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- 1. A Theoretical Analysis of ”Flight to Safety”byRasmus Sørensen & Ole Bugge Frederiksenat the University of CopenhagenDepartment of EconomicsMay, 2013Thesis Supervisor:Prof. Henrik Jensen1AbstractThis paper focuses on the phenomenon ﬂight to safety where empirical trends will beshown theoretically through a model, emphasizing ﬁnancial system risk and the Knightianuncertainty related to these episodes. The theoretical framework is mainly based on theCaballero and Krishnamurthy (2005) paper.The theoretical framework is based on max-min expected utility theory, which allows usto conclude on agents’ equilibrium decisions. This is used to support the investigation,of investors going for safer assets when the risk in market increases. By implementingrobust control theory when solving the agents preferences, the paper ends up causing ascenario which could cause inﬂexibility and thereby worsen the aggregated utility. Thepaper applies a possible solution to these episodes by implementation of a ”lender of lastresort” given as a central bank providing resources to the limited collateral.The result indicates that ﬂight-to-safety episodes are caused by the protective investors,who seek to minimize their expected losses during highly volatile periods. The paperpresents a case where investors, given by agents, protect themselves by purchases of ﬁnan-cial insurances when the risk of being hit by a shock is high. The Knightian uncertaintyprovide evidence of wasted collateral when assuming robust decisions among agents.1Thanks to Henrik Jensen for his help and advice regarding this paper.Censor: Niels Blomgren-Hansen
- 2. DeclarationWe declare that this thesis was written by the authors, that the work contained hereinis our own, except where explicitly stated otherwise, and that this work has not beensubmitted to any other degree or professional qualiﬁcation except as speciﬁed.We state that the thesis was carried out by a well-functioning group, where all involvedcooperated through out the entire period, although the following shows a rough distributionof the thesis per author:Rasmus Sørensen wrote: 1, 2, 2.2, 2.2.2, 2.2.4, 3, 3.2, 3.4, 4.1, 5, 5.4Ole Bugge Frederiksen wrote: 1.1, 2.1, 2.2.1, 2.2.3, 2.3, 3.1, 3.3, 4, 4.2, 5.1, 5.2, 5.3Copenhagen, May 2013Ole Bugge Frederiksen Rasmus SørensenI
- 3. Contents1 Introduction 11.1 Flight-to-Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Theoretical Approach 22.1 Setting Up the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Implying the Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 The Intermediaries Perspective . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 The Benchmark Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3 The Agents’ Decision Problem . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Concluding the Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . 163 Flight-to-Safety Episodes 163.1 Allocating Collateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Run on Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 When Liquidity Suppliers Become Demanders . . . . . . . . . . . . . . . . . . . . . 183.4 Changes in Liquidity Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Consequences of Flight-to-Safety 204.1 The Adverse Selection Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Implementation of Savage Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Lender of Last Resort 235.1 Lender of Last Resort with Robust Agents . . . . . . . . . . . . . . . . . . . . . . . 245.2 The Fully Informed Central Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 The Partially Informed Central Bank . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Should the Central Bank Intervene? . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Discussion 297 Conclusion 30Appendices iII
- 4. 1 IntroductionThis paper seeks to analyse the phenomenon ﬂight to safety by taking a theoretical approach toexplain the trend among investors during risky periods. The paper imposes a lender of last resortto accommodate the costs following a ﬂight-to-safety episode.The structure of this thesis is as follows: Section 1 gives an overview of the phenomenon ﬂightto safety while providing a short walkthrough of historical events, which motivated us to analyseﬂight-to-safety episodes theoretically. Section 2 describes the economical environment the modelset-up. Section 3 is connected to real life events, demonstrating that the theoretical frameworksupports empirical ﬁndings. Section 4 summarizes some of the consequences related with theseepisodes, whereas Section 5 covers the intervention of a central bank as a lender of last resort.In Section 6, the empirical and theoretical evidence of ﬂight to safety are discussed. Section 7concludes.1.1 Flight-to-SafetyHistorically the ﬁnancial markets have often experienced the phenomenon ﬂight to safety, wheresome participants of the market instantly want to reduce risk in their portfolios, by ﬂeeing frommarkets with credit risks (e.g. corporate bond, stocks) towards markets with highly liquid assets(e.g. Treasury bills, gold and government bonds). Examples of episodes where this occurred arethe Penn Central default in 1970, the ”IT-Bubble” in 2001 and most recently the Global FinancialCrisis in 2008. Every time a ﬂight-to-safety episode takes place there is a risk that the economywill lead into prolonged deep recessions, resulting in a signiﬁcant macroeconomic cost. This issueis the focus of this thesis and will be discussed throughout the paper.The TED-spread during the Global Financial Crisis is a great empirical illustration of this trendamong investors. In this case, the TED-spread increased signiﬁcantly reﬂecting the increase inperceived credit risk due to the crisis. Figure 1.1 illustrates the spread between three-monthfutures contracts for US Treasuries and three-month contracts for Eurodollars, shown as thethree-month LIBOR rate, with identical expiration dates (i.e. TED-spread).The TED-spread can be used as an indicator for credit risk in the market, as the US Treasurybill is considered a risk free asset while the Eurodollar indicates the credit ratings of corporateborrowers. If the TED-spread increases, the overall default risk increases, due to the fact thatbanks are less likely to lend each other money. The spread can increase in two ways. One way isthrough the increase in the demand for US T-bills, which leads the interest rate to decrease. This1
- 5. inverse correlation showcases that when the yield goes up, the interest rates go down. Anotherway is through the increase in the LIBOR, which reﬂects the increase in the default risk as a resultof banks demanding a higher internal borrowing rate. Appendix A.1 shows that the TED-spread,went in just two months, from 1,28 pct. pt. in 2008(8) to 4,64 pct. pt. in 2008(10). This can beconsidered as a signiﬁcant increase and a clear empirical support of the concept of ﬂight to safety(Cecchetti, 2009).Figure 1.1: TED Spread 2002-2013. Source: Federal Reserve SystemThe empirical evidence of ﬂight to safety is clearly identiﬁable through several events. Yet ourinterest and focus for this thesis rely on the theoretical framework supporting such events. Un-covering this macroeconomic trend through a theoretical model, which can be used to generateand predict these episodes, is our aim. We will also discuss the potential consequences derivedfrom the model and episodes.2 Theoretical ApproachAs indicated in Section 1.1, ﬂight-to-safety episodes can be seen as possible sources to ﬁnancialinstability. The primary risk during these episodes are ﬁnancial system risk. In this section, Ca-ballero and Krishnamurthy’s model (2005) is used to create and analyse ﬂight-to-safety episodesand to showcase how higher risk and demand for safety are compelling factors in ﬂight-to-safetyepisodes. For simplicity, from now on Caballero and Krishnamurthy will referred to as C&K.The idea is to study a model where the risk of unexpected events (e.g. liquidity shocks), causeagents to re-evaluate their asset positions. We ignore what triggers these events but focus on themechanisms that play out during the liquidity episode. The model contains a set of competitiveagents, who can be deﬁned as running ﬁrms in two diﬀerent markets, and ﬁnancial intermediaries2
- 6. 2 Theoretical Approachwho can be seen as suppliers of credit insurances. There are two diﬀerent markets, in order tocreate a scenario where the shock is market speciﬁc, and therefore not by default hits the wholeeconomy.The model focuses on the agents’ awareness of the ﬁnancial intermediaries’ ability to deliver onpromises of credit. Agents write contracts with ﬁnancial intermediaries to cover from liquidityshocks that may arise in their markets. However, agents are uncertain about the probability ofother markets being subject to shocks, treating such uncertainty as Knightian, i.e. when uncer-tainty cannot be modelled with known probability distributions (Stranlund & Ben-Haim, 2006).This Knightian uncertainty is a key point due to the assumption that ﬁnancial intermediarieshave limited collateral available in the economy. If they have limited collateral, agents will beconcerned that shocks may arise in other markets than their own, which will deplete the resourcesof intermediaries and compromise their promises of credit.We will place focus on the case where agents shield themselves from the increased risk by for-mulating plans that are robust to their uncertainty regarding other markets, i.e. being robust,means taking the worst-case scenario into consideration when making decisions (Sargent & Elli-son, 2009). Robustness will be thoroughly described later through out the paper.Higher risk in the market can therefore arise from two sources: an increase in the Knightian un-certainty or a fall in intermediation collateral, and that is what we will show given this model.2.1 Setting Up the ModelWe consider a continuous time economy with a single consumption good. The economy entersthe model at date 0 where liquidity shocks may occur, and it exits at some random date τ,distributed exponentially with hazard rate δ. We take these as being exogenously given. Shocksare sequentially ordered, beginning at date s and then at date t conditional to the fact that theﬁrst shock at date s has occurred. Figure 2.1 gives a simpliﬁed overview of the model’s timeline.Figure 2.1: Overview of the models timeline.The investigated model is presented by C&K (2005), which is a variant of a bank-run model(Diamond and Dybvig, 1983).3
- 7. 2 Theoretical ApproachDemand for Liquidity We have two groups of agents, types A and B. These agents represent,in this model, ﬁnancial specialists who focus on a given market. To simplify, we choose to haveA and B represent both the agents and the market they belong to. An example, agents could beﬁrms in a speciﬁc industry susceptible to shocks in cash-ﬂows; or they could be hedge funds whoare susceptible to shocks to their trading capital. Each agent derives utility from the consumptionfunction below when the model exits at date τ:Ui= u(ciτ ) = lnciτ , i ∈ {A, B} (2.1)where cτ is consumption at date τ, and where the log-utility function show diminishing return.At date 0, the agent’s initial endowment is given by ω0 which can be seen as consumption goodsthat are perishable and can be used to purchase ﬁnancial claims. Since we are looking at ﬁrm-speciﬁc agents, when these agents exit the model, they can expect to earn a proﬁt Π, i.e. cτ = Π.However, during the liquidity phase, the agents may receive a liquidity shock that will cause theproﬁt Π to drop to zero, unless they use some of the initial endowment to purchase ﬁnancial claimsin form of liquidity insurances. With such liquid funds, agents can insure themselves against adrop in their proﬁts.Liquidity shocks can be interpreted as a market-speciﬁc shock in either market A or B and theseshocks occur to each market at most once between 0 and τ. The probability of an arrival isPoisson distributed with the intensity λ, and there is no correlation across markets. Agents willtherefore belong to either market A or B.Supply of Liquidity The supply of liquidity comes from the ﬁnancial sector, represented byintermediaries who sell insurances to agents at date 0. These intermediaries can be ﬁnancialinstitutions oﬀering futures, option contracts etc. The ﬁnancial intermediaries are required tocollateralize all sales of liquidity insurance. The supply of liquidity insurances is given by thetotal collateral in the economy, which we assume to be limited given by the parameter Z.Given their utility function, the intermediaries seek to maximize the revenue at date 0 frominsurance sales less a cost of the insurance resources, paid out at time s multiplied with β perunit:UI= cI0 + βcIs, β ≥ 0and subject to the collateral constraint, which states that the insurer do not write insurancesexceeding Z. I indicates that the utility covers the intermediaries.4
- 8. 2 Theoretical ApproachFinancial Claims Each individual is uncertain about whether he will be hit by a shock or not,and in which order the shocks will take place. We deﬁne x(s) as a claim that pays one at date s,conditional on the shock at date s being the ﬁrst shock. Likewise, y(t) is a claim paying one atdate t, conditional on the ﬁrst shock having occurred at date s < t, cf. Figure 2.1.Robustness Based on Sargent and Ellison (2009), agents are considered robust when individ-uals take the worst case scenario into consideration when buying insurances. When agents maketheir insurance decisions, they have a probability model of liquidity shocks in mind, but whilebeing sure about the intensity of the shock in their own market, they are unsure of the intensity ofthe shock in the other agent-type’s market. This uncertainty is Knightian. We will later providethe case where agents formulate decisions that are robust to the intensity of the other agents’shock.Earlier we deﬁned that λ was the probability of any given shock that might occur. To distinguishbetween the agents’ markets, we built the model from agent A’s perspective, thereby referring tothe other agent as agent B. That is why we denote A’s assessment of B’s intensity as λB.In Section 2.2.4 we express agent A’s decision problem as a max-min problem when we analysethe robust solution. This is where agent A seeks to maximize his expected utility by purchasesof ﬁnancial claims, and then ’nature’ seeks to minimize the expected utility by λB.2.2 Implying the Theoretical FrameworkIn this Section we will look at the benchmark economy, where we assume the existence of abenevolent social planner who knows all the distributions and seeks to maximize the aggregateutility of the economy. This optimization will be used as a contrast to the agent decision’sproblem. Finally, we will analyse the robustness case.2.2.1 The Intermediaries PerspectiveThe ﬁnancial intermediary sell ﬁrst and second shock claims denoted by ˆx(s) and ˆys(t), respec-tively. The intermediary’s problem is to maximize revenue less the cost of providing liquidityinsurance, subject to the collateral constraint. The price of a ﬁrst-shock claim is p(s) and theprice of a second-shock claim is qs(t). For simplicity, we denote φ1(s) as the true probability of theﬁrst shock, and we denote the true probability of the second shock as φs(t). These probabilities5
- 9. 2 Theoretical Approachare given by:φ1(s) = λe−(δ+2λ)sφs(t) = λ2e−λse−(δ+λ)t(2.2)We assume that the intermediary knows the true probabilities and that he does not have anyrobustness concerns, which gives us the intermediary’s objective function:max{ˆx(s)},{ˆy(s)}∞0p(s) − βφ1(s))ˆx(s) +∞s(qs(t) − βφs(t))ˆys(t)dt dssubject toˆx(s) ≤ Z (2.3)ˆx(s) + ˆys(t) ≤ Z (2.4)Consider the constraint denoted by equation (2.3). This is a standard collateral constraint, sincethe agent considering buying ﬁrst-shock insurance has access to the intermediary’s balance sheet.In this case, we might assume that the agent does not want to buy ﬁrst-shock insurance if theintermediary does not have enough collateral, Z, to cover the agents liabilities, or else the agentcannot cover his losses, which gives us constraint (2.3).One can assume that the agent who seeks to buy second-shock insurance has the same possibilitiesas the ﬁrst agent, though he also has to take into consideration that the second shock insuranceis paid after the ﬁrst shock insurance. This is represented by constraint (2.4). If the collateralconstraints do not restrict the objective function, one can ﬁnd that ˆx(s) and ˆys(t) are interior,due to linearity, only if:p(s) = βφ1(s)qs(t) = βφs(t)When β = 1 we get actuarially fair prices as an interior solution. This means that under normal-ization, claims are sold for prices corresponding to the true probability of receiving a shock, whichin turn demonstrates that intermediaries do not have arbitrary incentives for selling insurances.Assuming normalization, this means that when β > 1, ﬁnancial claims are sold for more thanwhat the true probabilities indicate, thus aﬀecting negatively the amount of claims sold. Theopposite happens when β < 1.Our interest lies where the objective function is constrained by (2.3) and (2.4), i.e. when theintermediary has limited resources. We assume that there are groups of agents interested in6
