Credit Rating Game
Mutlu Celik (10207295)
Supervisor: dr. M.A.L. (Maurice) Koster Financial Econometrics
15-01-2018
Abstract
Credit ratings in the financial sector can be modelled and analysed with game theory. The decisions of the players in this credit rating game can be optimized by using game theory. The aim of this paper is to find the best strategies for the investor, firm and credit rating agency (CRA) in this game. In this paper a simple signalling game is analysed without a CRA, to analyse under which conditions the companies and investors can coordinate themselves. The extended model includes a CRA where two refinements are made in the game. Firstly, the CRA can make mistakes and/or inflating ratings. Secondly, the investor has not full information concerning the market. The investor does not know the actual state of the company based on the report, but makes beliefs by making use of Bayesian updating. The objective of this paper is to analyse whether these refinements in the extended model influence the outcome in the simple model. The conclusion is that CRAs are unnecessary in the market when reputation cost is sufficiently high. The firms communicate with the investors by signalling their true state. The CRA should be included in the game to help coordinate the firms and investors when reputation costs are low.
2 Statement of Originality
This document is written by Student Mutlu Celik who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
3
Table of Contents
1. Introduction 4 2. Literature Review 5 2.1 General Information 5 2.2 Conflict of Interest 6 2.3 Equilibria 7 3. Model 83.1 Game with Firm and Investor, without CRA 8
3.2 Extended Game with CRA included 10
4. Results 14
4.1 Game with Firm and Investor, without CRA 14
4.1.1 Separating equilibria 14
4.1.2 Pooling equilibria 16
4.2 Extended Game with CRA included 20
4.2.1 Partially Separating equilibria 20
4.2.2 Pooling equilibria 30
5. Conclusion 37
4 1 Introduction
Credit rating agencies (CRAs) analyse the creditworthiness of companies and their financial obligations. After the Financial Crisis, the CRAs have been criticized concerning their inflated ratings. The accounting and auditing scandals led to many questions concerning the ratings and their competence. One of the most famous example is the case of Enron (2001), which was caused by the conflict of interest in the CRA industry.
The main problem that arises concerning the CRAs, is that their revenue comes from the firms they are rating. The ratings are inflated to stimulate the rated firms buying the ratings, and thereby the CRA collects fee. CRAs focus on rating debt products and credit derivatives, which generate the biggest share of their revenues. Almost 90 percent of credit rating agencies’ revenues comes from company fees (Benmelech & Dlugosz, 2010). The fees that CRAs collect vary depending on the complexity and size of the issue.
The CRAs acknowledges that conflict of interest arises from the fact that firms have to pay for the ratings. According to the CRAs, they are well capable to manage the conflicts. Before the Financial Crisis, the CRAs profits rapidly increased, for instance Moody’s profits almost tripled between 2002 and 2006 (Bolton et al., 2012). After the Financial Crisis, there were large number of downgrades by the CRAs. The CRAs relaxed the rating standards before the crisis, which was tightened after the crisis by the large number of downgrades. The downgrades were caused by the higher rating standards implemented by the CRAs.
Another conflict of interest is the issuers’ ability to purchase the highest rating. The firm has the possibility to buy the report or try another CRA, which causes back and forth negotiations (Bongaerts et al., 2012).
Lastly, an important conflict of interest is the trusting of the investors concerning the ratings. Some of the investors do not understand the conflict of interest of the CRAs. They follow the rating of the CRA, which is not always advisable. CRAs credit models may vary in precision, which leads to imperfect analysis of default risk (Bolton et al., 2012). Most of the investors were not able to see through imperfect ratings, as in the Financial Crisis (Taylor, 2009).
This paper formalizes the situation in a game and the focus will be on analysing theoretical focal points, i.e. equilibria of this game. This paper is organized as follows; the second section gives an introduction into the subject, reviews related literature and equilibria. The third section
5 describes the games followed by the fourth section which gives the results of the models. Finally, the fifth section gives the conclusion.
2 Literature review
This section starts with providing information concerning CRAs, investors and firms. Furthermore, literature related to conflict of interest in credit rating will be reviewed. Lastly, equilibria found in literature concerning credit rating games will be analysed.
2.1 General information
There is substantial literature on CRAs in microeconomics. The CRA has an important role in the financial industry. CRAs basically provide two services. First, they offer an independent analysis of the capability of companies to meet their debt obligations. The rating of a CRA only refers to the credit risk, other risks such as market and liquidity are not covered. CRAs frequently provide different ratings for the same company. In De Haan (2011) there are three explanations mentioned for these differences. The first explanation is the difference in usage of factors and weights in the models for the CRAs. Second explanation, the rating agencies may disagree more about speculative-grade rated companies. Last explanation, some CRAs rate more favourable in their home region compared to other areas (De Haan, 2011).
The second service of the CRA relates to the monitoring services, where the companies are informed about potential downgrades. The CRA points out the aspects that the company should change, otherwise it will get a downgrade. The CRA signals in advance their plans to change the rating. These signals are considered to be strong predictors of rating changes relative to other data. These ratings play an important role in financial markets as investors use them to determine the credit risk of the related company. The ratings are considered as a good tool to measure credit quality and are positively associated with ratings accuracy (Frost, 2007). Analysing these credit risks is time consuming, so investors mainly rely on these ratings. However, the CRAs may vary in precision in rating a firm (Bolton et al, 2011). The CRAs provide imperfect analysis of default risk. The CRAs also inflate ratings in good economic environment. Reputational costs discipline the CRAs when inflating the ratings. However, the value of reputation depends on economic indicators that vary over the business cycle. Heski and Joel (2012) conclude that rating accuracy is countercyclical, CRAs are more likely to issue less accurate ratings when revenues are high from
6 fees. The competition in the labor market for analysts is then tough and the default probabilities for the investments are low.
The rating has an important effect on the interest rate that borrowers have to pay. Downgrading leads to a higher interest rate on the loan. Portfolios are sometimes benchmarked against standard indices that are constructed on credit ratings. This implies that a downgrade to below the investment-grade threshold triggers immediate liquidation, which leads to selling the downgraded firm in the portfolio. This behaviour may increase market volatility and may cause a downward spiral of asset prices with negative performance of the firm as consequence. Credit ratings have a significant effect on financial markets, as they affect bond prices (Hill et al., 2010), and also affect the stock markets. Economic indicators, including ratings, are priced in the stock price as suggested in Chen et. Al (1986). Brook et al. (2004) analyse that rating downgrades have a strong negative impact on stock returns, but limited upside potential related to upgrades.
