Chpt 9 Ruin theory

[Ark raw]

In the solution given for question 9.30, it says in the very last line of that
"The adjustment coefficient R is the smallest positive solution of this equation" now I wanna know why is R the smallest positive solution of that equation? because as per my knowledge a higher value of R would give more security to the insurer so why is R the smallest positive solution of that equation?

Thank You

 

[John Lee]

Because the MGF only exists for \(R < \beta \).

 

[Ark raw]

But how can we know from the last equation that the smallest value of R would ensure R<β holds true?

 

[John Lee]

Well earlier in the chapter we say that R is the unique positive root. So negative roots are excluded and if there's more than one positive root then it must be the case that only one of them satisfies \(r < \beta\). Which will be the smallest one.