The value of risk reduction: new tools for an old problem

Abstract

The relationship between willingness to pay (WTP) to reduce the probability of an adverse event and the degree of risk aversion is ambiguous. The ambiguity arises because paying for protection worsens the outcome in the event the adverse event occurs, which influences the expected marginal utility of wealth. Using the concept of downside risk aversion or prudence, we characterize the marginal WTP to reduce the probability of the adverse event as the product of WTP in the case of risk neutrality and an adjustment factor. For the univariate case (e.g., risk of financial loss), the adjustment factor depends on risk aversion and prudence with respect to wealth. For the bivariate case (e.g., risk of death or illness), the adjustment factor depends on risk aversion and cross-prudence in wealth.

Appendices

Appendix 1

This appendix justifies the claim that an increase in risk aversion typically increases downside risk aversion.

If one decision maker ($v)$ is more risk averse than another one ($u)$, then $v$ can be written as $v=k\left( u \right)$ with ${k}^{\prime }>0$ and ${k}^{{\prime }{\prime }}<0$. Easy computations reveal:

$$\begin{aligned} {v}^{\prime }&= {k}^{\prime }\cdot {u}^{\prime }\\ {v}^{{\prime }{\prime }}&= {k}^{{\prime }{\prime }}\cdot \left( {{u}^{\prime }} \right) ^{2}+{k}^{\prime }\cdot {u}^{{\prime }{\prime }}\\ {v}^{{\prime }{\prime }{\prime }}&= {k}^{{\prime }{\prime }{\prime }}\cdot \left( {{u}^{\prime }} \right) ^{3}+3{k}^{{\prime }{\prime }}{u}^{{\prime }{\prime }}{u}^{\prime }+{k}^{\prime }{u}^{{\prime }{\prime }{\prime }}. \end{aligned}$$

Since $-\frac{{v}^{{\prime }{\prime }}}{{v}^{\prime }}=-\frac{{k}^{{\prime }{\prime }}}{{k}^{\prime }}{u}^{\prime }-\frac{{u}^{{\prime }{\prime }}}{{u}^{\prime }}$, the degree of absolute risk aversion of $v$ exceeds that of $u$.

Moreover,

$$\begin{aligned} \frac{{v}^{{\prime }{\prime }{\prime }}}{{v}^{\prime }}=\frac{{k}^{{\prime }{\prime }{\prime }}}{{k}^{\prime }}\cdot \left( {{u}^{\prime }} \right) ^{2}+3\frac{{k}^{{\prime }{\prime }}}{{k}^{\prime }}{u}^{{\prime }{\prime }}+\frac{{u}^{{\prime }{\prime }}}{{u}^{\prime }}. \end{aligned}$$

Since ${k}^{{\prime }{\prime }}\cdot {u}^{{\prime }{\prime }}>0, {k}^{{\prime }{\prime }{\prime }}\ge 0$ implies that $\frac{{v}^{{\prime }{\prime }{\prime }}}{{v}^{\prime }}>\frac{{u}^{{\prime }{\prime }{\prime }}}{{u}^{\prime }}$. If ${k}^{{\prime }{\prime }{\prime }}$ is negative then $\frac{{v}^{{\prime }{\prime }{\prime }}}{{v}^{\prime }}$ may, but need not, be smaller than $\frac{{u}^{{\prime }{\prime }{\prime }}}{{u}^{\prime }}$.

This result justifies the assertion that an increase in risk aversion tends to increase downside risk aversion, so that it will have an ambiguous effect on WTP. Notice finally that if one assumes that ${k}^{{\prime }{\prime }{\prime }}\ge 0$ (while ${k}^{\prime }>0$ and ${k}^{{\prime }{\prime }}<0)$, then an increase in risk aversion always increases downside risk aversion.

Appendix 2

The purpose of this appendix is to formally prove the interpretation of the term $\frac{u_{122} }{u_1 }$ that appears in the expression for WTP under bivariate utility (Eq. (2.7)).

A preference for ‘combining good with bad’ (see Eeckhoudt et al. 2009) induces in the bivariate case a preference for lottery A over B:

where $\tilde{\varepsilon }$ is a zero-mean risk (a ‘bad’ for a risk-averse decision maker) and $-l$ is a sure loss (a ‘bad’ if marginal utility is positive).