Financial managers take actions concerning capital structure to target minimum credit rating levels over time. This is done for the benefits related to the higher credit rating. The capital structure decisions are more based on firm’s credit rating change, than by change in profitability and leverage (Kisgen, 2009). In Goh and Ederington (1993) the effect of downgrades on stock returns is analysed. The researchers conclude that downgrades associated with deteriorating financial prospects affect stock returns negatively. However, downgrades associated to change in firm’s leverage do not.
2.2 Conflict of interest
The CRA has an important role for the functioning of the financial market, the ratings therefore should be objective and reliable. The first conflict of interest is that the CRAs’ revenue mainly comes from the firms positively rated. The main source of revenue comes from the companies which are rated by the CRAs. This leads to overstating the creditworthiness of a company in order to build a good relationship with the entity. On the other hand, the CRAs must guard their reputation in the market, otherwise the ratings would be worth less, which leads to a trade-off for the CRA between a short-term gain from the fee and the long-term reputation of the CRA. By underrating the risk, the CRA attracts business and therefore his revenue increases.
This conflict of interest caused many crises such as failures of Enron (2001), Worldcom (2002) and the recent Financial Crisis (2008). The profits of the CRAs grow extraordinary with
7 the growth of structured products, for example Moody’s profits tripled between 2002 and 2006. From 2007 onwards there were large number of downgrades, which created distrust in rating standards that had been relaxed during the years before (Brunnermeier, 2009). CRAs may inflate the rating of the investment when CRAs’ expected reputation cost are lower (Bolton et al., 2012). CRAs have been criticised for responding with a great time lag. The ratings were not downgraded when the problems in the sub-prime market became clear, the agencies were slow in adjusting during the Financial Crisis in 2008.
The second conflict of interest is that the company can purchase the most favourable ratings. The company can make a choice whether to buy the rating after having analysed the report. There are back and forth negotiations when CRAs are making reports. The company wants to get the highest rating possible from the CRAs, this leads to competition between the agencies to give the highest rating which results in earning the fee. The CRAs are criticized for being biased in the rating, for selling the report in favour of companies (Bolton et al., 2012).
The last conflict of interest is that the investors do not know the true state of the company. CRAs play an important role in the functioning of the capital markets. Investors in general do not have the resources and expertise to analyse complex financial products. The investors rely on the information provided by the CRA (Bolton et al., 2012).
2.3 Equilibria
Bolton et al. (2009) analyse a credit rating game, whereby is concluded that competition between CRAs can reduce efficiency. The reduction in efficiency is caused by the stimulation of ratings shopping. Bolton et al. (2009) also conclude that ratings are more likely to be inflated in good economic environments and when investors are more relying on the ratings. These aftereffects are created by the conflict of interest arising in these credit rating games. The first conflict of interest is that the CRA is understating the risk to earn the rating fee. Secondly, companies are able to purchase the most favourable ratings. Lastly, the relying nature of the investors causes these aftereffects.
In Boot et. al (2005) the CRAs are seen as coordination mechanisms. The CRAs provide a focal point for firms and investors. The CRAs help realizing the desired equilibrium and therefore they have economically an important role. Boot et. Al (2005) also analyse whether investors could replicate the role of the CRA. The problem is that the costs are high, when single investors replicate
8 the CRA role. Collective investors are also not able to fulfil this role, because of the free-rider problem that would arise.
In Mariano (2008) the behaviour of the CRA is analysed. The analysis is about incentives to issue a rating which is not correct according to the information received by the CRA. In the model the reputational concerns are not enough to prevent the CRA inflating the report. The CRA in this setting issues too many false ratings which are biased upwards, ignoring the true state of the companies. CRAs are more biased when they are acting in a competitive market. The ratings in this competitive market are more biased, making sure the rating fee is earned by the agency.
3 Model
In this paper, the elements explained earlier are combined in a game. The advanced and realistic model for credit rating is a more complicated game and therefore we start with a simpler game. The first game we are going to consider, is a game where we leave out the CRA. This game is simpler and the equilibrium in this game gives important applications. This could be compared with the game where the CRA is included. By comparing the equilibria of the models, one can conclude what the role of the CRA is in a game with a firm and investor. In section 3.1 the simple game without a CRA will be formally introduced. Section 3.2 will contain an extension of the simple game, where the CRA will be included.
3.1 Game with firm and investor, without CRA
In the first game, there are two players: a firm (F) and an investor (I) with two strategies each. Formally the set of players is denoted as follows: N={F,I}. The game has a Bayesian set up and the Harsanyi model common prior assumption is followed (Harsanyi, 1967). By the move of the player ‘Nature’, there is a good quality company or a bad quality company in this game. The firm then decides whether it distributes a report with good quality (G) or poor quality (P). The investor analyses this report and decides if not to invest in this company. However, the investor does not know the true type of the firm. The decision of the investor is based on the report. The firm knows his true type and distributes a report. The payoff of the investor depends on whether the investment product is purchased or not and the quality of the firm. In this game, it is assumed that the investment costs are T or W depending on the outcome of the report. It is assumed that π and the investment costs are greater than zero. The default probability of the bad quality firm is 1-p, where
9 0 < p < 1. In case that the investor did not purchased the investment, the payoff for both players in this game is zero.
The assumptions on the payoffs and parameters are justified by the following reasons:
- The cost of production of the investment is normalized to zero. The price T and W can be interpreted as the profit of the firm. The investor can only buy one investment product. In this game it is assumed: π-W > 0, π-T > 0 and pπ-W < 0, pπ-T < 0, where T > W.
- Investment in a good quality company defaults with probability zero, where an investment in a bad quality company defaults with probability 1-p > 0. Both types of investments yield the return π when not default, and 0 in default.
- The cost K are so called ‘’reputation cost’’. These costs are caused by punishment of other players related to playing strategies that are not desirable, as discussed in Nelissen (2008). In case of this game, the company gets punished by reputation cost K, when the firm is of bad quality but distributes a good report and the investor buys this investment product. The firm is not punished for distributing a poor report when the actual state is good quality. The firm in this case is punishing himself, by not signaling that the firm is of good quality. The investors in this case are positively surprised by buying a good quality investment for the bad quality investment price W.
