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 . In equilibrium all markets must clear
1 h h (c0 + w0 )dh = 1 0 Z 1 h y0 dh = 1 0 Z 1 h ϕ0dh = 0 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 25 / 78 Equilibrium Interest Rate Is Zero
Lenders are not impatient A large supply of durable consumption goods It is supposed to be less than 40% optimists who hold all the assets since 0.2/0.6 < 1 per person In equilibrium, there is no default
Zhe Li (SUFE) Leverage Cycle 26 / 78 Equilibrium Price
Equilibrium price
p = h 1 + (1 h) 0.2 Market clears
Money of optimists = asset of pessimists 0.2 1 1 1 (1 h) + = p 1 h 1 + r Additional money: the optimists hold all assets and can promise 0.2 1
Zhe Li (SUFE) Leverage Cycle 27 / 78 Equilibrium
Price rises modestly from 0.68 to 0.75, because there is a modest amount of borrowing
(p, r) = (0.75, 0) (ch, y h, ϕh, w h, ch , ch ) = (0, 3.2, 0.64, 0, 2.6, 0) for h 0.69 0 0 0 0 U D (ch, y h, ϕh, w h, ch , ch ) = (0, 0, 0.3, 1.45, 1.75, 1.75) for h < 0.69 0 0 0 0 U D Fewer agents hold all the assest, because they can a¤ord to buy on borrowed money
Zhe Li (SUFE) Leverage Cycle 28 / 78 Endogenous Leverage
Exogenously …xed the maximal promise at 0.2 Why wouldn’tthe most optimistic buyers be willing to borrow more, defaulting in the bad state of course, but compensating the lenders by paying a higher interest rate?
Zhe Li (SUFE) Leverage Cycle 29 / 78 Contracts as Ordered Pairs of Promises and Collateral
Contracts as ordered pairs of promises and collaterals
Contractj = (Promisej , Collateralj ) = (Aj , Cj )
The same promise j = 0.2 with di¤erent collateral levels are di¤erent contracts a promise 0.2 backed by one house delivers 0.2 in both states a promise 0.2 backed by 2/3 of a house delivers 0.2 in good state but 0.2 2/3 in bad state The ordered pairs are homogeneous of degree one A promise of 0.2 backed by 2/3 of a house is simply 2/3 of a promise of 0.3 backed by a full house Normalize collateral Cj to one unit of asset Each contract trades in a separate market, with its own price
Zhe Li (SUFE) Leverage Cycle 30 / 78 Budget Set with Endogenously Collateralized Borrowing
Promises j : quantity of contract (collateral) ϕj , price of contract πj
ϕj > 0 : sell promises, increase current disposable income πj ϕj , but giving up j ϕj in good state, 0.2ϕj in bad state in the future
ϕj < 0 : buy promises, reduce expenditures in period 0 by πj ϕj , but enable him to consume more in the future
2 J 3 (c0, y0, (ϕj )j J , w0, cU , cD ) R+ R R+ : 2 2 J c0 + w0 + py0 = 1 + p + ∑j=1 ϕj πj h 8 J 9 B (p, π) = > ∑j=1 max(ϕj , 0) y0 collateral (leverage) constraint > > J > <> cU = w0 + y0 ∑j=1 ϕj min(1, j) => J cD = w0 + 0.2y0 ∑j=1 ϕj min(0.2, j) > > > > :> ;>
Zhe Li (SUFE) Leverage Cycle 31 / 78 Equilibrium
Equilibrium is de…ned by the prices (p, π) and agent choices h h (c0, y0, (ϕj )j J , w0, cU , cD ) in B (p, π) that maximizes his utility u . In equilibrium all2 markets must clear
1 h h (c0 + w0 )dh = 1 0 Z 1 h y0 dh = 1 0 Z 1 h ϕj dh = 0, j J 0 8 2 Z 1 1 jh 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 32 / 78 Unique Equilibrium Margin
Everybody will voluntarily trade only the 0.2 loan, why? 0.69 Price for loan contract π0.4 = 0.69 0.4 + 0.31 0.2 = 0.337 Give up 0.337 at time 0, get 0.4 in good state and 0.2 in bad state Lenders are people with h < 0.69 h 0.4 + (1 h) 0.2 < 0.337 0.69 Expected return < Price π0.4 = Cost 0.337 Why should they give up more money at time 0 to get more money in a state U they do not think will occur?