In lottery A, the two bad consequences never occur together; they are spread between the two states of the world. In lottery B, on the contrary, either everything is bad (in the first state of the world, where the decision maker faces both $-l$ on $x$ and $\tilde{\varepsilon }$ on $y$), or nothing is bad (in the second state).

For an expected utility maximizer, a preference for A over B implies

$$\begin{aligned} \frac{1}{2}u\left( {x-l,y} \right) +\frac{1}{2}E\left[ {u\left( {x,y+\tilde{\varepsilon }} \right) } \right] >\frac{1}{2}E\left[ {u\left( {x-l,y+\tilde{\varepsilon }} \right) } \right] +\frac{1}{2}u\left( {x,y} \right) ,\quad \end{aligned}$$

(5.1)

which is equivalent to

$$\begin{aligned} u\left( {x,y} \right) -E\left[ {u\left( {x,y+\tilde{\varepsilon }} \right) } \right] <u\left( {x-l,y} \right) -E\left[ {u\left( {x-l,y+\tilde{\varepsilon }} \right) } \right] . \end{aligned}$$

(5.2)

Using the concept of ‘utility premium’ (Friedman and Savage 1948), it is shown in Eeckhoudt et al. (2007, p. 121) that inequality (5.1) holds for risk-averse expected utility maximizers if $u_{122}>0$ and this is termed ‘cross-prudence in wealth.’

While this result indicates a direction for preferences as in (5.1) and relates it to the sign of a third cross-derivative of $u$, one may also be interested in the intensity of this preference. In order to do so, we define a positive change in wealth $m$ such that one obtains

$$\begin{aligned} \frac{1}{2}u\left( {x-l,y} \right) +\frac{1}{2}E\left[ {u\left( {x,y+\tilde{\varepsilon }} \right) } \right] =\frac{1}{2}E\left[ {u\left( {x-l,y+\tilde{\varepsilon }} \right) } \right] +\frac{1}{2}u\left( {x+m,y} \right) .\nonumber \\ \end{aligned}$$

(5.3)

In (5.3), the right hand side of (5.1) is increased through the addition of $m$ in the best state of the world until equality prevails.⁶

Rearranging terms in (5.3), one obtains

$$\begin{aligned} u\left( {x+m,y} \right) -E\left[ {u\left( {x,y+\tilde{\varepsilon }} \right) } \right] =u\left( {x-l,y} \right) -E\left[ {u\left( {x-l,y+\tilde{\varepsilon }} \right) } \right] . \end{aligned}$$

(5.4)

Because $u\left( {x+m,y} \right) \approx u\left( {x,y} \right) +mu_1 \left( {x,y} \right)$, (5.4) becomes

$$\begin{aligned} u\left( {x,y} \right) +mu_1 \left( {x,y} \right) -E\left[ {u\left( {x,y+\tilde{\varepsilon }} \right) } \right] \approx u\left( {x-l,y} \right) -E\left[ {u\left( {x-l,y+\tilde{\varepsilon }} \right) } \right] .\nonumber \\ \end{aligned}$$

(5.5)

Applying the methodology of Eeckhoudt et al. (2007) to the terms other than $mu_1 \left( {x,y} \right)$, one is left with

$$\begin{aligned} mu_1 \left( {x,y} \right) \approx \frac{\sigma _\varepsilon ^2 }{2}\Bigg [ {u_{22} ( {x,y} )-u_{22} ( {x-l,y} )} \Bigg ]. \end{aligned}$$

Approximating $u_{22} \left( {x-l,y} \right)$ to first order around $u_{22} \left( {x,y} \right)$,

$$\begin{aligned} mu_1 \left( {x,y} \right) \approx \frac{\sigma _\varepsilon ^2 \cdot l}{2}u_{221} \left( {x,y} \right) , \end{aligned}$$

(5.6)

so that finally

$$\begin{aligned} m\approx \frac{\sigma _\varepsilon ^2 \cdot l}{2}\frac{u_{221} \left( {x,y} \right) }{u_1 \left( {x,y} \right) }. \end{aligned}$$

(5.7)

As a result, $m$ measures in monetary terms the intensity of the pain induced by the misallocation of the losses in B as compared with A. From (5.7), this monetary intensity depends upon $\frac{u_{122} }{u_1 }$ as intuitively claimed in the discussion of (2.7).