Adverse Selection and Costly External Finance

Adverse Selection and Costly External Finance This section is based on Chapter 6 of Tirole. Investors have imperfect knowledge of the quality of a firm s collateral, etc. They are thus worried that they might purchase overvalued claims. Akerlof (1970) showed how markets for used wares (e.g. cars) may even disappear when sellers are better informed about their quality than buyers. The application of Akerlof to credit markets is that entrepreneurs may raise less funds or raise them less often when there is asymmetric information. The informed side of the market is likely to accept distortions in contracting so as to signal their good type. Theoretically, the models are typically plagued by a multiplicity of (perfect Bayesian) equilibria. There will be no asymmetric information among investors. 1 A Simple Model of Adverse Selection (T6.2) Tirole calls this the privately known prospects model. 1.1 Environment A unit measure of firms (or entrepreneurs or borrowers) have projects which each require a fixed level of investment I. Firms have no cash (A = 0). Project either yields R > 0 if it succeeds or 0 if it fails. Adverse Selection. Borrowers can be one of two unobservable types. A fraction α of good (G) borrowers who have probability of success p A fraction (1 α) of bad (B) borrowers who have probability of success q < p Let m = αp + (1 α)q be the investor s prior probability of success. Firms (borrowers) and investors (lenders) are risk neutral β = 1 so expected rate of return equals 0. Limited Liability on part of firm (can t pay anything in failure state). 1

There are a large number of lenders (loans make zero profit) Loan contract specifies R b goes to borrower and R l goes to lender (this is known as a pooling contract). Timing: Lenders post contract (R b, R l ) Investment I Outcome R (observable) Parameters (I, R, p, q, α). We will assume pr > I but will consider both I > qr (so it is not profitable to invest with a known bad type) or I < qr (both types are creditworthy) 1.2 Full Info Equilibrium (First Best) In this case, type is known so we can condition payments on it. Let R G b be the good borrowers compensation in the case of success. Investors participation constraint p(r R G b ) I The solution to the first best problem maximizes the borrower s utility subject to lender participation. Perfect competition drives the lenders participation constraint down to their zero profit condition: R G b = R I p If qr < I, then the bad borrower does not even want to receive funding since it is a negative NPV project. If qr > I, then R B b = R I q and since p > q, RG b > RB b. 1.3 Asymmetric Info Equilibrium (Second Best) With asymmetric information the bad borrower has an incentive to pretend to be the good borrower and receive Rb G with probability q. Of course, with asymmetric information it is no longer possible to condition payments on type, so the contract must be simply R b in a successful state and 0 otherwise (we will show that 0 is in fact optimal). 2

The IR constraint for this pooling contract is m (R R b ) I (1) where m = αp + (1 α)q is the prior mean probability of success. Consider two cases (note (1) implies mr I mr b ): mr < I (= 0 > R b ). Since there is limited liability (R b 0), individual rationality by the lender implies there is no lending. Given that pr > I this can only happen if qr < I and α is suffi ciently small. 1 The good borrower is therefore hurt by the suspicion he is bad. There is under investment (no lending is an extreme form of underinvestment relative to the effi cient lending level of α projects financed). mr I. In this case there is lending (either since qr > I or α suffi ciently large). R b is pinned down by the lender s IR constraint (1) R b = R I m Hence the good borrower is hurt since Rb G > R b as p > m If qr < I, then there is overinvestment (lending to unit measure of borrowers instead of α). Aside: The condition mr I can be re-written [ ( )] p q 1 (1 α) pr I (1 χ)pr I (3) p where χ is a measure of adverse selection. Notice that χ is the product of the measure of bad types in the economy (1 α) times the likelihood ratio (p q)/p. This has the same interpretation as B (p H p L )/p H. On optimality of the pooling contract: Could an investor make money by offering a more sophisticated contract? When qr > I, the answer is no. When qr < I, the answer is yes. More on this in Sections 6.3 and 6.5 (appendix). This has important implications for existence of pooling contracts. 1 That is, when α < α where α (pr I) + (1 α )(qr I) = 0. (2) 3

