Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Reviewing the Leverage Cycle A. Fostel (GWU) J. Geanakoplos (Yale) Federal Reserve Bank Dallas, 2013 1 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Outline 1 Introduction 2 A Simple Model 3 Simple Example 4 Results 5 Leverage Cycle 6 Multiple Leverage Cycles 7 Volatility 2 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility We review the theory of leverage developed in collateral equilibrium models with incomplete markets. Geanakoplos (1997) Collateral Equilibrium Geanakoplos (2003) Leverage Cycle Fostel and Geanakoplos (2008) Multiple Leverage Cycles We explain how leverage tends to boost asset prices, and create bubbles. We show how leverage can be endogenously determined in equilibrium, and how it depends on volatility. 3 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Time Series properties: Leverage Cycle. leverage ()volatility () asset prices. 4 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Repo Market Leverage Securities Leverage Cycle Margins Offered and AAA Securities Prices - 100.0 10 20 90.0 30 80.0 40 Price 50 70.0 Reversed Scale Reversed 60 60.0 70 80 50.0 6/1/98 10/14/99 2/25/01 7/10/02 11/22/03 4/5/05 8/18/06 12/31/07 5/14/09 Margin % (Down Payment Required to Purchase Securities) - Securities) Purchase to Required Payment (Down % Margin Average Margin on a Portfolio of CMOs Rated AAA at Issuance Estimated Average Margin Prime Fixed Prices Note: The chart represents the average margin required by dealers on a hypothetical portfolio of bonds subject to certain adjustments noted below. Observe that the Margin % axis has been reversed, since lower margins are correlated with higher prices. The portfolio evolved over time, and changes in average margin reflect changes in composition as well as changes in margins of particular securities. In the period following Aug. 2008, a substantial part of the increase in margins is due to bonds that could no longer be used as collateral after being downgraded, or for other reasons, and hence count as 100% margin. 20 5 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility 6 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility VIX Index 90 80 70 60 50 40 30 20 10 0 1/1/98 1/1/99 1/1/00 1/1/01 1/1/02 1/1/03 1/1/04 1/1/05 1/1/06 1/1/07 1/1/08 1/1/09 1/1/10 7 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Cross Sectional Properites: Multiple Leverage Cycles: Flight to Collateral Contagion Drastic swings in the volume of trade of high quality assets. 8 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Lessons from the Leverage Cycle Theory 1 Increasing leverage on a broad scale can increase asset prices. 2 Leverage is endogenous and fluctuates with the fear of default. 3 Leverage is therefore related to the degree of uncertainty or volatility of asset markets. 4 The scarcity of collateral creates a collateral value that can lead to bubbles in which some asset prices are far above their efficient levels. 5 Booms and busts of the leverage cycle can be smoothed best not by controlling interest rates, but by regulating leverage. 6 Multiple leverage cycles can explain important phenomena like Flight to Collateral, Contagion and violent swings in volume of trade 9 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Outline 1 Introduction 2 A Simple Model 3 Simple Example 4 Results 5 Leverage Cycle 6 Multiple Leverage Cycles 7 Volatility 10 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Time and Uncertainty Time t = 0, ..., T . Uncertainty: s 2 S including a root s = 0. ST , the set of terminal nodes of S. Binomial tree: each state s 6= 0 has an immediate predecessor s∗, and each nonterminal node s 2 S n ST has a set S(s) = fsU, sDg of immediate successors. binomial tree is simplest model in which uncertainty plays a role in determining leverage; also general theorems can be proved. 