- The payoff in the game is as following: ( 𝐹𝑖𝑟𝑚 𝑝𝑎𝑦𝑜𝑓𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟 𝑝𝑎𝑦𝑜𝑓𝑓)
I formalize the elements described earlier in a signaling game below, as described in Harsanyi
10 Short description of the strategies of both players, in the signaling game:
G: The firm signals that it is a good firm (F), distributing a good report.
P: The firm signals that it is a poor firm (F), distributing a poor report.
B: The investor (I) buys the investment.
N: The investor (I) does not buy the investment.
3.2 Extended Game with CRA included
In the second game, there are two strategic players where we include the effect of the CRA in the game. In this game there are four types of firms, namely good quality or bad quality firms and those firms can have good or bad report as outcome of the analysis of the CRA. Summarizing, there are four types of firms:
FG,G = good quality firm with good report.
FG,B = good quality firm with bad report.
FB,B = bad quality firm with bad report.
FB,G = bad quality firm with good report.
The firm has the possibility to analyse the report, before buying and distributing this to the investor. By doing this we include the negotiating effect, where the firm and CRA meet with each
11 other to come to a report which the firm is willing to buy. The firm can choose to rate (R) or not rate (U), depending on the report the CRA is offering. When the report is not bought, the only signal the investor gets, is that the firm did not buy the report and distributed. The investor in this case, does not know which report is not bought, good or bad quality since buying a report is costly by the firm. In the next stage the investor then decides to buy the investment or not. This is done by analysing the report if available to the investor.
Summarizing, the timing of the game is as following:
1. By nature there is a good quality company or bad quality company. The company is also being analysed by the CRA with a report as outcome.
2. The company decides whether to buy and distribute this report or not. The firm does know his actual state.
3. The investor observes the report if distributed and decides to buy the investment or not. 4. The return is realized.
In this game, it is assumed that the investment costs are T or W, depending on the quality of the report that is distributed. The company can ask price Q for an unrated investment product from the investor, where T > W > Q. It is assumed that π and investment costs are greater than zero and 0 < p < 1, where 1-p is the default probability of a bad quality firm. When the report is bought the firm pays 𝜙 to the CRA. In case the investor did not purchased the investment, the payoff for the investor is zero.
In this game, there are assumptions made concerning the strategies that are available to the firm. Firstly, when the firm decides not to buy the report the only information that the investor observes, is that the firm did not buy the report. The investor has no report and could be facing one of the four types of firms. When the firm buys a report and distributes this, the investor must take into account that the firm could be of good quality or bad quality, despite the nature of the report. The report serves as a signal, but there is no a priori 1-1 relationship.
The good quality firm rationally wants to signal that it is a good quality company. This company wants to distinguish himself from a bad quality company and is willing to buy the good report, depending on the price. Namely, when the price of the report is high, the signal is not worth buying. The report is then too expensive to be profitable for the firm, even for the good quality firm. When the price of the report decreases slightly, it will serve as strong signal since poor firms cannot afford the report. The price of the report can drop to such a level, that the poor-quality
12 company can hide behind a good report. The good quality companies also signal their true state through a report. The signal becomes less important when the price of the report becomes very cheap. When the price of the report becomes too cheap, the signal is not distinguishing between types of firms. Buying the report has no added value for the firms to distinguish themselves from other types, namely every type of company can buy the report. Lastly, the firm knows α, the distribution of good and bad quality companies.
Another important feature in this model are the type I (𝜀) and type II (𝜁) errors. In this game the type I error, means that a bad quality firm distributes a good report. There are two reasons why the type I error could occur, namely by mistake of the CRA. The firm could have been hiding bad numbers to present himself as a good quality firm (Gurau and Grigore, 2016). Another possibility is that the CRA is inflating the rating on purpose to sell the report. This means that the error of type I could be higher than expected in this game, because of lower reputation costs in high economic grow environment (Bolton et al., 2012). These lower costs incentivize the CRAs to inflate the ratings and increasing the type I error in this game. The type II error occurs when the CRA is cautious in rating, meaning that the good quality firm is rated as a poor firm (Dittrich, 2007).
The believes of the investor are not the same as in the first simple model. We need to take into account the type I and type II errors to calculate the expected returns of both players. In case of a good report there are two scenarios possible. Firstly, the company analyzed by the CRA is truly a good quality company. This means that the report reveals the state of the company. Secondly, the company could be of bad quality but distributed a good report, which leads to type I errors. This could be the case when bad quality firms hide bad numbers to present themselves as a good company (Gurau and Grigore, 2016). Good quality companies can also get a bad quality report, which leads to type II errors. This is the case when CRAs are cautious in their rating (Dittrich, 2007).
Short description of the strategies of both players, in the signaling game:
R: The firm (F) signals the report received by the CRA to the investor.
U: The firm (F) signals no report. The investor receives no report.
13 N: The investor (I) does not buy the investment.
The payoff in the game is as following: ( 𝐹𝑖𝑟𝑚 𝑝𝑎𝑦𝑜𝑓𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑜𝑟 𝑝𝑎𝑦𝑜𝑓𝑓)
I formalize the elements described earlier in a signaling game below, as described in Harsanyi (1967):
14 4 Results
Signalling games can have three types of equilibria: pooling equilibria, separating equilibria and hybrid equilibria. We are not looking for hybrid equilibria since it is outside the scope of this research. In the sections that follow, the possible equilibria in the different games are discussed.
4.1 Game with firm and investor, without CRA 4.1.1 Separating equilibria
A separating equilibrium is an equilibrium in which the different types of firms send out different signals. For the first game, there are two possible equilibria, namely the intuitive and unintuitive separating equilibrium.
The intuitive separating equilibrium is established when the good quality firm sends out a good report (G) and a poor report (P) when it is a bad quality company:
𝛽𝐹(𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝐺, and 𝛽𝐹(𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝑃.
This induces that consistent beliefs: = 1 and q = 0. This equilibrium holds if both types of firms have no incentive to deviate from their signal. The intuitive separating equilibrium is illustrated in figure 3, by the green lines.
Theorem 1:
The intuitive separating equilibrium exists if T-K < 0.
Proof of Theorem 1:
Given that the company is of good quality, the firm sends out a good report. The best response of the investor is to buy the investment, because π-T > 0. When a bad quality company distributes a poor report the best response for the investor is to not buy the investment, because pπ-W < 0, resulting in a payoff of zero for both players in this part of the game.