Borrowers believe that h > h, so they only like to promise 0.4 if they can use 0.337
Zhe Li (SUFE) Leverage Cycle 33 / 78 Endogenous Leverage
Shocks to the economy that a¤ect asset prices, also typically change equilibrium leverage Will the change in equilibrium leverage multiply the e¤ect on asset prices, or dampen the e¤ect? Aggents become more optimistic An Increase in endowment of the consumption good Shock to the tail of the distribution of asset payo¤: 0.2 rises to 0.3
Zhe Li (SUFE) Leverage Cycle 34 / 78 Aggents Become More Optimistic
Suppose for all h (0, 1) 2 γh = (1 h)2 < 1 h D γh = 1 (1 h)2 > h U The equilibrium promise 0.2; the equilibrium interest rate 0 The price of asset Y rises to 0.89, and the marginal buyer is h = 0.63 h h 2 p = γ 1 + γ 0.2 = 1 0.8(1 h) U D Money of optimists = asset of pessimists
1 1 (1 h) + 0.2 = p 1 h The leverage falls to 1.29 = 0.89/(0.89 0.2) from 1. 36 Positive news has been dampened by tightening of leverage: if the leverage was …xed, the optimists would borrow more money; more money chasing the same asset causing the asset price to rise more
Zhe Li (SUFE) Leverage Cycle 35 / 78 Shock to the Tail of the Distribution of Y Payo¤s
If the tail payo¤ 0.2 is increased to 0.3 That will have a positive e¤ect on the expected payo¤ of Y The e¤ect on the price of Y will be reinforced by the expansion of equilibrium leverage
Zhe Li (SUFE) Leverage Cycle 36 / 78 Legacy Assets Versus New Assets
As a result of the leverage crunch of 2007 2009, asset prices plummeted. One critical e¤ect was that it became very di¢ cult to support asset prices for new ventures that would allow for new activity (No bank provides loan to some activities) Who would buy a new mortgage at 100 when virtually the same old asset could be purchased on the secondary market at 65?
Zhe Li (SUFE) Leverage Cycle 37 / 78 Prop Up the Price of New Assets
Suppose the government wants to prop up the price of new assets, by providing leverage beyond what the market will provide Given a …xed upper bound in defaults, would the government do better to provide lots of leverage on just the new assets or by providing moderate leverage on all the assets, new and legacy? 2009, the government appears to have adopted the strategy of leveraging only the new assets. Yet all the asset prices are rising
Zhe Li (SUFE) Leverage Cycle 38 / 78 Equilibrium with Legacy (Red) and New (Blue) Assets Blue Asset could be Used as Collateral, but Red could not
The de…nition of equilibrium consists of h h h h h h h (r, pB , pR , (c0 , y0B , y0R , ϕ0, w0 , cU , cD )h H ) such that the individual choices are optimal in the budget sets 2
7 (c0, y0B , y0R , ϕ , w0, cU , cD ) R : 0 2 + c0 + w0 + pB y0B + pR y0R = 8 1 9 h > 1 + pB β + pR (1 β) + 1+r ϕ0 > B0.2(p, r) = > > > ϕ0 0.2y0B > <> => cU = w0 + y0B + y0R ϕ 0 > cD = w0 + 0.2y0B + 0.2y0R ϕ0 > > > > > and markets clear :> ;>
1 1 h h y0B dh = β, y0R dh = 1 β Z0 Z0
Zhe Li (SUFE) Leverage Cycle 39 / 78 Buyers with Blue and Red Assets
Blue asset will have a higher price Only the most optimistic agents buy the blue assets There will be an agent a indi¤erent between buying blue assets with leverage at a high price, and red assets without leverage at a low price The agents with b < h < a will exclusively hold red assets There will be an agent b who will be indi¤erent between buying red assets and selling all his assets The agents with h < b will hold no assets and lend
Zhe Li (SUFE) Leverage Cycle 40 / 78 Equilibrium Price of Red Assets
pR = b 1 + (1 b) 0.2 agent b is indi¤erent between buying red or not buying (a b) + pB β(a b) pR = (1 β)(1 (a b)) the agents between a and b can just a¤ord to buy all red that is being sold by the other (1 (a b)) agents, their expenditure consists of one unit of conumption good and the revenue they get from selling o¤ their blue asset
Zhe Li (SUFE) Leverage Cycle 41 / 78 Equilibrium Price of Blue Assets
a(1 0.2) a 1 + (1 a) 0.2 = marginal gain pB 0.2 pR agent a is indi¤erent between buying blue with leverage and red without (1 a) + pR (1 β)(1 a) + 0.2β pB = βa the top 1 a agents can just a¤ord to buy all the blue by spending their endowment of the consumption good plus the revenue from selling their red plus the amount they can borrow using blue as collateral
Zhe Li (SUFE) Leverage Cycle 42 / 78 Equilibrium with Blue and Red Assets
Suppose we begin with the situation more or less prevailing in the early 2009, with β = 0% and no leverage
fraction blue 1 0.5 0.05 0 a 0.686 0.842 0.984 0.597 b 0.680 0.637 0.600 pR 0.749 0.709 0.680 0.677 pB 0.749 0.747 0.742
By setting β = 5% and thereby leveraging the 5% new assets, the government can raise their price from 0.677 to 0.742 Interestingly this also raises the price of the red assets which remain without leverage, from 0.677 to 0.680
Zhe Li (SUFE) Leverage Cycle 43 / 78 Equilibrium with Blue and Red Assets
fraction blue 1 0.5 0.05 0 a 0.686 0.842 0.984 0.597 b 0.680 0.637 0.600 pR 0.749 0.709 0.680 0.677 pB 0.749 0.747 0.742
Extending β to 0.5 or 1 and thereby leveraging some of the legacy assets, raises the value of all the assets! Thus if one wanted to raise the price of new assets, the government should leverage all the assets, (0.2 promises no default)
Zhe Li (SUFE) Leverage Cycle 44 / 78 Remark
This analysis holds some lessons for the current discussion about TALF (Term Asset-Backed Securities Loan Facility) program, the government program designed to inject leverage into the economy in 2009 (Business in the Asset Backed Securities market practically ceased in October 2008 to just $500 million, down from $51 billion in October 2007) The introduction of leverage for new assets did raise the price of new assets substantially It also raised the price of old assets that were not leveraged One might think that the best way to raise new asset prices is to give them scarcity value as the only leveraged assets in town But on the contrary, the analysis shows that the price of the new assets could be boosted further by extending leverage to all the legacy assets, without increasing the amount of default
Zhe Li (SUFE) Leverage Cycle 45 / 78 Repo Market
Example The collateralized loan markets we have studied so far are similar to the Repo markets that have played an important role on Wall street for decades. In these markets borrowers take their collateral to a dealer and use that to borrow money via non-contingent promises due one day later.
