Derivation of the Beta Risk Factor

Calculate portfolio variance Split into market proportional variance and firm specific variance

ij

m

1jji

m

1i

2P σwwσ

)σσβ(βwwσijε

m

1j

2Mjiji

m

1i

2P

2Mjiijε

ε2Mjiij

σββσσ

σσββσ

ij

ij

ijε

m

1jji

m

1i

m

1j

2Mjiji

m

1i

2P σwwσββwwσ

1

m: number of assets in portfolio

M: ‘market’

Derivation of the Beta Risk Factor2

2Mmm

2M2m

2M1m

2Mm2

2M22

2M12

2Mm1

2M21

2M11

M

σββ...σββσββ............

σββ...σββσββσββ...σββσββ

C

2ε

2ε

2ε

2ε

2ε

2ε

2ε

2ε

2ε

ε

mm2m1m

m22221

m11211

σ...σσ............

σ...σσσ...σσ

C

εM

ε2Mjiij

CCC

σσββσij

Derivation of the Beta Risk Factor

Split

Firm specific covariance is assumed zero. Split the variances and covariances

ijε

m

1jji

m

1i

m

1j

2Mjiji

m

1i

2P σwwσββwwσ

iji ε

m

ij1j

ji

m

1i

2ε

m

1i

2i

m

ij1j

2Mjiji

m

1i

m

1i

2M

2i

2i

2P σwwσwσββwwσβwσ

Market proportional Firm specific

variance covariance variance covariance

m

ij1j

2Mjiji

m

1i

2ε

m

1i

2M

2i

2i

2P σββww)σσβ(wσ

i

3

Derivation of the Beta Risk Factor4

2ε

2ε

2ε

ε

mm

22

11

σ...00............0...σ00...0σ

C

Firm specific covariance is assumed zero

Derivation of the Beta Risk Factor

m

ij1j

2Mjiji

m

1i

2ε

m

1i

2M

2i

2i

2P σββww)σσ(βwσ

i

2ε

2M

2i

2i i

σσβσ

2MMiiM σββσ

2MiiM σβσ

2M

iMi σ

σβ

2Mjiij σββσ

5

Systemic and

non-systemic

(firm specific)

Contribution to variance for asset i

Systemic risk contribution

only

Portfolio covariance is a linear function

of beta

Derivation of the Beta Risk Factor

)r(rσσ

r

)r(rβr r

FM2M

iMF

FMiFi

Substitute into CAPM formula

M

iiM

2M

MiiM

2M

iMi

MiiMiM

σσ

ρ

σσσρ

σσ

β

σσρσ

6

The two assumptions in the derivation are:• A market proportional covariance matrix can be

constructed and • Covariances between firm specific risks are zero

Return other than from taking market driven risk is captured by α which in the CAPM model is a normally distributed random variable with an expected value of zero

iFMiFi α)r(rβrr

Portfolio Beta Derivation

m

1iiiP

m

1i2M

iMiP

2M

m

1iiMi

P

2M

Mi

m

1ii

P

2M

Mi

m

1ii

P

2M

MPP

2M

PMP

βwβ

σσ

wβ

σ

)σ(wβ

σ

)r,cov(rwβ

σ

r,rwcovβ

σ)r,cov(r

β

σσ

β

7