A good quality company has no incentive to change his signal to poor report (P) when T > 0, which is the case given the assumptions in the game. The good quality firm is strictly better off to send good report than bad report. When the good quality company distributes the signal poor report, the investor beliefs he is facing a bad quality company and plays strategy N. This results in a payoff 0 for the firm, where the signal G would result in payoff T for the firm. For all
15 strategies of the investor, the strategy of sending good report results in a higher payoff for the good quality firm then a poor report.
The bad quality firm has no incentive to change his signal. When the bad quality firm sends a good report, the investor buys this investment, assuming he is facing a good quality firm. This strategy results in a payoff of T-K for the firm. The intuitive equilibrium holds if T-K< 0, meaning that the punishment of deceiving the investor should be at such a level, that the firm is better off not selling the investment when signaling P. The firm in this case is getting punished by the fact that he was lying about his true state, by K the reputation costs. This means that the intuitive equilibrium holds if: T-K < 0. In this case the CRA would not be efficient, namely the equilibrium: 𝛽𝐹(𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝐺 and 𝛽𝐹(𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝑃 holds, without interruption of
the CRA. The companies are signalling their true states and the good investments are bought, so the CRA is unnecessary in the intuitive separating equilibrium. The firms and investors
coordinate themselves to desirable equilibria in which the good quality investments are bought.
Theorem 2:
16 The unintuitive equilibrium is established when the good quality firm sends out a poor report (P) and a good report (G) when it is a bad quality company: 𝛽𝐹(𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝑃 and
𝛽𝐹(𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝐺. The firm is constantly lying about his true state and is sending a different signal to the investor than his true state. This indicates with consistent beliefs that = 0 and q = 1. The unintuitive separating equilibrium is illustrated in figure 4, by the green lines.
Proof of Theorem 2:
Given that a good quality company sends out a poor report (P). The best response of the investor is to buy the investment, because π-W > 0. This strategy results in a payoff W for the firm. In case of a bad quality firm, the company distributes a good report (G). The best response of the investor in this case, is not to buy the investment, namely pπ-T < 0. A good quality firm has no incentive to change his strategy, namely when signaling a good report, the investor assumes that the company is of bad quality and plays strategy N. This results in a payoff of zero for both players, which leads to a decrease in payoff for the firm from W to 0 when deviating from the original strategy.
The bad quality firm has an incentive to change his signal from G to P because this increases his payoff. When the bad quality firm plays strategy P, the investor beliefs that he is facing a good quality firm and plays strategy B. This strategy results in a payoff of W instead of zero for the firm, which means that the firm is better off deviating from the original strategy. This means that the unintuitive separating equilibrium does not hold.
4.1.2 Pooling equilibria
A pooling equilibrium is an equilibrium in which the different types of firms always send the same signal, good or poor report. This signal therefore provides no information to the investor, because the signal is always the same. The investor adapts his belief to the probabilities of in which Nature provides good or bad quality firms.
The first pooling equilibrium to be considered is when the firm sends out good report independent of his true state. Consistent beliefs for the receiver = and 1- = 1-. The pooling equilibrium, where the firm signals good report is illustrated in figure 5, by the green lines.
17 Theorem 3:
The pooling equilibrium where the firm signals good report exist if T-K > W and (𝜋 − 𝑝𝜋) + 𝑝𝜋 − 𝑇 > 0.
18 Proof of Theorem 3:
The investor buys the investment if the following equation holds:
(𝜋 − 𝑇) + (1 − )(𝑝𝜋 − 𝑇) > 0, which is rewritten as: (𝜋 − 𝑝𝜋) + 𝑝𝜋 − 𝑇 > 0.
This is the case when (𝜋 − 𝑝𝜋) is more positive then the negative term 𝑝𝜋 − 𝑇. The bad quality firm has a payoff T – K when the investment is bought, where the payoff from the poor signal (P) would lead to payoff W, which makes the firm better off deviating, if W > T-K. The bad quality firm changes his strategy if the reputation cost punishes him enough. This means that this pooling equilibrium holds if the reputation cost are low enough, namely then T-K > W and the bad quality firm has no incentive to deviate from his original strategy. The parameter , proportion good quality companies, should be high enough for the equation above to be satisfied. The good quality firm has no incentive to deviate from his original strategy, namely T > W. Concluding, the pooling equilibrium holds where the firm signals good report if T-K > W and equation:
(𝜋 − 𝑝𝜋) + 𝑝𝜋 − 𝑇 > 0 is satisfied.
Theorem 4:
The pooling equilibrium where the firm signals poor report does not exist if T > W.
The second possible pooling equilibrium is that the firm always sends out poor report (P). Consistent beliefs for the receiver are q = and 1-q = 1-. The pooling equilibrium, where the firm signals poor report is illustrated in figure 6, by the green lines.
Proof of Theorem 4:
The investor buys the investment if the following equation holds: (𝜋 − 𝑝𝜋) + 𝑝𝜋 − 𝑊 > 0,
which is the case when (𝜋 − 𝑝𝜋) is more positive then the negative term 𝑝𝜋 − 𝑊. This equilibrium does not hold, namely the good quality firm has an incentive to change his strategy to G, given that T > W, which is in common with Chen et. al (1986). The price of a good rated investment vehicle should be higher than a bad rated investment, which is confirmed by the
19 condition T > W. The good quality company can ask a higher price for the investment when signalling G to the investors. Concluding, the pooling equilibrium where the firm signals poor report does not hold.
Conclusion equilibria simple model without CRA:
The intuitive separating equilibrium holds where the good quality firm sends out a good report (G) and a poor report (P) when it is a bad quality company:
𝛽𝐹(𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝐺, and 𝛽𝐹(𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦) = 𝑃.
In this game, the intuitive separating equilibrium holds where the CRA seems inefficient to include. In this game where we have two players, the investor and firm, reach an equilibrium where the true state of the company is signalled to the investor. There is no need for an CRA to clarify the state of the company, that would be inefficient in this case. The unintuitive
equilibrium does not hold given that W > 0, which was one of the assumptions made beforehand the game.
The pooling equilibrium holds where the firm signals good report if T-K > W and equation:
20 is satisfied. For this equilibrium to hold it is of great importance that the reputation cost is low enough and the proportion of good quality companies is high enough. The pooling equilibrium where the firm signals bad report does not hold. The criteria for this equilibrium to hold is that W > T, which contradicts the model assumption that T > W.
Concluding, in this simple model without CRA the intuitive separating equilibrium holds if the reputation cost is high, namely then T-K < 0. The pooling equilibrium with good report signalling holds when the reputation cost is low, namely then T-K > W.
