Risk Analysis in Finance and Insurance, 2nd Edition

Insurance and Reinsurance Risks 199

and since X(t) is a compound Poisson process, then

E

Q



X(1)



= E

Q



N(1)



E

Q



X

1



,

where

E

Q



N(1)



= λE

P



exp{β(X

1

)}



,

E

Q



X

1



= E

P



X

1

exp{β(X

1

)}

*

E

P



exp{β(X

1

)}



.

Finally, we deduce

c = λE

P



X

1

exp{β(X

1

)}



.

Choosing appropriately β, we then obtain all the traditional actuarial prin-

ciples of premium calculations. For example, the expectation principle corre-

sponds to β(x)=ln(1+a), and we have

c = λE

P



X

1

exp{β(X

1

)}



= λE

P



X

1

(1 + a)



=(1+a) E

P



X

1



=Π(X) .

7.2 Risks transfers via reinsurance

Reinsurance is a mechanism that insurance companies use to transfer some

or all of their risks to reinsurance companies. The primary aim of reinsurance ...