1.3.1 Application 3: Pecking-Order Hypothesis Myers and Majluf (1984, JFE) provide a framework which rationalizes why firms first use internal finance and if they don t have enough, then issue debt, and as a last resort equity. The idea is to issue low information intensity claims when there are information asymmetries. As in Chapter 3, there is no distinction between debt and equity when the profit is either R or 0.Hence as in Section 3.3, we assume R F > 0 in the failure state and R S > R F in the success state with R = R S R F. Otherwise the model is the same as above. Parameterization: Assume that mr S + (1 m)r F > I so even a pooling equilibrium is profitable where m = αp + (1 α)q. 2 Let {R S b, RF b } R2 + be the borrower returns in success and failure states. Recall that in a pooling equilibrium, the good borrower subsidizes the bad borrower. Since pooling does not require incentive compatibility, consider the problem of maximizing the good borrower s utility subject to the investor s breakeven constraint in order to minimize the subsidy. The good borrower s utility is U G b pr S b + (1 p)r F b The individual rationality constraint for the lender is m(r S R S b ) + (1 m)(r F R F b ) I. (4) Thus, the problem is s.t. (4). max {R S b,rf b } U G b Since the objective is increasing in {Rb S, RF b } and the IR constraint (4) is decreasing in {Rb S, RF b }, we know the IR constraint will be binding. 2 Note that after adding and subtracting p, then m can be re-written m = p (1 α)(p q) and again after adding and subtracting p, then (1 m) can be re-written 1 m = 1 p + (1 α)(p q). 4

Using the binding IR constraint it is possible to re-write the objective function as: 3 U G b pr S +(1 p)r F I (1 α)(p q) {[ R S Rb S ] [ R F Rb F ]} (5) where the first term is the good borrower s NPV and the second term is the adverse selection discount. Notice that from (5): du G b dr F b du G b dr S b = (1 α)(p q) < 0, = (1 α)(p q) > 0 so that to maximize the good borrowers utility, Rb F is given by the IR constraint (4): = 0. In that case, RS b R S + (1 m)rf I m = R S b. In summary, Since Rb F = 0, the good borrower commits the entire salvage value R F to the lender. That is, he issues debt of D Rl F = R F since D can be paid back in both successful and failure states (i.e. debt is noncontingent). Further, the good borrower issues equity Rl S = I D in the case of success and 0 otherwise. Note that the lower is m (i.e. the worse the adverse selection problem), the higher the equity that must be offered. Since this is a pooling equilibrium, the bad borrower must mimic this behavior to be pooled with the good borrower. 3 To see this, use the previous footnote s m algebra: [p (1 α)(p q)] (R S R S b ) + [1 p + (1 α)(p q)] (RF R F b ) I = 0 or [p (1 α)(p q)] R S + [(1 α)(p q)] Rb S + [1 p + (1 α)(p q)] R F [(1 α)(p q)] Rb F I = prb S + (1 p)rf b Since the rhs is just the objective function, we can rewrite it as Ub G [p (1 α)(p q)] R S + [(1 α)(p q)] Rb S + [1 p + (1 α)(p q)] R F [(1 α)(p q)] Rb F I which can again be re-arranged to (5). 5

