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

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31
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12answers
14k 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 ...
18
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 ...
15
votes
5answers
3k 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 ...
13
votes
4answers
4k 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 ...
12
votes
3answers
2k views

Role of skewness in portfolio optimization?

What is the role of skewness in portfolio optimization?
11
votes
5answers
1k 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
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 ...
10
votes
2answers
1k 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 ...
10
votes
1answer
357 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 ...
9
votes
1answer
367 views

Models crumbling down due to negative (nominal) interest rates

Given that the negative interest rates on a lot of sovereign bonds with maturity under 10 years are trading in the negative (nominal) interest rate territory (recently also the short term EURIBOR has ...
8
votes
1answer
269 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
735 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
4answers
670 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
1answer
179 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 ...
7
votes
2answers
199 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 ...
6
votes
4answers
752 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 ...
6
votes
2answers
1k 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 ...
6
votes
2answers
356 views

Which algorithms do robo-advisors use?

Some pundits claim that there is a revolution in portfolio management under way: The rise of the robots, a.k.a. robo-advisors. The most well known are Betterment.com, FutureAdvisor, Schwab Intelligent ...
6
votes
0answers
68 views

portfolio optimization averaging weights, what are benefits?

I'm playing around with different portfolio optimization techniques. Amongst others I was also looking at the resampling method, especially the one described in Meucci. I have two general questions ...
6
votes
0answers
670 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 ...
5
votes
2answers
127 views

Why is the variance of a portfolio a quadratic form?

I was reading about MPT http://en.wikipedia.org/wiki/Modern_portfolio_theory and notices that the total variance of a portfolio is $x' \Sigma x$, where x is the weighting of the assets and $\Sigma$ ...
5
votes
2answers
305 views

R package for portfolio

In the context of modern portfolio theory, one often wishes to minimise $\mathbf{w}^{\mathrm{{\scriptstyle T}}}\boldsymbol{\Sigma}\mathbf{w}$ subject to $\mathbf{w}^{T}\boldsymbol{\mu}=c_{1}$, ...
5
votes
2answers
168 views

Beta Constrained Markowitz Minimum Variance Portfolio - Closed Form Solution

This question is related to recent rule changes in the Quantopian Open. I am trying to figure out a closed form solution to a beta constrained minimum variance portfolio problem but it doesn't seem ...
5
votes
2answers
179 views

Intuitive explanation of stochastic portfolio theory

Fernholz and Karatzas have published various papers about so called stochastic portfolio theory. Basically they say that the return to be expected from a portfolio on the long run is rather the ...
4
votes
4answers
506 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
2answers
187 views

Optimal Portfolios

In modern portfolio theory, one famous problem is the Markowitz mean variance optimal portfolio, defined by solving ...
4
votes
2answers
4k 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
173 views

Difference between Sharpe Ratio and Information Ratio

I am finding it difficult to understand the difference between the sharpe ratio and the information ratio and the relationship between the two, and cannot find a decent reference that breaks it down ...
4
votes
3answers
237 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 ...
4
votes
1answer
435 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
1answer
80 views

Calculate mean variance portfolio

I am trying to calculate the mean variance portfolio using the plug-in approach. First I generate some artificial data: x <- replicate(10,rnorm(1000)) Then I ...
4
votes
1answer
221 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 ...
4
votes
2answers
9k 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?
4
votes
1answer
359 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 : ...
4
votes
1answer
1k 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
357 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 ...
3
votes
3answers
8k 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 $N$ ...
3
votes
4answers
225 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 ...
3
votes
1answer
169 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
1answer
140 views

Portfolio with lots of subportfolios

An account manager has $N$ distinct, equally-sized pots of money, which will be used to make $N$ distinct subportfolios, each of which is drawn from a slightly different (but potentially overlapping) ...
3
votes
2answers
254 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 ...
3
votes
3answers
241 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
2answers
205 views

Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions form 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} ...
3
votes
1answer
199 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 ...
3
votes
1answer
64 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
149 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
3answers
190 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
115 views

Combining modern portfolio theory and Kelly betting?

I'm using modern portfolio theory to compute the frontier of efficient portfolios. I'd like to pick the best one in the spirit of Kelly betting, ie. maximising expected growth. I'm looking for a ...
3
votes
3answers
385 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 ...
3
votes
0answers
44 views

Finding mean vector and covariance matrix for annual returns given quarterly returns

I am currently trying to calculate a vector for the mean annual returns of 4 different asset classes along with their 4x4 covariance matrix in excel. However, I am having problems since the data I ...