Zhe Li (SUFE) Leverage Cycle 46 / 78 The Leverage Cycle
A crash in the fundamentals: if in the example bad news occurs and the value plummets to 0.2, there will be a crash in the fundamentals. There is nothing the government can do to avoid it The point of the leverage cylce is that excess leverage followed by excessive deleveraging will cause a crash even before there had been a crash in fundamentals
Zhe Li (SUFE) Leverage Cycle 47 / 78 Leverage Cycle
Now suppose the asset Y pays o¤ after two periods instead of one period. Only with two pieces of bad news does the asset pay 0.2 The state space is now S = 0, U, D, UU, UD, DU, DD f g
Zhe Li (SUFE) Leverage Cycle 48 / 78 Bad news
The move of nature from 0 to D lowers the expected payo¤ of the asset Y in every agent’seyes, the probability of bad news increases from (1 h)2 to (1 h), the variance of the expected prices also increase
We de…ne the dividend of the asset by dUU = dUD = dDU = 1, and dDD = 0.2, and d0 = dD = dU = 0
Zhe Li (SUFE) Leverage Cycle 49 / 78 Preference and Endowment
h u (c0, cU , cD , cUU , cUD , cDU,cDD )
= c0 + hcU + (1 h)cD + 2 2 h cUU + h(1 h)cUD + (1 h)hcDU + (1 h) cDD
h h h h h h h h (eo , y0 , eU , eD , eUU , eUD , eDU , eDD ) = (1, 1, 0, 0, 0, 0, 0, 0)
Zhe Li (SUFE) Leverage Cycle 50 / 78 Prior
The agents are now more optimistic than before, since agent h assigns only a probability of (1 h)2 to reaching the only state DD, where the asset pays o¤ 0.2 The marginal buyer from before, b = 0.69, for example, thinks the chances of DD are only 0.312 = 0.09. Agent h = 0.87 thinks the chance of DD are only 0.132 = 1.69% But more importantly, if buyers can borrow short term, their loan at 0 will come due before the catastrophe can happen. It is thus much safer than a loan at D
Zhe Li (SUFE) Leverage Cycle 51 / 78 Repo Loans
Assume that repo loans are one-period loans, so that loan sj promises j in states sU and sD, and requires one unit of Y as collateral, for s = 0, U, D
The budget set can now be written iteratively, for each state s (s is the predecessor of s)
2 J 1+S (cs , ys , (ϕsj )j J , ws )s S (R+ R R+) : s 2 2h 2 8 h cs + ws es + ps (ys ys ) = B (p, π) = 8 J J 9 > ys ds + ∑ ϕ πsj ∑ ϕ min(ps + ds , j), > > j=1 sj j=1 s j > <> J => ∑ max(ϕ , 0) ys j=1 sj > > :> ;>
Zhe Li (SUFE) Leverage Cycle 52 / 78 Repo Loans
2 J 1+S (cs , ys , (ϕsj )j J , ws )s S (R+ R R+) : s 2 2h 2 8 h cs + ws es + ps (ys ys ) = B (p, π) = 8 J J 9 > ys ds + ∑ ϕ πsj ∑ ϕ min(ps + ds , j), > > j=1 sj j=1 s j > <> J => ∑ max(ϕ , 0) ys j=1 sj > > > > In each state: s the price of consumption is normalized to 1, and the; price of the asset is ps and the price of loan sj promising j is πsj Income: dividends from last period’sholdings plus sales revenue from selling promises, less the payments on previous loans The agents who borrow must hold the required collateral
Zhe Li (SUFE) Leverage Cycle 53 / 78 Leverage
In every state s, the only promise that will be actively traded is the one that makes the maximal promise on which there will be no default Since there will be no default on this contract, it trades at the riskless rate of interest rs per dollar promised
We can simplify the notations by rede…ning ϕs as the amount of consumption good promised at state s for delivery in the next period, in states sU and sD
4(1+S ) (cs , ys , ϕs , ws )s S R+ : s h 2 2 8 h cs + ws es + ps (ys ys ) = B (p, π) = 8 1 9 ys ds + ϕ ϕ , > s 1+rs s > <> => ϕ ys min(psU + dsU , psD + dsD ) s > > :> ;>
Zhe Li (SUFE) Leverage Cycle 54 / 78 Equilibrium
Equilibrium occurs at price (p, r) such that when everyone optimizes in his h h h h budget set by choosing (cs , ys , ϕs , ws )s S the markets clear in each state s 2
1 1 1 h h h h (cs + ws )dh = es dh + ds ys dh 0 0 0 Z 1 Z 1 Z h h ys dh = ys dh 0 0 Z 1 Z h ϕs dh = 0 Z0
Zhe Li (SUFE) Leverage Cycle 55 / 78 Repo Loans
It will turn out in equilibrium that the interest rate is zero in every state Thus at time 0, agents can borrow the minimum of the price of Y at U and at D, for every unit of Y they hold at 0 At U agent can borrow 1 unit of the consumption good, for every unit of Y they hold at U. At D they can borrow only 0.2 units of the consumption good, for every unit of Y they hold at D In normal times, at 0, there is not very much bad that can happen in the short run. Lenders are therefore willing to lend much more on the same collateral, and leverage can be quite high
Zhe Li (SUFE) Leverage Cycle 56 / 78 Equlibrium Prices
Solving the example gives the following prices
Zhe Li (SUFE) Leverage Cycle 57 / 78 Prices
The prices of Y at time 0 of 0.95 occurs because the marginal buyer is h = 0.87 Assuming the price of Y is 0.69 at D and 1 at U, the most that can be promised at 0 using Y as collateral is 0.69 With an interest rate r0 = 0, that means 0.69 can be borrowed at 0 using Y as collateral Hence the top 13% of buyers at time 0 can collectively borrow 0.69, and by adding their own 0.13 of money they can spend 0.82 on buying the 0.87 units that are sold by the bottom 87%. The price is 0.95 0.82/0.87.
