Impact of Pay-for-Performance on Rating Accuracy

Abstract

This study analyses whether the pay-for-performance scheme can encourage Credit Rating Agencies (CRAs) to issue accurate ratings under an investor-pay model. In our model, a CRA individually sets disclosure rules between biased rating and the full disclosure regime; an investor who solicits ratings, decides to acquire information accuracy. The CRA’s information production cost is compensated by a fixed fee, and incentive pay is tied to the portfolio outcome. Finding shows that the pay-for-performance scheme can efficiently motivate the CRA to adopt the full disclosure regime. Sufficiently high information acquisition level requested by the investor and relatively low incentive pay can induce the CRA to fully disclose rating information. The results reveal how the pay-for-performance scheme and the investor’s decision effectively influence the CRA’s behaviour of selecting rating policy.

Appendix

Proof of Proposition 1

To maximize The CRA’s expected profit from Eq. 4 knowing that the investor invests in the portfolio only if it gets a high rating, the CRA chooses to conduct full disclosure or a bias rating regime after observing the signal type of each loan portfolio:

Case 1: when the CRA observes $\emptyset = h$, the CRA decides whether to report a low rating (bias rating) or report a high rating (full disclosure). The CRA’s expected profit when $\emptyset = h$ is:

$$\mathop {\Pi }\limits_{CRA}^{\emptyset = h} = \left[ {\alpha_{g} q(i)\left( {1 - I_{h} (i)} \right)x_{g} (R) + \alpha_{b} \left( {1 - q(i)} \right)\left( {1 - I_{h} (i)} \right)x_{b} (R)} \right]\left( {1 - \beta } \right)$$

The CRA’s expected profit when a high rating is reported (Full disclosure $( I_{h} (i) = 0 )$):

$$\mathop{\Uppi }\limits_{\text{CRA}}^{{{\text{h}},{\text{FD}}}} = \left[ {{\upalpha }_{\text{g}} {\text{q}}( {\text{i}}){\text{x}}_{\text{g}} \left( {\text{R}} \right) + {\upalpha }_{\text{b}} \left( {1 - {\text{q}}( {\text{i}})} \right){\text{x}}_{\text{b}} \left( {\text{R}} \right)} \right]\left( {1 - {\upbeta }} \right)$$

The expected profit of CRA when a low rating is reported (Bias rating $(I_{h} (i) = 1 )$):

$$\mathop{\Pi }\limits_{CRA}^{h,BR} = C(i) - C(i) = 0$$

The CRA adopts the full disclosure regime when ${\Pi }_{CRA}^{h,FD} \ge {\Pi }_{CRA}^{h,BR}$ so that:

$$\left[ {\alpha_{g} q(i)x_{g} (R) + \alpha_{b} \left( {1 - q(i)} \right)x_{b} (R)} \right]\left( {1 - \beta } \right) \ge 0$$

Thus, $i \ge \bar{i}^{h} = - \left[ {\frac{{\alpha_{b} x_{b} (R)}}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} + \alpha_{m} } \right]$

The CRA adopts the bias rating regime when ${\Pi }_{CRA}^{h,FD} < {\Pi }_{CRA}^{h,BR}$ so that:

Thus, $i < \bar{i}^{h} = \overbrace {{\frac{{ - \alpha_{b} x_{b} (R)}}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}}}^{ \le 1} - \alpha_{m}$ while $\,\,\bar{i}^{h} \le 1 - \alpha_{m}$

Thus, the CRA observes $\emptyset = h$ and reports a high rating (the full disclosure regime) only if $i \ge \bar{i}^{h} = - \left[ {\frac{{\alpha_{b} x_{b} (R)}}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} + \alpha_{m} } \right]$. Otherwise, the CRA implements the bias rating regime.

