Revision Lecture. MSc Finance: Theory of Finance I MSc Economics: Financial Economics I

Transcription

1 Revision Lecture Topics in Banking and Market Microstructure MSc Finance: Theory of Finance I MSc Economics: Financial Economics I April 2006 PREPARING FOR THE EXAM ² What do you need to know? All the models we studied in the lectures. ² How should you prepare for the exam? 1. Use the texts and handouts to review the material. 2. Try to summarise each model with and without formal algebra. 3. Can you solve through a model using special cases and /or numerical values? ² Any tips for the examination? 1. Read the questions carefully. Sometimes the material sounds familiar but di ers slightly from the class material. 2. Pay special attention to notation: it may not be the same as in handouts. 3. Answer questions as completely as possible. Partial answers get little credit. 4. Explain your answers carefully. You can be brief AND precise. 1

2 1 Topics in Banking 1.1 Inter-temporal Optimization ² Can you analyse a simple, two-period, model of inter-temporal consumption choice? In the handout for lecture 1, review the intertemporal budget constraint, the Lagrangean and the conditions for the optimal consumption pro le. ² Can you solve the model for standard utility functions, such as logarithmic utility? See exercises 1 and 2 in the handout for lecture Expected utility ² Can you de ne expected utility and compute it in simple contexts? ² Do you recall Jensen's inequality? ² What is the link between risk aversion and curvature (concavity/convexity) of the utility function? ² How do we measures of risk aversion (absolute risk-aversion, relative risk aversion)? ² Can you solve for the optimal portfolio in simple contexts, with given utility functions? See exercises 3 and 4 in the handout for lecture 1. ² Demonstrate that a risk averse who is o ered insurance at acturially fair rates will buy complete insurance. 1.3 Liquidity Insurance & Bank Runs Review the material in the handout for lecture 2 and 6. ² Can you provide a brief summary of the Bryant /Diamond-Dybvig models of liquidity insurance and bank runs? You should be able to outline the basic structure { what kind of shocks do individuals face, the investment technology, their optimal choices in various environments (in autarky, with securities or bond markets, with fractional banking), and be able to rank these by in terms of e±ciency. ² De ne a bank run. Explain why fractional reserve banking is vulnerable to bank runs. 2

3 You should be able to demonstrate understanding of the `bigger picture'. In this case, the idea is that nancial markets and institutions provide pools of liquidity. This liquidity is valuable when individuals face idiosyncratic liquidity shocks. Liquidity provision may make is easier to nance investment in illquid assets, increasing welfare. Fractional reserve banking is one way to implement this. RECENT EXAM QUESTIONS ON THIS TOPIC (June 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 nanced by deposits from consumers/investors). ANSWER: This requires a careful summary of the model. (June 2005) Consider a three-period economy with a continuum of agents. Each agent is endowed with one unit of a good in period 0, which she must use to nance consumption c 1, c 2 in periods 1 and 2, respectively Agents have access to a storage technology that allows costless transfer of the good from one period to another. They can also invest in a two-period productive technology: each unit invested in this technology in period 0 returns R>1 in period 2 while premature liquidation in period 1 returns L<1. The agents are ex-ante identical but subject to consumption uncertainty: with probability ¼ an agent will want to consume only in period 1, and with probability 1 ¼ she will want to consume only in period 2. The ex ante utility function of all agents is U = ¼u(c 1 )+ (1 ¼)u(c 2 ); where <1 is a discount factor. Assume that R > 1. (a) What is the outcome if agents cannot trade with each other (i.e., in autarky)? Explain why this outcome is ine±cient. ANSWER: In the absence of trade, each agent much choose how much to invest in the two-period investment technology and how much to keep in the more liquid storage technology. If she invests an amount I and stores the rest, 1 I, the consumption possibilities are c 1 = LI +1 I; 3

4 if she has to liquidate and consume early, and c 2 = RI +1 I<R; if she does not liquidate early. The agent will choose I to maximise expected utility subject to these constraints. Even without solving for this, it is easy to see that the outcome is ex-post ine±cient. (Here c 1 < 1andc 2 <Ras long as 0 <I<1. In the event of perference for early consumption, the agent would have preferred zero investment in the productive technology and consumed 1; in the event of preference for later consumption, she would have preferred to invest everything in the productive technology and consumed R. The actual consumptions levels are strictly lower in each event. Hence the ine±ciency. (b) Obtain a condition for the optimal symmetric allocation in this economy. ANSWER: The optimal symmetric allocation maximizes subject to ¼u(c 1 )+ (1 ¼)u(c 2 ) ¼c 1 = 1 I (1 ¼)c 2 = RI The rst constraint requires that the aggregate consumption of the early consumption types is met through storage. The second constraint requires that the consumption of the patient types be met from the proceeds of the productive technology. Substituting fromtheconstraints,wecanwritetheobjectivefunctioninterms of I alone: ¼u( 1 I RI )+ (1 ¼)u( ¼ 1 ¼ ); with rst-order condition u 0 (c 1)+ Ru 0 (c 2)=0: (c) Explain how a system of fractional reserve banking can implement the optimal outcome. Identify any assumptions. ANSWER: Under a fractional reserve system, the bank collects depostions from all agents: it invests a fraction in the two-term 4

5 producitve asset and stores the rest. A deposit contract speci es the amounts (c 1 ;c 2 )thatcanbewithdrawnatvariousperiodsfor 1 unit deposited at time 0. Typically this outcome is sustanaible only when R > 1: if not, we have c 1 >c 2,sothatevenpatient agents would like to withdraw in period 1 and store the proceeds till they consume it. (d) What is a bank run? Why is fractional reserve banking sometimes vulnerable to bank runs? ANSWER: A bank run refers to a situation in which depositors seek to withdraw their funds due to the fear that others' withdrawals causing the bank to fail. Such situations may arise as Nash equilibria, so are entirely rational from an individual point of view. Details of the various possibilities are as in the handout. (Christmas 2005) Consider a three-period economy with a continuum of agents. Each agent is endowed with one unit of a good in period 0, which she must use to nance consumption c 1, c 2 in periods 1 and 2. Agents have access to two technologies. Technology is a one-period technology that returns A for every unit invested in the previous period. Technology is a two-period technology: each unit invested in this technology returns B after two periods, while premature liquidation after one period returns nothing. The agents are ex-ante identical but subject to consumption uncertainty: with probability ¼ an agent will want to consume only in period 1, and with probability 1 ¼ she will want to consume only in period 2. The ex ante utility function of all agents is U = ¼ ln(c 1 )+±(1 ¼)ln(c 2 ); where ±<1 is a discount factor. Assume that ±B > 1. Note that this is a slightly altered version of the standard model we covered in class. The classroom version had di erent notation, assumed A =1, and premature liquidation yielded L>0. These minor changes are designed to test your understanding. (a) What is the autarkic outcome (that is, if agents cannot trade with each other)? Identify any assumptions you make. Explain why this outcome is ine±cient. 5

6 ANSWER: In the absence of trade, each agent invest in each technology. If she invests an amount I in technology and 1 I in, the consumption possibilities are if she consumes `early', and c 1 = A(1 I); c 2 = BI + A 2 (1 I); if she consumes late. The agent will choose I to maximise expected utility. ¼ ln(a(1 I)) + ±(1 ¼)ln(BI + A 2 (1 I)) The rst-order condition for an interior maximum is A¼ A(1 I) + ±(1 ¼)(B A2 ) BI + A 2 =0; (1 I) assuming B>A 2. [Note that if B A 2,itisoptimaltoset I = 0. Investment in technology has both a higher return and greater liquidity.] We solve this to get andthensolvefor I = (B A2 )(1 ¼)± ¼A 2 (B A 2 )[¼ +(1 ¼)±] ; c 1 = A(1 I ) c 2 = BI + A 2 (1 I ) We can see that this is ine±cient even without solving explicity,. (In the event of perference for early consumption, the agent would have preferred all investment in technology and consumed A; in the event of preference for later consumption, she would have preferred to invest everything in the technology and consumed B. The actual consumptions levels are strictly lower in each event, so this outcome is ine±cient. (b) Specialise the analysis to the case where A =1: Obtain the optimal symmetric allocation in this economy. 6

7 ANSWER: Set A =1: Then c 1 = 1 I c 2 = BI +(1 I) The optimal summetric allocation maximizes ¼ ln(c 1 )+±(1 ¼)ln(c 2 ) subject to ¼c 1 +(1 ¼) c 2 B =1: The rst-order condition requires c 2 = ±Bc 1 Substituting this in the budget constraint requires or ¼c 1 +(1 ¼)±c 1 =1 c 1 = 1 ¼ +(1 ¼)± and c ±B 2 = ¼ +(1 ¼)± : (c) Explain how a system of fractional reserve banking can implement the optimal outcome. Identify any assumptions. ANSWER: Under a fractional reserve system, thebankcollects depostions from all agents and invests appropriate amounts. A deposit contract speci es the amounts (c 1 ;c 2 )thatcanbewithdrawn at various periods for 1 unit deposited at time 0. Typically this outcome is sustanaible only when ±B > 1: if not, we have c 1 >c 2, so that even patient agents would like to withdraw in period 1 and reinvest in technology. (d) What is a bank run? Why is fractional reserve banking vulnerable to bank runs? ANSWER: A bank run refers to a situation in which depositors seek to withdraw their funds due to the fear that others' withdrawals causing the bank to fail. Such situations may arise as Nash equilibria, so are entirely rational from an individual point of view but ine±cient. See the handout for details 7

8 1.4 Debt contracts and Risk Sharing Optimal Risk Sharing with Symmetric Information Review Section 2.1 in handout for lecture 6. Explain why standard debt contracts do not imply optimal risk-sharing arrangements when lenders (large banks) are less risk-averse than borrowers and all information is symmetric. E±cient Incentive-Compatible Contracts with Asymmetric Information Review Section 2.2 in handout for lecture 6. Summarise the basic model with aysmmetric information and costly state veri cation. Can you explain why the standard debt contract is the e±cient, incentive-compatible contract? 1.5 Moral Hazard in Debt Markets Incentives to Repay: The Threat of Termination Review Section 2.3 in handout for lecture 6. Outline a two-period model that shows how the threat of termination of future lending can create incentives to repay debt. Sovereign Debt Review Section 2.4 in handout for lecture 6. Assume, in an in nite horizon model, that a sovereign borrower who defaults will not be able to borrow again. If the return to borrowing, given by the function V (L), has decreasing returns to scale, show that the borrower faces a credit limit (i.e., there exists some b L such that borrowing must be less than b L to preserve the incentive for repayment. Moral Hazard, Direct and Intermediated Lending, 1.6 Adverse Selection What is adverse selection? How does it a ect debt markets? Review Section 2.5 in handout for lecture 6. Outline a simple model to show how collateral can solve the adverse selection problem Review Section 1 in handout for lecture 7 (this is the Leland-Pyle model). Outline a model to how partial self- nancing can serve as a signalling device. 8

9 Review Section 2 in handout for lecture 7. What is credit rationing? Explain the central argument of the Stigltiz-Weiss model. EXAM QUESTION (2002) Suppose the return of an investment project, er(µ); is normally distributed with mean µ and variance ¾ 2. The project costs 1. An entrepreneur has mean-variance preferences represented by the utility function u. The absolute risk aversion coe±cient is ½>0, the investor has initial wealth W and, therefore, end of period wealth W 0 + e R(µ) after investment. The expected utility is Eu(W 0 + e R(µ)) = u(w 0 + µ ½ 2 ¾2 ) Although the entrepreneur has su±cient funds to self- nance the project, he can also seek outside nancing from a risk-neutral investor. (a) What potential gains can you see from external nancing of the project? ANSWER: External nancing 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 nances the project himself, his expected utility is Eu(W 0 + e R(µ)) = u(w 0 + µ ½ 2 ¾2 ): If he sells the project at price P, he trades the uncertain return e R(µ) 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 + µ ½ 2 ¾2 ); or that W 0 + P W 0 + µ ½ 2 ¾2 ; 9

10 or that P µ ½ 2 ¾2 : Hence the minimum price at which the entrepreneur would be willing to sell is µ ½ 2 ¾2. The investor is risk neutral, so values each project by its mean return. Since the project R(µ) e has expected return µ, that is the maximum he is willing to pay. (c) Now suppose the investor can observe the variance of the project (¾ 2 ) but not the expected return (µ), so the price cannot depend on µ: Assume that the investor knows the probability distribution of µ,where µ is high (µ 1 )orlow(µ 2 ) with probability ¼ 1 and ¼ 2 respectively. Outline the condition for when it is optimal for the entrepreneur to seek outside nancing. When is the investor more likely to self- nance the project? Explain your answer. ANSWER: Note the change in notation in this exam question from that in the textbook. You need to be careful about notation in exam questions. If an investor self nances the project, he gets expected utility Eu(W 0 + R(µ e i )) = u(w 0 + µ i ½ 2 ¾2 ); wherehewillknowifµ i = µ 1 (high) or µ i = µ 2 (low). If he can sell the project for a xed price P,hegets Selling is a good idea as long as Eu(W 0 + P )=u(w 0 + P ): P>µ i ½ 2 ¾2 : It follows directly that entrepreneurs who know their project to be bad (µ 2 ) willbemoreinclinedtosell. (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- nance a part of the project. 10

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 nance at least a proportion of their project and sell the rest at a unit price P 1 = µ 1. Entrepreneurs with bad projects will sell their entire project for a lower unit price P 2 = µ 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 + µ 2 ) Eu(W 0 + R(µ e 2 )+(1 )µ 1 ) = u(w 0 + µ 2 ½ 2 2 ¾ 2 +(1 )µ 1 ); µ 2 µ ½¾2 2 +(1 )µ 1 ; 2 1 2(µ 1 µ 2 ) ½¾ 2 : EXAM QUESTION (2005) Consider a population of investors, each with a project that has two possible outcomes: a positive return y in the event of success, a zero return in the event of failure. Projects di er in their risk of failure: if µ denotes the probability of failure, a fraction ¼ of projects have high risk of failure µ H, while the rest have lower risk of failure µ L <µ H. Investors have no resources of their own, so must borrow to invest and must repay R if the project succeeds. The lender cannot observe the probability of failure of a project directly but knows the fraction of each type in the population. (a) Develop the above structure to explain how the market for loans may be vulnerable to adverse selection. Note any assumptions you make. ANSWER Adverse selection: If the lender could observe the risk type, it would o er separate contract for each type to re ect 11

12 their risk cahracteristics. Assume that lenders have the bargaining power but cannot sqeeze borrowing investers below their reservation utility. If U k denotes reservation utility of risk-type k, the highest sustainable interest rate must satisfy (1 µ k )(y R k )=U k ; so that high-risk types pay R H = y U H 1 µ H while low-risk types pay R L = y U L 1 µ L : U We assume that L > U H so that R H >R L. 1 µ L 1 µ H If the lender cannot observe the risk types it must o er the same contract to each. If it nds it pro table to lend to each type, it must o er a contract that is just sweet enough for the low risk type (i.e set R = R L ): if so, high-risk types would extract an informational rent. If lending to low risk types is not pro table, only the high risk types borrow and pay R H. (b) How can collateral requirements reduce the problem of adverse selection? ANSWER: Collateral, C: something that the borrow loses in the event of failure. Often ine±cient, in that the lender recovers only some fraction of the value. That is recovers ±C, with±<1 The lender could o er two contracts (C H ;R H )and(c L ;R L ) such that each type self-selects into the right type. It turns out that given that collateral is ine±encient, the rst contract will a high interest rate but no collateral requirement (so that C H = 0); the second contract has lower interest rate with a collateral requirement C L > 0. High risk types will select the contract without collateral (1 µ H )(y R H ) (1 µ H )(y R) µ H C L Low risk types will select the contract with collateral (1 µ L )(y R) µ L C L U L The role of collateral here is to allow for self selection. 12