A quadratic programming model has the important concavity convexity property that avoids the optimiz

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A quadratic programming model has the important
concavity (convexity) property that avoids the
optimization difficulties inherent with more
generalized NLPs.
Whereas Linear programming strives to maximize or
minimize the value of a linear objective function
subject to a set of linear constraints, the
Quadratic programming model strives to maximize
or minimize the value of a quadratic objective
function subject to a set of linear constraints.
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Quadratic Functions Here are some examples of
quadratic functions
9x12 4x1 7
3x12 4x1x2 15x22 20x1 13x2 14
In general, a quadratic function in N variables
is
Note that when Ai and Bij are 0, then the
function is linear.
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Geometric Representation Consider the following
symbolic QP model
Min (x1- 6)2 (x2- 8)2
s.t. x1 lt 7
x2 lt 5
x1 2x2 lt 12
x1 x2 lt 9
x1, x2 gt 0
The objective function can be rewritten as
x12 - 12x1 36 x22 - 16x2 64
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The contours of the objective function are
concentric circles around the point (6,8).
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Comparison with LP Like NLP models in general,
there need not be an optimal corner solution.
As a direct result, there may be more positive
variables in the optimal solution than there are
binding constraints.
Solver Solution of QP Problems
There are 2 approaches to optimizing QP models
1. Use a general nonlinear programming
optimizer, such as Solver.
2. Use a specially written quadratic
programming optimizer.
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Here is an example of an QP model optimized using
Solver
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Here is the resulting Sensitivity Analysis
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Portfolio selection is a fundamental model in
modern finance.
The Portfolio Model
An investor has P dollars to invest in a set of n
stocks and would like to know how much to invest
in each stock.
The chosen collection is called the investors
portfolio.
There are conflicting goals in this model a
large expected return and a small risk
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An example of a return is
Suppose an investment of Di dollars is put into
asset i. Over some specified time period, Di
dollars grows to 1.3Di. Then the return over
that period is (1.3Di - Di)/Di 0.3.
Risk is measured by the variance of the return on
the portfolio.
Since the portfolio manager seeks low risk and
high expected return, one way to frame the model
is to minimize the variance of the return (i.e.,
risk) subject to a given lower bound on expected
return.
This model turns out to be a quadratic
programming model.
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Formulating the Portfolio Model
Let
xi be the proportion of the portfolio invested
in stock i
P be the amount in dollars to invest
si2 variance of yearly returns from stock i,
i 1, 2
s12 covariance of yearly returns from stocks
1 and 2
Ri expected yearly return from stock i,
i 1, 2
G lower bound on expected yearly return
from total investment
Si upper bound on investment stock i, i
1, 2
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Model Facts
1. The variance of the yearly returns from
stock i is a number describing the
variability of these returns from year to
year.
2. The covariance of the yearly returns from
stock i is a number that describes the
extent to which the returns of the two
stocks move up or down together.
3. The expected return of the portfolio is
defined as the number x1R1 x2R2.
4. The variance of the return of the portfolio
is defined as the number 2s12x1 s22x22.
5. The standard deviation of the portfolio
is the square root of the variance.
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For a two-stock example, the symbolic model is
Min s12x12 2 s12x1x2 s22x22 (Variance of
return)
s.t. x1 x2 1 (All funds must be
invested)
x1R1 x2R2 gt G (Lower bound on the
expected
return of the portfolio)
x1 lt S1 (Upper bound on investments in
stock 1)
x2 lt S2 (Upper bound on investments in
stock 2)
x1, x2 gt 0 (nonnegativity implies
that short
selling of a stock is not allowed)
Let s12 0.09 R1 0.06 S1 0.75 G
0.03 s22 0.06 R2 0.02 S2 0.90
s12 0.02
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This graph shows the feasible set for the model
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Here is the spreadsheet model for the problem
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In this three-asset problem, data will be used to
estimate the parameters in this model. Solver
will be used to optimize the model.
Formulating the Model
Let
X fraction of asset x in the portfolio
Y fraction of asset y in the portfolio
Z fraction of asset z in the portfolio
In the real world, the expected returns,
variances, and covariances must be estimated with
historical data.
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In general, if n periods of data are available,
there will be, for each asset i, an actual
historical return Ri associated with period t
where t ranges from 1 to n.
The expected periodic return from asset i (i.e.
the average of the assets historical returns) is
estimated with
The expected periodic historic returns Rit, are
used to estimate variances and covariances.
Estimate of the variance of return for asset i
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Estimate of the covariance of returns for
assetsi and j
G lower bound on expected return of portfolio
Si upper bound on the fraction of asset i that
can be in the portfolio
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The quadratic programming formulation of the
three-asset model is as follows
Min sx2X2 sy2Y2 sz2Z2 2 sxyXY 2 sxzXZ
2 syzYZ
s.t. RxX RyY RzZ gt G
X Y Z 1
X lt Sx
Y lt Sy
Z lt Sz
X, Y, Z gt 0
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Solver Solution
Consider the following historical stock returns
The return for year n is defined by
(closing price, n) (closing price, n-1)
(dividends, n) (closing price, n-1)
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The spreadsheet model is given below
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Here are the Solver parameters and Sensitivity
Report
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Assuming that the portfolio returns are Normally
distributed with a mean of 15 and standard
deviation of 14.33, then (at a 95 confidence
level) the expected return would roughly be
between 13.7 and 43.7 (i.e., 15 214.33).
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The returns over the ensuing three years for the
three stocks and the actual portfolio returns are
The Lagrangian multiplier indicates that a 1
increase in expected return would lead to an
increase of 0.00324 in variance. Hence, the new
portfolio variance would be about 0.0238.
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This graph (a piecewise quadratic convex
function) shows that tightening the expected
return constraint (i.e., increasing the expected
return, b) hurts the OV more and more.