Case 2: when the CRA observes $\emptyset = l$, the CRA decides whether to report a high rating (bias rating) or report a low rating (full disclosure). The CRA’s expected profit when $\emptyset = l$ is:

$$\mathop{\Pi }\limits_{CRA}^{\emptyset = l} = \left[ {\alpha_{g} \left( {1 - q(i)} \right)I_{l} (i)x_{g} (R) + \alpha_{b} q(i)I_{l} (i)x_{b} (R)} \right]\left( {1 - \beta } \right)$$

The expected profit of CRA when a low rating is reported (Full disclosure $I_{l} (i) = 0$):

$$\mathop{\Pi }\limits_{CRA}^{l,FD} = C(i) - C(i) = 0$$

The expected profit of CRA when a high rating is reported (Bias rating $I_{l} (i) = 1$):

$$\mathop{\Pi }\limits_{CRA}^{l,BR} = \left[ {\alpha_{g} \left( {1 - q(i)} \right)x_{g} (R) + \alpha_{b} q(i)x_{b} (R)} \right]\left( {1 - \beta } \right)$$

The CRA adopts the full disclosure regime when ${\Pi }_{CRA}^{l,FD} \ge {\Pi }_{CRA}^{l,BR}$ so that:

$$0 \ge \left[ {\alpha_{g} \left( {1 - q(i)} \right)x_{g} (R) + \alpha_{b} q(i)x_{b} (R)} \right]\left( {1 - \beta } \right)$$

$$i \ge \bar{i}^{l} = \frac{{\alpha_{g} x_{g} (R)}}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} - \alpha_{m}$$

The CRA adopts the bias rating regime when ${\Pi }_{CRA}^{l,FD} < {\Pi }_{CRA}^{l,BR}$ so that:

$$i < \bar{i}^{l} = \overbrace {{\frac{{\alpha_{g} x_{g} \left( R \right)}}{{\left[ {\alpha_{g} x_{g} \left( R \right) - \alpha_{b} x_{b} \left( R \right)} \right]}}}}^{ \le 1} - \alpha_{m} \quad {\text{while}}\,\bar{i}^{l} \le 1 - \alpha_{m}$$

Thus, the CRA observes $\emptyset = l$ and reports a low rating only if $i \ge \bar{i}^{l} = \underbrace {{\frac{{\alpha_{g} x_{g} \left( R \right)}}{{\left[ {\alpha_{g} x_{g} \left( R \right) - \alpha_{b} x_{b} \left( R \right)} \right]}}}}_{ \le 1} - \alpha_{m}$. Otherwise, the CRA adopts the bias rating regime.

Proof of Proposition 2

We will show the results for two cases: (1) a bad market $( \alpha_{g} x_{g} (R) + \alpha_{b} x_{b} (R) < 0)$ and (2) a good market $( \alpha_{g} x_{g} (R) + \alpha_{b} x_{b} (R) \ge 0 )$.

• (1) Suppose $\alpha_{g} x_{g} (R) + \alpha_{b} x_{b} (R)$ < 0. We learned from Proposition 1 that in this case, $\bar{i}^{l} < min\left\{ {0,\bar{i}^{h} } \right\}$. We will first show the result assuming that $\bar{i}^{h} \ge 0$, and will later on discuss that the same result also holds when $\bar{i}^{h} < 0.$ To characterize the optimal increase in accuracy, $i^{*}$, we will consider two intervals: (1.1) $0 \le i \le \bar{i}^{h}$ and (1.2) $\bar{i}^{h} < i \le 1 - \alpha_{m} .$

  • (1.1) Consider $0 \le i \le \bar{i}^{h}$. In this case, the CRA adopts the full disclosure regime if observing a low signal since $i \ge 0 > \bar{i}^{l}$, and adopts the biased rating regime if observing a high signal since $i \le \bar{i}^{h} .$ Hence, the investor’s expected profit from Eq. 5 is given by

    $$\mathop{\Pi }\limits_{INV}^{{\tilde{r} = h}} (i) = - C(i)$$

    where $\frac{{d{\Pi }_{INV}^{{\tilde{r} = h}} }}{di} = - C^{\prime}(i) < 0$ since $C(i)$ is an increasing function of $i$.