2 Separation (T6.3) While the previous section focused on pooling equilibrium, this section considers ways that the good type can separate himself from bad types by offering signals which would not be beneficial for bad types to mimic. 2.0.2 Application 8: Diversification and Incomplete Insurance Leland and Pyle (1977, Journal of Finance), in one of the pioneering paper in the signaling literature, consider a situation in which a risk averse enterpreneur has a substantial stake in his firm and wants to diversity his portfolio. See also Rothchild and Stiglitz (1976, QJE). Diversification may, however, be costly due to adverse selection. Environment Assume I = 0 (so this is just a matter of diversification or insurance) While the investors are risk neutral, the enterpreneur is risk averse with preferences U(w) which are strictly increasing and concave. With risk averse preferences, even if the technology returns R in successful state or 0 in the failure state, in general there will also be payouts to the risk averse borrower in the failure state. Symmetric Info For a good borrower, the optimal contract solves max {R S b,rf b } pu(r S b ) + (1 p)u(r F b ) subject to p(r R S b ) + (1 p)(0 R F b ) 0 (6) Consider the lagrangian associated with the above problem: = pu(r S b ) + (1 p)u(r F b ) + λ [ p(r R S b ) + (1 p)(0 R F b ) ] The FOC are The ratio of the 2 foc gives. R S b : pu (R S b ) λp = 0 R F b : (1 p)u (R F b ) λ(1 p) = 0 pu (Rb S) (1 p)u (Rb F ) = p (1 p) (7) 6

Solving this we have U (R S b ) = λ = U (R F b ) R S b = R F b = R G b which says that in the first best there is perfect consumption smoothing. From (6), we have p(r R G b ) + (1 p)(0 R G b ) = 0 pr = R G b. That is, the good entrepreneur receives a constant income equal to the firm s expected income. Similarly, for a bad borrower R B b = qr < RG b. This can be illustrated in Figure T63SI: Indifference curves are given by (say for the good type): U G = pu(r S b ) + (1 p)u(r F b ) du G = pu (R S b )dr S b + (1 p)u (R F b )dr F b = 0 dr F b dr S b = The IR constraint is given by pu (R S b ) (1 p)u (R F b ) (8) p(r Rb S ) + (1 p)(0 Rb F ) = 0 Rb F = pr 1 p p 1 p RS b (9) Notice that the equilibrium condition in (7) is just that the slope of the indifference curve in (8) is tangent to the slope of the IR constraint in (9). Asymmetric Info Since Rb B < RG b, it is impossible to implement the first best when type is unobservable because a type B will always lie and report he is type G. In that case, investors would lose (1 α) ( Rb G ) RB b on bad borrowers. Consider the following contract designed to separate the good from the bad. max {(R S b,rf b )} pu(r S b ) + (1 p)u(r F b ) s.t. p(r R S b ) + (1 p)(0 R F b ) 0 (10) qu(r S b ) + (1 q)u(r F b ) U(R B b ) (11) R B b = qr (12) 7

That is, choose (Rb S, RF b ) to maximize the good borrower s utility subject to a lender breaking even on that contract (10) and (Rb B, RB b ) is a zero profit, perfect risk sharing (12) which satisfies incentive compatibility for the bad types (11). Both constraints (10) and (11) must be binding in equilibrium. That can be seen in Figure T63AI. In particular, the highest utility for a type G borrower under asymmetric information U G A which his consistent with the IR constraints for lenders ((10) and (12)) and incentive compatibility for the bad borrower (11) occurs at point S (for separating) where (10) and (11) are binding. At point S, that is allocation (Rb S, RF b ) the good borrower is less eager to obtain insurance against the failure state because he has a higher probability of success than a bad borrower. Notice that in the separating equilibrium, the bad borrower sells out or completely diversifies, while the good borrower doesn t completely diversify. Determinants of diversification: Keeping p constant, as q decreases, point B moves down along the diagonal and so point S moves away from the full insurance point G (i.e. the good borrower gets less and less insurance as the adverse selection effects get worse). Notice that limited diversification is good news about the firms prospects. Thus, if the entrepreneur initially owns some equity, the news that the entrepreneur sells his entire staken in the firm generates negative stock price reaction. Put differently, a limited equity offering creates a positive stock price reaction. Appendix shows that the allocation {S, B} is the unique perfect Bayesian equilibrium iff the proportion of good borrowers lies below some threshold α. 2.0.3 Application 5: Costly Collateral Pledging Here explore the possibility of signalling by pledging collateral in the failure state. Intuition: Given that the good borrower has a low probability of failure, he can separate himself from the bad type by giving up shares in the failure state which is costly to the bad borrower since that state happens more frequently for him. Back to risk neutral borrrowers. Suppose that while the borrower still has no cash ( A = 0), she has shares that can be pledged to investors. 8

