In this section we discuss the optimization problems of financial institutes. Both banks and underwriters maximizes their profits by providing financial services.
Commercial Banks
Commercial banks take in deposits as the source of funds and lend funds to firms through loan contracts. We assume that banks are price-takers in deposit markets.
This assumption deviates from Boyd and de Nicol´o (2005)’s set up. They assume that banks have monopoly power in both loan transactions and deposit business.
This assumption allow them to analyze strategic behaviors of financial firms in both markets of their outputs(loans) and inputs(deposits).
In our setup of banking optimization we only allow banks have monopoly power in loan markets, but not in deposit markets (or more precisely funds
mar-kets). When a firm comes to a bank and applies for a loan, the bank applies its screening technology at a cost of c to identify the types of the firm and its pro-duction project. After identifying the types, the bank determines a loan rate RL to charge for its loan provision. Since the bank can identify loan applicants and their project types, loan rates are firm- and project-specific, denoted by RL(e). We assume that the bank has monopoly power in loan markets. When a bank chooses a loan rate, it takes into account the borrower’s reaction to the loan rate - the first order condition of the loan applicant’s optimization problem. Banks take in de-posits at a given deposit rate RDand the cost side of a loan business only has two terms: the screening cost c and the deposit rate RD. Thus the bank’s optimization problem for a loan application is written as
max
Since c and RDare exogenous to the optimization, one can solve this problem by maximizing the expected revenue. Substituting the constraint into the objective function, the maximization problem is rewritten as
max In our parametric model with P(S, e) = 1 − AS/e, the probability of success of each project is the same: P(S∗(RL(e)), e) = 1 − (A/e)(3/4)(e/A) = 1/4.
Notice that every firm who obtains loan finance has the same probability of success. Commercial banks optimally respond to firms choice rule by setting loan rates such that all loan-financed projects will have the same probabilities of failure but with different levels of productivity. This is due to our setup of P(S, e) = 1 − AS/e in which Pss(S, e) = 0. With a more general form with Pss(S, e) < 0, we can show that firms with greater skill levels of management choose more productive projects (greater S) and have greater probabilities of success when the direct effect of skill level of management dominates the indirect effect through the selection of projects (see Chiang (2008)).
The loan rates increases as e increases. The productivity of projects also creases as the skill level of management increases. The value of firms also in-creases as the skill level of management inin-creases. The bank’s profits from loan business also increase as the loan applicant’s skill level of management increases.
A profitable loan transaction requires a positive expected profit:(1/4)(e/(2A)) − (c + RD) > 0 or e > (8A)(c + RD); that is the firm’s skill level of management cannot be too low.
We summarize the findings on loan business in Proposition 3:
Proposition 3
[1] The loan rate is increasing in the skill level of loan applicants’ management.
[2] The bank’s profit from loan business is increasing in the level of loan appli-cants’ skill levels of management.
[3] Firms with higher skill levels of management choose more productive projects.
[4] Firms that obtain loans have the same default probability, regardless of their skill levels of management.
[5] Firms with sufficiently low skill level of management will be rejected for loans;
more specifically for firms with e< (8A)(c + RD)(≡ ˜e).
Investment Banks
Investment banks provide firms with underwriting services. When a firm decides to raise funds directly from markets, it needs an investment bank to underwrite its securities for public offerings. Investment banks underwrite firms’ securities in exchange of service fees. Unlike commercial banks, investment banks do not face default risks when they underwrite securities. Investors who buy the securities face default risks. However, investment banks earn profits by providing convinc-ing information to investors such that the performance of underwritconvinc-ing activities depend upon how well the information perceived by investors. The performance of underwriting affects how much an investment bank can charge firms who ask for underwriting services. A underwriter needs put efforts in order to make infor-mation convincing to investors, those efforts are costly.
How much a firm is willing to pay for underwriting services depends upon the performance of underwriting; more specifically, the interest rate required by investors. There are a number of factors affect prices of securities (and the inter-est rates firms are required to pay), including macroeconomic conditions, issuers’
reputation, tightness of funds in markets, ..., and so on. How an investment bank helps firms to obtain funds is certainly one of these factors. This paper focuses on the analysis of how market-oriented finance affects the risk exposure of the entire economy. For simplicity we abstract from all factors which cannot con-trolled by investment banks and to focus on only the performance of underwriting technology (θ).
In loan services the cost of funds faced by firms are determined in the loan contract, completely under the control of banks and firms. When raising funds through issuing bonds, the cost of funds faced by firms differs to those raised through loan contracts. Both the buyer and the seller of underwriting services have no “complete” control over the cost of funds. Market investors’ belief plays an important role. The contingency on investors’ belief makes the analysis of underwriting activities different from that of loan activities. We will elaborate what are differences before we analyze the strategic behaviors of both firms and underwriters.