    Thus, the investor’s profit is monotonically decreasing in $i$ for $0 \le i \le \bar{i}^{h} .$

  • (1.2) Consider $\bar{i}^{h} < i \le 1 - \alpha_{m} .$ In this case, the CRA adopts the full disclosure regime if observing a low signal since $i \ge \bar{i}^{h} > \bar{i}^{l}$, and also adopts the full disclosure regime if observing a high signal since $i > \bar{i}^{h} .$ The investor’s expected profit from Eq. 5 is given by

    $$\mathop{\Pi }\limits_{INV}^{{\tilde{r} = h}} (i) = \mathop {\hbox{max} }\limits_{i} \left[ {\left[ {\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\left[ {i + \alpha_{m} } \right] + \alpha_{b} x_{b} (R)} \right]\beta - C(i)} \right]$$

    where

    $$\frac{{d{\Pi }_{INV}^{{\tilde{r} = h}} }}{di} = \left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\beta - C^{\prime}(i)$$

    and

    $$\frac{{d^{2} {\Pi }_{INV}^{{\tilde{r} = h}} }}{{di^{2} }} = - C^{\prime\prime}(i) < 0$$

    This shows that the investor’s profit is concave in $i$, with a unique solution to the first-order condition given by $i_{0} \text{ := }\frac{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\beta }}{2a}$.

Next, we will show that $\underline{\beta }$ = $\frac{{4a\bar{i}^{h} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}$ and $\bar{\beta }$ = $\frac{{2a\left( {1 - \alpha_{m} } \right)}}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}$ characterize the investor’s optimal $i$ as described in the proposition.

Suppose $\beta \le \frac{{2a\bar{i}^{h} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}.$ Notice that $\frac{{2a\bar{i}^{h} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} \le \underline{\beta }$ since $\overline{i}^{h} \ge 0$. With some algebra, one can check that $i_{0} \le \overline{i}^{h} .$ This implies that the investor’s expected profit for $\overline{i}^{h} < i \le 1 - \alpha_{m}$ is monotonically decreasing in $i$. It is also simple to check that the investor’s profit function is continuous at $i = \overline{i}^{h}$. Together with what we learned from (1.1) that the investor’s profit is monotonically decreasing in $i$ for $0 \le i \le \overline{i}^{h} ,$ we have that the optimal increase in accuracy, $i^{*} ,$ is 0. In other words, it is optimal to not acquire additional information accuracy in this case.

Suppose $\frac{{2a\overline{i}^{h} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} < \beta \le \underline{\beta } .$ It follows that $i_{0} > \bar{i}^{h}$, hence $i_{0}$ is the interior maximizer of the investor’s profit function for $\bar{i}^{h} < i \le 1 - \alpha_{m}$. However, notice that ${\Pi }_{INV}^{{\tilde{r} = h}} \left( {i = 0} \right) \ge {\Pi }_{INV}^{{\tilde{r} = h}} \left( {i = i_{0} } \right)$ since $\beta \le \underline{\beta }$. Hence, it is optimal to not acquire additional information accuracy $(i^{*} = 0)$.

Suppose $\underline{\beta } < \beta < \bar{\beta }.$ One can check that $\bar{i}^{h} < i_{0} < 1 - \alpha_{m}$. This implies that $i_{0}$ is the interior maximizer of the investor’s profit function for $\bar{i}^{h} < i \le 1 - \alpha_{m}$. We obtain that ${\Pi }_{INV}^{{\tilde{r} = h}} \left( {i = 0} \right) < {\Pi }_{INV}^{{\tilde{r} = h}} \left( {i = i_{0} } \right)$ since $\beta > \underline{\beta }$. Hence, it is optimal for the investor to choose $i^{*} = i_{0}$.

Lastly, suppose $\beta \ge \bar{\beta }$. It follows that $i_{0} \ge 1 - \alpha_{m}$. This implies the investor’s profit is monotonically increasing in $i$ for $\bar{i}^{h} < i \le 1 - \alpha_{m}$, and is maximized at $i = 1 - \alpha_{m} .$ Comparing the investor’s profit at $i = 0$ and $i = 1 - \alpha_{m}$, we obtain that ${\Pi }_{INV}^{{\tilde{r} = h}} \left( {i = 0} \right) < {\Pi }_{INV}^{{\tilde{r} = h}} \left( {i = 1 - \alpha_{m} } \right)$ since $\beta \ge \bar{\beta }.$ Thus, it is optimal for the investor to choose $i^{*} = 1 - \alpha_{m}$.

