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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 194Source: Financial Models Using Simulation and Optimization by Wayne Winston
CHAPTER 16:
PORTFOLIO OPTIMIZATION USING SOLVER
PROBLEM DEFINITION:
Given a set of investments (for example stocks) how do we find a portfolio thathas the lowest risk (i.e. lowest volatility or lowest variance) and yields an
acceptable expected return?
• Harry Markowitz solved the above problem in 1950’s. For this work and
other investment topics he received the Nobel Prize in Economics in 1991.
• Ideas discussed in this session are the basis for most methods of asset
allocation used by Wall Street firms.
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 195Source: Financial Models Using Simulation and Optimization by Wayne Winston
WEIGTHED SUM OF RANDOM VARIABLES
• A fixed number of N investments (or Stocks) are available.
• iS : Random return during a calendar year on a dollar invested in investment
I, where i=1, …, N.
Examples:
• 0.10i s = , dollar invested at the beginning of the years is worth $1.10 at the
end of the year
• 0.20i s = − , dollar invested at the beginning of the years is worth $0.80 at the
end of the year
Definition:
• i x : the fraction of one dollar that is invested in investment i. Note that:
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 196Source: Financial Models Using Simulation and Optimization by Wayne Winston
,11 =∑=
N
ii x N i xi ,,1,0 =≥
• R : be the random annual return on our investments. Then
∑=
∗=n
i
ii S x R1
Note that:
• Because the returns on our investments iS are random the annual return R
on our portfolio of investments, defined by ( ) N S S ,,1 and the specified
weights ( ) N x x ,,1 is random as well.
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 197Source: Financial Models Using Simulation and Optimization by Wayne Winston
• The expected (or average) annual return for R can be calculated as:
][][1
∑=
∗=n
i
ii S E x R E
WHY?
• We know that if X is a random variable, a and b are constants and Y=a*X+bthat:
E[Y]=E[a*X+b]= a*E[X]+b
• We know that if X and Y are random variables and Z=X+Y that:
E[Z]=E[X+Y]=E[X]+E[Y]
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 198Source: Financial Models Using Simulation and Optimization by Wayne Winston
WHAT ABOUT THE UNCERTAINTY IN OUR ANNUAL RETURN R?
(or What about the variance of our Annual Return R?)
• We know that if the returns on the investments iS are independent random
variables that
][)(][
1
2∑=
∗=n
i
ii S VAR x RVAR
WHY?
•
• We know that if X is a random variable, a and b are constants and Y=a*X+b
that:
VAR[Y]=VAR[a*X+b]= a2*VAR[X]
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 199Source: Financial Models Using Simulation and Optimization by Wayne Winston
• We know that if X and Y are independent random variables and Z=X+Y that:
VAR[Z]=VAR[X+Y]=VAR[X]+VAR[Y]
BUT?
Can we reasonably say that the annual return iS
are independent random variables?
Answer: NO!
• We know that if the market (e.g. Dow Jones Index) is in an upward trend,
that all stocks have a better chance of doing well.
• We know that if the market (e.g. Dow Jones Index) is in a downward trend,
that all stocks have a higher chance of doing worse.
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 200Source: Financial Models Using Simulation and Optimization by Wayne Winston
SO WHAT IS THE CORRECT FORMULA FOR THE VARIANCE OF THE
ANNUAL RETURN R WHEN THE INVESTMENTS ARE DEPENDENT?
INTERMEZZO: COVARIANCE
Let X, Y be two random variables.
• If X and Y are independent there is no relationship between X and Y, and:
VAR(X+Y)=VAR(X) +VAR(Y)
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 201Source: Financial Models Using Simulation and Optimization by Wayne Winston
• If X and Y are positively dependent large values of X tend to be associated
with large values of Y
• If X and Y are positively dependent small values of X tend to be associated
with small values of Y
• If X and Y are negatively dependent large values of X tend to be associated
with small values of Y
• If X and Y are negatively dependent small values of X tend to be associated
with large values of Y.
WHAT IS THE SIMPLEST RELATIONSHIP
THAT YOU THINK OF BETWEEN X AND Y? ANSWER: A Linear Relationship!
