A Study of Efficiency in CVaR Portfolio Optimization chris bemis Whitebox Advisors
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A Study of Efficiency in CVaR Portfolio Optimization chris bemis Whitebox Advisors
A Study of Efficiency in CVaR Portfolio Optimization chris bemis Whitebox Advisors January 5, 2011 chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization ”The ultimate goal of a positive science is the development of a ‘theory’ or ‘hypothesis’ that yields valid and meaningful (i.e., not truistic) predictions about phenomena not yet observed.” Milton Friedman chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Many are familiar with the following optimization problem, minimize w ′ Σw subject to µ ′w ⩾ α 1 ′w = 1 w ⩾ 0, suggested by Markowitz in 1952. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Many are familiar with the following optimization problem, minimize w ′ Σw subject to µ ′w ⩾ α 1 ′w = 1 w ⩾ 0, suggested by Markowitz in 1952. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Financial data are (most likely) nonstationary, though: chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Financial data are (most likely) nonstationary, though: chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization For a single variable, the variance of the error in sample mean, µ̄ converges at a rate of n1 . And the variance of the error in sample variance, σ̄ converges at a rate of √1n . So that, disregarding correlation, we need very large sample sizes to obtain realistic estimates of first and second moments. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization For a single variable, the variance of the error in sample mean, µ̄ converges at a rate of n1 . And the variance of the error in sample variance, σ̄ converges at a rate of √1n . So that, disregarding correlation, we need very large sample sizes to obtain realistic estimates of first and second moments. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Markowitz’ formulation for optimal portfolios also presupposes I Every investor has the same utility over a fixed horizon I That utility is quadratic in risk; viz., variance I This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Markowitz’ formulation for optimal portfolios also presupposes I Every investor has the same utility over a fixed horizon I That utility is quadratic in risk; viz., variance I This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Markowitz’ formulation for optimal portfolios also presupposes I Every investor has the same utility over a fixed horizon I That utility is quadratic in risk; viz., variance I This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Markowitz’ formulation for optimal portfolios also presupposes I Every investor has the same utility over a fixed horizon I That utility is quadratic in risk; viz., variance I This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Markowitz’ formulation for optimal portfolios also presupposes I Every investor has the same utility over a fixed horizon I That utility is quadratic in risk; viz., variance I This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz’ framework include Goldfarb and Iyengar’s (2003) robust portfolio optimization method. We will pursue another avenue... chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz’ framework include Goldfarb and Iyengar’s (2003) robust portfolio optimization method. We will pursue another avenue... chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz’ framework include Goldfarb and Iyengar’s (2003) robust portfolio optimization method. We will pursue another avenue... chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz’ framework include Goldfarb and Iyengar’s (2003) robust portfolio optimization method. We will pursue another avenue... chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization For a vector of portfolio weights, w, and a ’scenario’, y, define the function f , f (w, y) : Rn × Rm → R to be the loss of the portfolio allocated according to w under scenario y. We will call a positive value from f a loss. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization For a vector of portfolio weights, w, and a ’scenario’, y, define the function f , f (w, y) : Rn × Rm → R to be the loss of the portfolio allocated according to w under scenario y. We will call a positive value from f a loss. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming that the scenarios have probability density function p, the cumulative distribution function of losses, given portfolio weights w, is ∫ p(y)dy Ψ(x, γ) = f (x,y)<γ Notice, our framework is about as general as possible. This is intentional chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming that the scenarios have probability density function p, the cumulative distribution function of losses, given portfolio weights w, is ∫ p(y)dy Ψ(x, γ) = f (x,y)<γ Notice, our framework is about as general as possible. This is intentional chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming that the scenarios have probability density function p, the cumulative distribution function of losses, given portfolio weights w, is ∫ p(y)dy Ψ(x, γ) = f (x,y)<γ Notice, our framework is about as general as possible. This is intentional chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization We next define the value at risk for a given threshold, α: VaRα (w) = min{γ ∈ R | Ψ(w, γ) ⩾ α} We have that VaRα (w) is the smallest amount of loss that we can expect with probability 1 − α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization We next define the value at risk for a given threshold, α: VaRα (w) = min{γ ∈ R | Ψ(w, γ) ⩾ α} We have that VaRα (w) is the smallest amount of loss that we can expect with probability 1 − α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization We next define the value at risk for a given threshold, α: VaRα (w) = min{γ ∈ R | Ψ(w, γ) ⩾ α} We have that VaRα (w) is the smallest amount of loss that we can expect with probability 1 − α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization We next define the value at risk for a given threshold, α: VaRα (w) = min{γ ∈ R | Ψ(w, γ) ⩾ α} We have that VaRα (w) is the smallest amount of loss that we can expect with probability 1 − α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization The VaR construction ignores tail behavior. Conditional value at risk, or CVaR, incorporates the tail past the VaR value; viz., ∫ 1 CVaRα (w) = f (w, y)p(y)dy 1 − α f (w,y)⩾VaRα (w) We can discretize this in a natural way by sampling our scenarios discretely according to p chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization The VaR construction ignores tail behavior. Conditional value at risk, or CVaR, incorporates the tail past the VaR value; viz., ∫ 1 CVaRα (w) = f (w, y)p(y)dy 1 − α f (w,y)⩾VaRα (w) We can discretize this in a natural way by sampling our scenarios discretely according to p chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization The VaR construction ignores tail behavior. Conditional value at risk, or CVaR, incorporates the tail past the VaR value; viz., ∫ 1 CVaRα (w) = f (w, y)p(y)dy 1 − α f (w,y)⩾VaRα (w) We can discretize this in a natural way by sampling our scenarios discretely according to p chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min CVaRα (w), w∈ A linear programming problem. However, a problem that increases linearly with the number of scenarios used. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min CVaRα (w), w∈ A linear programming problem. However, a problem that increases linearly with the number of scenarios used. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min CVaRα (w), w∈ A linear programming problem. However, a problem that increases linearly with the number of scenarios used. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min CVaRα (w), w∈ A linear programming problem. However, a problem that increases linearly with the number of scenarios used. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Based on what we observed in convergence of mean and variance, we will need many, many scenarios to reflect even the first two moments. The LP problem may not be stable for large numbers of scenarios, however We therefore consider other formulations of the CVaR objective problem. In particular I A smoothed approximation as in Alexander, Coleman, and Li (2004) I A fast gradient descent method proposed by Iyengar and Ma (2009) I A convolution smoothing model constructed in my IMA workshop (2010) chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Based on what we observed in convergence of mean and variance, we will need many, many scenarios to reflect even the first two moments. The LP problem may not be stable for large numbers of scenarios, however We therefore consider other formulations of the CVaR objective problem. In particular I A smoothed approximation as in Alexander, Coleman, and Li (2004) I A fast gradient descent method proposed by Iyengar and Ma (2009) I A convolution smoothing model constructed in my IMA workshop (2010) chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Based on what we observed in convergence of mean and variance, we will need many, many scenarios to reflect even the first two moments. The LP problem may not be stable for large numbers of scenarios, however We therefore consider other formulations of the CVaR objective problem. In particular I A smoothed approximation as in Alexander, Coleman, and Li (2004) I A fast gradient descent method proposed by Iyengar and Ma (2009) I A convolution smoothing model constructed in my IMA workshop (2010) chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Based on what we observed in convergence of mean and variance, we will need many, many scenarios to reflect even the first two moments. The LP problem may not be stable for large numbers of scenarios, however We therefore consider other formulations of the CVaR objective problem. In particular I A smoothed approximation as in Alexander, Coleman, and Li (2004) I A fast gradient descent method proposed by Iyengar and Ma (2009) I A convolution smoothing model constructed in my IMA workshop (2010) chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization Based on what we observed in convergence of mean and variance, we will need many, many scenarios to reflect even the first two moments. The LP problem may not be stable for large numbers of scenarios, however We therefore consider other formulations of the CVaR objective problem. In particular I A smoothed approximation as in Alexander, Coleman, and Li (2004) I A fast gradient descent method proposed by Iyengar and Ma (2009) I A convolution smoothing model constructed in my IMA workshop (2010) chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization We will be mainly interested in I Run time of the various methods as a function of assets and as a function of scenarios I Accuracy I Out of sample performance chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization fin. chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization