Revision Lecture Microeconomics of Banking MSc Finance: Theory of Finance I MSc Economics: Financial Economics I April 2005 PREPARING FOR THE EXAM What models do you need to study? All the models we studied in the lectures. This revision handout covers only some of the models you need to know. I picked these only because they match recent examination or course work questions. How should you prepare a model for the exam? 1. Use the textbook to understand the key elements: ask yourself if you can provide a summary of the model. 2. See if you can solve through a model using special cases and /or numerical values. Any tips for the examination? 1. Answer questions as completely as possible. Partial answers get little credit. 2. Explain your answers carefully. Aim to be brief AND precise. 1
Liquidity Insurance and Bank Runs [Bryant (1980), Diamond-Dybvig (1983)] The idea: Financial markets and institutions provide pools of liquidity. Liquidity is valuable when individuals face idiosyncratic liquidity shocks. Liquidity provision may make is easier to finance investment in illquid assets, increasing welfare. Fractional reserve banking is one way to implement this. 1. The key elements of the Bryant model. One good, with three periods. Continuum of ex-ante identical individuals, each with one unit of the good in period 0, which they use to finance their consumption (C 1, C 2 ) in periods 1 and 2. 2. Liquidity shocks: at t = 1 each consumer finds out whether she will want to consume early, in which case her utility function is u(c 1 ), or if she would like to consume late, in which case her utility function is ρu(c 2 ). Here ρ < 1 is the discount factor, and u(.) is increasing and concave (that is, the consumers are risk averse). The consumer s utility function is written as U(.) = π 1 u(c 1 ) + π 2 ρu(c 2 ), where π i is the probability that you will care about period i consumption. 3. The consumer has a choice of two technologies: simple storage (which returns 1 unit for every unit invested) and an investment technology that returns R > 1 after two periods or L < 1 after one period. Autarky: no trade Consumers choose their level of investment in the illiquid technology. If an individual chooses to invest I and store 1 I, the resulting level is C 1 (I) = 1 I + LI 1 if she wants to consume in period 1, and C 2 (I) = 1 I + RI R 2
if she is willing to wait till period 2. In autarky, each consumer will choose I to maximise expected utility. If you know the functional form of u(.), you can proceed further. A special case: u(c i ) = ln C i Suppose u(c i ) = ln C i. The autarkic consumer chooses I to maximise U(.) = π 1 u(c 1 ) + π 2 ρu(c 2 ) = π 1 ln(1 I + LI) + π 2 ρ ln(1 I + RI) The first-order condition for an interior maximum is which implies 1 π 1 1 I + LI (L 1) + π ρ 2 (R 1) = 0, 1 I + RI 1 π 1 1 + I(L 1) (1 L) = π ρ 2 (R 1). 1 + I(R 1) This can be solved for I. We have I = π 1(L 1) + π 2 ρ(r 1) (π 1 + π 2 ρ)(r 1)(1 L) (You should check that the second-order condition for a maximum holds.) Thus, optimal autarkic consumption is C autarky 1 = 1 + I (L 1) = 1 π 1(L 1) + π 2 ρ(r 1) (π 1 + π 2 ρ)(r 1) C autarky 2 = 1 + I (R 1) = 1 + π 1(L 1) + π 2 ρ(r 1) (π 1 + π 2 ρ)(1 L) Market Economy Suppose consumers can trade in period 1 by buying or selling a securitised assets (call it a bond). A riskless bond sold at price p in period 1, pays one unit of the good in period 2. Suppose an individual holding illiquid investment I discovers she wants to consume early, that is, in period 1. She expects to get investment returns RI in period 2. If so, she can sell RI bonds (remember each bond must pay one unit, so her future investment 3
returns will support the required payment) at price p. This generates pri to supplement her consumption in period 1 (rather than the LI under autarky) C 1 (I) = 1 I + pri. Consider consumers who want to consumer only in period 2. In period 1, they can buy bonds with their liquid assets 1 I. Since each bond costs p, each such consumer can buy 1 I p bonds and consumption in period 2 equals C 2 (I) = 1 I p + RI Market clearing in the bond market requires that p = 1 R. To understand this, note that if pr > 1, both C 1 (I) and C 2 (I) are increasing in I. Every investor would want to invest all their wealth in the illquid technology, and there will