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

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+50

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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1answer
33 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 ...
3
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2answers
145 views

Optimal Portfolios

In modern portfolio theory, one famous problem is the Markowitz mean variance optimal portfolio, defined by solving ...
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0answers
36 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 ...
2
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4answers
188 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 ...
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0answers
100 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
47 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 ...
5
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4answers
358 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 ...
0
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1answer
75 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. ...
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1answer
74 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
35 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 ...
3
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1answer
63 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 ...
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1answer
90 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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4answers
279 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 ...
2
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2answers
160 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 ...
3
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1answer
123 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 : ...
0
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1answer
303 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
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1answer
105 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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3answers
364 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
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1answer
92 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?
3
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3answers
123 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 ...
5
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1answer
120 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
242 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
112 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
37 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 ...
2
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2answers
485 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
40 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 ...
2
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2answers
365 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 ...
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43 views

Compute the average efficient frontiers with estimated parameters from generated time series

My overall objective is to analyse the impact of error in mean-variance analysis from historical data. I am given the returns and standard deviation for the five assets under consideration, as well as ...
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1answer
120 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
votes
1answer
227 views

How are the Hamilton–Jacobi–Bellman 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
188 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 ...
7
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0answers
198 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 ...
4
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390 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 ...
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1answer
114 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 ...
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203 views

Easier references to understand “The Asset Pricing and Portfolio Choice Theory” of Back Kerry

I'm trying to read the excellent book "The Asset Pricing and Portfolio Choice Theory" of Back Kerry, but find it too much difficult. I really need to read it but before I assume that I may need to ...
2
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1answer
311 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 ...
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0answers
133 views

Portfolio optimization with absolute position constraints

I'm looking to optimize a portfolio maximizing expected return for a particular risk budget, but with absolute constraints on the individual instrument positions. I've been experimenting with QP, ...
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1answer
339 views

VaR Calculation - Covariance matrix is not positive semidefinite

This is a basic question. I have three assets, equally weighted, and all the mutual covariances are -1. Then, the covariance matrix looks like - ...
2
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3answers
3k views

Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz Portfolio Theory?

I have seen the following formula for the tangency portfolio in Markowitz portfolio theory but couldn't find a reference for derivation, and failed to derive myself. If expected excess returns of $N$ ...
4
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2answers
2k views

Typical risk aversion parameter value for mean-variance optimization?

What are typical values for risk aversion parameters $\lambda$ used in mean-variance optimization? Please provide references. Just to be clear, I'm talking about the $\lambda$ in $U(w) = w'\mu - ...
2
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2answers
196 views

Portfolio risk-return when assets have limited and inconsistent historical data / time series?

Lets say we have "today's" snapshot of asset allocation and need to determine the 6mo, 1 yr and 5 yr risk and returns of this portfolio. If the time series for every asset is very long, longer than ...
1
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1answer
123 views

Find a paper about portfolio management

Where to find the following paper of the noble prize Paul Samuelson (2003) “When and Why Mean-Variance Analysis Generically Fails,”. I was looking for it desperately on Google and Google Scholar but ...
2
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2answers
192 views

Comparing Cash Equivalent of risky portfolios

To compare two risky portfolios, Mean-Variance (M-V) portfolios for example, many compare their Cash Equivalent ($CE$). $CE$ is defined as the amount of cash that provides the same utility as the ...
3
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2answers
4k views

What do the terms in-sample and out-of-sample estimates mean in MVO?

How do the in-sample estimates and out-of-sample estimates I so often hear authors refer to in emperical analysis of MVO differ?
3
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3answers
176 views

How to see if a set of asset returns corresponds to a known correlation matrix?

Let's say I have an arbitrary set of $n$ period returns for $k$ assets, and a given $k \times k$ correlation matrix (of asset returns), which is known a priori. Does it makes sense, or is it even ...
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3answers
301 views

Why is the CAPM securities market line straight?

Let $\gamma$ be the expected return, in terms of its exponential growth rate, of the market asset. If we set $\gamma=\mu-\sigma^2/2$ as explained by the Doléans-Dade exponential, then the expected ...
2
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1answer
572 views

Is it possible to derive the “risk tolerance” from the portfolio efficient frontier?

I am trying to solve the Portfolio Optimization Problem using a "Multi-objective Evolutionary Algorithm". After obtaining the efficient frontier, I would like to know if we can infer for each point of ...
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2answers
653 views

Why do low standard deviation stocks tend to have superior future returns?

I've recently stumbled on something that really surprised me. These papers (1, 2) find that past standard deviation of returns is inversely related to future returns. That is, portfolio of low ...
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1answer
153 views

Resampled efficient frontier length of simulation

I was provided with a VBA program from my lecturer that applies the resampled efficient frontier. We have an investment horizon $T$ (6 years) and he uses a multivariate normal distribution with ...