Leverage Cycle John Geanakoplos
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Leverage Cycle John Geanakoplos Zhe Li SUFE Zhe Li (SUFE) Leverage Cycle 1 / 78 Leverage A homeowner takes a loan using a house as collateral, he negotiates: (a) interest rate (b) how much he can borrow Example house costs $100, loan $80, he pays $20 in cash margin or haircut: 20% loan to value: 80/100 = 80% collateral rate: 100/80 = 1.25 leverage: 1/margin = 5 Zhe Li (SUFE) Leverage Cycle 2 / 78 Motivation In times of crisis: Interest rate (main policy instrument) Collateral rate (equivalently margins, leverage) E¤ectiveness of two instruments: During a crisis, leverage can fall by 50% overnight, and by more over a few days or months A homeowner who brought a house in 2007 by taking out a subprime mortgate with only 5% down payment cannot take out a similar loan in 2009 without putting 30% The odds are great that he won’thave the cash to do it, and reducing the interest rate by 1 or 2% won’tchange his ability to act Zhe Li (SUFE) Leverage Cycle 3 / 78 Heterogeneous beliefs For many assets there is a class of buyer for whom the asset is more valuable than it is for the rest of the public (standard economic theory, in contrast, assumes that asset prices re‡ect some fundamental value). These buyers are willing to pay more, perhaps because they are more sophisticated and know better how to hedge their exposure to the assets, or they are more risk tolerant, or they simply like the assets more Zhe Li (SUFE) Leverage Cycle 4 / 78 Distribution of wealth If the optimistic buyers can get their hands on more money through more highly leveraged borrowing (that is, getting a loan with less collateral), they will spend it on the assets and drive those prices up If they lose wealth, or lose the ability to borrow, they will buy less, so the asset will fall into more pessimistic hands and be valued less Zhe Li (SUFE) Leverage Cycle 5 / 78 Leverage cycle: meaning De…nition In the absence of intervention, leverage becomes too high in boom times, and too low in bad times (leverage is procyclical). As a result, in boom times asset prices are too high, and in crisis times they are too low. This is the leverage cycle. Leverage here (a ratio of collateral values to the downpayment that must be made to buy them) is di¤erent from the usuall de…nition: debt+equity/equity, which is misleading since equity is wiped out during crisis and it seems like that the leverage increases Zhe Li (SUFE) Leverage Cycle 6 / 78 Leverage cycle: U.S. boom Leverage dramatically increased in the United States from 1999 to 2006 A bank that in 2006 wanted to buy a AAA-rated mortgage security could borrow 98.4% of the purchase price, using the security as collateral, and pay only 1.6% in cash. The leverage was thus 100 to 1.6, or about 60 to 1 The average leverage in 2006 across all of the US$2.5 trillion of so-called ‘toxic’mortgage securities was about 16 to 1, meaning that the buyers paid down only $150 billion and borrowed the other $2.35 trillion Home buyers could get a mortgage leveraged 20 to 1, a 5% down payment. Security and house prices soared Zhe Li (SUFE) Leverage Cycle 7 / 78 Leverage cycle: U.S. current crisis Today leverage has been drastically curtailed by nervous lenders wanting more collateral for every dollar loaned Those toxic mortgage securities are now leveraged on average only about 1.5 to 1 Home buyers can now only leverage themselves 5 to 1 if they can get a government loan, and less if they need a private loan De-leveraging is the main reason the prices of both securities and homes are still falling Zhe Li (SUFE) Leverage Cycle 8 / 78 What causes crises? Bad news More than one bad news For example: output is 1 unless two things go wrong, in which case output becomes 0.2 Zhe Li (SUFE) Leverage Cycle 9 / 78 Optimist Suppose that the chance of each thing going wrong is independent and equal to 10% 1 At the beginning: 1 The chance of ultimate breakdown is 1% 2 The expected output is 0.99 1 + 0.01 0.2 = 0.992 2 2 3 The variance of …nal output is 0.99 0.008 + 0.01 (0.992 0.2) = 6. 336 10 3 2 After the …rst piece of bad news, 1 The expected output drops to 0.9 + 0.1 0.2 = 0.92 2 2 2 The variance jumps to 0.9 0.08 + 0.1 0.72 = 0.057 6 Zhe Li (SUFE) Leverage Cycle 10 / 78 Pessimist Suppose that the chance of each thing going wrong is independent and equal to 20% 1 At the beginning: 1 The chance of ultimate breakdown is 4% 2 The expected output is 0.96 1 + 0.04 0.2 = 0.968 3 The variance of …nal output is 0.96 (1 0.968)2 + 0.04 (0.968 0.2)2 = 2. 457 6 10 2 2 After the …rst piece of bad news, 1 The expected output drops to 0.8 + 0.2 0.2 = 0.84 2 2 2 The variance jumps to 0.8 (1 0.84) + 0.2 (0.84 0.2) = 0.102 4 Zhe Li (SUFE) Leverage Cycle 11 / 78 Disagreement Expectations di¤ered Originally by 0.992 0.968 = 0.024 After the bad news: 0.92 0.84 = 0.08 This kind of bad news increases uncertainty and disagreement — scary news The news in current crisis, forecasts on subprime losses ranged from 30% to 80% Zhe Li (SUFE) Leverage Cycle 12 / 78 Heterogeneity between optimists and pessimists Have di¤erent priors A continnum of agents uniformly arrayed between 0 and 1 Agent h on that continnum thinks the probability of good news (Up) is γh = h, and the probability of bad news (down) is γh = 1 h. U D The higher the h, the more optimistic the agent, the more natural a buyer he is Zhe Li (SUFE) Leverage Cycle 13 / 78 Sides of the market The more optimistic the agent, the more natural a buyer he is. There will be some break point h such that those more optimistic with h > h are on one side of the market and those less optimistic, with h < h, are on the other side. But h is endogenous. Zhe Li (SUFE) Leverage Cycle 14 / 78 Borrowing and asset pricing One consumption good C One asset Y Two time periods 0, 1 Two states of nature U and D in the last period Zhe Li (SUFE) Leverage Cycle 15 / 78 Borrowing and asset pricing Each unit of Y pays either 1 or 0.2 of the consumption good, in the two states U and D, respectively Imagine the asset as a mortgage that either pays in full or defaults with recovery 0.2 Or an oil well that might be a gusher or small Zhe Li (SUFE) Leverage Cycle 16 / 78 Endowment and consumption Endowment: eh = (eh , eh , eh , eh ) = (1, 1, 0, 0) C0 Y0 CU CD Preference: h u (c0, y0, w0, cU , cD ) = c0 + hCU + (1 h)CD c is canned food, can be consumed at time 0, c0, or stored costlessly, w0 Zhe Li (SUFE) Leverage Cycle 17 / 78 Asset pricing suppose the price of the asset Y at time 0 is p : Buyer if h 1 + (1 h) 0.2 > p Seller if h 1 + (1 h) 0.2 < p Zhe Li (SUFE) Leverage Cycle 18 / 78 Budget set without borrowing The price of the consumption good in each period to be 1 The price of the asset Y at time 0 is p 5 (c0, y0, w0, cU , cD ) R+ : c0 + w0 + py0 = 1 + p Bh (p) = c2= w + y N 8 U 0 0 9 < cD = w0 + 0.2y0 = : ; Zhe Li (SUFE) Leverage Cycle 19 / 78 Markets clear In equilibrium all markets must clear: 1 h h (c0 + w0 )dh = 1 0 Z 1 h y0 dh = 1 0 Z 1 1 h h cU dh = 1 + w0 dh 0 0 Z 1 Z 1 h h cD dh = 0.2 + w0 dh Z0 Z0 Zhe Li (SUFE) Leverage Cycle 20 / 78 Equilibrium In equilibrium, p = h 1 + (1 h) 0.2 Markets clear: Optimists spend all consumption goods (money) today to buy the asset Money of optimists = asset of pessimists 1 1 (1 h) = p 1 h h = 0.597, p = 0.677 Zhe Li (SUFE) Leverage Cycle 21 / 78 Equilibrium In equilibrium, agents are indi¤erent to storing or consuming right away, so we can describe equilibrium as if everyone warehoused and postponed consumption. (c0, y0, w0, cU , cD ) = (0, 2.5, 0, 2.5, 0.5) for h 0.597 (c0, y0, w0, cU , cD ) = (0, 0, 1.677, 1.677, 1.677) for h < 0.597 Zhe Li (SUFE) Leverage Cycle 22 / 78 Collateral A borrower can use the asset itself as collateral, so that if he defaults the collateral can be seized If the promise is j in both states, he will only receive min(j, 1) if good news min(j, 0.2) if bad news The biggest promise that is sure to be covered by the collateral: 0.2y0 Zhe Li (SUFE) Leverage Cycle 23 / 78 Budget set with collateralized borrowing Leveraging: using collateral to borrow, gives the most optimistic agents a chance to spend more 6 (c0, y0, ϕ0, w0, cU , cD ) R+ : 2 1 c0 + w0 + py0 = 1 + p + ϕ 8 1+r 0 9 Bh (p, r) = ϕ < 0.2y 0.2 > 0 0 > > cU = w0 + y0 ϕ > < 0 = cD = w0 + 0.2y0 ϕ0 > > > > Subscript 0.2 : we have arbitrarily:> …xed the maximum promise;> that can be made on a unit of collateral Zhe Li (SUFE) Leverage Cycle 24 / 78 Equilibrium Equilibrium is de…ned by the price and interest rate (p, r) and agent h h choices (c0, y0, ϕ0, w0, cU , cD ) in B0.2(p, r) that maximizes his utility u .