nothing available to consume in period 1. If pr < 1, both C 1 (I) and C 2 (I) are decreasing in I, and investors would want to invest none of their wealth in the illquid technology, and there will be no bond market. Evaluating C 1 and C 2 at this value, p = 1 R, we get CM 1 = 1 and C2 M = R. This is better (in the Pareto dominance sense) than the autarky allocation, but may not be Pareto Optimal (the most efficient solution). Pareto Optimal Allocation To find the optimal allocation, we consider the following problem max π 1 u(c 1 ) + ρπ 2 u(c 2 ) subject to the aggregate budget constraint π 1 C 1 + π 2 C 2 R = 1. The optimal allocation (C1, C 2 ) must satisfy the first-order condition, u (C 1) = ρru (C 2). The special case again: If u(c) = ln C, we have u (C) = 1/C so that the first-order condition reduces to 1 C 1 = ρr C 2 4
which implies C2 = ρrc 1. Substituting this in the budget constraint, we get so that π 1 C 1 + π 2 ρrc 1 R = C 1(π 1 + π 2 ρ) = 1, C 1 = C 2 = 1 π 1 + π 2 ρ ρr π 1 + π 2 ρ Note that this efficient allocation differs from the market allocation which was C M 1 = 1 and C M 2 = R. In other words, the market solution is not generally Pareto efficient. Fractional Reserve Banking The optimal allocation can be implemented through financial intermediation. Everyone deposits all their endowment with the financial intermediary ( bank ), in exchange for the following contract. You get either C1 in period 1 or C2 in period 2, where these are given as above. The bank invests a fraction of the deposits in the long-term illquid project and keeps the rest in a liquid form to allow depositors to withdraw on demand. Is this fractional reserve system stable? The answer depends on the behaviour and expectations of the patient consumers. We consider two cases. Consider the case where C1 > C 2. This obtains whenever ρr < 1. In this case, stability cannot be achieved because even type-2 consumers will want to withdraw at period 1. They will obtain C1 which they can store for consumption in period 2, and thus get more than they would from the bank if they waited till period 2. Hence, it is more interesting to consider the case where C1 < C 2. This obtains if ρr > 1. Here there are two sub-possibilities. One, if every type-2 consumer expects the bank to honour its commitments, they will be willing to wait. Only type-1 will withdraw (so the porportion of population withdrawing is π 1. As long as the bank holds π 1 C1 in liquid assets, it will meet demand for withdrawals. Two, suppose one type-2 consumer worries that other type-2 consumers may be tempted to withdraw. If so, the bank will be forced to liquidate its long-term assets to meet this demand. In extreme cases, it may not be 5
able to meet the entire demand for withdrawals and will fail. If so, it makes sense for our consumer to withdraw too. Thus there are two Nash equilibria: one in which all type-2 consumers are willing to wait till period 2, and the second in which all depositors, type 1 and type 2, withdraw at t = 1. This second case can be described as a bank run. Note that a bank run is inefficient. RECENT EXAM QUESTION(2004) Derive the Diamond-Dybvig model of banks as providers of liquidity insurance. Describe in detail the market allocation (when real investments are securitized and traded in an intermediate stock market), and the allocation with intermediaries (when real investments are carried out by a bank that is financed by deposits from consumers/investors). 6
Adverse Selection Basic idea: If projects are risky, and the true level of risk is known to risk averse entrepreneurs but not to the lenders ( investors ), financial markets may suffer from adverse selection. Entrepreneurs with better quality projects may choose to (partially) self-finance the project in order to signal their confidence in their projects. 1. Investors are risk neutral. Entrepreneurs are risk averse: assume their utility function is given by u(w) = e ρw. 2. The return R(θ) to any projects is normally distributed, with mean θ and variance σ 2. The mean return θ varies across entrepreneurs but only they know its true value. The variance is common across all entrepreneurs. 