4.2 Extended Game with CRA included 4.2.1 (Partially) Separating equilibria
The first equilibria that are going to be analyzed are the partially separating equilibria. In this extended game the different types of firms are going to send different signals. We need to take into consideration that these equilibria are partially separating equilibria, namely there are 4 types of firms and 2 signals. In this game we are going to focus on two partially seperating equilibria, namely the intuitive and the unintuitive equilibrium.
The intuitive partially separating equilibrium is established when the different types of firms send the following signals:
𝛽𝐹𝐺,𝐺( 𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑔𝑜𝑜𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑅 𝛽𝐹𝐺,𝐵( 𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑏𝑎𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑈
𝛽𝐹𝐵,𝐵( 𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑏𝑎𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑈 𝛽𝐹𝐵,𝐺( 𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑔𝑜𝑜𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑅
The intuitive partially separating equilibrium equilibrium is illustrated in figure 7, where the red lines are the strategies of the firms. The company FG,G intuitively wants to play the strategy R
and buys the report. The company is of good quality and is prepared to buy the report to signal that the company is trustworthy to invest in. The good quality firm which is rated as a poor company by the CRA will not buy the report. The good quality company will not signal that the firm is of bad quality, which is counterintuitive from a rational point of view. The bad quality firm will not buy a bad report to ensure the investors, that the entity is of bad quality and make costs by paying the fee. The bad quality firm considers buying a report, when this states that the
21 company is of good quality. The decision of this type of firm depends on the reputation costs and report costs.
The company of type FG,G plays the strategy R in the intuitive equilibrium. The same is
the case for type FB,G , which also plays the strategy R in the intuitive equilibrium. We are going
to analyse under which circumstances this equilibrium holds. The problem the investor is facing, is that he can be in two different states. The good report could be a result of a good quality company or by error of type I a bad quality company, which is illustrated in figure 7 by the green dotted line trough node 1 and 3.
22 Theorem 5:
The firms FG,G and FB,G are signalling that they are of good quality and have no incentive to deviate
if the following two equations are satisfied:
𝛼(1−𝜀) 𝜀(1−𝛼)+𝛼(1−𝜀)(𝜋 − 𝑇) + 𝜀(1−𝛼) 𝜀(1−𝛼)+𝛼(1−𝜀)(𝑝𝜋 − 𝑇) 0 (1) Q < T- 𝜙 –K. (2) Proof of Theorem 5:
The investor makes his choice based on the expected payoffs calculated by making use of beliefs. Analysing figure 7 and by making use of Bayesian updating the investors belief in node 1:
P[FG | report states good firm]=
𝛼(1−𝜀) 𝜀(1−𝛼)+α(1−ε) .
The investors belief that he is in node 3:
P[FB | report states good firm]=
𝜀(1−𝛼) 𝜀(1−𝛼)+α(1−ε).
The investors best response in node 1 and 3 is to buy the investment product when the equation below is satisfied. This part of the equilibrium holds when the following equation is satisfied:
𝛼(1−𝜀)
𝜀(1−𝛼)+α(1−ε)(π − 𝑇) +
𝜀(1−𝛼)
𝜀(1−𝛼)+α(1−ε)(𝑝π − 𝑇) 0.
where the equation above can be simplified further given that 𝜀, 𝛼 ∈ (0,1) to 𝛼(1 − 𝜀)(π − 𝑇) + 𝜀(1 − 𝛼)(𝑝π − 𝑇) > 0
The equation definitely holds when the investor knows for sure that the CRA does not inflate ratings and does not make mistakes in these ratings, which means that 𝜀 = 0. The equation above then simplifies to π− 𝑇 > 0. When the investor buys the investment based on the good report, he knows for sure that this is a good quality company, a prior relationship of 1-1 when 𝜀 = 0. This results in an investment payoff π− 𝑇 for the investor.
When 𝜀 = 1, the CRA is inflating ratings and or making mistakes in ratings. In this case, only bad quality companies receive a good report from the CRA. The equation above for the equilibrium to hold, then simplifies to 𝑝π− 𝑇< 0. Now the investor decides not to buy the investment, which results in the payoff zero for the investor and – ϕ for the firm. The firm is then strictly better off to play strategy U.
23 In case of high 𝜀, the good quality firm buys the report, signalling that the company is a good investment. The bad quality company also buys the good report. The investor decides not to buy the investment because of the high 𝜀, fearing to face a bad quality company with a negative payoff. Therefore the 𝜀 should not be too high that equation 1 of Theorem 5 is not satisfied. The 𝜀 should be at such a level that it is lucrative for the bad quality firm to buy the good report, and profitable enough for the investor to buy the investment product.
The level of 𝛼 is also important for the equation to hold, namely for high 𝛼 the investor is incentivized to buy an investment which distributed a good report. The probability of truly facing a good quality company is high when 𝛼 is high. However when the 𝛼 is low, the probability of facing a good quality company is low. The investor consequently does not buy the investment. Further in the derivation we assume that the equation
𝛼(1−𝜀)
𝜀(1−𝛼)+α(1−ε)(π− 𝑇) +
𝜀(1−𝛼)
𝜀(1−𝛼)+α(1−ε)(𝑝π− 𝑇) 0
is satisfied. When this equation is not satisfied the investor is better off not buying the investment. Concluding, the investors buy the investment offered by types FB,G and FG,G after
signalling G, assuming equation 1 of Theorem 5 holds.
However, the firms FG,G and FB,G should have no incentive to deviate in equilibrium. For
the companies to deviate we assume that FG,G and FB,G send out the signal U instead of R. The
investors belief that they are facing the firms FG,B and FB,B when the companies FG,G and FB,G
play the strategy U, therefore the investors play the strategy B, which is proven later on in proof of Theorem 7.