Zhe Li (SUFE) Leverage Cycle 58 / 78 Why a Crash from 0 to D
First, there is a bad news. But the bad news is not nearly as bad as the fall in prices The marginal buyer of the asset at time 0, h = 0.87, thinks there is only a 0.132 = 1.69% chance of ultimate default and when he gets to D after the …rst piece of bad news he thinks there is a 13% chance for ultimate default The news for him is bad, accounting for a drop in price about 11 points, but it does not explain a fall in price from 0.95 to 0.69 of 26 points
Zhe Li (SUFE) Leverage Cycle 59 / 78 Why a Crash from 0 to D
In fact, no agent h thinks the loss in value is nearly as much as 26 points The biggest optimist h = 1 thinks the value is 1 at 0 and still 1 at D The biggest pessimist h = 0 thinks the value is 0.2 at 0 and still 0.2 at D The biggest loss attributable to the bad news of arriving at D is felt by h = 0.5, who thought the value was 0.8 at 0 and thinks it is 0.6 at D. But that drop of 0.2 is still less than the drop of 26 points in equilibrium
Zhe Li (SUFE) Leverage Cycle 60 / 78 Why a Crash from 0 to D
The second factor is that the leveraged buyers at time 0 all go bankrupt at D They spent all their cash plus all they could borrow at time 0, and at time D their collateral is con…scated and used to pay o¤ their debts: they owe 0.69 and their collateral is worth 0.69. Without the most optimistic buyers, the price is naturally lower Finally, and most important, the margins jump from (0.95 0.69)/0.95 = 27% at U to (0.69 0.2)/0.69 = 71% at D. In other words, leverage plummets from 3.6 = 0.95/(0.95 0.69) to 1.4 = 0.69/(0.69 0.2) All three of these factors working together explain the fall in price
Zhe Li (SUFE) Leverage Cycle 61 / 78 Equilibrium Price of Asset at D
Let b marginal buyer in state D, and a marginal buyer in state 0
pD is equal to the valuation of the marginal buyer b at D
pD = b + (1 b) 0.2
pD is equal to the ratio of all the money spent on asset Y at D, divided by the units of assets sold at D
(1/a)(a b) + 0.2 1.2a b pD = = (1/a) b b The top a investors are all out of business at D The remaining 1 a agents hold all consumption goods and Y between them, in equal amounts At D the new optimistic buyers in the interval [b, a) spend all they have, which is 1/a(a b) dollars plus the 0.2 1 they can borrow The amount of Y sold at D is 1/a b Zhe Li (SUFE) Leverage Cycle 62 / 78 Equilibrium Price of Asset at 0
The top (1 a) agents have (1 a) + pD to spend on the a units of Y for sale at 0 1 a + pD p0 = a
Notice that at 0 it is possible to borrow pD using each unit of Y as collateral
Zhe Li (SUFE) Leverage Cycle 63 / 78 Equilibrium
To agent a : marginal utility of putting 1 dollar of cash down on a leveraged purchase of Y = marginal utility of warehousing the dollar
a(1 pD ) a = a 1 + (1 a) p0 pD b Marginal utility of leveraging a dollar by buying Y on margin at time 0: p0 pD downpayment, payo¤ of (1 pD ) in state U, to which a assigns probability a, nothing at D
Warehousing the dollar (w0 > 0) at time 0 gives higher utility than consuming it: With probability a, U will be reached and this dollar will be worth one utile. With probability 1 a, D will be reached and a will want to leverage the dollar to purchase Y . This will result in a gain at D of a(1 0.2) a(1 0.2) a = = pD 0.2 b 1 + (1 b) 0.2 0.2 b Zhe Li (SUFE) Leverage Cycle 64 / 78 Equilibrium Agent Behavior
