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

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26
votes
12answers
11k views

Why does the minimum variance portfolio provide good returns?

I've been a researching minimum variance portfolios (from this link) and find that by building MVPs adding constraints on portfolio weights and a few other tweaks to the methods outlined I get ...
15
votes
5answers
2k views

What methods do you use to improve expected return estimates when constructing a portfolio in a mean-variance framework?

One of the main problems when trying to apply mean-variance portfolio optimization in practice is its high input sensitivity. As can be seen in (Chopra, 1993) using historical values to estimate ...
15
votes
1answer
1k views

Portfolio optimization with monte carlo sampling from predictive distribution

Let's say we have a predictive distribution of expected returns for N assets. The distribution is not normal. We can interpret the dispersion in the distribution as reflection of our uncertainty (or ...
11
votes
3answers
1k views

Role of skewness in portfolio optimization?

What is the role of skewness in portfolio optimization?
10
votes
5answers
857 views

How can higher co-moments be applied to portfolio optimization in an asset allocation context?

Traditional portfolio optimization involves mean variance optimization, where only the mean and covariance matrix of returns are estimated. What asset allocation and portfolio optimization techniques ...
10
votes
4answers
2k views

What is a “coherent” risk measure?

What is a coherent risk measure, and why do we care? Can you give a simple example of a coherent risk measure as opposed to a non-coherent one, and the problems that a coherent measure addresses in ...
9
votes
2answers
816 views

What is the relationship between risk aversion and preference for skewness and kurtosis in portfolio optimization?

Is there any relationship between the risk aversion coefficient in an individual's utility function (commonly used in portfolio optimization) and the preference for higher moments such as skewness and ...
8
votes
1answer
247 views

Do weights from portfolio theory contain bias?

I want to experiment with some portfolio modelling and I was wondering if you guys could help me with something. If I try to estimate and implement the traditional two-fund portfolio consisting of one ...
8
votes
2answers
654 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 ...
7
votes
1answer
2k views

Why is the first principal component a proxy for the market portfolio, and what other proxies exist?

Let's say that I have a universe of stocks from a certain sector. I want to compute the market portfolio of this sector. Beta is the covariance between each stock and the market. But how do you ...
7
votes
4answers
280 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 ...
7
votes
2answers
185 views

Is there an optimal covariance one would want forecasts to have?

Often in a quant process, one will generate a time series of return forecasts and use them in some sort of optimization to generate a portfolio. Generally, there will be a covariance matrix of market ...
7
votes
0answers
201 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 ...
6
votes
2answers
920 views

Comparing MVO with Resampled Efficient Frontier

My question: How can I compare the Resampled Frontier (REF) to the standard MVO frontier when I have been provided with $\mu$, $\Omega$, and don't have access to true future data to test real out of ...
5
votes
4answers
368 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 ...
5
votes
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
votes
4answers
486 views

Given two portfolios with identical correlation matrices, which one will have a better risk/reward ratio?

I have one portfolio with high beta stocks, and one with low beta stocks. Is it better to have higher expected return with high volatility, or medium expected return with medium volatility? (All from ...
4
votes
1answer
245 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 ...
4
votes
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 - ...
4
votes
1answer
940 views

Michaud's Resampled Efficient Frontier - Out of Sample Simulation Testing

I will be putting ALL my account points on bounty to whoever answers this question [if your answer is crap but it's the only answer, you're getting the 165 points]. You will have to wait 2 days or so ...
4
votes
1answer
228 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 ...
4
votes
0answers
391 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 ...
3
votes
2answers
146 views

Optimal Portfolios

In modern portfolio theory, one famous problem is the Markowitz mean variance optimal portfolio, defined by solving ...
3
votes
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 ...
3
votes
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 ...
3
votes
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 ...
3
votes
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
votes
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 ...
3
votes
1answer
56 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 ...
3
votes
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 : ...
2
votes
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 ...
2
votes
1answer
380 views

What are some “Must Know” investment/portfolio management theories out there?

What are the most important portfolio management theories you must know in order to competently manage an investment portfolio? In order to keep the topic focused, I would like to narrow down the set ...
2
votes
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 ...
2
votes
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 $N$ ...
2
votes
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 ...
2
votes
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?
2
votes
2answers
491 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 ...
2
votes
1answer
573 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 ...
2
votes
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 ...
2
votes
2answers
214 views

Proxy for risk in portfolio theory when return can take only two values

I'm trying to adapt tools from portfolio theory for another use, and I have a question about how I might do so. Suppose that instead of having normally distributed returns, the return $R_i$ is ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
2answers
47 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 ...
2
votes
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, ...
1
vote
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 ...
1
vote
1answer
340 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 - ...
1
vote
1answer
92 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 ...
1
vote
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 ...
1
vote
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 ...