The firms earn in case of type FG,G, payoff Q instead of T- 𝜙. The company FB,G earns
the payoff Q instead of 𝜙. The firms deviate when Q > T- 𝜙, which also implies Q > 𝜙. The report cost and reputation cost should be at such a level that the equation Q < T- 𝜙 -K is satisfied, for the firms FG,G and FB,G not to deviate. This states that for type FB,G the cost of the
report and reputation cost should be low enough to inflate the ratings, which is in accordance with Bolton et al. (2012). Inflated ratings are more likely when the reputation cost is low. If the reputation cost is high, the firm is better off not buying the report and asking the unrated
investment price Q. The cost of the report could also imply that the firm FG,G buys the report, but
it is too expensive for the FB,G to buy the good report. This would mean that this equilibrium
24 Theorem 6:
The firms FG,B and FB,B are signalling no report and have no incentive to deviate if the following
two equations are satisfied:
(𝜋 − 𝑝𝜋) + 𝜌𝜋 − 𝑄 > 0 (1)
Q > W – 𝜙 (2)
Proof of Theorem 6:
The investors receive no report, from the companies FB,B and FG,B. The believes are
somewhat different from the believes in node 1 and 3, namely the investors only know the 𝛼. The investor believes that he could be in the nodes passing through the orange dotted line, namely node 2,4,6 and 8 in figure 7. The investor believes he could be facing one of the four types. The beliefs in this case are:
P[FG | U] = 𝛼 , and
P[FB | U] = 1 − 𝛼,
which are the beliefs in respectively node 6 and 8. The investor buys the investment when the following equation holds:
(𝜋 − 𝑝𝜋) + 𝑝𝜋 − 𝑄 > 0.
The equation is satisfied if term (𝜋 − 𝑝𝜋) is more positive than the negative term 𝑝𝜋 − 𝑄. The is very important for this equation to hold. The is the only information the investor has concerning the investment, when there is no report of the firm. The higher the , the more the investor is incentivized to buy the investment. For the equation above to hold, the should be high enough and the default rate on bad quality firms should be low enough to make the equation above positive. Namely, we can write the equation above as the following equation:
> 𝑄 − 𝑝𝜋 (1 − 𝑝)𝜋
which is the threshold for the equation to hold such that the investors buy the investment from the types FG,B and FB,B.
The firms FG,B and FB,B should have no incentive to deviate for the intuitive equilibrium
25 motivated to buy the bad report. The companies FG,B and FB,B are better off not rating because of
the report cost.
Concluding, the partially intuitive separating equilibrium does hold if Theorem 5 and 6 holds, which is the case. The companies FG,G and FB,G have no incentive to change their signal
from R to U. The firm FB,G buys the report to ask the higher investment price if 𝑇 − 𝐾 − 𝜙 > 𝑄.
The companies FB,B and FG,B have no incentive to change their strategy from U to R if the
condition 𝑄 > 𝑊 − 𝜙 holds. The report costs in this case are at a level, that the reports are affordable for the firms FG,G and FB,G, but not interesting for firms FB,B and FG,B. The actions of
both players in the partially intuitive separating equilibrium are illustrated in figure 8 by the redlines.
The next possible equilibrium that can hold is the partially unintuitive separating equilibrium. This equilibrium is called unintuitive because the companies are playing strategies that are counterintuitive. For example, the company FG,G should buy the report to reveal his true
state from a rational point of view. In this equilibrium, the company FG,G is not buying the report
to reveal this to the investors. The company FB,B reveals his true state and buys the report to
ensure the investors that the firm is of bad quality. The bad quality company does not buy the good report when an error of type I is made, so the company FB,G is never lying about his state. However, the good quality company is buying a bad report, which is not rational. This could be seen as a cautious rating, with upside potential for the investor. This equilibrium is established when the different companies send the following signals:
26 𝛽𝐹𝐺,𝐺(𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑔𝑜𝑜𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑈
𝛽𝐹𝐺,𝐵(𝑔𝑜𝑜𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑏𝑎𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑅 𝛽𝐹𝐵,𝐵(𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑏𝑎𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑅 𝛽𝐹𝐵,𝐺(𝑏𝑎𝑑 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑓𝑖𝑟𝑚 𝑎𝑛𝑑 𝑔𝑜𝑜𝑑 𝑟𝑎𝑡𝑒𝑑) = 𝑈
The unintuitive equilibrium is illustrated in figure 9, where the red lines are the strategies of the firms.
27 Theorem 7:
The firms FG,G, and FB,G are signalling no report if the following two equations hold:
(𝜋 − 𝑝𝜋) + 𝜌𝜋 − 𝑄 > 0 (1)
𝑇 − 𝜙 < 𝑄 (2)
Proof of Theorem 7:
In nodes 2 and 4 the investor is in the same situation, namely the company did not distribute a report. In the nodes 2 and 4 the investor believes he could be in node 2,4,6 or 8, which is illustrated by the orange dotted line in figure 9. The investor has no additional information received from the company. The beliefs in the nodes 2 and 4 are respectively:
P[FG | U] = 𝛼, and
P[FB | U] = 1 − 𝛼.
This part of the equilibrium is stable when the following equation holds: (𝜋 − 𝑝𝜋) + 𝜌𝜋 − 𝑄 > 0
The equation is satisfied if term (𝜋 − 𝑝𝜋) is more positive than the negative term 𝑝𝜋 − 𝑄. The is very important for this equation to hold. The is the only information the investor has concerning the investment, when there is no report available of the firm. The higher the more the investor is incentivized to buy the investment product. For the equation above to hold, the should be high enough and the default rate on bad quality firms should be low enough to make the equation above positive. The firms FG,G and FB,G do not deviate when 𝑇 − 𝜙 < 𝑄.
28 Theorem 8:
The firms FG,B and FB,B are signalling that they are of bad quality if the following two equations
hold: 𝛼(𝜁) (1−𝜁)(1−𝛼)+𝛼(𝜁)(𝜋 − 𝑊) + (1−𝜁)(1−𝛼) (1−𝜁)(1−𝛼)+𝛼(𝜁)(𝑝𝜋 − 𝑊) 0 (1) 𝑊 − 𝜙 > 𝑄 (2)
29 Proof of Theorem 8:
Next in the unintuitive case, the focus is on the companies FG,B and FB,B. The investor in
this case cannot distinguish whether the company has a bad report because of the poor quality or because of an error of type II. Good quality companies can also get a bad quality report, which leads to type II errors. This is the case when CRAs are cautious in their ratings. The belief of the investor in node 5:
P[FG | report states bad firm]= 𝛼(𝜁) (1−𝜁)(1−𝛼)+α(ζ).
The belief of the investor in node 7:
P[FB | report states bad firm]=
(1−𝜁)(1−𝛼) (1−𝜁)(1−𝛼)+𝛼(𝜁).
The investor in node 5 and 7 are buying the investment if the following equation holds:
𝛼(𝜁)
(1−𝜁)(1−𝛼)+α(ζ)(π − 𝑊) +
(1−𝜁)(1−𝛼)
(1−𝜁)(1−𝛼)+𝛼(𝜁)(𝑝π − 𝑊) 0.