To formalize that such assets are more valuable to the borrower than to the investor assume that a transfer of assets valued at C to the borrower has value βc to the investor with β < 1. Parameterization: 0 < V qr I < V pr I. Hence even the bad borrower does not need to pledge collateral to receive funding. Symmetric Info From lender IR constraint again we have p(r R G b ) = I = pr G b = pr I V R G b = V p > V q = RB b Asymmetric Info The good borrower cannot obtain Rb G mimic him and expect to obtain since then a bad borrower would q V p = qr q p I > qr I = V. Can the good borrower signal her type by pledging costly collateral C to be seized by the lender in the event of failure? It can be shown it is not optimal to offer collateral in the event of success. This is because posting collateral in the event of success is more costly to the good borrower than the bad borrower because the good borrower is more likely to succeed. Separating equilibrium solves max pr b (1 p)c (R b,c) subject to individual rationality by the lender p(r R b ) + (1 p)βc I (13) and a (no-mimic) incentive compatibility constraint qr b (1 q)c V. (14) Note that the outside option of the no-mimic constraint is the full info bad type utility since if he reveals himself to be bad, there is no risk in lending to a known bad borrower given knowledge of q (i.e. the risk is rationally priced into R l ). Both constraints must be binding. If the incentive constraint (14) were not binding, then the good borrower s objective could be raised by lowering C and raising R b. 9

The solution to the problem is thus simply 2 equations (13) and (14) in two unknowns (R b, C ). In particular, from (13) we have p(r R b) + (1 p)βc I = 0 R b = R + (1 p)βc I p into (14) using V = qr I gives [ qrb (1 q)c (qr I) = 0 = q R + (1 ] p)βc I (1 q)c (qr I) = 0 p [ q(1 p)βc ] qi (1 q)c + I = 0 p q(1 p)βc p(1 q)c + pi qi = 0 (p q) [p(1 q) q(1 p)β] I = C C = I 1 + q(1 p)(1 β) (p q) where in the last I added and subtracted q to the denominator. Given that the contract maximizes the good borrower s utility, it is not surprising that the good borrower is at least weakly better off offering these costly contracts than being thought of as a bad borrower since R b = Rb B, C = 0 satisfies (13) and (14). Signaling helps because it is relatively more costly for a bad borrower to pledge collateral than for a good one since p > q. Determinants of collateralization: From (15) we have the following comparative statics: dc dq < 0. If q falls, then the adverse selection effect is worse (for a given p). Alternatively, as q p, C 0. A testable implication is that good borrowers post more collateral than bad ones. But Berger and Udell (1990) suggests the opposite. On uniqueness: It can be shown (Section 6.7) that the allocation (Rb, C ) chosen by the good type and (Rb B, 0) chosen by the bad type is the unique perfect Bayesian outcome when there are few good borrowers (i.e. α < α in (2)). Notice that this separating equilibrium exists independent of α. However, uniqueness follows when α is low so there is not also a pooling equilibrium. (15) 10

3 Appendix for T6.2 3.0.4 Application2: Negative Stock Price Reaction Data: There are frequent falls in stock prices upon a seasoned equity offer. For simplicity assume the entrepreneur already owns all the shares, which guarantees him return pr or qr depending on his type. Suppose he is considering a deepening investment. At cost I, the probability of success can be raised by τ and that τr > I, so that investment is effi cient for both types of borrowers. Since the entrepreneur has no cash, he must issue new shares (this is known as a seasoned equity offering), thereby reducing the fraction of shares he owns. The key insight is that relinquishing shares to investors is relatively less costly to the borrower with overvalued assets in place (the bad borrower) than to the borrower with undervalued assets (the good borrower) To attract an investor, the borrower must offer (m + τ)r l = I. (16) The good borrower, though, can guarantee himself pr by not diluting his stake in the shares he already owns). Thus, the good borrower is willing to issue new shares only if ( (p + τ)(r R l ) pr (p + τ) R I ) pr m + τ (p + τ) τr (m + τ) I (17) where the first follows from substituting in the IRL constraint (16). Aside: The latter expression can be re-written ( ) χτ τr I I 1 χ τ where [(p + τ) (q + τ)] χ τ (1 α) p + τ so χ 0 = χ in (3). Thus, there is a positive hurdle to undertake investment which depends on the extent of adverse selection. If (17) holds, then it is possible to find a pooling equilibrium where both types issue seasoned equity. 11