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 202Source: Financial Models Using Simulation and Optimization by Wayne Winston
Let X,Y be two random variables such that: Y=a*X+b
Y=a*X+b, a>0
X
Y
Large Values of X tend to be
associated with Large Values of Y
Small Values of X tend to be
associated with Small Values of Y
POSITIVE DEPENDENCE
Y=a*X+b, a<0
X
Y
Large Values of X tend to be
associated with Small Values of Y
Small Values of X tend to be
associated with Small Values of Y
NEGATIVE DEPENDENCE
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 203Source: Financial Models Using Simulation and Optimization by Wayne Winston
CAN WE THINK OF A MEASURE IN THE SAME LINE OF THINKING AS E[X]
AND VAR[X] THAT “MEASURES” POSITIVE DEPENDENCE OR NEGATIVE
DEPENDENCE? WHAT ABOUT : COV(X,Y) = E[(Y-E[Y])(X-E[X])]
E[X]
E[Y]
E[(Y-E[Y])(x-E[X])]
(-,-)= +
(+,+)= +
(+,-)= -
(-,+)= -
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 204Source: Financial Models Using Simulation and Optimization by Wayne Winston
CONCLUSION :
• Covariance is positive when there are more (+,+) points and (-,-) points then
(+,-) points and (-,+) points.
• Covariance is negative when there are more (+,-) points and (-,+) points then
(+,+) points and (-,-) points.
THEREFORE:
Covariance is a measure of positive dependence or negative dependence.
If Y=a*X+b, what is the relationship between COV(X,Y)and the slope of the line a?
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 205Source: Financial Models Using Simulation and Optimization by Wayne Winston
• E[Y]=a*E[X]+b
• Y-E[Y]= a*X+b- a*E[X]-b=a*(X-E[X)
• Cov(X,Y)=E[(Y-E[Y])(X-E[X])] =
E[a*(X-E[X])(X-E[X])] = a
a*E[(X-E[X])(X-E[X])] = a*Var[X]
or
( , )
( )
COV X Y a
VAR X =
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 206Source: Financial Models Using Simulation and Optimization by Wayne Winston
Y=a*X+b, a>0
X
Y
Y=a*X+b, a>0
X
Y
^ ^
a =COV(X,Y)
VAR(X)
a =COV(X,Y)
VAR(X)
=
^^
^
∑=
−−−
n
i
ii x x x xn 1
))((1
1
∑=
−−−
n
i
ii x x y yn 1
))((1
1
PERFECT LINEAR RELATIONSHIP IMPERFECT LINEAR RELATIONSHIP
Best FitThrough
Linear
Regression
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 207Source: Financial Models Using Simulation and Optimization by Wayne Winston
CONCLUSION:
• If you have a set of points ),( ii y x , i=1,…,n you can estimate positive
dependence or negative dependence by calculating COV(X,Y).
• If you have a set of points ),( ii y x , i=1,…,n you can estimate the slope of the
“linear relationship” by calculating a .
HOW CAN WE DISTINGUISH BETWEEN A PERFECT LINEAR
RELATIONSHIP AND AN IMPERFECT LINEAR RELATIONSHIP
GIVEN THE SET OF POINTS ),( ii y x i=1,…,n?
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 208Source: Financial Models Using Simulation and Optimization by Wayne Winston
Let X,Y be two random variables such that
Y=a*X+b,
• Cov(X,Y)= a*Var[X]
• Var[Y]= a2*Var[X]
CORRELATION BETWEEN
TWO RANDOM VARIABLES
1)(**)(
)(*)(*)(
),(),(2
±=== X Var a X Var
X Var aY Var X Var
Y X CovY X Cor
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 209Source: Financial Models Using Simulation and Optimization by Wayne Winston
Y=a*X+b, a>0
X
Y
Y=a*X+b, a>0
X
Y
^ ^
a =COV(X,Y)
VAR(X)
a =COV(X,Y)
VAR(X)
^^
^
PERFECT LINEAR RELATIONSHIP IMPERFECT LINEAR RELATIONSHIP
Best FitThrough
Linear
Regression
COR(X,Y) = 1
,VAR(Y)>^,VAR(Y)>^ a2 VAR(X)^
a2 VAR(X)^
COR(X,Y) = < 1^ COV(X,Y)
VAR(X)
^
^*VAR(Y)
^
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 210Source: Financial Models Using Simulation and Optimization by Wayne Winston
• If X and Y are dependent random variables
VAR(X+Y) = VAR(X) + VAR(Y) + 2*COV(X,Y)
CONCLUSION :
• If X,Y are positive dependent variables the level of variation of X+Y is
amplified by the dependency.
• If X,Y are negative dependent variables the level of variation of X+Y is
reduced by the dependency (the random variables tend to “cancel each
other out”).
BACK TO OUR ORIGINAL PROBLEM
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 211Source: Financial Models Using Simulation and Optimization by Wayne Winston
• iS : Random returns during a calendar year on a dollar invested in
investment I, where i=1, …, N.