Agent a is indi¤erent to buying Y on margin at 0 or buying it for cash
a a(1 pD ) a 1 + (1 a)pD = b p0 pD p0 The right hand side shows that by spending p0 dollars to buy Y at 0, agent a can get a payo¤ of 1 with probability a, and with probability a (1 a) a payo¤ of pD dollar at D, which is worth pD to a b
Zhe Li (SUFE) Leverage Cycle 65 / 78 Equilibrium
By guessing a value of b, and then iterating through all the equations, one ends up with all the variables speci…ed, and a new value of b. By searching for a …xed point in b, one quickly comes to the solution just described, with the crash from 0.95 to 0.69
Zhe Li (SUFE) Leverage Cycle 66 / 78 Remark
In recent times, there has been bad news, but according to most modelers the price of assets today is much lower than would be warranted by the news. There have been mumerous bankruptcies of mortgate companies, and even of great investment banks. And the drop in leverage has been enormous The current cirsis is so bad because of double leverage cycles
Zhe Li (SUFE) Leverage Cycle 67 / 78 The Double Leverage Cycles
Time periods: 0, 1, 2 Two markets: housing market and mortgage securities market States of nature s = 0 in period 0, s = U, D in period 1, with objective probabilities (1 ε) and ε, s = UU, UD, DU, DD in period 2 with probability (1 ε)2, (1 ε)ε, ε(1 ε), ε2 Two types of contracts: depending on the collateral
1 A mortgage loan of type j: agents can promise the …xed amount j of good F in every one of the last states UU, UD, DU, DD, using one house as collateral 2 Contract (k, j) called a Repo: agents can promise k units of good F in every immediate successor state, using contract j as collateral
Zhe Li (SUFE) Leverage Cycle 68 / 78 Two Types of Agents
Type A agents with h [0, 1], where h is subjective probability of a good state 2 Preference of type A with h
uh((x , x , x ), (x , x ), E 0F 0H 0L UF DF 0 (x , x , x , x )) UUF UDF DUF DDF = x0F + cx0L + hxUF + (1 h)xDF 2 +h xUUF + h(1 h)xUDF 2 +(1 h)hxDUF + (1 h) xDDF Endowment of type A
h h h h e = (e0F , e0H , e0L) = (15, 0, 15)
Zhe Li (SUFE) Leverage Cycle 69 / 78 Type B Agents
A continuum of type B agents who like canned goods, leisure and house Type B agents perceive the objective probability of bad state ε The expected utility of type B:
B u (x0F , x0H , x0L, E (x , x ), (x , x ), 0 8 UUF UUH UDF UDH 9 (xDUF , xDUH ), (xDDF , xDDH )) ε < j = = 9x 2x2 + 15x + cx 0:F 0F 0H 0L ; 2 +(1 ε) (xUUF + 15xUUH ) + (1 ε)ε(xUDF + 15xUDH ) 2 +ε(1 ε)(xDUF + 15xDUH ) + ε (xDDF + 15xDDH )
Zhe Li (SUFE) Leverage Cycle 70 / 78 Type B Agents’Fortune
Type B agents could become rich in period 2 The endowment of a type B agent is
B B A B (e0F , e0H , e0L)t=0, e = B B B B B B B B ((e , e ), (e , e ), (e , e ), (e , e ))t=2 UUF UUH UDF UDH DUF DUH DDF DDH = ((1, 0, 3.15)t=0, ((50, 0), (50, 0), (50, 0), (3, 0))t=2)
Technology: type B agents can transform 18.15 units of labor into a house at time 0
Zhe Li (SUFE) Leverage Cycle 71 / 78 Financial Exchange:
The type B agents will borrow money at time 0 by issuing a mortgage, using the house they will be constructing at the same time as collateral The most optimistic type A agents in [0, 1] will buy those mortgages, thereby lending the B agents the money Since the mortgages will default in state DD (but not until then), they will be risky. Hence the pessimists will not want to buy those mortgages But the optmists don’thave enough money to buy them all. So they will borrow money from the pessimists by selling repo loans against the mortgages they hold These safer repo loans will be held by the pessimists. The Repos are one-period loans, unlike the mortgage, which is a two-period loan
Zhe Li (SUFE) Leverage Cycle 72 / 78 Agents B Borrow Money to Pay Wage
Suppose ε = 0.25 We normalize by taking the price of canned food to be 1 in every state 15 Maximum promise of one house (j = 15): A15 = ( 3 ) 2 2 supply price is π15 = 14.25 = 0.25 3 + (1 0.25 ) 15 price is determined by competition between type B agents Optimistic agents of type A who expect to get more than 14.25 will pay π15 = 14.25 to get the asset Agents B cannot pay the wage to agents A, so they borrow 14.25 from type A Agents to pay the wage, w = 14.25/15 = 0.95 (suppose c 0.95 such that all the agents provide all their labor to produce houses )
Zhe Li (SUFE) Leverage Cycle 73 / 78 Build a House
By constant returns to scale, the price of the house will be 18.15 0.95 = (3.15 + 15)(.95) = 3 + 14.25 = 17.25 The B agents will each buy a house by putting 3 dollars down and borrowing the remaining 14.25 by issuing a mortgage j = 15 promising 15 in every state in the last period
Zhe Li (SUFE) Leverage Cycle 74 / 78 Repo Market
The expected return of an agent A from buying a mortgate contract with promise (j = 15)
The terminal payo¤s of the mortgage security is (15, 15, 15, 3)
Zhe Li (SUFE) Leverage Cycle 75 / 78 Repo Market
Equivalent to 15 contracts with terminal payo¤s (1, 1, 1, 0.2)
Zhe Li (SUFE) Leverage Cycle 76 / 78 Repo Market
Repo market: safe assets Pessimistists do not want to hold those two period contracts, the optimistists buy them by borrowing money from pessimistists in the Repo market The marginal buyer expected value h 15 + (1 h ) 3 = pD 15 = 0.69 15 = 10. 35 1 1 h1 = 7.35/12 = 0.61 is the marginal buyer at D Let h be the marginal buyer at 0, h 1 + (1 h ) 0.69 = 0.95, so 0 0 0 h0 = 0.87 In state 0, the top 13% of agents h will buy the mortgages by issuing repos promising ϕ = 15(.69) = 10. 35 In state D these optimists will be wiped out, and the mortgages will fall into the hands of more pessimistic investors Agents h [.61,.87) will buy the mortgage securities at D, issuing contract (2j, k) = (15, 3) in the repo market to obtained borrowed money to help them buy it
Zhe Li (SUFE) Leverage Cycle 77 / 78 Asset Market Crash
Part of the reason the price of mortgages is so low at D is that the payo¤s are so bad at DD, which limits not just the value but the amount of leverage that can be obtained at D At 0 homeowners leveraged too much in the mortgage market, and investors leveraged their mortgage purchases too much in the repo markets. At D there is massive deleveraging in the securities market, and no opportunity for leverage in the housing market, since the homeowners cannot sell their houses
Zhe Li (SUFE) Leverage Cycle 78 / 78