"The Limits of Arbitrage"
The Limits of Arbitrage Andrei Shleifer & Robert Vishny (JF 1997) Presented by Steven Laufer December 4, 2007 Motivation Absence of arbitrage has been an underlying assumption in most/all of papers so far: Existence of SDF, Compact budget sets. In reality, many funds practice professional arbitrage. Why don’t the actions of these funds eliminate arbitrage opportunities? Answer: Limited funds, agency Overview Agency Model of Limited Arbitrage Model Results Authors’ Discussion My Discussion Model: General Assumptions Arbitrage requires capital. Arbitrage practiced by managers who manage investors’ money. Investors may withdraw capital. Investors do not understand the arbitrage mechanism and can only evaluate managers based on past performance, not future expected returns. Arbitrageurs highly specialized. Only have expertise to practice in one market. Take large enough positions to aﬀect prices. Risk-free arbitrage: arbitrage strategy makes money with probability one. Mis-pricing may deepen before it is resolved. Model: Setup Market for one security Agents: Arbitrageur, Investors, Noise traders Three periods: Period 3: Price goes to fundamental value V (p3 = V ). V known to arbitrageur from start. Period 2: With probability (1-q), nose traders realize the mis-pricing and price goes to fundamental value V (p2 = V ). Otherwise . . . Periods 1 & 2: Prices pt determined in equilibrium. Noise traders have “pessimism shock” St which causes them to underprice asset. If p2 = V , S2 > S1 so mis-pricing worsens. Aggregate demand from noise traders: QN(t) = [V − St ]/pt Model: Setup Arbitrageur in periods 1 & 2 Arbitrageur has available funds Ft . F1 given exogenously. F2 depends on t=1 returns. Investors may withdraw funds between periods 1 and 2. Arbitrageurs invest Dt ≤ Ft . Security in unit supply so equilibrium prices given by pt = V − St + Dt Assume F2 < S2 so p2 < V . (Mis-pricing survives because of lack of capital available for arbitrage.) If p2 = V , D2 = Ft because prices surely rise to p3 = V . If p2 = V , arbitrageur invests in cash. The interesting period is t=1. Performance-based Arbitrage Investors use gross returns from t=1 to t=2 to judge whether to withdraw or invest more capital (even though lower returns predict increased future returns). F2 = F1 · G ([D1 · (p2/p1) + (F1 − D1 )]/F1 ) G (x) = ax + 1 − a, a ≥ 1 a controls sensitivity of investment to past performance. G (1) = 1, G = a ≥ 1, G ≤ 0 If mis-pricing worsens at t = 2, investors see loss and withdraw funds. F2 = F1 − aD1 (1 − p2 /p1 ). Arbitrageurs maximize expected third-period wealth D1 · V EW = (1 − q) a + F 1 − D1 + (1 − a)F1 p1 V D1 · p2 +q a + F 1 − D1 + (1 − a)F1 p2 p1 Model Results First order condition for arbitrageur (1 − q) (V /P1 ) + q (p2/p1 − 1) V /p2 ≥ 0 Equality if D1 < F1 . Inequality if D1 = F1 . Proposition 1: For a given V ,S1 , S2 ,F1 , and a, there is a q ∗ s.t. for q > q ∗ , D1 < F1 and for q < q ∗ , D1 = F1 . Proposition 2: At the corner solution (D1 = F1 ), dp1 /dS1 < 0, dp2 /dS2 < 0, and dp1 /dS2 = 0. At the interior solution dp1 /dS1 < 0, dp2 /dS2 < 0, and dp1 /dS2 < 0. No general results about eﬀect of a on pricing. Depends on whether prices can approach true value in t = 2, which determines whether arbitrageurs might be given more or less funds in period 2. Proposition 3: If D1 = F1 and S2 > S1 , then for a > 1, F2 < D1 and F2 /p2 < D1 /p1 . (D1 = F1 is a suﬃcient but not necessary condition). Proposition 4: If D1 = F1 , dp2 /dS2 < −1 and d 2 p2 /dadS2 < 0. Author’s Discussion Why is PBA plausible? Contractual restrictions on withdrawls are not common. Creditors demand immediate repayment when value of collateral falls. Arbitrageurs may liquidate positions following losses if fear that future deepening mis-pricings would cause dramatic future investment withdrawls. Arbitrageurs may be uncertain about proﬁtability of strategy. Author’s Discussion Less arbitrage in equity and foreign exchange markets than in bond markets because true value in more uncertain. PBA model explains why arbitrage is harder in more volatile markets. If traders are specialized, they cannot diversify away from idiosyncratic risk, no risk premium for extra volatility. Alternatively, in segmented market, idiosyncratic risk may be priced. Arbitrageurs would earn higher expected returns in more volatile assets, even if volatility not correlated with systematic risk. Not in data. Arbitrage strategies more attractive in markets where resolution of mis-pricing is faster and more certain. Example: Stocks of commercial banks were underpriced in 1990-91 but funds had to liquidate before could realize returns. Liquidation put further downward pressure on prices. Discussion Important to understand why arbitrage opportunities persist. Limited capital available for arbitrage and uncertainty about timing of resolution of mis-pricings are interesting explanations. BUT These are not the main mechanisms of this paper. Discussion: PBA Model Investors evaluate arbitrageurs on time scale shorter than the time expected to proﬁt from arbitrage. Investors judge based on past performance even though past and future returns are perfectly negatively correlated. Investors are not behaving rationally! Do we really expect mis-pricings to become worse? Mechanism of PBA reducing ability of arbitrageurs to aﬀect prices relies on this assumption! Discussion: Importance of PBA Limited capital reduces ability of arbitrageurs to correct mis-pricings. PBA further reduces that ability. Evidence for this additional mechanism? Why is PBA more appealing than risky arbitrage? What about just uncertainty in timing of resolution of mis-pricing? Authors were going to consider a=1 but don’t. Discussion of volatility doesn’t rely on PBA. If large probability of future deepening of mis-pricing, capital-limited ﬁrms will hold more cash to take advantage of future opportunities. Conclusion Limited capital available for arbitrage and uncertainty about timing of resolution of mis-pricings are interesting explanations. Not convinced about importance of PBA. Particularly with modeling of investors and assumption of deepening mis-pricing. Need better description/model of evolution of mis-pricings. If the price gets further from true value, is that evidence that the mis-pricing will persist? Compare that to PBA model.