- 10. 2 Theoretical Approachsecond-shock claims, thus considering the constraint (2.4) which binds in equilibrium for all (s, t)pairs (ˆx(s) + ˆys(t) = Z). Further we denote shadow values µs(t) > 0, corresponding to themarginal eﬀect of a change in the constraint. The calculations of ﬁnding the ﬁrst order conditionis shown in Appendix A.2:qs(t) = βφs(t) + µs(t)p(s) = βφ1(s) + µ(s); µ(s) ≡∞sµs(t)dt. (2.5)Given that µ > 0, the prices increase to reﬂect the scarcity of collateral. To ﬁnd the price for thesecond-shock claims for all t > s, one has to integrate the expression for qs(t):q(s) ≡∞sqs(t)dt = βφ2(s) + µ(s) (2.6)whereφ2(s) ≡∞sφs(t)dt = φ1(s)λλ + δ(2.7)We integrate for the interval s → ∞, therefore getting an expression of s and not t. The lastterm φ2(s) is the true probability of the economy receiving two shocks before the date τ which isgiven by the probability of receiving the ﬁrst shock multiplied by the probability of receiving thesecond shock, given that the ﬁrst shock has occurred. This is, denoted by λλ+δ . The expressionφ2(s) is solved in Appendix A.3. From this, we ﬁnd that the multiplier is common to both q(s)and p(s) and does not depend on the probabilities of these events, which evidences that a tightcollateral constraint may push up the prices on the insurances, no matter how the probabilitiesof the events might look, i.e. if ˆx(s) is high then the availability of the insurance on the secondshock is restricted for all t > s.One can conclude that the opportunity cost of writing ﬁrst-shock claims is to write less second-shock claims, which results in prices being equal to reﬂect the common cost of locking collateral.This can be shown by solving for µ(s) (see Appendix A.4), and writing p(s) as a function ofq(s):p(s) = q(s) + β(φ1(s) − φ2(s))Since we are looking at a sequential-shock structure to the economy in this model, the intermediaryhas to lock the capital bound to ˆy(s). That is, why he cannot use the same collateral to backup both events, ct. ˆys(t) ≡ ˆy(s) = Z − ˆx(s). This leaves us with a common cost of locking thecollateral. If ˆy(s) and ˆx(s) corresponded to two mutual events, then the intermediary could use7
- 11. 2 Theoretical Approachthe same collateral for backing up either event, thus not being limited to having a common costof locking collateral. To simplify our expression we assume for now that β = 0, which gives us:p(s) = q(s) (2.8)Equation (2.8) captures the equilibrium cost of locking collateral in the model, and representsthe main result of modelling the supply side of the economy. By having equal prices, we geta expression which tells us that p(s) and q(s) are related to each other by a common cost oflocking collateral. This expression will be used later on to solve the agents’ decisions problemwhen choosing the optimal level of x(s) and y(s).2.2.2 The Benchmark EconomyHaving found the equilibrium prices from the supply side of the economy, we will now establish abenchmark for the model. We introduce a benevolent social planner, a planner that seeks to ﬁndthe most eﬃcient social optimum. The goal is to ﬁnd the benevolent planners optimum whenthere are no robustness concerns.x(s) is deﬁned as the amount of liquidity insurance delivered to the ﬁrst agent who receives ashock. Likewise, we deﬁne ys(t) as the amount of liquidity insurance delivered to the second agentwho receives a shock. The benevolent planner’s problem can be deﬁned as:max{x(s)},{ys(t)}δλ + δu(Π) +∞0φ1(s)u(x(s)) + φ2(s)∞sη(t − s)u(ys(t))dt dswhere η(t − s) = (λ + δ)e−(λ+δ)(t−s) is the probability that the second shock takes place at timet, given that a second shock takes place at some time t > s.The ﬁrst term in the objective function is the agent’s expected utility when the market does notreceive a shock. This term is independent of the decision problem and therefore we will omit itwhere we ﬁnd convenient. The second term is the expected utility that the agent receives if themarket is hit by a shock before the model exits the liquidity phase, and the utility if the marketis hit by a second shock.Given that the intermediaries have a limited amount of liquidity, Z, and we assumed that β = 0,the constraints are then given by:0 ≤ x(s) ≤ Z0 ≤ x(s) + ys(t) ≤ Z8
- 12. 2 Theoretical Approachwhere the planner allocates the scarce resources across time and agents.The second collateral constraint binds, because it includes the ﬁrst constraint and there is noother cost of providing insurance other than the collateral limitation i.e. since the intermediariesare proﬁt maximizing, they will keep providing insurances until:ys(t) ≡ y(s) = Z − x(s). (2.9)This leads us to ﬁnd that ys(t) is not a function of the time interval between the ﬁrst shockand the exit date τ, but a function of the residual from the binding constraint. Z can either beused to back up insurance x(s) or ys(t), but not both at the same time. The opportunity cost ofproviding x(s) is less provision of ys(t). We can then rewrite the planner’s maximization problem,as only being dependent on s due to equation (2.9):max{x(s)},{y(s)}δδ + λu(Π) +∞0[φ1(s)u(x(s)) + φ2(s)u(y(s))]ds,We solve this by using the same argument as in Appendix A.2 and inserting equation (2.9) weﬁnd the ﬁrst order conditions:w.r.t. x(s) : φ1(s)u (x(s))w.r.t. y(s) : φ2(s)u (y(s))To analyse the benchmark economy further, we will ﬁnd relative relation between the two derivedutility functions:u (x(s))u (y(s))=φ2(s)φ1(s)=λλ + δwhere y(s) = Z − x(s). This is solved in Appendix A.5, together with the optimal quantity ofinsurances interpreted below.For the log utility case, given the utility function (2.1), we ﬁnd the optimal quantity of insuranceclaims in the benchmark economy are given by:xbe=λ + δ2λ + δZ ybe=λ2λ + δZ;and satisﬁes:xbeybe=λ + δλwhere be denote the benchmark quantities. Assuming that one market is hit by a shock and δ > 0,9
- 13. 2 Theoretical Approachthe planner has an incentive to allocate more than half of the liquidity to the market receivingthe ﬁrst shock, since the chance that the second shock never occurs is increasing with δ. We areinterested in liquidity crisis episodes with a large δ, where the phenomena is temporary and notlong lasting. In cases like these, the incentive to inject more funds to the ﬁrst market hit by ashock increases, due to the small probability of the second shock occurring.In the benchmark economy we found that when δ > 0, the agents will demand more ﬁrst-shockinsurances. This is due to the fact that the higher δ the smaller the possibility of a second shockoccurring, and that, due to the common cost of insurance (see equation (2.8)), the incentive ofbuying ﬁrst-shock insurance increases relative to the second-shock insurance, because of equalprices.2.2.3 The Agents’ Decision ProblemWhen analysing the benchmark of the economy, we assumed a benevolent social planner. In thissection we will remove the planner and implement two individual agents in two separate marketsas described in Section 2.1. A new parameter λB is also introduced and like λ it measures theprobability of a market receiving a shock, though λB is only related to market B.We analyse the decision problem from agent A’s perspective, who will try to choose paths of x(s)and ys(t) while being robust to any value of λB.The expected utility function for A, given choices of {x(s)} and {ys(t)} and some value of λB,is:V ({x(s)}, {ys(t)}, λB) =∞0φA1 (s)u(x(s)) + φA2 (s)∞sη(t − s)u(ys(t))dt dswhere, from the exponential distribution function, A’s probabilities are:φA1 (s) ≡ P(First shock is A’s, at s) = λe−(δ+λ+λB)sφA2 (s) ≡ P(Second shock is A’s, at t > s) =λBλ + δφA1 (s)We want to maximize V (·) subject to a budget constraint. The budget constraint reﬂects purchaseof x(s) and ys(t), given prices of p(s) and qs(t) and the initial endowment of w0. Now we writethe agent’s maximization problem:max{x(s)≥0},{ys(t)≥0}V ({x(s)}, {ys(t)}, λB) st.∞0p(s)x(s) +∞sqs(t)ys(t)dt ds ≤ w0 (2.10)In Appendix A.6 we use Lagrange optimization to solve the agent’s problem, where ψ is the La-grange multiplier on the budget constraint, and ﬁnd the ﬁrst order condition for the optimization10
- 14. 2 Theoretical Approachproblem.φA1 (s)u (x(s)) = ψp(s) (2.11)φA2 (s)η(t − s)u (ys(t)) = ψqs(t) (2.12)To simplify we integrate both sides of (2.12) with respect to t, and by using both equation (2.8)and (2.6), we get:φA2 (s)∞sη(t − s)u (ys(t))dt = ψp(s)This solution is shown in Appendix A.7. Then we insert (2.11) and solve u (x(s)):u (x(s)) =φA2 (s)φA1 (s)∞sη(t − s)u (ys(t))dt (2.13)From the benchmark, we know that ys(t) ≡ y(s) = Z − x(s) and inserting the expression of y(s)we get a function only given by s and not of t, which simpliﬁes the relation.u (x(s)) =φA2 (s)φA1 (s)∞sη(t − s)u (y(s))dt (2.14)In Appendix A.8 we show that∞s η(t − s)dt = 1 and using this by solving for u (x(s)) andu (y(s)), we get the true probability of receiving a second shock conditional on having receivedthe ﬁrst shock:u (x(s))u (y(s))=φA2 (s)φA1 (s)=λBλ + δ(2.15)where the log-utility ease the interpretation since we get this equation, cf. eq. (2.1):x(s)y(s)=λ + δλBIntuition:First of all, by solving for agent A is the same as solving for all agents belonging to market A,why we can interpret for all agents by only looking at one.The more likely the other agent is to be hit by a shock, the more agent A demands second-shockinsurances. By that, agent A ensures that he still receives payments if he is hit by a liquidityshock, i.e. the ratio between x and y claims is decreasing with respect to λB. If λB = λ, theagent’s decision problem correspond with the benchmark model.11
- 15. 2 Theoretical Approach2.2.4 RobustnessIn the previous section we deﬁned an expected utility function V ({x(s)}, {ys(t)}, λB) and derivedequation (2.15) that characterized the decision problem for agent A in choosing x(s) relative toys(t), given a particular value of intensity λB.The robustness step for agent A is to make these choices while being robust to alternative valuesof λB.The full problem for agent A is given by:max{x(s)≥0},{ys(t)≥0}minλB∈ΛV ({x(s)}, {ys(t)}, λB)given equilibrium prices p(s) and qs(t) and the budget constraint. The idea is to let agent Achoose x(s) and ys(t) and then let ”nature” choose λB ∈ Λ to minimize the utility of agent A,given his maximization choice.To add robustness to the model, we assume a set of alternative models over which agents wouldlike to be robust, in an interval deﬁned by:max{0, λ − K} ≤ λB ≤ λ + Kwhere K indexes the robustness preferences of agents, meaning that with a larger K, agentsprotect themselves against a more extreme worst-case scenario. One could conclude, that therobustness preferences is separated in two groups, either being partially robust or fully robustgiven the relative relation between the sizes of K and δ. The following expressions are built onour ﬁndings in the proof stated below.• For K < δ, agent’s decisions are as if λB = λ + K:xprypr=λ + δλ + KThis is referred to the ”partially robust” case.• For K ≥ δ, agent’s decisions are as if λB = δ:xfryfr= 1This equal claim is referred to the ”fully robust” case.12
- 16. 2 Theoretical Approach• In both cases, agent’s decisions are time invariant1When K = 0, the ’robustness’ part of the model vanishes and we get a model that correspondto the benchmark model described in Section 2.2.2. Here, the decisions are as if λB = λ, andtherefore we do not interpret further.As K rises, agents become more concerned in regards to receiving a second shock, in which casethe ﬁrst-shock insurance will be useless. This results in a reduction of their purchases of ﬁrst-shock insurances and leads to an increase in purchases of second-shock insurances. If K gets largeenough, agents will equate their purchases of ﬁrst and second-shock claims, and thereby insulatethemselves against their uncertainty over the likelihood of receiving a shock. This intuition comesfrom the proof provided below.Proof. C&K (2005, p. 16) derive the prices in the proposed equilibrium, denoting λB whenagents base decisions in equilibrium as ˜λ, and imply that the market clearing conditions are givenby:xA(s) + yBs (t) = ZxB(s) + yAs (t) = Zusing the fact that the collateral availability of Z determines the supply of liquidity insurance,which is to be understood as the insurer saving up his limited collateral by backing the twopossible shock sequences of A then B, and B then A.Relative demand satisﬁes:x(s)y(s)=λ + δ˜λ(2.16)which can be substituted when solving for x(s) and y(s). To ﬁnd prices, the derived quantitiesare substituted in Appendix A.9 into the ﬁrst order conditions for agent A, eq. (2.11) and (2.12),to yield:p(s) =w0Z(λ + δ + ˜λ)e−(λ+δ+˜λ)s(2.17)qs(t) =w0Z(λ + δ + ˜λ)(λ + δ)e−(λ+δ+˜λ)s−(λ+δ)(t−s)(2.18)Below the robustness step is taken into precaution:For small K: The idea is to show that, at equilibrium prices, the highest utility attainable when1Decisions does not depend explicitly on time13
- 17. 2 Theoretical ApproachsolvingV = max{x(s)≥0},{ys(t)≥0}minλB∈ΛV ({x(s)}, {ys(t)}, λB) (2.19)happens at xpr and ypr, and given these choices, ”nature” always chooses λB = λ + K. No otherchoice of {x(s)} {y(t)} will make ”nature” choose a λB that results in a higher utility for theagent.Assuming the worst case of λ + K, and having the equilibrium prices given by eq. (2.17) and(2.18), the unique optimum to the agents’ max problem, (2.10), is:x(s) =λ + δλ + δ + ˜λe−(λB−˜λ)sZ and y(s) ≡ ys(t) =λBλ + δ + ˜λe−(λB−˜λ)sZThe expression of x(s) and y(s) is solved in Appendix A.10. At ˜λ = λB, x(s) and y(s) areconstant functions of time. With this, the expected utility function for the agent can be derivedas:V (xpr, ypr, λB) = φA1 u(xpr) + φA1 u(xpr), (2.20)where φA1 and φA2 denote the probability of a ﬁrst and second shock at any time (showed throughagent A), and the notation pr is due to the partially robust case, deﬁned by:φA1 ≡∞0φA1 (s)ds =λ(δ + λ + λB); φA2 ≡∞0φA2 (s)ds = φA1λB(λ + δ)(2.21)Given the optimal values of x(s) and y(s), the ’nature’ now seeks to minimize via λB, so we insert(2.21) into (2.20):V (·) =λ(δ + λ + λB)u(xpr) +λ(δ + λ + λB)λBλ + δu(ypr)We will derive this expression using the envelope theorem2:dVdλB=−λ(δ + λ + λB)2u(xpr) +λ((λ + δ)(δ + λ + λB)) − (λλB)(λ + δ)(δ + λ + λB)2(λ + δ)2u(ypr)+λ(λ + δ)(δ + λ + λB)3ZλBλ + δu (ypr) − u (xpr)2The envelope theorem states that the change in the optimal value of a function with respect to aparameter of that function can be found by partially diﬀerentiating the objective function while holdinge.g. x (or several x’s) at its optimal value.14
- 18. 2 Theoretical ApproachApplying the log utility from equation (2.1), one ﬁnds that the last expressions cancel out:=−λ(δ + λ + λB)2u(xpr) +λ(δ + λ + λB)2u(ypr)⇓VλB=λ(δ + λ + λB)2[u(ypr) − u(xpr)] (2.22)From the envelope theorem, we ﬁnd that u (xpr) and u (ypr) cancels out. This is because they areat their optimal values, not having any marginal eﬀect. This leaves us with the residual (2.22),which is strictly negative as long as xpr > ypr. For small values of K one ﬁnds that the agent isfairly susceptible to second shocks if the ’nature’ choose to increase λB, since a higher lambdaBmakes ﬁrst shock insurances less valuable. The constraint set in choosing λ ∈ Λ is linear for smallK, resulting that the utility for the agent given choices of (xpr, ypr) is minimized at the highestpossible value when λB = λ + K.We deﬁne the partially robust expected utility as V pr ≡ V (xpr, ypr, λB = λ + K). One canargue through contradiction that V pr is the highest possible utility in (2.19). Consider choicesof x(s) and y(s) which could result in V > V pr: such choices can only increase utility if naturechoose λB < λ + K. The problem in (2.10) is a strictly concave function, therefore the ﬁrst orderconditions deﬁne a unique global maximum. If the agent chooses to deviate from xpr and ypr,and ’nature’ keeps choosing the worst case scenario λB = λ + K, then we ﬁnd that the inducedutility is strictly smaller than V pr, wherefore the agent can do no better than choosing xpr andypr.For large K: The intuition is that for large values of K, the gap between ﬁrst-shock and second-shock claims narrows down until it disappears, once K = δ. This stage is called ”fully robust”with respect to any λB (since x = y at this point), which implies:VλB= 0At K = δ, the agent reaches the utility of V fr ≡ V (Z2 , Z2 , λB = λ + δ). If K increases furtherthe constraint set for ’nature’ will be expanded, and thereby weakly decreases the highest utilityattainable in eq. (2.19). To guarantee the utility V fr, the agent has to choose x = y = Z2 . Thus,the fully robust choices continue to attain the highest possible utility in eq. (2.19) for all K ≥ δ.15
- 19. 2.3 Concluding the Theoretical FrameworkWe have shown that the model manages to generate episodes similar to a ﬂight to safety phe-nomenon. As described in the introduction of this section, higher risk in the market, given byK, can come from two sources, namely, an increase in the Knightian uncertainty or a fall in theintermediation collateral. A rise in K will lead to a rise in the probability of receiving a secondshock, given by the intensity λB, which will then result in an increase of purchases of second-shockinsurances. Supported by the robustness case, we found that the more robust agents seem to be,the more ineﬃcient the market will be, due to the hazard rate δ, i.e. the higher δ, the more likelythe model is not to be hit by a second shock, and the collateral bound to the y(s) insuranceswill go wasted. This behaviour is the essence of a ﬂight-to-safety episode, i.e. higher demand forcertainty and thereby safety.3 Flight-to-Safety EpisodesIn the following, we will apply the model described in the previous chapter to analyse givenepisodes of ﬂight to safety. We have chosen only to look at episodes that involve a ﬁxed amountof Z, i.e. a ﬁxed amount of collateral available in the economy, and robust agents that may aﬀectthe available capital negatively.First, we will assess episodes with robust actions, showcasing how this sort of allocation willreduce the aggregate liquidity. Then, we will look at the possible macroeconomic costs of theseevents.Most ﬂight-to-safety episodes occur due to unexpected events in the economy. For instance in2008 the Lehman Brothers bankruptcy was the ﬁnal trigger that led to market failure, as theFederal Reserve chose not to intervene by increasing the bank’s collateral. Thus, the LehmanBrothers could not uphold their maintenance margin, ﬁling the bank for bankruptcy. This re-sulted in a crisis that led to a ﬂight-to-safety episode, as one can note in Figure 1.1, where theinterbank oﬀered rate increased due to the rising market risks1. The increased default risk mighthave led agents to recalculate their models. In the model, a reassessment can be thought of as arise in K, which implies a rise in λB, i.e. the agent’s perception of market B’s default probabilitywill worsen.Given a ﬁxed K, one can, by reducing Z, create a ﬂight-to-safety episode. A drop in Z in-creases the ﬁnancial intermediary risk, due to the fact that the agents get concerned with the1http://www.guardian.co.uk/business/2009/sep/03/lehman-collapse-us-uk-blame16
- 20. 3 Flight-to-Safety Episodesintermediaries not having enough liquidity to cover their ﬁnancial claims if shocks occur. Theagents become worried that they might be hit by a second shock and thereby they increase theirperception of market B failure, i.e. an increase in λB. This is a clear sign of a ﬂight-to-safetyepisode.3.1 Allocating CollateralLooking at equilibrium in market A and B, one can note that they are both guaranteed liquidityof:y = min{x, y}i.e. given the second shock occurs, the markets can expect at least y capital to their disposal dueto the insurance, which gives x − y ≤ 0 available for the intermediary to allocate to the marketreceiving the ﬁrst shock. This can be interpreted as the intermediaries’ allocation decision. Sincey is the amount of insurance delivered to the second agent who receives a shock, this capital islocked to service the second agent if the second shock should occur, which results in that x − ycapital is free to move to the ﬁrst market who receives a shock. As the probability (density of λB)of occurrence of second shock increases, the gap between x and y falls and thereby the ﬂexibilityof the market diminishes, where the fully robust case is given by x − y = 0, leaving the marketssegmented and rigid. Segmentation is the markets’ response to the agents’ robustness concerns.Since the agents’ ﬁnancial claims are independent of the shocks on the external markets, theintermediaries are bound to keep a certain amount of capital allocated at their disposal, given byx and y.3.2 Run on BanksIn our model, we have so far a limited amount of collateral available, and due to this, one caninterpret the ﬁnancial claims of A and B as credit lines from intermediaries. The agents knowthat there are boundaries on how much capital intermediaries have available to back up thesecredit lines. Hence, the robust agent A is worried that the robust agent B will erode the banks’limited resources, given that agent B is the ﬁrst to experience a shock. That is, why the robustagent A sets x = y, to ensure that he will have available capital if the shock occurs. This meansthat agent A’s actions can be interpreted as pre-emptive behaviour when the fear of large Bshocks increases. Since the markets in the model are complete, the future action of agent A isprearranged at date 0, by setting x = y.17
- 21. 3 Flight-to-Safety EpisodesThe actions of the agents in the model can be inferred as run on banks’ credit facility, though it isalso related to the more commonly term ”run on banks’ deposits”. As Diamond and Dybvig (1983)concludes, a deposit contract induces an optimal shock-contingent allocation of liquidity. In thefully robust equilibrium in the model, each agent chooses to allocate their resources independentlyof each other’s shocks. The agents decide to hoard Z/2 units of capital to ensure for shocks intheir own markets, independently of shocks in other markets. Given the benchmark economy,this may not be the most eﬃcient way to allocate resources, since the agents are not faced withthe same probability of shocks occurring. In this sense, the robust agents’ behaviour can be seenas the behaviour of panicked depositors in a common bank run episode. For example, one couldrelate this to the present situation in Cyprus. There is a great probability that the banks are hitby a run on the banks’ credit facilities. The agents know there is a probability that their bankdeposits may be hit by an external shock, a one-time tax payment of 6, 45% on deposits belowe100.000 and a 9, 9% tax payment on anything above the amount, thus compelling the agents towithdraw their deposits to avoid this tax payment2. To prevent this bank-run, the banks haveshut down their deposit stores, thereby preventing the agents to make a bank-run. This kind ofﬂight-to-safety episode occurs when the agents’ demand for certainty increases. The alternativeto an un-contingent insurance as λB rises is that the agents seek more well collateralized ﬁnancialinstitutions, as they fear that their present bank has a greater default probability, and largerﬁnancial institutions are less likely to be aﬀected by the other market’s shock.3.3 When Liquidity Suppliers Become DemandersOne can think of the above robustness concern as corresponding to hedge funds’ worries thatthe liquidity in the market will fall, triggering the funds to overprotect against sudden drops inthe market, which further reduces the liquidity available in the market. Consider a hedge fundin market A that needs other liquidity providers to act in its market to make the business moreproﬁtable. The robust hedge fund grows concerned that a shock in market B may cause theother liquidity providers to pull back from market A, thereby reducing the liquidity available andmaking the market unproﬁtable. To avoid this, the hedge fund reduces the amount of capital xcurrently used for liquidity in market A, allocating its resources to increase the buﬀer y to preventa possible B shock, which in turn reduces the available ”trading” capital in both markets A andB.2http://borsen.dk/nyheder/oekonomi/artikel/1/254643/boersen i nicosiaeurolande slagter cyperns ststorbank.html?hl=Q3lwZXJuOzksOQ18
- 22. 3 Flight-to-Safety EpisodesJohn C. Hull (2012) gives an example of how the hedge fund Long-Term Capital Management(LTCM) caused an increase in the liquidity risk. LTCM followed a hedge strategy known as con-vergence arbitrage, which means that they where highly leveraged. Due to the Russian defaultthis strategy resulted in huge losses for LTCM, leading to a serious ﬂight-to-safety episode. Thiscrisis of Fall of 1998 caused hedge funds to retreat from risky markets, thus liquidating assetsand reducing the supply of liquidity, i.e. liquidity suppliers became demanders.The outlined model does not distinguish between the diﬀerent liquidity providers that can beobserved in reality; instead the model shows how the behaviour of robust agents reduce the eﬀec-tive Z in the economy and removes the ﬂexibility of the markets, leaving the economy relativelyrigid.3.4 Changes in Liquidity PricesAnother circumstance under which ﬂight to safety may take place is when changes in the priceof liquidity provision occurs. One could substitute the values of x(s) and y(s) in the equilibriuminto the ﬁrst order conditions for agent A, i.e. eq. (2.11) and (2.12), to ﬁnd the price of the ﬁrstand second shock liquidity insurance, as shown in Section (2.2.3):p(s) = q(s) =w0Z(λ + δ + λB) exp−(λ+δ+λB)s(3.1)One can note that as λB rises, the prices in equilibrium change as well. As s goes to 0, i.e.the agent expects an early ﬁrst shock, the exponential term in eq. (3.1) gets more and moredominated by the linear term, implying that an increase in K will lead to an increase in pricesof both the ﬁrst and second shock claims:p(0) = q(0) =w0Z(λ + δ + λB) (3.2)The increase in price of the ﬁrst shock insurance is indirectly driven by the increased price of thesecond shock insurance. Like earlier, when λB increases, robust agents decrease their demandfor ﬁrst shock insurance, due to the increased probability of occurrence of the second shock.This means that, they increase demand for second shock insurance, forcing the insurers to lock-up collateral for y, which in turn decreases the credit supply available to ﬁrst-shock insurance,leading to an increase in price for ﬁrst-shock insurance.p and q are the marginal cost of liquidity provision in the economy. The liquidity providers have alimited amount of capital at the aggregate level, given by the parameter Z. As the robustness ofthe agents rises, so does the cost of liquidity provision. This is a result of the fact that the more19
- 23. robust the agents are, the more the eﬀective part of Z reduces, i.e. at the extreme, they insulatethemselves against their uncertainty over the likelihood of receiving a second shock, x = y.4 Consequences of Flight-to-SafetyIn this section, we outline and discuss some of the macroeconomic consequences of ﬂight-to-safetyepisodes. In practice, system risk caused by these episodes is the important macroeconomicconcern derived from these episodes, as the ﬁnancial system becomes inﬂexible. This inﬂexibilitycreates bottlenecks that can trigger accelerators and create system-wide problems, leaving theeconomy overexposed to recessionary shocks. The eﬀect does not concern only the shift of assets,but it is also geographical. Empirically it is known that ﬂight-to-safety episodes have ﬁnancialand real eﬀects. An example is the euro spread of bond yields, in the EU’s debt market.4.1 The Adverse Selection EﬀectStieglitz and Weiss (1981) present a model analysing credit rationing in markets with imperfectinformation, focusing on the equilibrium in a supply/demand market for ﬁnancial lenders/bor-rowers.The model is a general equilibrium model of supply and demand for ﬁnancial credit with ﬁnan-cial intermediaries, where banks are the suppliers of loans and agents who are the demanders forthese loans. The suppliers seek to maximize their expected payoﬀ by increasing and decreasingthe interest rates of the loans. The agents seek to minimize their expected costs by signallingtheir liquidity, i.e. there is a signalling value attached to the agent’s probability of paying theloan back to the bank.The objective of their paper is to show that, in equilibrium, a loan market may be characterizedby credit rationing (Stiglitz and Weiss, 1981 p. 393), where the interest rate set by the bankreﬂects two things:1. Sorting potential borrowers (the adverse selection eﬀect)2. Aﬀecting the actions of borrowers (the incentive eﬀect)We choose to focus on the adverse selection eﬀect, which can be perceived as a consequence of dif-ferent borrowers with diﬀerent credibilities. This eﬀect is very important to consider when lookingat the macroeconomic costs during ﬂight to safety episodes, since the extreme case results in theclassic lemons problem (Akerloﬀ, 1970) - The bad ”driving” out the good), where good borrowers20
