Supplemantary material to: Leverage causes fat tails and clustered volatility

Transcription

1 Supplemantary material to: Leverage causes fat tails and clustered volatility Stefan Turner a,b J. Doyne Farmer b,c Jon Geanakoplos d,b a Complex Systems Researc Group, Medical University of Vienna, Wäringer Gürtel 18-20, A-1090 Vienna, Austria b Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe NM 87501, USA c LUISS Guido Carli, Viale Pola 12, 00198, Roma, Italy d Department of Economics, Yale University, New Haven, Connecticut, USA Contents 1 Model description Overview Price formation Noise traders Hedge funds Hedge fund investors Setting maximum leverage Banks extending leverage to funds Defaults Return to edge fund investors Simulation procedure Summary of parameters and teir default values How leverage increases volatility 7 Corresponding autor, jdf@santafe.edu 1

2 1 Model description 1.1 Overview In our model tere is a single financial asset wic does not pay a dividend. Tere are two types of agents wo buy and sell te asset, noise traders and value investors wic we refer to as edge funds. Te noise traders buy and sell more or less at random, but wit a sligt bias tat makes te price weakly mean-reverting around a fundamental value. Te edge funds use a strategy tat exploits mispricings by taking a long-position (olding a net positive quantity of te asset) wen te price is below its perceived fundamental value. A pool of investors wo invest in edge funds contribute or witdraw money from edge funds depending on teir istorical performance relative to a bencmark return; successful edge funds attract more capital and unsuccessful ones lose capital. Te edge funds can leverage teir investments by borrowing money from a bank, but tey are required to maintain teir leverage below a fixed value tat is determined in te model. Prices are set using market clearing. We now describe te components of te model in more detail. 1.2 Price formation We use a standard market clearing mecanism in wic prices are obtained by self-consistently solving te demand equation. Let D nt (p(t)) be te noise trader demand and D (p(t)) be te edge fund demand. N is te number of sares of te asset. Te asset price p(t) is found by solving D nt (p(t)) + D (p(t)) = N, (1) were te sum extends over all edge funds in te system. 1.3 Noise traders We construct a noise trader process so tat in te absence of any oter investors te logaritm of te price of te asset is a weakly mean-reverting random walk. Te central value is cosen so tat te price reverts around fundamental value V. Te dollar value of te noise traders oldings is defined by ξ nt (t), wic follows te equation log ξ nt (t + 1) = ρ log ξ nt (t) + σχ(t) + (1 ρ) log(v N). (2) Te noise traders demand is D nt (t) = ξ nt(t) p(t). (3) Substituting into equation (1), and letting χ be normally distributed wit mean zero and standard deviation one, tis coice of te noise trader process guarantees tat wit ρ < 1 te price will be a mean reverting random walk wit E[log p] = log V. In te limit as ρ 1 te log returns r(t) = log p(t + 1) log p(t) are normally distributed wen tere are no edge funds. For te purposes of tis paper we fix V = 1, σ = and ρ = Tus in te absence of te edge funds te log returns are close to being normally distributed, wit tails tat are sligtly truncated due to te mean reversion. 2

3 1.4 Hedge funds At eac time step eac edge fund allocates its wealt between cas C (t) and its demand for te asset, D (t). To avoid dealing wit te complications of sort selling we require D (t) 0, i.e. te edge funds are long-only. Tis means tat wen te mispricing is zero te edge funds are out of te market. Tus, to study teir affect on prices we are only interested in situations were tere is a positive mispricing. Te edge fund s wealt is te value of te asset plus its cas, W (t) = D (t)p(t) + C (t). (4) On any given step te edge fund may buy or sell sares of te asset and te cas C (t) canges according to C (t) = C (t 1) [D (t) D (t 1)] p(t). (5) If te edge fund uses leverage te cas may become negative, and te edge fund is forced to take out a loan wose size is L (t) = max[ C (t), 0]. Te leverage of fund is te ratio of te value of te assets it olds to its wealt, i.e. λ (t) = D (t)p(t) W (t) = D (t)p(t) D (t)p(t) + C (t). (6) Te bank tries to limit te size of its risk by enforcing a maximum leverage λ MAX. In purcasing sares a fund spends its cas first. If te mispricing is sufficiently strong, in order to purcase more sares it takes out a loan, wic can be as large as permitted by te maximum leverage. Suppose te fund is using te maximum leverage on timestep t and te price decreases at t + 1. If te fund takes no action its leverage at te next time step will exceed te maximum leverage. It is tus forced to sell sares and repay part of te loan in order to reduce te leverage to λ MAX. Tis is called making a margin call. We require tat te edge funds attempt to stay below te maximum leverage at eac time step. We normally keep te maximum leverage constant, but we also investigate policies tat adjust te maximum leverage dynamically based on time dependent factors suc as price volatility, see Section 1.6. Our edge funds are value investors wo base teir demand on a mispricing signal m(t) = V p(t), (7) were as before V is te perceived fundamental value, wic is eld constant to keep tings simple. All edge funds perceive te same fundamental value V. Eac edge fund computes its demand D(t) based on te mispricing at time t. Te edge fund s demand function is sown in dollar terms in Fig. 1. As te mispricing increases te dollar value of te fund s position increases linearly until it reaces te maximum leverage, at wic point it is capped. It can be broken down into tree regions: 1. Te asset is over-priced. In tis case te fund olds only cas. 2. Te asset is under-priced wit λ (t) < λ MAX. In tis case te dollar value of te asset is proportional to te mispricing and proportional to te wealt. 3. Te asset is under-priced wit λ (t) = λ MAX. In tis case its oldings of te asset are capped to remain under te maximum leverage. 3

4 D (t) p(t) (a) D (t+1), N D nt (t+1) (b) D N D NT m(t) p(t+1) Figure 1: (a) Demand function of a fund as a function of te mispricing signal m defined in equation (7). (b) Market clearing: price and demand are determined by te intersection of te demand functions of noise traders and funds. Expressing all quantities at time t, te edge fund demand can be written: m < 0 : D = 0 0 < m < m crit : D p = β mw m m crit : D p = λ MAX W. (8) We call β > 0 te aggressiveness of te edge fund. It sets te slope of te demand function in te middle region, i.e. it relates te size of te position fund is willing to take for a given mispricing signal m. m crit is defined as m crit = λ MAX /β. Tis is te critical mispricing beyond wic te fund cannot take on more leverage. For larger mispricings te leverage stays constant at λ MAX. If te price decreases tis may require te fund to sell assets even toug te mispricing is ig. Tis is wat we mean by making a margin call 1. To compute te demand it is convenient to substitute equation (5) into equation (4), wic gives W (t) = C(t 1) + D(t 1)p(t). (9) Tis is useful because it means tat in te expression for te demand everyting except te price is known, and te price can be found using market clearing, equation (1). 1.5 Hedge fund investors A pool of edge fund investors (representative investor) contribute or witdraw money from eac fund based on a moving average of its recent performance. Tis kind of beavior is well documented 2, and guarantees a steady-state beavior wit well-defined long term statistical 1 A more realistic margin policy would set a leverage band, wic as te effect of making margin calls larger but less frequent. For example, if te leverage band is (5, 7), wen te edge fund reaced 7 it would need to make a margin call to reduce leverage to 5. To avoid introducing yet anoter free parameter, we simply ave te edge funds make continuous margin calls, so tat as long as te mispricing is sufficiently strong, tey constantly adjust teir leverage to maintain it at λ MAX. Introducing a leverage band exaggerates te effects we observe ere. 2 Some of te references tat document or discuss te flow of investors in and out of mutual funds include [Busse (2001); Cevalier and Ellison (1997); Del Guercio and Tka (2002); Remolona et al. (1997); Sirri and Tufano (1998)]. 4