It remains to show that the same result holds when $\bar{i}^{h} < 0$. Suppose $\bar{i}^{h} < 0$. Then, we only need to consider one interval of $0 \le i \le 1 - \alpha_{m}$, where the investor’s profit is the same as that given in (1.2). Hence, the investor’s profit is concave in $i$ with $i_{0} = \frac{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\beta }}{2a}$ as the solution to the first-order condition. Notice that $i_{0} \ge 0$ since $\alpha_{b} x_{b} (R) < 0$. We have also seen earlier that $i_{0} < 1 - \alpha_{m}$ if and only if $\beta < \bar{\beta }$. This implies, $i^{*} = i_{0}$ if $\beta < \bar{\beta }$, and $i^{*} = 1 - \alpha_{m}$ if $\beta \ge \bar{\beta }$. Note also that $\underline{\beta } < 0$ since $\bar{i}^{h} < 0.$ Hence, the same result holds.

• (2) Suppose $\alpha_{g} x_{g} (R) + \alpha_{b} x_{b} (R) \ge 0$. We learned from Proposition 1 that in this case, $\bar{i}^{h} < min \left\{ {0,\bar{i}^{l} } \right\}$. We will first show the result assuming that $\bar{i}^{l} \ge 0$, and will later on discuss that the same result also holds when $\bar{i}^{l} < 0.$ To characterize the optimal increase in accuracy, $i^{*}$, we will consider two intervals: (1.1) $0 \le i \le \bar{i}^{l}$ and (1.2) $\bar{i}^{l} < i \le 1 - \alpha_{m} .$

  • (2.1) Consider $0 \le i \le \bar{i}^{l}$. In this case, the CRA adopts the full disclosure regime if observing a high signal since $i \ge 0 > \bar{i}^{h}$, and adopts the biased rating regime if observing a low signal since $i \le \bar{i}^{l} .$ Hence, the investor’s expected profit from Eq. 5 is given by

    $${\Pi }_{INV}^{{\tilde{r} = h}} = \mathop {\hbox{max} }\limits_{i} \left[ {\left[ {\alpha_{g} x_{g} (R) + \alpha_{b} x_{b} (R)} \right]\beta - C(i)} \right]$$

    where $\frac{{d{\Pi }_{INV}^{{\tilde{r} = h}} }}{di} = - C^{\prime}(i) < 0$ since $C(i)$ is an increasing function of $i$.

    Thus, the investor’s profit is monotonically decreasing in $i$ for $0 \le i \le \bar{i}^{l} .$

  • (2.2) Consider $\bar{i}^{l} < i \le 1 - \alpha_{m} .$ In this case, the CRA adopts the full disclosure regime if observing a low signal since $i \ge \bar{i}^{l} > \bar{i}^{h}$, and also adopts the full disclosure regime if observing a high signal since $i > \bar{i}^{h} .$ The investor’s expected profit from Eq. 5 is given by

    $${\Pi }_{INV}^{{\tilde{r} = h}} (i) = \mathop {\hbox{max} }\limits_{i} \left[ {\left[ {\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\left[ {i + \alpha_{m} } \right] + \alpha_{b} x_{b} (R)} \right]\beta - C(i)} \right]$$

    where

    $$\frac{{d{\Pi }_{INV}^{{\tilde{r} = h}} }}{di} = \left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\beta - C^{\prime}(i)$$

    and

    $$\frac{{d^{2} {\Pi }_{INV}^{{\tilde{r} = h}} }}{{di^{2} }} = - C^{\prime\prime}(i) < 0$$

    This shows that the investor’s profit is concave in $i$, with a unique solution to the first-order condition given by $i_{0} \text{ := }\frac{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\beta }}{2a}$.

Next, we will show that $\underline{\beta }$ = $\frac{{4a\bar{i}^{l} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}$ and $\bar{\beta }$ = $\frac{{2a\left( {1 - \alpha_{m} } \right)}}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}$ characterize the investor’s optimal $i$ as described in the proposition.

