Risk adjusted premium principle

Question

I had to do : Show that the risk-adjusted premium principle with $g(x) = x^{1/\rho}$ is consistent, scale-invariant and satisfies the no-ripoff property.

I know from my course that if $g(x) = x^{1/\rho}$ then $$\pi(x) = \int_0^\infty (1-F_X(x))^{1/\rho}\,\mathrm d x$$

But here i don't know what is $F_X(x)$.

And to show that it is consistent i have to show $\pi(x+c)$. Is it showing $$\pi(x+c) = \int_0^\infty (1-F_X(x+c))^{1/\rho}\,\mathrm d x\qquad ?$$

Thank you

Answer 1

You don't need to know $F_X(x)$. Let's call $S_X(x)=1-F_X(x)$ the survival function of $X$. So you have $$ \pi(X)=\int_0^{\infty}\Big(S_X(u)\Big)^{\frac{1}{\rho}}\;\mathrm d u $$

You have to prove the followings:

1. scale invariance or positive homogeneity: for any $a>0$, $\pi(aX)=a\pi(X)$
2. traslation invariance: for any $b>0$, $\pi(X+b)=\pi(X)+b$
3. subadditivity: for any loss variables $X$ and $Y$, $\pi(X+Y)\le \pi(X)+\pi(Y)$
4. monotonicity: for any loss variables $X$ and $Y$ such that $X\le Y$, $\pi(X)\le \pi(Y)$
5. no rip-off: if $X$ has a finite support with maximum value $x^*$, then $\pi(X)\le x^*$

They are very simple to prove. For example

1. scale invariance $$ \pi(aX)=\int_0^{\infty}\left(S_X\left(\frac{u}{a}\right)\right)^{\frac{1}{\rho}}\;\mathrm d u=a\int_0^{\infty}\Big(S_X\left(t\right)\Big)^{\frac{1}{\rho}}\;\mathrm d t=a\pi(X) $$
2. traslation invariance $$ \pi(X+b)=\int_0^{b}1\;\mathrm d t+\int_b^{\infty}\Big(S_X\left(u-b\right)\Big)^{\frac{1}{\rho}}\;\mathrm d u=b+\pi(X) $$ and so on.