5 averages of te wealt of te edge funds. Te performance of a fund is measured in terms of its Net Asset Value (NAV), wic can be tougt of as te value of a dollar initially invested in te fund. Letting F (t) be te flow of capital in or out of te fund at time t, and initializing NAV(0) = 1, te NAV is computed as Let NAV(t + 1) = NAV(t) W (t + 1) F (t). (10) W (t) r NAV (t) = NAV(t) NAV(t 1) NAV(t) be te fractional cange in te NAV. Te investors make teir decisions about weter to invest in te fund based on r perf (t), an exponential moving average of te NAV, defined as r perf (t) = (1 a) r perf (t 1) + a r NAV (t). (11) Te flow of capital in or out of te fund, F (t), is F (t) = b [r perf (t) r bm ] W (t), (12) were b is a parameter controlling te fraction of capital witdrawn and r bm is te bencmark return of te investors. Te parameter r bm plays te important role of determining te relative size of edge funds vs. noise traders. Funds are initially given wealt W 0 = W (0). At te end of eac timestep te wealt of te fund canges according to W (t + 1) = W (t) + [p(t + 1) p(t)]d (t) + F (t). (13) In te simulations in tis paper, unless oterwise stated we set a = 0.1, b = 0.15, r bm = 0.005, and W 0 = Setting maximum leverage In most of te work described ere we simply set te maximum leverage at a constant value. However, we explicitly test te effect of policies tat adapt leverage based on market conditions. A common policy for banks is to monitor volatility, increasing te allowable leverage wen volatility as recently been low and decreasing it wen it as recently been ig. We assume te bank computes a moving average of te asset price volatility, στ, 2 measured as te variance of p of over an observation period of τ time steps. Here we use τ = 10. Te bank adjusts te maximum allowable leverage according to te relationsip λ max (t) = max [ 1, λ MAX 1 + κσ 2 τ ]. (14) Tis policy lowers te maximum leverage as te volatility increases, wit a floor of one corresponding to no leverage at all. Te parameter κ sets te bank s responsiveness to canges in volatility. For most of te work presented te maximum leverage is constant, corresponding to κ = 0, in Fig. 4 of te main paper we compare te effects to κ =

6 1.7 Banks extending leverage to funds Here we assume tat eac fund as only one bank wic extends leverage to it. Tere are no connections (influences) from one bank to anoter, oter tan te fact tat bot migt be invested in te same asset troug (different) funds. 1.8 Defaults If a fund s wealt falls below zero it defaults, i.e. it can not repay its loans. A fund can default because of redemptions or because of trading losses, or a combination of bot. Te fund is ten removed from te simulation. After a waiting period of T wait a new fund is introduced wit wealt W 0 and wit te same parameters as te original fund. Furter, wenever a fund falls below a non-zero tresold, somewat arbitrarily set to W (t) < W 0 /10, i.e. 10% of te initial endowment, it will be removed and reintroduced after T wait. Using tis tresold to reintroduce funds avoids te problem of zombie edge funds, i.e. funds wose wealt is very close to but not zero, wo take a very long time to recover. 1.9 Return to edge fund investors Since te investor pool actively invests and witdraws money from funds, te NAV does not properly capture te actual return to investors. For example, by witdrawing money troug time an investor may make a good return from a edge fund tat eventually defaults. To solve tis accounting problem we compute te effective return r inv to investors from teir witdrawals by discounting te present value of te flows in and out of te fund. For any given period from t = 0 to t = T tis is done by solving te equation F (0) + F (1) + F (2) 1 + r inv (1 + r inv ) + + F (T ) = 0. (15) 2 (1 + r inv ) T Based on te sequence F (t) of investments and witdrawals tis can be solved numerically for r inv. Note tat if te simulation ends and te fund is still in business ten F (T ) is computed under te assumption tat all te oldings of te fund are liquidated at te current price. If te fund defaults ten F (T ) = 0. During te course of a simulation a fund may default and be re-introduced several times, and it becomes necessary to compute an average performance for te full simulation. Suppose it defaults n times at times T i over te total simulation period, i.e. it existed for n + 1 time periods. For eac period were te fund remains in business witout defaulting we compute te corresponding return r inv[ti,t i+1 ], and ten average tem, weigted by te time over wic eac existed, according to n+1 r inv = r inv[ti 1,T i ]/(T i T i 1 ), (16) i=1 were T 0 = 1 is te first timestep in te simulation, and by definition T n+1 is te ending time of te simulation Simulation procedure Te numerical implementation of te model on te t t timestep proceeds as follows: 6

