# Relative risk aversion and the demand for insurance (addendum to Problem Set #5)

In Problem Set #5, we studied the effect that premium loadings have upon the demand for insurance (see http://fin4335.garven.com/spring2018/ps5solutions.pdf for the solutions to this problem set).  One of the important takeaways from Problem Set #5 was that other things equal, a more risk averse decision-maker will select a higher coinsurance rate for a given premium loading than a less risk averse decision-maker.  Thus, if you have logarithmic utility; i.e., U = ln W, then your optimal coinsurance rate for a 20% premium loading is $\alpha$  = 5/6, but if your utility is $U = - {W^{ - 1}}$, then your optimal coinsurance rate for a 20% premium loading is $\alpha$  = .9139.

The fact that $U = - {W^{ - 1}}$ is a more risk averse utility than U = ln W is apparent by comparing Arrow-Pratt risk aversion coefficients for these utilities (see pp. 16-26 of http://fin4335.garven.com/spring2018/lecture6.pdf for more on this topic).  Specifically, the Arrow-Pratt risk aversion coefficient for U = ln W is equal to 1/W, and the Arrow-Pratt risk aversion coefficient for  $U = - {W^{ - 1}}$ is equal to 2/W.  Both of these utilities feature decreasing absolute risk aversion, which implies that if we were to increase initial wealth without making any further changes, then both of these utilities would imply lower coinsurance rates (although the decision-maker whose  $U = - {W^{ - 1}}$  will still select a higher coinsurance rate then the decision-maker whose U = ln W).  For example, suppose that we double initial wealth for both utilities without making any other changes; i.e., increase initial wealth from $1,000 to$2,000.  Then, the optimal coinsurance rate for a 20% premium loading for = ln W falls to $\alpha$  = .614, and if $U = - {W^{ - 1}}$, then the optimal coinsurance rate for a 20% premium loading falls to $\alpha$  = .8007.

However, suppose that instead of only doubling initial wealth, we also double the state-contingent loss from $1,000 to$2,000, holding everything else constant.  This will cause the coinsurance rates for both utilities to revert back to their original values of 5/6 and .9139 respectively.  The reason why this occurs is because not only are these utilities characterized by decreasing absolute risk aversion, they also feature constant relative risk aversion (see the discussion on p. 25 of the http://fin4335.garven.com/spring2018/lecture6.pdf lecture note as to how relative risk aversion is calculated).  Constant relative risk aversion implies that the proportion of  wealth which the decision-maker is willing to put at risk does not change as wealth changes.