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

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135 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 ...
5
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2answers
265 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 ...
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1answer
53 views

Understanding portfolio weights and purchasing stock in modern portfolio theory

Recently I've been learning about the markowitz algorithm. It's pretty interesting, but I'm curious how we apply this in practice. Lets say I have some optimal portfolio: $R_p = x_aR_a + x_bR_b$ ...
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1answer
67 views

calculating portfolio volatility [closed]

Given: vector of portfolio weights $W = [w_1 w_1 ]$ correlation matrix $C = \left( \begin{array}{ccc} a & b \\ d & e \end{array} \right) $ standard deviation of the asset returns $S = [s_1 ...
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1answer
106 views

What Exactly is Expected Return

Consider the following plot, courtesy of this page: Regarding the $y$-axis, how does this "expected return" relate to the "instantaneous expected return" in a geometric Brownian motion (GBM)? E.g., ...
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1answer
81 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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3answers
351 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 ...
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2answers
121 views

When to adjust portfolio weights?

In portfolio allocation literature there is lot of effort made in obtaining 'better' portfolio weights, for example via improving parameter estimates, introducing Bayesian approaches, incorporating ...
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0answers
62 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....
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0answers
70 views

Residual Covariance Matrix, and MVO for Residual Variance and Alpha

My overall goal is to find an efficient frontier using QP in terms of $\alpha$ and residual variance ($\omega^2$) for a portfolio $P$ given a benchmark $B$. We know the equation for residual variance ...
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1answer
60 views

Calculating Fees (Kane, Marcus, and Trippi)

Having read a chapter in Bodie, Kane and Marcus' Investment, I came across a formula I do not quite understand. It states that the percentage fee in excess of what an index fund would charge on active ...
10
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1answer
412 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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3answers
356 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 ...
5
votes
2answers
150 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 $\Sigma$ ...
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0answers
37 views

Hypothesis Testing for Portfolio Weights

Investigating international diversification is an ongoing topic in portfolio allocation literature. Britten-Jones and Kempf-Memmel , for example, use derived properties of the distribution of ...
5
votes
2answers
375 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 $\mathbf{w}^{T}\boldsymbol{\mu}=c_{1}$, $\left\...
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1answer
72 views

Expected Utility and $\log$

I've just started reading about expected utility and utility functions and have the following question. $\textbf{Question:}$ An investor has an initial wealth of 100 and a utility function of the ...
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1answer
171 views

Markowitz portfolio optimization question

I am studying the Markowitz portfolio optimization theory, and I just wanted to ask if I understood this correctly. For a stock portfolio we distinguish two kinds of risks: an unsystematic risk, which ...
2
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2answers
158 views

Calculate efficient frontier using fPortfolio with incomplete set of returns

I want to calculate the efficient frontier for a set of 140 assets using returns from the past 10 years. However, some of these assets came into existence only more recently, so for some assets I have ...
16
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4answers
5k 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 ...
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0answers
50 views

Investing in all assets with positive expected return and allowing for positive correlation

How does the answer to this question Risk minimization by investing in all assets with positive expected return change if assets can be positively correlated (but not perfectly) and short sales are ...
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0answers
183 views

Tangent portfolio weights without short sales?

Consider a mean-variance investor in a world with a risk-free asset. Let $R_f>0$ be the return of the risk-free asset, $\mathbb{E}(R_i)>R_f$ the expected return of the risky asset $i$ and $SD(...
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1answer
101 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 $\...
4
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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
272 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 ...
3
votes
2answers
309 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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1answer
213 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 ...
3
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2answers
274 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} ...
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3answers
493 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 ...
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0answers
212 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 ...
3
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0answers
571 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 ...
3
votes
1answer
245 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 ...
2
votes
1answer
447 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
66 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 ...
0
votes
1answer
467 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 ...
4
votes
2answers
202 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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237 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
84 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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0answers
1k 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
73 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 ...
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4answers
830 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
votes
1answer
314 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*σ(M)^2=...
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1answer
202 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. ...
4
votes
3answers
306 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 ...
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1answer
50 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,\...
3
votes
1answer
187 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
274 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
829 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
votes
2answers
185 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 ...
4
votes
1answer
420 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 : -...