The equation above can be simplified further if 𝜉, 𝛼 ∈ (0,1):
𝛼(𝜁)(π − 𝑊) + (1 − 𝜁)(1 − 𝛼)(𝑝π − 𝑊) > 0.
Given that 𝑝π− 𝑊 < 0, the equation only holds if π− 𝑊 > 0 is positive enough. If we take 𝜁=1, meaning that companies that are rated poor, are all caused by type II errors in the reports. This means that the poor ratings come from companies which are of good quality. The equation would then simplify to π− 𝑊 > 0. The investor in this case would buy the investment and makes a great profit because of the low price it had to pay, namely W instead of T for a good quality investment. On the other hand, when 𝜁=0, the CRA makes no mistakes and all poor rated companies are of bad quality. The investor in this case is better off not buying the investment, because of the bad quality firm. When 𝜁=0, all of the companies that are rated poor are then truly of bad quality. The 𝜁 should be high enough for the investor to take a risky decision on a bad report and buy the investment. Important here is, that 𝛼 is large enough so that the probability of facing a good quality company is high enough to make equation 1 of Theorem 8 satisfy.
Concluding, when the poor rated companies FG,B and FB,B deviate andsend the signal no report.
The investor beliefs that he is facing the firms FG,G andFB,G, and buys the investment, following
Theorem 7. The firms FG,B and FB,B in this case earn the payoff Q instead of W – ϕ. The firms
do not deviate if Q < W – ϕ. Secondly, according to Theorem 7, T − ϕ < Q should be satisfied for the companies FB,G and FG,G not to deviate. Given that the equations W − ϕ > Q and T −
30 ϕ < Q should be satisfied simultaneously, the partially unintuitive separating equilibrium cannot hold, given that T > W. The firms have an incentive to change the signal from the original equilibrium. Theorems 7 and 8 can hold separately, but not simultaneously. This means that the
partially unintuitive separating equilibrium cannot hold.
4.2.2 Pooling equilibria
In this extended game another possible equilbrium is that the firm sends out a report independent of his true state. The investor in that case analyzes the outcome of the report, but does not know the true type of the firm. The investors make their beliefs by making use of Bayesian updating. In case that the investor receives a good report the beliefs of the investor regarding the true quality of the firm is:
P[FG | report states good firm]= 𝛼(1−𝜀)
𝜀(1−𝛼)+α(1−ε), and
P[FB | report states good firm]=
𝜀(1−𝛼) 𝜀(1−𝛼)+α(1−ε).
When the investor receives a bad report the beliefs of the investor is: P[FG | report states bad firm]=
𝛼(𝜁)
(1−𝜁)(1−𝛼)+α(ζ), and
P[FB | report states bad firm]=
(1−𝜁)(1−𝛼) (1−𝜁)(1−𝛼)+𝛼(𝜁).
The investor does not know in which situation the investment decision is taken. The investor takes the decision based on expected payoff by making use of the beliefs as described above.
The first possible pooling equibrlium that is going to be considered is when the investor always plays the strategy R. The investor in that case can get a good report or bad report
depending on the state of the company. This pooling equilibrium is illustrated in figure 10, where the red lines are the strategies of the firms.
31 Theorem 9:
The firms FG,G and FB,G are signalling a good report if the following two equations hold:
𝛼(1−𝜀) 𝜀(1−𝛼)+α(1−ε)(π − 𝑇) + 𝜀(1−𝛼) 𝜀(1−𝛼)+α(1−ε)(𝑝π − 𝑇) 0 (1) 𝑇 − 𝐾 − 𝜙 > 𝑄 (2) Proof of Theorem 9:
In case that the investor gets a good report the following equation must hold for the investor to invest in the company.
𝛼(1−𝜀)
𝜀(1−𝛼)+α(1−ε)(π − 𝑇) +
𝜀(1−𝛼)
𝜀(1−𝛼)+α(1−ε)(𝑝π − 𝑇) 0
which holds assuming that the term (𝜋 − 𝑇) is more positive than the negative term 𝑝𝜋 − 𝑇. The is here very important for this equation to hold. The level of 𝛼 is also important for the
equation to hold, namely for high 𝛼 the investor is incentivized to buy an investment which distributed a good report. The probability of truly facing a good quality company is high when 𝛼
32 is high. The equation definitely holds when the investor knows for sure that the CRA does not inflate ratings and does not make mistakes in these ratings, which means that 𝜀 = 0. The equation above then simplifies to π − 𝑇 > 0. When the investor buys the investment based on the good report, he knows for sure that this is a good quality company, a prior relationship of 1-1. This results in an investment payoff π − 𝑇 for the investor. When 𝜀 = 1, which means that only bad quality firms earn a good report. The CRA is consistently making mistakes and inflating ratings. The equation above then simplifies to 𝑝π − 𝑇< 0. However, the 𝜀 could be higher then expected, not equal to 1, but high enough to not satisfying the equation above. This would lead to the investor playing the strategy not buying, resulting in a payoff 0 for the investor and - 𝜙 for the firm. The firm is then strictly better off to play strategy U. We need to assume that the report cost and reputation cost are low enough for the firm not to deviate to strategy U. This means that the firm has no incentive to change his strategy from R to U when equations 𝑇 − 𝜙 > 𝑄 and 𝑇 − 𝐾 − 𝜙 > 𝑄 are satisfied.
Theorem 10:
The firms FG,B and FB,B are signalling a bad report if the following two equations hold:
𝛼(𝜁) (1−𝜁)(1−𝛼)+𝛼(𝜁)(𝜋 − 𝑊) + (1−𝜁)(1−𝛼) (1−𝜁)(1−𝛼)+𝛼(𝜁)(𝑝𝜋 − 𝑊) 0 (1) 𝑊 − 𝜙 > 𝑄 (2) Proof of Theorem 10:
We need to analyse the part of the game where the firm distributes a bad report. The investor buys an investment with a bad report, if the following equation holds:
𝛼(𝜁)
(1−𝜁)(1−𝛼)+α(ζ)(π − 𝑊) +
(1−𝜁)(1−𝛼)
(1−𝜁)(1−𝛼)+𝛼(𝜁)(𝑝π − 𝑊) 0.
The equation above can be simplified further if 𝜉, 𝛼 ∈ (0,1) to:
𝛼(𝜁)(π − 𝑊) + (1 − 𝜁)(1 − 𝛼)(𝑝π − 𝑊) > 0.