11 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility UU U UD 1 DU D DD 12 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Goods and Assets There is a single perishable consumption good c. Numeraire. k K = f1, ..., K g assets k which pay dividends ds of the consumption good in each state s 2 S n f0g. Price psk . Financial assets: it gives no direct utility to investors, and it pays the same dividends no matter who owns it. 13 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Financial Contracts: Collateral and Debt A debt contract j 2 J is a one-period non contingent bond Issued in state s(j) 2 S Promise: b(j) > 0 units of the consumption good in each immediate successor state state s0 2 S(s) Collateral: one unit of asset k(j) 2 K as collateral We denote the set of contracts with issue state s backed by k S k one unit of asset k by Js ⊂ J; we let Js = k Js and J S J . = s2SnST s 14 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Financial Contracts: Collateral and Debt Price of contract j is pj . An investor can borrow pj today by selling the debt contract j in exchange for a promise of b(j) in each s0 2 S(s(j)). Actual delivery of debt contract j in each state s0 2 S(s(j)) is (no-recourse loan) k minfb(j), ps0k(j) + ds0 g The rate of interest promised by contract j in equilibrium is (1 + rj ) = b(j)/pj . If promise is small enough, the same formula defines a riskless rate of interest. Let jj > 0(< 0) be the number of contracts j sold (bought). 15 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Leverage The Loan-to-Value LTVj associated to contract j in state s(j) is given by pj LTVj = . ps(j)k(j) 16 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Endogenous Leverage We follow Geanakoplos (1997) methodology. Agents have access to a menu of contracts J. In equilibrium every contract, as well as the asset used as collateral, will have a price. Each contract has a well defined LTV . The key is that even if all contracts are priced in equilibrium, because collateral is scarce, only a few will be actively traded. In this sense, leverage becomes endogenous. 17 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Asset and Investor Leverage k Leverage for asset k in state s, LTVs , as the trade-value weighted average of LTVj across all actively traded debt k contracts j 2 Js by all the agents h 2 H h k max(0, ) k ∑h ∑j2Js jj pj LTVs = h . k max(0, )p ∑h ∑j2Js jj sk h Leverage for investor h in state s, LTVs , is defined analogously as h k max(0, ) h ∑k ∑j2Js jj pj LTVs = h . k max(0, )p ∑k ∑j2Js jj sk 18 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Investors Each investor h 2 H is characterized by a utility function h h t(s) h h U = u (c0) + ∑ dh g¯s u (cs ). s2Sn0 h Endowment of the consumption good: es 2 R+, s 2 S. h K Endowment of assets: as 2 R+ ,s 2 S. 19 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Budget Set Given asset prices and contract prices (p, p), each agent h 2 H choses consumption, c, asset holdings, y, and contract sales/purchases j in order to maximize utility (4) subject to the budget set defined by h S SK Js B (p, p) = f(c, y, j) 2 R+ × R+ × (R )s2SnST : 8s h h (cs − es ) + ps · (ys − ys∗ − as ) ≤ k k ≤ ∑k2K ds ys∗k + ∑j2J jj pj − ∑k2K ∑ k jj min(b(j), psk + ds ); s j2Js∗ k k max(0, ) ≤ y , 8kg. ∑j2Js jj s 20 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Collateral Equilibrium h h h A Collateral Equilibrium is ((p, p), (c , y , j )h2H ) 2 K Js S SK Js H (R+ × R+ )s2SnST × (R+ × R+ × (R )s2SnST ) such that 1 h h h k ∑h2H (cs − es ) = ∑h2H ∑k2K ys∗k ds , 8s. h h h 2 ∑h2H (ys − ys∗ − as ) = 0, 8s. h 3 ∑h2H jj = 0, 8j 2 Js , 8s. 4 (ch, y h, jh) 2 Bh(p, p), 8h (c, y, j) 2 Bh(p, p) ) Uh(c) ≤ Uh(ch), 8h. Geanakoplos and Zame (1998) equilibrium exists. 21 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility Outline 1 Introduction 2 A Simple Model 3 Simple Example 4 Results 5 Leverage Cycle 6 Multiple Leverage Cycles 7 Volatility 22 / 90 Introduction A Simple Model Simple Example Results Leverage Cycle Multiple Leverage Cycles Volatility The Economy T = 2, and S = f0, U, Dg.