Introduction to Computational Finance and Financial ...

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Introduction to Computational Finance and Financial Econometrics

Portfolio Theory with Matrix Algebra

Eric Zivot

Spring 2015

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Outline

1 Portfolios with Three Risky Assets Portfolio characteristics using matrix notation Finding the global minimum variance portfolio Finding efficient portfolios Computing the efficient frontier Mutual fund separation theorem again

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Example

Example: Three risky assets

Let Ri (i = A, B, C) denote the return on asset i and assume that Ri follows CER model:

Ri iid N (?i, i2)

cov(Ri, Rj) = ij

Portfolio "x":

xi = share of wealth in asset i xA + xB + xC = 1 Portfolio return:

Rp,x = xARA + xBRB + xC RC .

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Example cont.

Stock i A (Microsoft) B (Nordstrom) C (Starbucks)

?i 0.0427 0.0015 0.0285

i 0.1000 0.1044 0.1411

Pair (i,j) (A,B) (A,C) (B,C)

ij 0.0018 0.0011 0.0026

Three asset example data.

In matrix algebra, we have:

?A 0.0427

?=

?B

=

0.0015

?C

0.0285

A2 AB AC (0.1000)2 0.0018 0.0011

=

AB

B2

BC

=

0.0018

(0.1044)2

0.0026

AC BC C2

0.0011 0.0026 (0.1411)2

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Outline

1 Portfolios with Three Risky Assets Portfolio characteristics using matrix notation Finding the global minimum variance portfolio Finding efficient portfolios Computing the efficient frontier Mutual fund separation theorem again

Eric Zivot (Copyright ? 2015)

Portfolio Theory

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Matrix Algebra Representation

RA

?A

1

R=

RB

,

?=

?B

,

1=

1

RC

?C

1

xA

A2

x=

xB

,

=

AB

xC

AC

AB B2 BC

AC

BC

C2

Portfolio weights sum to 1:

1

x 1 = ( xA

xB

xC

)

1

1

= x1 + x2 + x3 = 1

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Portfolio return

RA

Rp,x = x R = ( xA

xB

xC

)

RB

RC

= xARA + xBRB + xC RC

Portfolio expected return:

?A

?p,x = x ? = ( xA

xB

xX

)

?B

?C

= xA?A + xB?B + xC ?C

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Computational tools

R formula: t(x.vec)%*%mu.vec crossprod(x.vec, mu.vec)

Excel formula: MMULT(transpose(xvec),muvec) --

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