• Random returns iS are dependent random variables
• i x : the fraction of one dollar that is invested in investment i. Note that:
• R be the random annual return on our investments. Then
∑= ∗=
n
iii S x R 1
1
1, N
i
i
x=
=∑ 0, 1, ,i x i N ≥ =
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 212Source: Financial Models Using Simulation and Optimization by Wayne Winston
• The expected (or average) annual return for R can be calculated as:
][][1
∑=
∗=n
i
ii S E x R E
• The variance of the annual return for R can be calculated as:
∑ ∑∑= ≠==
+∗=n
i
n
i j j
ji ji
n
i
ii S S Cov x xS Var x RVar
1 ,11
2),(][][
or
2
1 1 1,
[ ] [ ] ( , ) [ ] [ ]n n n
i i i j i j i j
i i j j i
Var R x Var S x x Cor S S Var S Var S
= = = ≠
= ∗ +∑ ∑ ∑
EQUATIONS LOOK COMPLICATED!
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 213Source: Financial Models Using Simulation and Optimization by Wayne Winston
INTERMEZZO: MATRIX - VECTOR MULTIPLICATION
M N matrix:
=
−
−
nm N M M
N M
N
aaa
a
aa
aaa
A
,1,1,
,1
2,21,2
,12,11,1
M rows, N Columns
N-Vector (or N 1 matrix) :
=
N x
x
x 1
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 214Source: Financial Models Using Simulation and Optimization by Wayne Winston
Matrix-Vector Product:
=
=
∑
∑
∑
=
=
=
−
−
j
N
j
j M
j
N
j
j
j
N
j
j
N
nm N M M
N M
N
xa
xa
xa
x
x
aaa
a
aa
aaa
x A
1
,
1
,2
1
,1
1
,1,1,
,1
2,21,2
,12,11,1
(M N-Matrix)*(N-vector) =M-Vector
(M N-Matrix)*(N 1-Matrix) =M 1-Matrix
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 215Source: Financial Models Using Simulation and Optimization by Wayne Winston
EXAMPLE :
=
++
++=
36
20
4*53*42*2
4*33*22*1
4
3
2
542
321
VECTOR - MATRIX MULTIPLICATION
M N matrix:
=
−
−
nm N M M
N M
N
aaa
a
aa
aaa
A
,1,1,
,1
2,21,2
,12,11,1
M rows, N Columns
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 216Source: Financial Models Using Simulation and Optimization by Wayne Winston
TRANSPOSED M-Vector (or M
1 matrix) :
( ) M
T x x x 1=
Vector - Matrix Product:
( ) =
=
−
−
nm N M M
N M
N
M
T
aaa
a
aa
aaa
x x A x
,1,1,
,1
2,21,2
,12,11,1
1
= ∑∑∑
=== N j
N
j
j
N
j
j j
N
j
j j a xa xa x ,
11
2,
1
1,
(Transposed M –Vector)*(M N-Matrix) =N-Vector
(1 M-Matrix)*(M N-Matrix) =1 N-Matrix
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 217Source: Financial Models Using Simulation and Optimization by Wayne Winston
EXAMPLE :
( ) =
542
321
54
( ) =+++ 5*53*44*52*42*51*4
( )372814
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 218Source: Financial Models Using Simulation and Optimization by Wayne Winston
MATRIX – MATRIX MULTIPLICATION: (M
N-Matrix)*(N
P-Matrix) =M
P-Matrix
1,1 1,2 1, 1,1 1,2 1,
2,1 2,2 2,1 2,2
1, 1,
,1 , 1 , ,1 , 1 ,
N P
N N P
M M N M N N N P N P
a a a b b b
a a b b AB
a b
a a a b b b
− −
− −
=
1, ,1 1, ,2 1, ,
1 1 1
2, ,1 2, ,2
1 1
1, ,
1
, ,1 , , 1 , ,
1 1 1
N N N
j j j j j j P
j j j
N N
j j j j
j j
N
j j P
j N N N
j j M j j P M j j P
j j j
a b a b a b
a b a b
a b
a b a b a b
= = =
= =
=
−= = =
=
∑ ∑ ∑
∑ ∑
∑
∑ ∑ ∑
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 219Source: Financial Models Using Simulation and Optimization by Wayne Winston
EXAMPLE:
=
65
43
21
542
321
=
++++
++++
5039
2822
6*54*42*25*53*41*2
6*34*22*15*33*21*1
NOTE THAT:
(M N-Matrix)*(N P-Matrix) =M P-Matrix
EMSE 388 Q tit ti M th d i C t E i i
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 220Source: Financial Models Using Simulation and Optimization by Wayne Winston
BACK TO OUR ORIGINAL PROBLEM:
2
1 1 1,
[ ] [ ] ( , ) [ ] [ ]n n n
i i i j i j i j
i i j j i
Var R x Var S x x Cor S S Var S Var S = = = ≠
= ∗ +∑ ∑ ∑
HOWEVER INTRODUCING WEIGHT VECTOR:
=
N x
x
x
1
AND TRANSPOSE OF WEIGHT VECTOR;
( )n
T
N
T x x
x
x
x 1
1
=
=
EMSE 388 Quantitative Methods in Cost Engineering
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 221Source: Financial Models Using Simulation and Optimization by Wayne Winston
AND VARIANCE – COVARIANCE MATRIX:
=
−
−
∑
),(),(),(
),(
),(),(
),(),(),(
11
1
2212
12111
N N N N N
N N
N
S S CovS S CovS S Cov
S S Cov
S S CovS S Cov
S S CovS S CovS S Cov
AND REALIZING THAT: COV(Si,Si)=VAR(Si)
HOMEWORK: SHOW THAT ABOVE ASSERTION IS CORRECT!
