Ambiguity preferences, risk taking and the banking firm

Abstract

This paper examines the risk taking behavior of a banking firm facing ambiguity and possessing smooth ambiguity preferences. Ambiguity is modeled by a second-order probability distribution that captures the bank’s uncertainty about which of the subjective beliefs govern the return on its loans. Ambiguity aversion is modeled by a concave transformation of the (first-order) expected utility of profit conditional on each plausible subjective distribution of the random loan return. Within this framework, we show that the bank finds it less attractive to take risk in the presence than in the absence of ambiguity. This result extends to the case of greater ambiguity aversion. Given that the bank’s smooth ambiguity preferences exhibit non-increasing absolute ambiguity aversion, imposing a more stringent capital requirement to the bank has the desired effect that limits the bank’s incentive to take on excessive risk.

Appendix

Proof of Proposition 2

Given that $R'(\theta )\ge (\le ) \ 0$ for all $\theta \in [\underline{\theta },\overline{\theta }],$ it follows from Eq. (10) that there exists a point, $\theta ^*\in (\underline{\theta },\overline{\theta }),$ at which $R(\theta ^*)=R_C+C'(L^*).$ Note that

$$\begin{aligned} \frac{\partial }{\partial \theta }K'\bigg (\phi \Big ( [R({\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ) \end{aligned}$$

$$\begin{aligned} =K''\bigg (\phi \Big ([R({\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ) \phi '\Big ([R({\theta })-R_C]L^{*}-C(L^{*})\Big )R'(\theta ), \end{aligned}$$

(21)

which is negative (positive) given that $R'(\theta )\ge (\le ) \ 0$ for all $\theta \in [\underline{\theta },\overline{\theta }]$ and $K''(\cdot )<0.$ Using Eq. (10), we have

$$\begin{aligned} \mathrm{Cov}_G\bigg [K'\bigg (\phi \Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ), \phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big ) \nonumber \\ \times [R(\tilde{\theta })-R_C-C'(L^*)]\bigg ] \nonumber \\ =\int _{\underline{\theta }}^{\overline{\theta }}\bigg [K'\bigg (\phi \Big ( [R({\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg )-K'\bigg (\phi \Big ( [R({\theta ^*})-R_C]L^{*}-C(L^{*})\Big )\bigg )\bigg ] \end{aligned}$$

(22)

$$\begin{aligned} \times \phi '\Big ([R({\theta })-R_C]L^{*}-C(L^{*})\Big ) [R({\theta })-R_C-C'(L^{*})]\mathrm{d}G(\theta )<0, \end{aligned}$$

(23)

where the inequality follows from $R'(\theta )\ge (\le ) \ 0$ for all $\theta \in [\underline{\theta },\overline{\theta }]$ and Eq. (21). From Eq. (10), the right-hand side of Eq. (10) is negative so that $L^{\dagger }<L^{*}$. $\Box$

For intuition we use Eq. (16) to compare the ambiguity premium under $\phi (U)$ and that under $K\Big (\phi (U)\Big )$, both of which are evaluated at $L=L^*$:

$$\begin{aligned} \frac{\mathrm{Cov}_G\Big [\phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big ),R(\tilde{\theta })\Big ]}{\mathrm{E}_G\Big [\phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\Big ]} \end{aligned}$$

$$\begin{aligned} -\frac{\mathrm{Cov}_G\bigg [K'\bigg (\phi \Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ) \phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big ),R(\tilde{\theta }) \bigg ]}{\mathrm{E}_G\bigg [K'\bigg (\phi \Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ) \phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ]} \end{aligned}$$

$$\begin{aligned} =-\frac{\mathrm{Cov}_G\bigg [K'\bigg (\phi \Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ), \phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\Delta ^* \bigg ]}{\mathrm{E}_G\bigg [K'\bigg (\phi \Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ) \phi '\Big ([R(\tilde{\theta })-R_C]L^{*}-C(L^{*})\Big )\bigg ]}, \end{aligned}$$

(24)

with $\Delta ^* = R(\tilde{\theta })-R_C-C'(L^*)$ where we have used Eq. (10). It follows from Eq. (10) that the covariance term on the right-hand side of Eq. (24) is negative given that $R(\theta )$ is monotonic in $\theta$. In this case, the ambiguity premium is more negative under $K\Big (\phi (U)\Big )$ than under $\phi (U)$. Greater ambiguity aversion as such reduces the certainty equivalent marginal revenue of loans. $\square$

Proof of Proposition 3

Consider the function:

$$\begin{aligned} \Phi (\theta )=-\frac{\phi '' \Big ([R(\theta )-R_C]L^*-C(L^*)\Big )}{\phi ' \Big ([R(\theta )-R_C]L^*-C(L^*)\Big )}. \end{aligned}$$

(25)

Given that $R'(\theta )\ge (\le ) \ 0$ for all $\theta \in [\underline{\theta },\overline{\theta }]$, it follows from Eq. (10) that there exists a point, $\theta ^*\in (\underline{\theta },\overline{\theta })$, at which $R(\theta ^*)=R_C+C'(L^*)$. Since $-\phi ''(U)/\phi '(U)$ is a non-increasing function of U, Eq. (25) implies that $\Phi '(\theta )\le (\ge ) \ 0$ for all $\theta \in [\underline{\theta },\overline{\theta }]$ given that $R'(\theta )\ge (\le ) \ 0$ for all $\theta \in [\underline{\theta },\overline{\theta }]$. Using Eq. (10), we have

$$\begin{aligned} \int _{\underline{\theta }}^{\overline{\theta }}[\Phi (\theta )-\Phi (\theta ^*)] \phi ' \Big ([R(\theta )-R_C]L^*-C(L^*)\Big )[R(\theta )-R_C-C'(L^*)]\mathrm{d}G(\theta )<0. \end{aligned}$$

(26)

It then follows from Eq. (10) that the second term on the right-hand side of Eq. (20) is negative so that $\mathrm{d}L^{*}/\mathrm{d}\alpha <0$. $\square$