Camp

UNIVERSITY OF CALIFORNIA, BERKELEY

UGBA 137-1

PROF. K. MAGIN

HANDOUT 2

: Let random variables

= ln

= ln( (

+1

+1

)− )

are bivariate normally distributed with expectations

h

ln( (

+1 ]

( [ln

+1

i

)− ) ) = (

and the variance-covariance matrix

⎛

where

⎞

⎜

V =⎜

⎝

=

[ln

[ln

+1

+1 ]

)

⎟

⎟

⎠

ln( (

+1 −

)

[ln( (

)]

+1 −

)

)]

is the correlation coefficient between random variables

That is random variables

and

and

have joint density function

1

(

)=

1

√

1−

2

−

(

1

2(1− 2 )

−

)2

2

(

−2

)(

−

−

)

+(

−

)2

2

2

Then

∞ ∞

Z Z

∞ ∞

Z Z

2

(

)

+

=

2

(−

·

+

) F

+

−∞

2

(

)

+

=

2

(−

·

+

) FF

+

−∞

and

∞ ∞

Z Z

e

+

f (x y)dxdy = e

+

+

( 2 +2

+ 2)

2

·N ( −

+

+

+

)

FFF

−∞

1−

Suppose also that random variables

: Let (c) = 1−

= ln

= ln( (

+1

+1

,

)− )

are bivariate normally distributed with expectations

( [ln

+1 ]

h

ln( (

and the variance-covariance matrix

2

+1

i

) ) )=(

−

)

⎛

⎞

⎜

V =⎜

⎝

where

⎟

⎟

⎠

[ln

=

[ln

+1 −

)

ln( (

+1

+1 ]

[ln( (

)]

+1 −

)

)]

is correlation coefficient between random variables

That is random variables

(

)=

2

1

√

1−

and

−

and

have joint density function

1

2(1− 2 )

(

−

)2

2

(

−2

−

)(

−

)

+

( −

2

then

ln [

+1 ]

− ln

=

[ln

+1

PROOF : We know from CCAPM that

[ (

+1

)− ] =

1

and

[ (

+1

)−

+1 ]

But

3

=1

ln(

+1

)] .

)2

2

+1

[ (

+1

2

)− ] =

+1

2

=⇒

2

2

=⇒

1

=

Also,

+1

[ (

)−

+1 ]

+

=⇒

=⇒

+

− ln

+

+

( 2 +2

( 2 +2

(

+ 2)

+

+

2 +2

+ 2)

2

2

+

+

+

=

+

= 1...