Given that 𝑝π− 𝑊 < 0, the only way this equation holds is if the term π− 𝑊 > 0 is positive enough. If we take 𝜁=1, meaning that the companies that are rated poor, are all type II errors. The investor in this case would buy the investment, and makes a great profit because of the low price it had to pay, namely W instead of T. When 𝜁=0, the CRA makes no mistakes and all poor rated companies are of bad quality. The investor in this case is better off not buying the
33 investment, because of the bad quality firm. When 𝜁=0, all of the companies that are rated poor are then truly of bad quality. The 𝜁 should be high enough for the investor to take a risky
decision on a bad report and buy the investment. There needs to be enough bad rated companies in the market, which are truly of good quality for the investor to buy the investment. Important here is, that 𝛼 is large enough so that the chance of facing a good quality company is high enough to make the equation above satisfy. The firms FG,B and FB,B have no incentive to change
their strategy from R to U when 𝑊 − 𝜙 > 𝑄.
Concluding, the pooling equilibrium holds where the firms sends signal R. For this equilibrium to hold, the cost of the report and reputation should be low enough for the Theorems 9 and 10 to hold. The reputation cost in this equilibrium are so low, that even bad quality
companies are buying the bad report, namely 𝑊 − 𝜙 > 𝑄. The bad quality companies are better off buying the report than not buying the report, and ending up with the revenue Q when the investment is bought. Concluding, this pooling equilibrium does holds. This pooling equilibrium is illustrated in figure 11 by the red lines
34 Another possible pooling equilibrium could be that firms signal no report. The investors in this case only know the 𝛼. The beliefs of the investor are then:
P[FG |U]= 𝛼, and
P[FB |U]=1- 𝛼.
The investor makes his choice based on the distribution of the quality of the companies. In figure 12 the pooling equilibrium is illustrated where the firm sends signal U.
Theorem 11:
All the firms are signalling no report if the following two equations hold:
(𝜋 − 𝑝𝜋) + 𝜌𝜋 − 𝑄 > 0 (1)
35 Proof Theorem 11:
The investor prefers to buy the investment if the following equation holds (𝜋 − 𝑝𝜋) + 𝜌𝜋 − 𝑄 > 0
𝜌𝜋 − 𝑄 is negative, assuming that the revenue 𝜌𝜋 is lower than the price of an unrated
investment vehicle Q. The is here very important for this equation to hold. The is the only information the investor has concerning the investment, when there is no report of the firm. The higher the the more the investor is incentivized to buy the investment. For the equation above to hold, the should be high enough and the default rate on bad quality firms should be low enough to make the equation above positive.
The firm has no incentive to change his strategy from U to R if the price of the report increases such that 𝑇 − 𝜙 < 𝑄 with consequence that 𝑊 − 𝜙 < 𝑄 is also satisfied. This means that the pooling equilibrium holds where the firm sends out no report, when the price of the report is high compared to the price of the investment. The pooling equilibrium is illustrated in
36 figure 13, by the red lines.
Concluding, the pooling equilibrium holds where the firms send no report. The price of the report is then too high for the companies. The companies therefore send the signal no report. This equilibrium only holds when the price of the report is too expensive for all the types in the game, However, this is not rational for the CRAs because they do not make revenue if the report is not bought. The best response of the CRA is to lower the price of the report so that the reports are bought by the firms. The pooling equilibrium where the companies send report also holds, if the price of the report is low enough.
37 5 Conclusion
The purpose of this research is to propose a more realistic model for credit rating games and analyse whether the extensions are a valuable addition to the simple game without a CRA. One can draw two important conclusions when comparing the model with CRA, and the simple model without CRA.
Firstly, in the simple model without CRA the intuitive separating equilibrium only exists if T-K < 0. This means that the reputation cost should be higher than the payoff of the firm. However, the reputation cost could be lower than expected. The investors could be forgiving, leading to lower reputation cost. This causes the intuitive separating equilibrium not to hold. The investors in this case cannot invest in the good quality firms without an intermediary, which distributes information concerning the true state of the companies.
Secondly, in the extended game with CRA the partially intuitive separating equilibrium holds. For this equilibrium to hold, the cost of the report should be low enough for the companies FG,G and FB,G purchase the rating. The report is a valuable signal in the market and the investor
can rely on the analysis of the CRA if the price of the report is sufficiently high. When the report is cheap the signal is not valuable in the market because every firm can buy the report. It is also important that the reputation cost is low enough for the company FB,G not to deviate to strategy U.
The main problem with the partially intuitive separating equilibrium is that could be higher than desired so that the equilibrium does not hold. The exists of actual mistakes and inflated ratings. The actual mistakes, the error of type I, cannot be changed by the CRA. However, the CRA can choose the number of ratings that it wants to inflate. The inflated ratings lead to more reports being sold and therefore the revenue of the CRA increases. However, the CRA cannot choose too high, otherwise the investor does not buy the investment. The expected payoff of the investor associated with buying the investment is low when is high, which leads to deviating of the investor not buying the investment.
Comparing the models, it can be concluded that in the simple model the reputation cost are important for the intuitive separating equilibrium to hold. However, the investors could punish less severe than expected, leading to lower reputation costs. This problem is solved in the partially intuitive separating equilibrium in the extended model, where the reputation costs and report cost play a role in reaching an equilibrium. For this equilibrium to hold, the reputation cost should not
38 be too high, where the same holds for the report cost. Moreover, the report cost should be high enough for the firms FB,G and FB,B not to buy the rating.
When the reputation cost is high enough the companies could signal their true state without the intervention of a CRA in the simple model. This equilibrium means that the CRA is inefficient to include in the game. There is no need for a CRA to clarify the state of the company. When the reputation cost is low, the CRA should be included in the game for the investors to be coordinated towards good quality companies. It is of importance that the is not too high, otherwise the CRA is not capable to coordinate between firms and investors. When the is too high the investor believes that the CRA is making too much mistakes and or inflating ratings. Consequently, the investor does not buy the investment and the firm is better off not buying the rating.
In further research, it would be interesting to analyse what the impact is on the equilibria when the investor has more information available concerning the market than assumed here. The investor could receive information from other investors or from other CRAs. This leads to competition between information providers in the market, which could affect the equilibria. Another field of research could be relating to the which exist of mistakes and/or inflating ratings by the CRA. It is interesting to analyse which components of the affects the equilibria more and under which circumstances the CRA influences the ratings.
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