Suppose $\beta \le \frac{{2a\bar{i}^{l} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}}.$ Notice that $\frac{{2a\bar{i}^{l} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} \le \underline{\beta }$ since $\bar{i}^{l} \ge 0$. With some algebra, one can check that $i_{0} \le \bar{i}^{l} .$ This implies that the investor’s expected profit for $\bar{i}^{l} < i \le 1 - \alpha_{m}$ is monotonically decreasing in $i$. It is also simple to check that the investor’s profit function is continuous at $i = \bar{i}^{l}$. Together with what we learned from (2.1) that the investor’s profit is monotonically decreasing in $i$ for $0 \le i \le \bar{i}^{l} ,$ we have that the optimal increase in accuracy, $i^{*} ,$ is 0. In other words, it is optimal to not acquire additional information accuracy in this case.

Suppose $\frac{{2a\bar{i}^{l} }}{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]}} < \beta \le \underline{\beta } .$ It follows that $i_{0} > \bar{i}^{l}$, hence $i_{0}$ is the interior maximizer of the investor’s profit function for $\bar{i}^{l} < i \le 1 - \alpha_{m}$. However, notice that ${\Pi }_{INV}^{{\tilde{r} = l}} \left( {i = 0} \right) \ge {\Pi }_{INV}^{{\tilde{r} = l}} \left( {i = i_{0} } \right)$ since β ≤ $\underline{\beta }$. Hence, it is optimal to not acquire additional information accuracy $(i^{*} = 0)$.

Suppose $\underline{\beta } < \beta < \bar{\beta }.$ One can check that $\bar{i}^{l} < i_{0} < 1 - \alpha_{m}$. This implies that $i_{0}$ is the interior maximizer of the investor’s profit function for $\bar{i}^{l} < i \le 1 - \alpha_{m}$. We obtain that ${\Pi }_{INV}^{{\tilde{r} = l}} \left( {i = 0} \right) < {\Pi }_{INV}^{{\tilde{r} = l}} \left( {i = i_{0} } \right)$ since $\beta > \underline{\beta }$. Hence, it is optimal for the investor to choose $i^{*} = i_{0}$.

Lastly, suppose $\beta \ge \bar{\beta }$. It follows that $i_{0} \ge 1 - \alpha_{m}$. This implies the investor’s profit is monotonically increasing in $i$ for $\bar{i}^{l} < i \le 1 - \alpha_{m}$, and is maximized at $i = 1 - \alpha_{m} .$ Comparing the investor’s profit at $i = 0$ and $i = 1 - \alpha_{m}$, we obtain that ${\Pi }_{INV}^{{\tilde{r} = l}} \left( {i = 0} \right) < {\Pi }_{INV}^{{\tilde{r} = l}} \left( {i = 1 - \alpha_{m} } \right)$ since $\beta \ge \bar{\beta }.$ Thus, it is optimal for the investor to choose $i^{*} = 1 - \alpha_{m}$.

It remains to show that the same result holds when $\bar{i}^{l} < 0$. Suppose $\bar{i}^{l} < 0$. Then, we only need to consider one interval of $0 \le i \le 1 - \alpha_{m}$, where the investor’s profit is the same as that given in (2.2). Hence, the investor’s profit is concave in $i$ with $i_{0} = \frac{{\left[ {\alpha_{g} x_{g} (R) - \alpha_{b} x_{b} (R)} \right]\beta }}{2a}$ as the solution to the first-order condition. Notice that $i_{0} \ge 0$ since $\alpha_{b} x_{b} (R) < 0$. We have also seen earlier that $i_{0} < 1 - \alpha_{m}$ if and only if $\beta < \bar{\beta }$. This implies, $i^{*} = i_{0}$ if $\beta < \bar{\beta }$, and $i^{*} = 1 - \alpha_{m}$ if $\beta \ge \bar{\beta }$. Note also that $\underline{\beta } < 0$ since $\bar{i}^{h} < 0.$ Hence, the same result holds.