On the properties of high-order non-monetary measures for risks

Abstract

This paper investigates how welfare losses for facing high-order risk increases change when the risk environment of the decision maker is altered. To that aim, we define the nth-order utility premium as a measure of pain associated with facing the passage of one risk to a more severe one and we examine some of its properties. Changes in risk are expressed through the concept of stochastic dominance of order n. The paper investigates more particularly welfare changes of merging increases in risk, first ignoring background risks, then taking them into account. Merging increases in risk may be beneficial or not, depending on whether background risks are considered and how. The paper also provides conditions on individual preferences for superadditivity of the nth-order utility premium. The results confirm the importance and usefulness of two analytical concepts: mixed risk aversion and risk apportionment.

Appendix

Proof of Proposition 1

We have $X_2 \preceq _{{n_2}-SD} Y_2$ and $Y_1 \preceq _{2-SD} 0$. Applying Eeckhoudt et al. (2009), we know that $E[u(x+Y_{2})]+E[u(x+X_{2}+Y_{1})]-E[u(x+X_{2})]-E[u(x+Y_{1}+Y_{2})]\le 0$ for all u such that $(-1)^{(1+k)}u^{(k)}\ge 0$ $\forall k=1,\ldots ,n_2+2$.

Analogously, we have $X_{1}\preceq _{{n_{1}}-SD}Y_{1}$ and $Y_{2}\preceq _{2-SD}0$. Applying Eeckhoudt et al. (2009), we know that $E[u(x+Y_{1})]+E[u(x+X_{1}+Y_{2})]-E[u(x+X_{1})]-E[u(x+Y_{1}+Y_{2})]\le 0$ for all u such that $(-1)^{(1+k)}u^{(k)}\ge 0$ $\forall k=1,\ldots ,n_{1}+2$. However, from Eeckhoudt et al. (2009) we get $E[u(x+Y_{1}+Y_{2})]+E[u(x+X_{1}+X_{2})]-E[u(x+X_{1}+Y_{2})]-E[u(x+Y_{1}+X_{2})]\le 0$ for all u such that $(-1)^{(1+k)}u^{(k)}\ge 0$ $\forall k=1,\ldots ,n_{1}+n_{2}$. Consequently, if u is such that $(-1)^{(1+k)}u^{(k)}\ge 0$ $\forall k=1,\ldots ,n_{1}+n_{2}$, $\forall n_{1}\ge 2$, $\forall n_{2}\ge 2$ then the following inequality holds: $\Bigl ( E[u(x+Y_{2})]+E[u(x+X_{2}+Y_{1})]-E[u(x+X_{2})]-E[u(x+Y_{1}+Y_{2})]\Bigr )+ \Bigl (E[u(x\!+Y_{1})]+E[u(x+X_{1}+Y_{2})]\!-E[u(x+X_{1})]-E[u(x+Y_{1}+Y_{2})] \Bigr )+\Bigl ( E[u(x+Y_{1}+Y_{2})]+E[u(x+X_{1}+X_{2})]\!-E[u(x+X_{1}+Y_{2})]-E[u(x+Y_{1}+X_{2})] \Bigr )\le 0$. It rewrites equivalently as $E[u(x+Y_{1})]-E[u(x+X_{1})]+E[u(x+Y_{2})]-E[u(x+X_{2})]\le E[u(x+Y_{1}+Y_{2})]-E[u(x+X_{1}+X_{2})$ that is equivalent to $w(x;Y_{1},X_{1})+w(x;Y_{2},X_{2})\le w(x;Y_{1}+Y_{2},X_{1}+X_{2})$ that ends the proof. $\blacksquare$

Proof of Corollary 1

Let us define $Y=Y_{1}+Y_{2}.$ Assuming $X_{1}\preceq _{n_{1}-SD}(Y_{1}+Y_{2})$ and $X_{2}\preceq _{n_{2}-SD}(Y_{1}+Y_{2})$, we want to prove that Proposition 1 implies Corollary 1. As $Y_{1}$, $Y_{2}$, $X_{1}$, $X_{2}$ are mutually independent, we have $Y\perp X_{1}$, $Y\perp X_{2}$. As $Y_1 \preceq _{2-SD} 0$ and $Y_2 \preceq _{2-SD} 0$, we have (using the convolution property) $Y_1+Y_2 \preceq _{2-SD} 0$. Assumptions of Corollary 1 hold.

Equation (17) (i.e. the claim of Corollary 1) rewrites then $w(x;Y_{1}+Y_{2},X_{1}+X_{2})\ge w(x;Y_{1}+Y_{2},X_{1})+w(x;Y_{1}+Y_{2},X_{2})$ that we label $(17^{\prime })$. Equation (16) (i.e. the claim of Proposition 1) writes as $w(x;Y_{1}+Y_{2},X_{1}+X_{2})\ge w(x;Y_{1},X_{1})+w(x;Y_{2},X_{2})$.

We show (see proof below) that $w(x;Y_{1},X_{1})\ge w(x;Y_{1}+Y_{2},X_{1})$ and $w(x;Y_{2},X_{2})\ge w(x;Y_{1}+Y_{2},X_{2})$. Using this result we obtain Eq. (16) $\Rightarrow$ Eq. (17$^{\prime }$).

To prove that $w(x;Y_{1},X_{1})\ge w(x;Y_{1}+Y_{2},X_{1}),$ we use the definition of the nth-order utility premium. Then $w(x;Y_{1},X_{1})\ge w(x;Y_{1}+Y_{2},X_{1})$ writes equivalently as $E[u(x+Y_{1}+Y_{2})]\le E[u(x+Y_{1})]$ which is true for all u such that $u^{\prime \prime }<0$ as $Y_{1}$ and $Y_{2}$ are zero-mean independent risks. The proof is the same for $w(x;Y_{2},X_{2})\ge w(x;Y_{1}+Y_{2},X_{2})$. $\blacksquare$

Proof of Corollary 2

Using the definition of w, $w(x+X_{2};Y,X)-w(x+Y_{2};Y,X)\ge 0$ rewrites equivalently as $E[u(x+Y+X_{2})]-E[u(x+X+X_{2})]\ge E[u(x+Y+Y_{2})]-E[u(x+X+Y_{2})]$, that is equivalent to $E[u(x+Y+X_{2})]+E[u(x+X+Y_{2})]\ge E[u(x+Y+Y_{2})]+E[u(x+X+X_{2})]$. Following Eeckhoudt et al. (2009), this last expression is equivalent to $(-1)^{(1+k)}u^{(k)}\ge 0$ for $k=1,\ldots ,n_{1}+n_{2}$. $\blacksquare$