- 24. 4 Consequences of Flight-to-Safetyleave the market, thus increasing the average risk and default rates of borrowers who remain inthe market. As a result, interest rates go up to compensate for the higher risk, more borrowersleave the market, causing the average risk to go up and more borrowers leave. At the end, thiscan lead to market failure. This phenomenon is more commonly known as ﬁnancial accelerators.A practical example is when companies’ ability to lend capital depends on the market value oftheir equity, which equals assets less liabilities. Using the asymmetric information theorem fromAkerlof (1970), one can argue that lenders have little information about the reliability of anygiven borrower, therefore lenders usually demand borrowers to display their ability to repay, i.e.their signalling value, often represented in form of collateralized assets. It follows that a fall inasset prices erodes the net worth of the company. The erosion of their ability to borrow hasa negative impact on their investment. Decreased economic activity further decreases the assetprices, which leads to a feedback cycle of falling asset prices, thus deteriorating the company’snet worth, tightening ﬁnancing conditions and lowering economic activity. This continous in aspiral.Figure 4.1: 10-year Government Bond Yields, 2006(1)-2013(3). Source: ECB StatsFigure 4.1 shows the yields of 10-year government bonds for 6 countries in the European bondmarket. One can clearly note that following a period with almost identical bond yields, the GlobalFinancial Crisis in 2008 caused the usual trend to stop, and in beginning 2012 led to a never beforeseen spread in the Eurozone. Combining Figure 4.1 and the imperfect information theorem can,therefore, be another way of analysing the macroeconomic costs resulting from a ﬂight-to-safetyepisode. In the Stieglitz and Weiss (1981) article, they investigate why credit is rationed. Theirconclusion is that credit is rationed as segmentation under the ”Allocating Collateral” episode,which means that, even though certain markets are willing to pay a relative higher price for the21
- 25. 4 Consequences of Flight-to-Safetycollateral, the intermediaries ﬁnd that the risks are too high and therefore choose not to invest inthese markets. Referring to Erik Jones (2012), one can argue that ﬂight to safety is not just seenas market speciﬁc asset movements, but also as geographical movements. Figure 4.1 emphasizesthis, where we see investors ﬂeeing from assumable risky Greek bonds towards safer Germanbonds, for instance.4.2 Implementation of Savage AgentsAll investors in the real world cannot be perceived as ﬁnancial specialists, therefore in a morerealistic model small number of rational savage agents can be incorporated. These agents will,as in the benchmark economy, ﬁnd that it is optimal to allocate resources towards the insurancefor the ﬁrst shock, rather than to ensure against the second shock, as there is a lower probabilityof occurrence of a second shock. These agents will face with the same inﬂated liquidity insur-ance prices as the robust agents. Given these high prices, the non-robust agents will reduce theamount of purchased insurance and leave themselves vulnerable to liquidity shocks. Consideringthe opposite of what was stated in Section 4.1, one can argue that an increase in robustness ofthe ﬁnancial specialist will lead to the rational savage agents being pushed out of the insurancemarket due to the increased prices. Under these circumstances, the ﬂight to safety in the modelis not just a problem aﬀecting the decisions for robust agents; it has an economy-wide cost.From 2.2.4, we found that the more robust the agents were, the more liquidity they chose to lockup in insurances to cover themselves from shocks. This left the ﬁnancial system inﬂexible andless able to react to shocks. Assuming that the agents are not fully robust, one can consider thesocial optimal, i.e. benchmark economy where xbe > ybe, leaving xbe − ybe of collateral to beallocated to the ﬁrst market receiving a shock. This is optimal as long as δ > 0, because at arelatively large value of δ, there is a great probability of the economy not being hit by a secondshock.Rewriting the collateral constraint, the total collateral waste associated with ﬂight-to-safetyepisodes is:x + y ≤ Zin terms of risky, x, and riskless claims, y, we rewrite the collateral constraint further:(x − y) + 2y ≤ Z22
- 26. The riskless claim ’consumes’ twice as much as the risky claim if all agents allocate their resourcesto the riskless claims. This will cause collateral to be locked up will therefore be unused whenδ > 0, i.e. due to the increased probability of the economy not being hit by a second shock.In practice this allocation leaves the economy inﬂexible towards market adjustments, which canlead to bottlenecks that might trigger accelerators and create system-wide problems, not onlyaﬀecting the ﬁnancial specialists but also the common rational savage agents.5 Lender of Last ResortIn the following section we will investigate how a public service can meet the macroeconomic costsof ﬂight-to-safety episodes. This can be accomplished by implementing a lender of last resort inthe model, i.e. a central bank.Consider a central bank that obtains resources ex-post, which it can lend to agents in the second-shock extreme case, i.e. the funds for these kinds of interventions may be obtained by increasinginﬂation, taxation, etc. In practice, the central bank is a lender of last resort that guaranteesthe liabilities of the ﬁnancial intermediaries, who have sold ﬁnancial claims to both markets andcannot meet their obligation. A classic case is the American insurance company AIG, who was,who was unable to meet its obligations due to the ﬁnancial crisis in 2008, which required theinfused of liquidity by the Federal Reserve. The main reason for the Federal Reserve to act inthis case was the opportunity cost of not doing so, i.e. the economic costs of not intervening wasgreater than the costs of the intervention1. We will in this section analyse the marginal beneﬁtof such a guarantee (C&K, 2005).To implement the central bank’s possible intervention, we include the parameter G to indicatethe increase in the collateral constraint:x + y ≤ Z + GSince the central bank is a lender of last resort the intervention, has no eﬀect on the constraintfor ﬁrst shock insurances.C&K (2005) indicates that they picture the robustness preferences as a realistic depiction of thedecision rules of ﬁnancial specialists. Thus they ﬁnd it unclear why a central bank should buildbiases into its objective function, which could result in a long-term loss due to agents exploitingthese biases. We will take three diﬀerent approaches to analyse the eﬀect a lender of last resort:First, we will analyze a welfare function based on the robust agents’ expected utility function.1http://dealbook.nytimes.com/2013/01/07/rescued-by-a-bailout-a-i-g-may-sue-its-savior/23
- 27. 5 Lender of Last ResortSecond, we will look at the classic model with a fully informed central bank. Finally, we will lookat the more realistic model where the central bank is partially informed.5.1 Lender of Last Resort with Robust AgentsAssuming agents are fully robust, the central bank uses the agents’ expected utility functionfrom equation (2.20) to evaluate welfare. From the proof in Section (2.2.4), we derived from theenvelope theorem that the indirect eﬀect is of second order. Therefore the only beneﬁt in thissituation is the direct eﬀect. The expected utility at optimal values of x and y is:V (x, y, λB) = φA1 u(x) + φA2 u(y) (5.1)Since the central bank only intervenes in extreme cases of second shock, the direct eﬀect liesupon y. However, as we are looking at fully robust agents, the indirect eﬀect on x comes from are-optimization of the agents’ portfolios. Thus, increasing the collateral available for y increasesboth x and y. Assuming the entire increase of Z is allocated to purchases of y, one concludesfrom the envelope theorem that the extra unit of Z gives an utility gain of:VG = φA2 u(y) =λλ + δ1Zsince y = λBλ+δ+λBZ. This is showed solved in Appendix A.11.We ﬁnd that the only beneﬁt of increasing Z is the increase in the availability of more insurances.This is equivalent to the assumption that the central bank can create liquidity to insure againsta shock where the private sector has a limited amount collateral, Z.5.2 The Fully Informed Central BankIn this section, we investigate the impact of the intervention where the central bank is fullyinformed, i.e. the central bank knows the true probabilities, φ1 and φ2, of the economy being hitby a ﬁrst and second shock, respectively. In this case, we consider a welfare criterion based onthe agents’ decisions, evaluated in the size of λ’s. The envelope theorem argument breaks downfor the robust economy and the reallocation of the ﬁrst and second shock insurances. Now theyhave ﬁrst order eﬀect, because agents overestimate the probability of the occurrence of secondshock.Due to the lender of last resort criterion, we only consider φ2. Thus, the expected value of the24