3. If θ were observable, all entrepreneurs would sell their projects to lenders/investors at a price P (θ) = θ. Investors would be willing to pay this price (being risk-neutral, they value each project by its mean return). Entrepreneurs dislike risk, and by selling the project they will eliminate all the risk. Note that as θ differs across entrepreneurs, so must the price. 4. If θ is not observable, projects must sell at the same price. At this price, entrepreneurs with low returns will sell while those with high returns will not: this is adverse selection. 5. To understand who will sell, note that given the preferences and normal distribution of returns, the expected utility of holding on to the project is Eu(W 0 + R(θ) = u(w 0 + θ 1 2 ρσ2 ), while the utility of selling the project at price P is u(w 0 +P ). It follows that all entrepreneurs with θ < ˆθ will sell, where ˆθ P + 1 2 ρσ2. Note that entrepreneurs will bad projects are keener to sell. 6. The market price will be depend on the average project quality offered for sale P = E[θ θ < ˆθ]. 7
7. Will the market deliver an efficient allocation? Efficiency requires that all entrepreneurs (who are risk averse) must transfer their entire risk to the investors. That is, all projects must be offered for sale. Consider a special case. Suppose projects are either good (θ = θ 2 ) with probability π 2 or bad (θ = θ 1 ) with probability π 1. If there is a ˆθ above which entrepreneurs do not sell their project, we would like θ 2 < ˆθ so that even owners of good projects choose to sell. (Of course, any price that persuades owners of good project to sell will be attractive to owners of bad projects too.) If everyone sells, the price equals the expected return across all projects P = π 1 θ 1 + π 2 θ 2 so that ˆθ P + 1 2 ρσ2 = (π 1 θ 1 + π 2 θ 2 ) + 1 2 ρσ2. For θ 2 < ˆθ, we require θ 2 < π 1 θ 1 + π 2 θ 2 + 1 2 ρσ2. Noting that π 2 = 1 π 1, this reduces to θ 2 < π 1 θ 1 + (1 π 1 )θ 2 + 1 2 ρσ2, or π 1 (θ 2 θ 1 ) < 1 2 ρσ2 In words, an efficient allocation will occur only if the adverse selection effect is not too large. 8. Suppose the adverse selection term is large, so that in general an efficient allocation will not be achieved. There may be a signalling equilibrium in which entrepreneurs who know their project to be good will self finance some amount α. Entrepreneurs who know their project to be bad must not be tempted to self finance in order to mimic the good type. The no-mimicking condition is θ 1 (1 α)θ 2 + αθ 1 1 2 ρσ2 α 2, Assume that there is separation of the two types at the signalling equilibrium, so that bad projects sell at a low price P 1 = θ 1 and bad 8
projects sell at a high price P 2 = θ 2. The left hand side in the above equation is what an owner of a bad project will get if he sells his project at a low price P 1 = θ 1. The right hand side is the payoff if she mimics the owner of a good project. She can realise a high price θ 2 for the share that she sells, namely (1 α), but the bit that she selffinances will provide a utility equivalent of αθ 1 1 2 ρσ2 α 2. The above no-mimicking condition requires that owners of bad projects must be better off selling the entire project. The rest is just algebra. The above condition can be rewritten as α 2 1 α 2(θ 2 θ 1 ) ρσ 2. For any given σ, the value of α that just satisfies this relation can be denoted as α(σ). 9. Informational cost is C = 1 2 ρσ2 α 2 = (θ 2 θ 1 )(1 α(σ)). The function α(σ) decreasing in σ, so C(σ) must be increasing in σ. Punchline: Larger coalitions reduce informational cost. 9
RECENT EXAM QUESTION(2002) Suppose the return of an investment project, R(θ), is normally distributed with mean θ and variance σ 2. The project costs 1. An entrepreneur has mean-variance preferences represented by the utility function u. The absolute risk aversion coefficient is ρ > 0, the investor has initial wealth W 0 + 1 and, therefore, end of period wealth W 0 + R(θ) after investment. The expected utility is Eu(W 0 + R(θ)) = u(w 0 + θ ρ 2 σ2 ) Although the entrepreneur has sufficient funds to self-finance the project, he can also seek outside financing from a risk-neutral investor. (a) What potential gains can you see from external financing of the project? ANSWER: External financing can allow the risk-averse entrepreneur to transfer risk (which he dislikes) to the risk neutral investor (who does not mind the risk). (b) If the entrepreneur can sell the project to an outside investor, what is the minimum price that he would accept and what is the maximum price the investor would pay? ANSWER: If the entrepreneur finances the project himself, his expected utility is Eu(W 0 + R(θ)) = u(w 0 + θ ρ 2 σ2 ). If he sells the project at price P, he trades the uncertain return R(θ) for the certain value P. The expected utility after sale is Eu(W 0 + P ) = u(w 0 + P ). The entrepreneur will be willing to sell as long as the utility after sale exceeds the utility prior to sale, that is as long as u(w 0 + P ) u(w 0 + θ ρ 2 σ2 ), or that or that W 0 + P W 0 + θ ρ 2 σ2, P θ ρ 2 σ2. 10
Hence the minimum price at which the entrepreneur would be willing to sell is θ ρ 2 σ2. The investor is risk neutral, so values each project by its mean return. Since the project R(θ) has expected return θ, that is the maximum he is willing to pay. (c) Now suppose the investor can observe the variance of the project (σ 2 ) but not the expected return (θ), so the price cannot depend on θ. Assume that the investor knows the probability distribution of θ, where θ is high (θ 1 ) or low (θ 2 ) with probability π 1 and π 2 respectively. Outline the condition for when it is optimal for the entrepreneur to seek outside financing. When is the investor more likely to self-finance the project? Explain your answer. (Note the change in notation from that in the textbook, where θ 1 was low, and θ 2 was high. You need to be careful about notation in exam questions.) ANSWER: If an investor self finances the project, he gets expected utility Eu(W 0 + R(θ i )) = u(w 0 + θ i ρ 2 σ2 ), where he will know if θ i = θ 1 (high) or θ i = θ 2 (low). If he can sell the project for a fixed price P, he gets Selling is a good idea as long as Eu(W 0 + P ) = u(w 0 + P ). P > θ i ρ 2 σ2. It follows directly that entrepreneurs who know their project to be bad (θ 2 ) will be more inclined to sell. (d) Leland and Pyle s (1977) paper describes a signalling equilibrium for this situation. In this signalling equilibrium, what is being signalled, and how? ANSWER: In the equilibrium, the quality of the project (that is, its expected return) is being signalled, by the entrepreneur being willing to self-finance a part of the project. 11
(e) Derive the Leland and Pyle equilibrium algebraically ANSWER: Reproduce the equilibrium described above, with the nomimicking condition. Recall that you need to adapt the notation. In the signalling equilibrium entrepreneurs of good project will self finance at least a proportion α of their project and sell the rest at a unit price P 1 = θ 1. Entrepreneurs with bad projects will sell their entire project for a lower unit price P 2 = θ 2. To ensure that entrepreneurs with bad projects do not want to mimic entrepreneurs with good projects, they must get a higher utility by selling their entire project at the lower price or equivalently or u(w 0 + θ 2 ) Eu(W 0 + α R(θ 2 ) + (1 α)θ 1 ) = u(w 0 + αθ 2 ρ 2 α2 σ 2 + (1 α)θ 1 ), θ 2 αθ 2 1 2 ρσ2 α 2 + (1 α)θ 1, α 2 1 α 2(θ 1 θ 2 ) ρσ 2. 12
Moral Hazard Basic idea? Try to provide a brief description of the model. 1. Firms seek to finance investment projects of unit size and can choose one of two possible technologies. The good technology produces G with probability π G, and zero otherwise, with π G G > 1. The bad technology produces B with probability π B, and zero otherwise, with π B B < 1. We have π G > π B (so that the good technology has higher probability of success) and B > G (so that a successful bad technology returns more than a successful good technology). Direct lending 2. The lender lends 1 unit for the investment and asks for repayment R. Repayment will be made only if the investment succeeds. 3. The lender s choice of R affects the firm s choice of technology. To see why, note that the expected profit of the firm is π G (G R) if it chooses the good technology and π B (B R) if it chooses the bad technology. The firm will choose the good technology if and only if π G (G R) π B (B R). If we define R C = π GG π B B, π G π B we find that the firm will choose the good technology if and only if R R C. 4. The fact that the firm switches to the bad technology for high values of R affects the probability of repayment and hence the expected return to the lender. In particular, the probability of repayment p(r) is { πg if R R p(r) = C π B if R > R C. and the expected return is p(r)r = { πg R if R R C π B R if R > R C. We assume that the market for direct lending is competitive, so that lenders make zero expected profit. If they lend 1 unit, the expected return is p(r)r, so that the break-even condition is p(r)r = 1. 13
5. When will such a direct credit market work? Note that the highest possible expected return is π G R C. If this is less than 1, the lender will never expect to break even so no loans will be made. In other words, direct lending requires π G R C > 1. Monitored lending ( bank lending ) 6. Assume now that the lender can observe a firm s choice of technology at some monitoring cost C. A bank that chooses to monitor can compel the firm to choose the good technology. The expected return for a bank that monitors is p(r)r = π G R If bank lending is competitive, the expected return just covers the loan amount and monitoring cost. Let R m be the repayment amount that allows a monitoring bank to just break even. π G R m = 1 + C. 7. When will bank lending work? Note that the highest possible expected return is π G G. If this is less than 1+C, banks will not expect to break even. Hence bank lending requires or equivalently if π G G > 1 + C π G > 1 + C G 8. There is a second condition that must be satisfied for bank lending. Bank lending is incurs monitoring costs C, so will be used only if direct lending is not possible. Recall that direct lending is not possible if or equivalently if π G R C < 1, π G < 1 R c 9. Combining the previous arguments we can determine ranges of values for π G for which direct lending will work and other ranges for which bank lending will work. 14
RECENT COURSEWORK QUESTION (December 2004) Recall the simple model of the credit market with moral hazard that we discussed in the lectures. Firms seek to finance an investment project through borrowing. Assume that the project requires an initial investment of 100. Firms must choose between two alternative technologies for the project. The good technology produces 200 with probability 0.7, and zero otherwise. The bad technology produces 400 with probability 0.2, and zero otherwise. A loan contract requires a repayment R when the output of the project is positive and nothing when the output is zero. (a) Consider, first, the possibility of direct lending (i.e., when a firm borrows directly from lenders). We assume that lenders cannot observe the firms choice of technology. Show how the value of the repayment obligation affects the firms choice of technology. Compute the range(s) of the interest rate for which good technology is chosen. ANSWER: This follows the model very closely, except that investment levels have been scaled up by a multiple of 100. Suppose the investor invests in the good technology. This yields G = 200 with probability π G = 0.7. The investor must repay R whenever the project succeeds. The net return to the investor is π G (G R) = 0.7(200 R) If the investor chooses the bad technology the net return is π B (B R) = 0.2(400 R) The good technology is preferable if and only if that is, if R < 120. 0.7(200 R) 0.2(400 R), (b) Is a competitive market in direct lending sustainable in this environment? Why or why not? ANSWER: Once we recognize that the choice of R affects the technology chosen, it is easy to see that the lender will get 0.7RifR 120 and 0.2Rif120 < R 400 15
and 0.2(400)ifR > 400 In none of these scenarios can the lender recover the original investment, 100, so competitive lending is not possible. (c) Assume now that firms can borrow from a bank that can monitor the firms choice of technology at a cost of 20. Monitoring allows the bank to enforce the choice of a particular technology. Is bank lending sustainable in this environment? ANSWER: Since monitoring allows the bank to enforce the choice of the good technology, the firm s expected return will be π G G = 140. With initial investment (100) and the monitoring cost (C = 20) adding up to 120, the bank can expect to break even. Note that we have already checked that direct lending is not possible. (d) What happens to the answer in the previous part if the cost of monitoring is 50? With initial investment (100) and the monitoring cost (C = 50) the bank must expect to get 150 to break even. Given the project has an expected return of 140, it cannot. 16