Risk Analysis in Finance and Insurance, 2nd Edition

136 Risk Analysis in Finance and Insurance

where C is a constant.

Since

X

T

= x exp



(1 − α)r + αμ −

α

2

σ

2

2

T + ασ W

t

%

,

then

E



X

T



= x exp





(1 − α)r + αμ



T



,

and we can prove the following result.

Theorem 4.2 If

min

α



CaR(x, α, T )



<C<xexp{rT},

then the optimization problem (4.38) has a unique solution α = εσ

−1

,where

ε =



μ −r

σ

+

z

λ

√

T



+

/



μ − r

σ

+

z

λ

√

T



2

− 2

ln



1 − C exp{−rT }/x



T

,

and

max E



X

T



= x exp



r + ε

μ −r

σ



T



.

Proof The assumption min

α



CaR(x, α, T )



<C<xexp{rT} implies that

the constraint CaR(x, α, T ) ≤ C is not redundant.

Since E



X

T



is an increasing function of α, we just need to find the max-

imum of α under the constraint CaR(x, α, T ) ≤ C.From

CaR(x, α, T )=