical" modern portfolio theory. However I have quite a lot of different sources. It seems that there is not a book which cover this topic in a rigorous way:
theory
application
...
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votes
4answers
3k views
Does Modern Portfolio Theory align with EMH?
I came to the conclusion that in literature Markowitz' Portfolio Theory is believed to be compliant with the Efficient Market Hypothesis. The weakest form states that the current price fully ...
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2answers
249 views
2 stocks, no shorting vs shorting. (concrete questions, mean-variance)
I'd appreciate help with the following questions.
Suppose there are two stocks $A$ and $B$ with expected returns $E_A, E_B >0$ and volatilities $v_A, v_B >0$, respectively . Also, suppose ...
5
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1answer
776 views
Electricity market : how to design an optimal hedging strategy using spot and futures markets for an industrial consumer?
Here is the problem : we should adopt the point of view of an industrial company which purchases electricity as an input in its production line and which wants to achieve the following two goals :
-...
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1answer
3k views
How do I do a mean variance optimization with constraints?
I am using python and the cvxopt library to calculate an efficient frontier, per the docs:
http://cvxopt.org/examples/book/portfolio.html
However, I cannot figure out how to add a constraint so that ...
3
votes
1answer
411 views
Create optimal portfolio by Treynor and Jensens Alpha
I would like to know which formula to use in order to optimize a portfolio based on highest Treynor and Jensens Alpha. I am aware that usually one optimize a portfolio by highest Sharpe ratio (the ...
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4answers
8k views
Calculate correlation between two sub portfolios and the combined portfolio
I have two sub portfolios (lets call them portfolio a & portfolio b - a portfolio is just a vector of weights that sum to 1) that combine to create a total portfolio. I also have a 2 x 2 ...
2
votes
1answer
709 views
What are the assumptions of portfolio optimisation with higher moments?
I was wondering whether there are a set of assumptions for portfolio optimisation with higher moments (including kurtosis and skewness) as there are for regular mean-variance optimisation?
4
votes
3answers
936 views
Capital Allocation for Portfolio of Multi-Strategy and Multi-Instrument
I would like to know if there is a way (or theory) to manage a multi-strategy, multi-instruments portfolio that would calculate the optimal weight to allocate capital for each combination of strategy ...
8
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1answer
423 views
Overview of robust/regularized portfolio selection
I am looking for either a review paper or individual papers on portfolio selection using robust statistics or regularization (e.g. LASSO, Ridge, etc.)
I.e. a review on methods along the lines of:
M ...
4
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1answer
2k views
Mean-variance portfolio & quadratic programming
I am somewhat confused when it comes to modern portfolio theory, mean-variance portfolio optimization and its quadratic programming formulation.
Issue 1: Formulation of mean-variance portfolio ...
2
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1answer
132 views
Is the volatility of a trader's wealth equal to the volatility of the underlying assets traded?
Assume that a trader trades in several stocks with different volatilities. The return of the trader's portfolio would be the weighted average of returns and the risk would be a function of the the ...
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0answers
51 views
evaluating portfolio performance without knowing the amount held on cash accounts
I would like to evaluate the performance of a portfolio mananger.
I know his trades, and the initial portfolio holdings. I do not know, however the amount held on his cash account.
That is, I ...
9
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2answers
8k views
Definition of Return of A Long/short Portfolio
This can either be a silly question or a question with no sure rigorous answer but defined with some convention. Any way, here it is.
What is the (industrial recognized) definition of the return of a ...
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0answers
56 views
How do I determine what is a separate objective in a multi-objective portfolio optimization?
Is there a general rule to determining when to separate objectives when developing a multi-objective portfolio optimization? For example, one might start with a standard portfolio optimization of ...
3
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2answers
500 views
Why does it “say” portfolio diversification not suitable during market turmoil?
Currently I am trying to get a hold of MPT, asset allocation and related applications.
While reading a particular resource, it says diversification works best for "normal" financial markets and ...
2
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1answer
400 views
How to score a portfolio's diversity based on security returns?
What is the best way to score a portfolio's diversity based on it's returns covariance matrix?
I know that if my portfolio has two securities and their returns' correlation coefficient is -1 that is ...
4
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1answer
668 views
How are the HamiltonāJacobiāBellman equations used to solve optimal control problems?
I would like to learn more on how optimal control problems are solved for financial applications.
The approach seems to have a lot of interesting applications such as
optimal consumption
choosing ...
0
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1answer
1k views
Step-by-Step PCA algorithm (checking correctness without math packages)
I would appreciate if someone could correct me if i am wrong in my suggestion.
I am using PCA to :
find measure of cointegration between selected assets
find the eigenvector and its portfolio with ...
13
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1answer
860 views
Portfolios from Sorts
Some time ago Almgren and Chriss proposed a method for portfolio optimization based on sorting criteria such as $r_1 > r_2 >... > r_N$ instead of explicit expected returns: see
portfolios ...
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0answers
1k views
Formula for the efficient portfolios (mean-variance optimisation)?
Consider the setting of mean-variance portfolio optimisation: $n$ assets with expected returns $\overline{r}_1,...,\overline{r}_n$ and standard deviations $\sigma_1,...\sigma_n$.
For a certain fixed $...
2
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1answer
191 views
What's wrong with this asset growth simulation?
Sorry if this is too basic, but I have this spreadsheet that simulates asset growth of a portfolio under a given return and risk using MPT.
Here is a plot of probability distribution of asset ...
4
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1answer
550 views
In Mean-Variance Analysis, why not the efficient frontier being pushed to the left near the axis?
I took some classes in portfolio theory, and learnt the Markowitz Mean-Variance Analysis.
If only two risky assets, the efficient frontier would be a hyperbola passing through the two points; now if ...