2
1
[ ] [ ]n
i i
i
Var R x Var S =
= ∗ +∑
1 1,
( , ) [ ] [ ]n n
T
i j i j i ji j j i
x x Cor S S Var S Var S x x= = ≠
+ =
∑ ∑ ∑
EMSE 388 Quantitative Methods in Cost Engineering
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EMSE 388 – Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 222Source: Financial Models Using Simulation and Optimization by Wayne Winston
PROBLEM DEFINITION:
Given a set of investments (for example stocks) how do we find a portfolio that
has the lowest risk (i.e. lowest volatility or lowest variance) and yields an
acceptable expected return?
• iS : Random return during a calendar year on a dollar invested in investment
I, where i=1, …, N.
• i x : the fraction of one dollar that is invested in investment i. Note that:
1
1, N
i
i
x=
=∑ 0, 1, ,i x i N ≥ =
EMSE 388 – Quantitative Methods in Cost Engineering
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EMSE 388 Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 223Source: Financial Models Using Simulation and Optimization by Wayne Winston
• The expected (or average) annual return for R can be calculated as:
)(][][1
x g S E x R E n
i
ii =∗= ∑=
• The variance of the annual return for R can be calculated as:
)(][ x f x x RVar T
== ∑
OPTIMIZATION PROBLEM:
Minimize : )( x f
Subject to: %)( r x g ≥
0, 1, ,i x i N ≥ = 1
1 N
i
i
x=
=∑
EMSE 388 – Quantitative Methods in Cost Engineering
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EMSE 388 Quantitative Methods in Cost Engineering
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 224Source: Financial Models Using Simulation and Optimization by Wayne Winston
WHAT TYPE OF OPTIMIZATION PROBLEM IS THIS?
• Variance – Covariance Matrix ∑ is known to be positive definite.
• Objective function is a multidimensional quadratic function. Hessian Matrix of
∑ x xT
is: ∑
• Conclusion: Objective Function is Convex
• Constraint function %)( r x g − linear function (and thus convex)
• Constraint function 11
=∑=
N
i
i x can be rewritten as:
1,1
11
≤≥ ∑∑==
N
i
i
N
i
i x x
both of which are linear function ( and thus convex).
EMSE 388 – Quantitative Methods in Cost Engineering
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g g
Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 225Source: Financial Models Using Simulation and Optimization by Wayne Winston
CONCLUSION: OPTIMIZATION PROBLEM IS CONVEX OPTIMIZATION
PROBLEM AND HAS A UNIQUE GLOBAL OPTIMUM.
EXAMPLE:
EMSE 388 – Quantitative Methods in Cost Engineering
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Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 226Source: Financial Models Using Simulation and Optimization by Wayne Winston
AFTER OPTIMIZATION
EMSE 388 – Quantitative Methods in Cost Engineering
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Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 227Source: Financial Models Using Simulation and Optimization by Wayne Winston
SUPPOSE WE INCREASE THE REQUIRED EXPECTED RETURN, WHAT
HAPPENS TO THE STANDARD DEVIATION OF THE PORTFOLIO?
Required Return Stock 1 Stock 2 Stock 3 Port stdev Exp return
0.100 0 0 1 0.0800 0.100
0.105 1/8 0 7/8 0.0832 0.105
0.110 1/4 0 3/4 0.0922 0.110
0.115 3/8 0 5/8 0.1055 0.1150.120 1/2 0 1/2 0.1217 0.120
0.125 5/8 0 3/8 0.1397 0.125
0.130 3/4 0 1/4 0.1591 0.130
0.135 7/8 0 1/8 0.1792 0.135
0.140 1 0 0 0.2000 0.140
EMSE 388 – Quantitative Methods in Cost Engineering
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Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 16 - Page 228Source: Financial Models Using Simulation and Optimization by Wayne Winston
THE EFFICIENT FRONTIER
Efficient Frontier (Risk Versus Expected Return)
0.0500
0.1000
0.1500
0.2000
0.050 0.100 0.150 0.200
Expected Return
S t a n d
a r d
D e v i a t i o
n