Posts Tagged ‘ Asset Allocation ’

Black-Litterman Model

November 15, 2011
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Black-Litterman Model

The Black-Litterman Model was created by Fisher Black and Robert Litterman in 1992 to resolve shortcomings of traditional Markovitz mean-variance asset allocation model. It addresses following two items: Lack of diversification of portfolios on the mean-variance efficient frontier. Instability of portfolios on the mean-variance efficient frontier: small changes in the input assumptions often lead to

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Resampling and Shrinkage : Solutions to Instability of mean-variance efficient portfolios

November 11, 2011
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Resampling and Shrinkage : Solutions to Instability of mean-variance efficient portfolios

Small changes in the input assumptions often lead to very different efficient portfolios constructed with mean-variance optimization. I will discuss Resampling and Covariance Shrinkage Estimator – two common techniques to make portfolios in the mean-variance efficient frontier more diversified and immune to small changes in the input assumptions. Resampling was introduced by Michaud in Efficient

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Geometric Efficient Frontier

November 9, 2011
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Geometric Efficient Frontier

What is important for an investor? The rate of return is at the top of the list. Does the expected rate of return shown on the mean-variance efficient frontier paints the full picture? If investor’s investment horizon is longer than one period, for example 5 years, than the true measure of portfolio performance is Geometric

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Maximizing Omega Ratio

November 3, 2011
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Maximizing Omega Ratio

The Omega Ratio was introduced by Keating and Shadwick in 2002. It measures the ratio of average portfolio wins over average portfolio losses for a given target return L. Let x.i, i= 1,…,n be weights of instruments in the portfolio. We suppose that j= 1,…,T scenarios of returns with equal probabilities are available. I will

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Minimizing Downside Risk

November 1, 2011
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Minimizing Downside Risk

In the Maximum Loss and Mean-Absolute Deviation risk measures, and Expected shortfall (CVaR) and Conditional Drawdown at Risk (CDaR) posts I started the discussion about alternative risk measures we can use to construct efficient frontier. Another alternative risk measure I want to discuss is Downside Risk. In the traditional mean-variance optimization both returns above and

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The Most Diversified or The Least Correlated Efficient Frontier

October 27, 2011
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The Most Diversified or The Least Correlated Efficient Frontier

The “Minimum Correlation Algorithm” is a term I stumbled at the CSS Analytics blog. This is an Interesting Risk Measure that in my interpretation means: minimizing Average Portfolio Correlation with each Asset Class for a given level of return. One might try to use Correlation instead of Covariance matrix in mean-variance optimization, but this approach,

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Controlling multiple risk measures during construction of efficient frontier

October 26, 2011
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Controlling multiple risk measures during construction of efficient frontier

In the last few posts I introduced Maximum Loss, Mean-Absolute Deviation, and Expected shortfall (CVaR) and Conditional Drawdown at Risk (CDaR) risk measures. These risk measures can be formulated as linear constraints and thus can be combined with each other to control multiple risk measures during construction of efficient frontier. Let’s examine efficient frontiers computed

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Expected shortfall (CVaR) and Conditional Drawdown at Risk (CDaR) risk measures

October 25, 2011
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Expected shortfall (CVaR) and Conditional Drawdown at Risk (CDaR) risk measures

In the Maximum Loss and Mean-Absolute Deviation risk measures post I started the discussion about alternative risk measures we can use to construct efficient frontier. Another alternative risk measures I want to discuss are Expected shortfall (CVaR) and Conditional Drawdown at Risk (CDaR). I will use methods presented in Comparative Analysis of Linear Portfolio Rebalancing

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Minimum Investment and Number of Assets Portfolio Cardinality Constraints

October 19, 2011
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Minimum Investment and Number of Assets Portfolio Cardinality Constraints

The Minimum Investment and Number of Assets Portfolio Cardinality Constraints are practical constraints that are not easily incorporated in the standard mean-variance optimization framework. To help us impose these real life constraints, I will introduce extra binary variables and will use mixed binary linear and quadratic programming solvers. Let’s continue with our discussion from Introduction

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130/30 Porfolio Construction

October 18, 2011
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130/30 Porfolio Construction

The 130/30 funds were getting lots of attention a few years ago. The 130/30 fund is a long/short portfolio that for each $100 dollars invested allocates $130 dollars to longs and $30 dollars to shorts. From portfolio construction perspective this simple idea is no so simple to implement. Let’s continue with our discussion from Introduction

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