Since investors know the environment, there is nothing unanticipated about such an offering. In that case, the pre-offer value of shares is V 0 = α [(p + τ)r I] + (1 α)[(q + τ)r I] = (m + τ)r I Since both issue, this is also the post-issue value. That is, there is no stock price reaction to the offering. If (17) does not hold, then there is a separating equilibrium where the good type entrepreneur does not raise funds but the bad type entrepreneur does raise funds, but at terms less favorable than the above pooling equilibrium since investors can always solve this problem too and must be compensated to take on the bad or overvalued offering. That is, to lend to bad borrowers, lenders must receive (q + τ)rl B = I Rl B = I q + τ > I m + τ = R l where the second term follows from the pooling IR constraint (16) and q < m. The pre-issue value of shares is The post-issue value of shares is V 0 = αpr + (1 α)[(q + τ)r I] V 1 = (q + τ)r I A negative stock price reaction occurs if V 0 > V 1 αpr + (1 α)[(q + τ)r I] > (q + τ)r I pr > (q + τ)r I (18) With the parameterization that we are considering, (18) holds. 4 4 To see this, since the good borrower does not issue, we know Then since (p + τ)(r R l ) < pr = (p + τ)(r max R l ) < pr ( = (p + τ) R I ) < pr. q + τ ( (p + τ)r ( p + τ q + τ ( 1 q + τ R ) ) I > (q + τ)r I ) I > 0 holds, we know (18) holds. Hence we have a negative stock price reaction. 12

Combining both cases, it is clear that the price reaction on average should be less negative in booms (i.e. (17) is more likely to hold and (18) is less likely to hold the higher is τ). This application sheds some light on the going public decision since it describes the conditions that an entrepreneur decides to tap further financing and dilutes his own stake. Entrepreneurs who feel that the assets in place are undervalued by the market (the good types above) tend to forgo profitable investment opportunities and remain private (see Chemmanur and Fulghieri (1999), RFS). 4 Appendix for T6.3 4.0.5 Application 4: Certification There is a large variety of certifying agents: underwriters, rating agencies, auditors, venture capitalists. The certifying agent must have an incentive to become well informed. Obvious for a venture caitalist who has a stake, but for rating agencies it must be reputation. Suppose there is a certification cost c that perfectly reveals a borrower s type. Everything else is the same as in Section T6.2 (i.e. R or 0). Parameterization: mr > I. A bad borrower has no incentive to pay a cost c to reveal it has success probability q. A good borrower can obtain R b G in the case of success ( p R R ) b G c = I The good borrower then certifies if R G b which can be manipulated to?: > R b R I + c p c < (1 α) I + c ( p q p > R I m Thus, if the certification cost relative to the total input is suffi ciently small relative to the adverse selction measure, the good borrower separates by becoming certified. ). 13

4.0.6 Application 6: Short-Term Maturities A good borrower can convey that he is confident about the firm s prospects and that she is not afraid of going back to the capital market at a later stage. The idea that short term debt can be used as a signal of high quality borrowing, which is in Diamond (1991), relates to a more general them in the economics of adverse selection. 4.0.7 Application 7: Payout Policy Most firms pay dividends at the same time they raise debt or equity. Payout announcements affect stock prices and convey information beyond that contained in earnings announcements; the firm s stock price substantially increases upon the announcement of an increase in payout. Bernheim and Wantz (1995, AER) provide evidence for this. 14