7 Noise traders compute teir demand for te time period t + 1 based on equation (3). Hedge funds compute teir demand for t+1 based on te mispricing signal m(t) according to equation (8). Note tat tis must be done in conjunction wit computing te new price p(t + 1) i.e. equations (1) and (8) are solved simultaneously. Tis includes computing te wealt W (t + 1), wic involves te new cas oldings C(t + 1) and te leverage λ (t + 1). Investors monitor te NAV of eac fund and make capital contributions or witdrawals. If te maximum leverage is not being eld constant, banks compute te new maximum leverage. If a fund s wealt W (t) falls too low it gets replaced as described in Section 1.8. Continue wit next time step Summary of parameters and teir default values Parameters eld fixed: number of assets: N = 1000 perceived fundamental value: V = 1 initial wealt (cas) of funds: W 0 = C (0) = 2. L(0) = D (0) = 0 noise trader parameters: ρ = 0.99, σ = bankruptcy level: 10% of initial wealt W 0 time to re-introcucing defaulted fund T wait = 100 time to compute variance for price volatility σ τ, τ = 10 (see Section 1.6) bencmark return for investors, r bm = moving average parameter for r perf, a = 0.1 investor witdrawl factor, b = 0.15 Parameters tat we vary: number of funds and teir banks: 1 or 10 aggressivity of funds, values range from β = 5 to 100 maximum leverage λ MAX = 1 to 15 volatility monitoring parameter κ = 0 or 100 (see Section 1.6) 2 How leverage increases volatility We now explain ow leverage increases volatility. Let us begin wit te case of noise traders alone, and assume for a moment V = 1 for simplicity. Market clearing requires tat D nt = N, and equation 3 implies ξ = Np. If we define x log ξ/n = log p, te noise trader process can ten be written in terms of log prices as x t+1 = ρx t + σχ t. (17) Tus te price process is a simple AR(1) process. Defining te log return as r t = x t+1 x t and te volatility in terms of te squared log returns as E[r 2 t+1], te volatility wit a pure noise 7

8 trader process is E[r 2 t ] = 2σ2 1 + ρ. (18) In te limit as ρ 1 tis converges to σ 2. Tus for ρ < 1 tere is a very mild amplification. Now assume te presence of a single edge fund wit aggressiveness β and assume a positive mispricing 0 < m 1, small enoug tat te price is sligtly below V and te edge fund is not at its maximum leverage, i.e. te edge fund demand is D p = βmw. Te market clearing condition can ten we written as NV ξ + βmw = Np. Wit te mispricing being m = V p togeter wit te definition W t = C t + D t p t, tis gives te quadratic equation in m βd t m 2 t + [N + β(c t + D t V )]m t + NV ξ t NV = 0. (19) At time t + 1 we can make use of equation (9), W t+1 = C t + D t p t+1, and write a similar equation for m t+1, wic is te same except for ξ t ξ t+1 : βd t m 2 t+1 + [N + β(c t + D t V )]m t+1 + NV ξ t+1 NV = 0. (20) Solving tese two quadratic equations (denoting te coefficients of equations (19) and (20) by a = βd t, b = N + β(c t + D t V ), c = NV ξ t NV and c = NV ξ t+1 NV ), te cange in price can be written p t+1 p t = m t m t+1 = ± b2 4ac b 2 4a c 2a c c b = NV (ξ t+1 ξ t ) N + β(c t + D t V ), (21) assuming tat ac/b 2 and a c/b 2 to be small, wic is certainly true for large N. Comparing to te pure noise trader case, were p t+1 p t = V (ξ t+1 ξ t ), we see tat te volatility is reduced by a factor (1 + β N (C t + D t V )) 1, wic is less tan 1 as soon as leverage is taken, i.e. λ > 1. At maximum leverage te market clearing condition is NV ξ + λ MAX W = pn. A similar calculation gives p t+1 p t = NV (ξ t+1 ξ t ) N λ MAX D t. (22) Bot D t and λ MAX are positive. Comparing to te pure noise trader case, we see tat now te volatility is amplified. Acknowledgements We would like to tank Barclays Bank, Bill Miller, te National Science Foundation grant and te Austrian Science Fund grant P17621 for support. We would also like to tank Duncan Watts for stimulating te initiation of tis project, and Alan Kirman for useful discussions. Any opinions, findings, and conclusions or recommendations expressed in tis material are tose of te autors and do not necessarily reflect te views of te National Science Foundation. 8

9 References Busse, J. A., Look at mutual fund tournaments. Te Journal of Financial and Quantitative Analysis, 36, Cevalier, J. and Ellison, G., Risk taking by mutual funds as a response to incentives. Te Journal of Political Economy, 105, Del Guercio, D. and Tka, P.A., Te determinants of te flow of funds of managed portfolios: Mutual funds vs. pension funds. Te Journal of Financial and Quantitative Analysis, 37, Remolona, E.M., Kleiman, P. and Gruenstein, D., Market returns and mutual fund flows. FRBNY Economic Policy Review, Sirri, E.R. and Tufano, P., Costly searc and mutual fund flows. Te Journal of Finance, 53,