(a) Investors’ belief and the costs of funds An investor decides to buy a se-curity mainly because he expects the returns from the investment on the sese-curity is worthwhile, and the information an investor receives determine how he expects the returns. Investors know that the type of production projects (S) and the en-trepreneur’s ability (e) are two of main factors determining the real returns of the investment. More specifically, investors know what P(S, e) is, but do not really know what S and e are. Their information about S and e is limited, and therefore,
investors are conservative about the probability of success and believe that the probability of success to be smaller than its true probability. Instead of perceiving P(S, e) · RU as the expected returns, an investor uses a smaller probability, ˜P(S, e), to predict his expected returns. This implies that an investor asks for higher returns to justify his investment and, thus, increases the costs of funds.
(b) Convincing power Delivering convincing information to investors is one of main functions investment banks serve. Investment banks collect and ana-lyze information about the projects in the early stage of the underwriting pro-cess. Gathering and analyzing information make investment banks be better in-formed of the security-issuing firms than those public investors. The investment bank’s statement about the underwritten case is the main source of information public investors rely on. The more convincing the information is, public in-vestors’ belief ˜P(S, e) will be closer to the true probability P(S, e). We assume that ˜P(S, e) =θ · P(S, e) to reflect the performance of underwriting efforts, and ignore the costs of underwriting for the simplicity of analysis. Given this setup, the underwriting technology is fixed and the investment bank does not have an optimization problem to solve. It simply runs underwriting business and collect fee incomesφ.
(c) Securities-issuing firms’ best response The required payments for funds raised through markets, RU depends uponφP(S, e). Appealing to arbitrage activ-ities, RU is determined by
θ·
where ˜PRM is the expected return of the average portfolio in the market, ˜P is the average probability of success of all investment projected which are market-financed, and RM is the market rate of returns determined by the supply and de-mand of funds in markets. The market rate RM is beyond the firm’s control and is given to the firm; however, the choice of S will affect the probability of success.
The firm’s optimization problem is written as
max the problem turns to choose a project with a maximal expected output, and has nothing to do with underwriting performance (θ). This is different from what we have in Section 2 and in bank finance.
The optimal choice of S, SU∗= e/(2A), is strictly increasing in e. The prob-ability of success is 1/2. Furthermore the security rate RU = 2RM/θ is con-stant over different levels of management skill (e). Define VU(e, RM,θ) as VU = P(SU∗, e)(SU∗− RU) = 4Ae −RθM. One can easily verify that VeU > 0, VRUM < 0 and VθU> 0. The value function of optimization by underwriting an investment project is VU(e, RM,θ) −φ. Whenφ and/or RMis large enough, firms with low skill level of management will have no incentive to obtain funds from securities markets. We can summarize the findings about underwriting activities in the following propo-sition:
Proposition 4
[1] Firms with higher e choose higher S.
[2] All firms with market finance have the same default probability and the same security rate.
[3] For sufficiently large RM and/or φ, firms with low e will not consider funds from markets.
In a more general setting we are able to drive the result that security rates are decreasing in the level of firms’ skill level of management, while loan rates are increasing. Just as in the analysis of loans, the reason for the constant security loan rates is due to the setting of Pss= 0. It is interesting that the probability of success in market finance (1/2) is greater than that in loan finance (1/4). Markets-financed projects have lower default rates than loan-financed projects.
Now we will derive demands for funds from demands for loan finance and security finance. The decision of funds-raising is transformed into the problem of max {V (RL, e),V (RU, e) −φ}. When P(S, e) has a particular form of 1 − AS/e, this problem is rewritten as
max { e 16A, e
4A−RM θ −φ}.
As one can see that both value functions are function of e, and V(RU, e) −φ has
a steeper slope; however the minus terms of RθM and φ make the curve have a negative vertical intercept as shown in Figure 4. There exists a critical value of
e
e 16A e
4A e
4A−RθM−φ
e∗
˜ e
loans securities
Figure 4: max {V (RL, e),V (RU, e) −φ}
e∗ = 16A3 (RθM+φ) such that firms with e > e∗ choose market finance and those with e< e∗choose loan finance.
From the commercial bank’s optimization we know that RL > c + RD, and from no arbitrage (7) we know RU = ˜PRM/(θ(1 − AS/e)). In a more general setting, we show that RL(e) > RU(e′) for all e ∈ [e, e∗] and e′∈ (e∗, ¯e] (see Chiang (2008) Proposition 7).
Recall that commercial banks do not lend their loans to firms with e< ˜e = (8A)(c + RD) (page. 14) Given ˜e and e∗ the demand for funds in the form of securities is
DS(RM) = n · Z e¯
e∗(RM) f(e)de (8)
where f(e) is the density function of the distribution of e. The demand for funds in the form of loans is
DL(RD, RM) = n ·
Z e∗(RM)
e(R˜ D) f(e)de. (9)
Notice that e∗is an increasing function of RM, implying that DS(RM) is decreasing in RM, while DL(RD, RM) is increasing in RM and decreasing in RD.