- 28. 5 Lender of Last Resortdirect impact of the intervention to a two shock event at the time of commitment, is given by:φ2u (y) (5.2)and the total expected value of the direct and indirect eﬀect of the intervention is:VG = φ1u (x)dxdG+ φ2u (y)dydG(5.3)where VG measures the expected value of intervening, and dxdG and dydG measure the direct andindirect eﬀect from G. In Appendix A.12 we ﬁnd that the ﬁrst order condition of the agent’soptimization is given by:u (x) =λBλ + δu (y) (5.4)Using the optimal value u (x) and the unit interventions that gives us the constraint dxdg + dydg = 1,we ﬁnd the value of the intervention, in Appendix A.13, by substituting the above into theexpected value of the central bank intervention:VG = φ2u (y) 1 +λBλ− 1dxdG(5.5)From this, we can ﬁnd a private sector multiplier of the lender of last resort commitment bydividing the total eﬀect in (5.5) by the direct eﬀect in (5.2), which gives us:M = 1 +λBλ− 1dxdGWe know from the benchmark economy that λB = λ. Inserting this into the multiplier gives us:Mbe= 1the multiplier in the benchmark economy is limited to the direct eﬀect. Thus, if we look intothe situation where λB > λ, we get the indirect eﬀect of the agents’ re-optimization of theirdecisions:Mr> 1The beneﬁt from the intervention in the robust economy is constrained to the reaction from theagents to the central bank’s guarantee. Agents making insurance decisions are overly concernedabout receiving the second shock and are overly insured against the incident. This breaks theenvelope theorem argument when the central bank considers their objective function. By oﬀeringto insure against the second shock, the central bank leads the agents to put a higher weight on25
- 29. 5 Lender of Last Resortthe insurances against the more probable ﬁrst shock event ( dxdG > 0).5.3 The Partially Informed Central BankIn this section, we loosen up the requirements we implemented in the former section, i.e. thecentral bank knows the true value of λ. We assume that agent A knows that λB ∈ [max{λ −K, 0}, λ+K], and the same applies in relation to agent B. Instead of the omniscient central bank,we assume that the central bank is uncertain about values of λ and only knows that λ is drawnfrom a non-degenerate symmetric joint distribution2 F(λcbA , λcbB ). We study the fully robust casewhere K > δ, where the agents’ decisions of x and y are independent of λ.C&K deﬁnes the distribution F(·) as symmetric, why we ﬁnd that the central bank focuses onsymmetric interventions, which means interventions that are equal no matter whether it is A orB who gets hit ﬁrst or second by a shock.We now deﬁne the total and direct expected eﬀect of intervention conditional on λA and λB asVG(λA, λB) and DE(λA, λB) which gives us the following multiplier:M ≡EλAλB[VG(λA, λB)]EλAλB[DE(λA, λB)]For the fully robust economy, we have:EλAλB[VG(λA, λB)] =12uZ2EλAλB[Pr(A; λA) + Pr(B; λB)] (5.6)From the robustness case, we have the optimal value where the agent allocates Z2 collateralbetween each market, and each agent allocates resources equally distributed between the twoinsurances. Pr(i; λi) is the probability that an agent i is hit by a second shock given intensitiesλA and λB.We now turn to the direct eﬀect case:EλAλB[DE(λA, λB)] = uZ2EλAλB[Pr(A | B; λA, λB) + Pr(B | A; λA, λB)] (5.7)where Pr(i | j; λA, λB) indicates the probability that agent i is hit second, given intensities of λAand λB. We will denote the probability that agent i is hit by a shock and the other agent is not,by Pr(i | Noj; λA, λB). The direct eﬀect shows that all the resources are spent in the event ofthe second shock.From the restriction imposed in the beginning of the section, i.e. the central bank knowledge ofthe market is limited to a non-degenerate symmetric joint distribution F(λA, λB) with ﬁnite and2a non comparable joint distribution26
- 30. 5 Lender of Last Resortpositive λcbA and λcbB , we left with the multiplier of the fully robust economy:Mr> 1 (5.8)Proof. C&K (2005, p. 25) use the assumed symmetry F(λabA , λcbB ) to deﬁne the multiplier as:Mr=EλAλB[Pr(A; λA)]2EλAλB[Pr(A | B; λA, λB)]In Appendix A.14 this is showed solved. The multiplier (5.8) then holds if:EλAλB[Pr(A; λA)] > 2EλAλB[Pr(A | B; λA, λB)]Expanding the left hand side of the expression, we ﬁnd:EλAλB[Pr(A; λA)] = EλAλB[Pr(A | B; λA, λB)] + Pr(A | NoB; λA, λB)] + Pr(B | A; λA, λB)= EλAλB[2Pr(A | B; λA, λB) + Pr(A | NoB; λA, λB)]We can substitute this expression into the inequality, which holds if and only if:EλAλB[Pr(A | NoB; λA, λB)] > 0which holds as long as F(λcbA , λcbB ) has some mass for ﬁnite λcbA and positive λcbB since δ > 0.We have found that the proof highlights the central bank’s knowledge of the symmetric distribu-tion. The two agents overestimate the probability of being second, thus the beneﬁt of the centralbank’s intervention reﬂects the overestimated probabilities of being second. Knowing that thedistribution of λ is symmetric indicates that the central bank can beneﬁt from the knowledge thatboth agents are not able to be second at the same time. The central bank uses this information tocalculate the beneﬁt from the direct eﬀect of intervening. That is why the symmetry informationis enough to conclude that the multiplier is greater than one.One can refer back to the AIG example: The Federal Reserve infused capital into AIG and guar-anteed that the company would not fail to meet its obligations. The Federal Reserve must havecalculated that the multiplier was greater than one to choose to intervene.5.4 Should the Central Bank Intervene?C&K (2005) rises two issue regarding interventions from a central bank - the welfare criterionand the moral hazard critique. Sims (2001) also questions the application of robust control to27
- 31. 5 Lender of Last Resortcentral bank policy-making. In his paper, he argues that max-min preferences are shortcuts togenerate observed behaviour of economic agents, meaning that these are not deep and do notcorrespond to the real world. Solutions to solve this might as well be the inclusion of all agents’preferences during evaluation policy, which one can argue is quite impossible.The moral hazard critique emphasises the complementarity between public and private provisionof insurance, since the model clearly argue against the normal moral hazard of having centralbanks intervene. The critique argues that agents will cut back on their own insurance activitiesfollowing from the promise of a lender of last resort.We assumed that the agents would allocate their resources away from the publicly insured shock.Consider a central bank that commits to intervene in the ﬁrst shock, though the policy would befar more expensive than extreme event intervention, due to the probability distribution of ﬁrstshock occurring. Under the same assumptions the agent would allocate their resources towardsthe second shock event. This is subject to the moral hazard critique. To show this by our modelwe consider the case of the fully informed bank. The direct eﬀect of the intervention in the eventof the ﬁrst shock is φ1u (x), while the total eﬀect is:VG = φ1u (x)dxdG+λBλ1 −dxdG(5.9)This is showed solved in Appendix A.15. From equation (5.9) it follows, that when we divideboth sides of the equation with φ1u (x), we get the moral hazard multiplicator Mmh:Mmh=VBφ1u (x)< 1since λB > λ and 0 < dxdG < 1. This concludes that the lender of last resort has to be a last, notan intermediate, resort (extreme case scenario) to be eﬀective and improve the private ﬁnancialmarket.28
- 32. 6 DiscussionEmpirical MethodsIn our empirical analysis we have chosen to use the TED-spread as a risk indicator. Thisdoes not come without implication due to the most recent ﬁndings, where Barclays werecaught in ﬁxing the interest rate during the boom in the 00’s. They manipulated the inter-est rate downwards to give the public an impression of Barclays being a riskless investment,and further the traders were making a proﬁt from the lower interest rate they had to payfor their gearing, i.e. the relationship between debt and equity1. Ceteris paribus this in-creases the residual of the empirical analysis.The publics’ perception of the bond market being a riskless market is also up to discussion,if we look at Figure 4.1 we see that the interest rates are equal until 2008. One can discusswhether this is a realistic perception of the bond market, mainly due to the boom in e.g.the Greek bond rate after the ﬁnancial crisis took its grasp. Erik Jones (2012) argues thatthe public saw the gathered Euro-zone as being one unit until 2008 and here after theﬂight-to-safety episode rigged this perception.Theoretical FrameworksWe have chosen to describe the ﬂight to safety phenomenon theoretically through C&K2005) paper mainly due to the model setup and the usage of insurances as describingagents decisions problems. Several others have build models describing ﬂight-to-safetyepisodes. Examples could be Briere et. al (2012) who analyse the geographic contagioneﬀect during ﬂight-to-safety episodes, or Guerrieri and Shimer (2012) who analyses ﬂight-to-safety episodes in a dynamic adverse selection environment.C&K (2005) chooses to include a variable Z into their model. A fall in Z increases theintermediaries risk due to the growing concern from agents, since they will assume theintermediaries does not have enough collateral to cover their liabilities, in a shock-event.We here provide the intuition from a variable Z which through the model would causean increase in λB. This would mean an increase in the agents robustness concern causingfree-trading capital to decrease, cf. an ”Allocating Collateral” episode.1http://www.bbc.co.uk/news/business-1867125529
