Stock Optimisation

7/17/2019 Stock Optimisation

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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.





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:





,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.





• 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]





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]





• 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.





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)





• 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!





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


Small Values of X tend to be


NEGATIVE DEPENDENCE





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])]

(-,-)= +

(+,+)= +

(+,-)= -

(-,+)= -





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?





• 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 =





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





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?





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





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)

^





• 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





• 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


• 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 ≥ =






][][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!





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





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





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





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





EXAMPLE :

( ) =

542

321

54

( ) =+++ 5*53*44*52*42*51*4

( )372814





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

= = =

= =

=

−= = =

=

∑ ∑ ∑

∑ ∑

∑

∑ ∑ ∑





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





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





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= = ≠

+ =

∑ ∑ ∑






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.


1

1, N

i

i

x=

=∑ 0, 1, ,i x i N ≥ =







)(][][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=

=∑






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).




g g


CONCLUSION: OPTIMIZATION PROBLEM IS CONVEX OPTIMIZATION

PROBLEM AND HAS A UNIQUE GLOBAL OPTIMUM.

EXAMPLE:





AFTER OPTIMIZATION





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





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

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