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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
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