- 33. 7 ConclusionHistorically the world has frequently been hit by shocks, panics and so on, which ledto ﬂight-to-safety episodes, i.e. scenarios where investors tries to cover themselves frommarkets with high risk of default, which could deplete their proﬁts. The investors protectsthemselves by investing in safe assets perceived as risk free such as gold, bonds or writinginsurances with intermediaries, this could be futures, swaps or options.The theoretical framework in the paper shows that due to the ﬁnancial system risk andKnightian uncertainty in the market, cause agents to take actions that are consistent withthe empirical evidence, saying that ﬂight-to-safety occur during events like the Russiandefault, the Global Financial Crisis, etc.The paper shows how one can describe ﬂight-to-safety episodes based upon robust deci-sions making - made by ﬁnancial specialists. Assuming that there is a limitation on theaggregate collateral, one ﬁnds that the agents take protective actions to cover themselvesfrom proﬁt depletion. Robust decisions can lead to two market distortions: First it couldcause ineﬃcient allocation of collateral. Second, it could potentially leave the economyoverexposed to recessionary shocks due to inﬂexibility, cf. the fully robust case where thefree ”trading capital” is given by x − y = 0, i.e. wasted collateral in the economy.One ﬁnding is that a lender of last resort can accommodate the costs following a ﬂight-to-safety episode. The paper also concludes that a lender of last resort is only valuableif the central bank intervenes during the extreme case scenario. If they intervene duringless extreme cases, one could conclude that the eﬀectiveness vanishes due to the welfarecriterion and the moral hazard critique.Looking at the geographical incident, one can conclude that the lender of last resort shouldonly intervene once contagion takes place. Here is the Euro debt crisis a good example,since there is clear empirical evidence of geographically ﬂight to safety. The contagioneﬀect is shown by the spread in the European bond market, cf. Figure 4.1.30
- 34. Appendicesi
- 35. A AppendixA.1Table A.1 shows explicitly how the TED-spread is found1. US Treasury bills given bythree-month futures contracts and Eurodollars shown as the three-month LIBOR rate aresubtracted to ﬁnd the exact TED-spread. Due to the amount of data the table is only abrief view of all data used for Figure 1.1.Year 2002(1) ... 2008(8) 2008(9) 2008(10) ... 2013(3)UST 1,65 ... 1,72 1,13 0,67 ... 0,09LIBOR 1,75 ... 3,00 3,95 5,31 ... 0,28TED-spread0,10 ... 1,28 2,82 4,64 ... 0,19Table A.1: US Treasuries and LIBOR rate, 2002(1)-2013(3)A.2Solving the maximization problem:max{ˆx(s)},{ˆy(s)}∞0p(s) − βφ1(s))ˆx(s) +∞s(qs(t) − βφs(t))ˆys(t)dt dssubject toˆx(s) + ˆys(t) = ZSince we are looking at a time interval from the period [0,∞], we can put the integralsaside when we optimize. The ﬁrst order condition for ˆys(t) and the corresponding shadowvalue is:qs(t) = βφs(t) + µs(t)where t lies between s → ∞.We do to same for ˆx(s) but since we are looking at a variable expire date for s shadow1Federal Reserve system: http://www.federalreserve.gov/releases/h15/data.htm#topii
- 36. A Appendixvalues diﬀer from the ﬁrst expression. The ﬁrst order condition for ˆx(s) is:p(s) = βφ1(s) + µ(s); µ(s) ≡∞sµs(t)dt.A.3To get expression (2.7) we integrate eq. (2.2) and ﬁnd the antiderivative:λ2e−λs 1−(δ + λ)e−(δ+λ)t∞sThen we insert the limitsλ2e−λse−(δ+λ)tδ + λ=λ2δ + λe−(δ+2λ)s= φ1(s)λλ + δA.4Rewrite eq. (2.6) to get:µ(s) = q(s) − βφ2(s)Insert in eq. (2.5) and factorize β:p(s) = βφ1(s) + q(s) − βφ2(s)= q(s) + β(φ1(s) − φ2(s))A.5The relative diﬀerence between u (x(s)) and u (y(s)) is given by:u (x(s))u (y(s))=φ2(s)φ1(s)=λe−(δ+2λ)sλe−(δ+2λ)s·λλ + δ=λλ + δiii
- 37. A Appendixusing the utility function from (2.1) and y(s) = Z −x(s), one can ﬁnd the optimal quantityof insurances:Z − x(s)x(s)=λλ + δ⇔ Z = x(s) · 1 +λλ + δ⇔ xbe= Z ·λ + δ2λ + δInserting the expression for xbeinto the relative demand, we ﬁnd:y(s)Z · λ+δ2λ+δ=λλ + δ⇔ ybe= Z ·λ2λ + δwhich gives us:xbeybe=λ + δλA.6Finding ﬁrst order conditions, by using the same method as in A.2.L1 : φA1 (s)u (x(s)) − ψp(s)L2 : φA2 η(t − s)(s)u (y(s)) − ψqs(t)= 0which gives us:φA1 (s)u (x(s)) = ψp(s)φA2 η(t − s)(s)u (y(s)) = ψqs(t)Lagrange optimization for the agent’s decision problem:L() = max −µL1 :∞0φ1(s)u (x(s)) − µL2 :∞0φ2(s)u (y(s)) − µ= 0L1L2:φ1(s)u (x(s))φ2(s)u (y(s))= 1 ⇔u (x(s))u (y(s))=φ2(s)φ1(s)iv
- 38. A AppendixA.7Here are the steps solved explicitly:φA2 (s)η(t − s)u (ys(t)) = ψqs(t)φA2 (s)∞sη(t − s)u (ys(t))dt = ψ∞sqs(t)φA2 (s)∞sη(t − s)u (ys(t))dt = ψp(s)A.8Solving:∞sη(t − s) = 1where η(t − s) = (λ + δ)e−(λ+δ)(t−s)∞s(λ + δ)e(λ+δ)(t−s)dt = 1(λ + δ)∞se(λ+δ)(t−s)dt = 1(λ + δ)∞ses(λ+δ)e−t(λ+δ)dt = 1(λ + δ)es(λ+δ)−1λ + δe−t(λ+δ)∞s= 1es(λ+δ)−e−t(λ+δ) ∞s= 1es(λ+δ)−0 + e−s(λ+δ)= 1es(λ+δ)−s(λ+δ)= 1e0= 1v
- 39. A AppendixA.9Finding the prices by solving equation (2.16) for x(s):x(s) =λ + δ˜λ· (Z − x(s)) ⇔x(s) · 1 +λ + δ˜λ=λ + δ˜λ· Z ⇔x(s) =λ + δλ + δ + ˜λ· ZThis expression is inserted in equation (2.11)φA1 (s)uλ + δλ + δ + ˜λ· Z = ψp(s)C&K deﬁnes the shadow price to be ψ = 1w0λλ+δand we know from earlier thatφA1 (s) = λe−(δ+λ+λB)sλe−(δ+λ+˜λ)s λ + ˜λ + δ(λ + δ)Z=1w0λλ + δ· p(s)Solving for p(s), we get:p(s) = e−(δ+λ+˜λ)s·w0Z(δ + λ + ˜λ)Using the found expression for x(s), and inserting this into equation (2.16), we get anexpression for qs(t):λ+δλ+δ+˜λ· Zy(s)=λ + δ˜λ⇔ y(s) =˜λλ + δ + ˜λ· ZWe insert this into equation (2.12)φA2 (s)η(t − s)u˜λλ + δ + ˜λ· Z = ψqs(t)From Subsection 2.2.2 we have η(t − s) = (λ + δ)e−(λ+δ)(t−s)and inserting the value forφA2 (s) we ﬁnd:λe−(δ+λ+˜λ)s˜λλ + δ(λ + δ)e−(λ+δ)(t−s)· uλ + δ + ˜λ˜λ·1Z=1w0λλ + δqs(t)vi
- 40. A AppendixSolving for qs(t) we ﬁnd:qs(t) =w0Z(δ + λ + ˜λ)(δ + λ)e−(δ+λ+˜λ)s−(λ+δ)(t−s)A.10To ﬁnd x(s) we use equation (2.11), by inserting the expression for qs(t):φA1 (s)u (x(s)) = ψw0Z(δ + λ + ˜λ)(δ + λ)e−(δ+λ+˜λ)s−(λ+δ)(t−s)inserting φA1 and ψ = 1w0λλ+δand isolating u (x(s)), gives us:u (x(s)) =1Z1λ + δ(λ + δ + ˜λ)e−(λB−˜λ)Using the log utility assumption, we get:x(s) =δ + λ(λ + δ + ˜λ)e−(λB−˜λ)sZTo ﬁnd the expression for y(s) we use equation (2.12). In that we insert φA2 ,η(t − s) = (λ + δ)e(λ+δ)(t−s)dt and ψ = 1w0λλ+δand then isolate for ys(t), to get:y(s) ≡ ys(t) =λBλ + δλe−(δ+λ+λB)s(λ + δ)e−λ+δt−s··1w0λλ + δw0Z(λ + δ + ˜λ)w0Z(δ + λ + ˜λ)(δ + λ)e−(δ+λ+˜λ)s−(λ+δ)(t−s)−1=λBδ + λ + ˜λe−(λB−˜λ)sZA.11Like in 2.2.4 we use the envelope theorem to analyze the eﬀect of an increase in Z, i.e.derive equation (5.1) w.r.t. Z. First we insert the value of y and φA2 :V (·) = φA1 u(x) +λλ + δ + λBλBλ + δuλBZλ + δ + λBvii
- 41. A Appendixand then x and φA1=λλ + δ + λBuλ + δλ + δ + λZ +λλ + δ + λBλBλ + δuλBZλ + δ + λBDiﬀerentiating with respect to Z, to get:V (·)Z =1ZλBλ + δλ + δλ + δ + λB+1ZλBλ + δλBλ + δ + λB=1ZλBλ + δλ + δλ + δ + λB+λBλ + δ + λB= VG =λBλ + δ1ZA.12Using Langrange optimization:L1 : φA1 u (x) − ξL2 : φA2 u (y) − ξ= 0Isolating u (x), to ﬁnd:u (x) =φA2φA1u (y) ⇔=λBλ + δu (y)A.13Inserting (5.4) this espression into equation (5.3), one ﬁnds:VG = φ1u (y)λBλ + δdxdG+ φ2u (y)dydG= u (y) φ1λBλ + δdxdG+ φ2dydGviii
- 42. A Appendixusing that dxdG+ dydG= 1, one ﬁnds:= u (y) φ1λBλ + δdxdG+ φ2 1 −dxdG= u (y) φ1λBλ + δ− φ2dxdG+ φ2Since we are looking at date 0 optimization, s = 0, we ﬁnd that:φ1 = λ and φ2 =λ2λ + δThis we insert into our found expression of VG:= u (y)λ2λ + δλBλ− φ2dxdG+ φ2VG = u (y)φ2 1 +λBλ− 1dxdGA.14Using the deﬁnition:M ≡EλAλB[VG(λA, λB)]EλAλB[DE(λA, λB)]Inserting equation (5.6) and (5.7), to ﬁnd the multiplier MrMr=12u Z2EλAλB[Pr(A; λA) + Pr(B; λB)]u Z2EλAλB[Pr(A | B; λA, λB) + Pr(B | A; λA, λB)]due to symmetry, we ﬁnd:=12EλAλB[2Pr(A; λA)]EλAλB[2Pr(A | B; λA, λB)]=12EλAλB[Pr(A; λA)]EλAλB[Pr(A | B; λA, λB)]ix
- 43. A AppendixA.15We use the ﬁrst order conditions, given by equation (5.4), to isolate u (y) and then insertit into (5.3), which gives us:VG = φ1u (x)dxdG+ φ2λ + δλBu (x)dydGWe had φ1 = λ, φ2 = λ2λ+δand dydG= 1 − dxdG, which gives us:VG = u (x) φ1dxdG+φ1λλ + δλ + δλB1 −dxdG= u (x) φ1dxdG+ φ1λλB1 −dxdG= φ1u (x)dxdG+λλB1 −dxdGx
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