Dynamic Speculative Attacks - An Example with Gaussian Noise

5 An example with Gaussian noise Assume that η has a normal distribution with variance ση2 . Let νt = Log(µt /(1 − µt )) be the Log likelihood ratio (LLR) between the two states. The Bayesian equation (5) takes the form νt = νt−1 + λt θ1 − θ0  θ1 + θ0  y − λ , t t ση2 2 with λt = λt−1 + ζt , and yt = λt θ + ηt . The expected change of belief from period t to period t + 1 is measured by E[νt+1 − νt ] =          θ1 − θ0 2 2ση2 (λt + ζt )2  2    θ1 − θ0    (λt + ζt )2 − 2 2ση in state θ1 , (12) in state θ0 . We have seen before that the signal to noise ratio in the demand yt increases with the fraction of active speculators λt + ζt . This property appears in the previous expression where the absolute value of the expected change of belief increases with the demand by speculators. 5.1 A numerical example The graphs of the payoff of an order and that of waiting in the first period of a three period game (T = 2) are represented in Figure 1. By assumption, no agent holds the asset at the beginning of the game, λ1 = 0, and the belief of the high state is µ1 = 0.05. The height of the band is γ = 0.025. The rate of the devaluation is A = 0.275. The parameter a is equal to 1 and the critical mass is therefore X̄ = γ/a = 0.025. The standard error of the noise trade is ση = 0.08X̄. Hence, the probability α that a devaluation is triggered by the noise traders is less than 10−12 . The values of the actual masses of speculators in the two states 23 are θ0 = 0.7X̄ and θ1 = 1.3X̄. The interest rate per period is 0.15 percent. The payoffs of an order and of delay are equal to u1 (ζ; 0, 0.05) and w1 (ζ; 0, 0.05), or u(ζ) and w(ζ) for short12 . There are three equilibria, but the middle equilibrium value of λ is unstable. The two stable equilibria are the zero-equilibrium and an equilibrium in which a fraction of speculators purchase the asset. (For other values of the parameters, there may be more than one stable equilibrium where the mass of speculation is strictly positive). When ζ is small, the value of a buy order is a decreasing function of ζ: a higher mass of speculation raises the price of the asset, but because this mass is still small, it does not provide much information on the state, and the gross payoff of an order remains low. -3 8 -3 x 10 8 6 6 w(ζ) 4 2 0 0 u(ζ) -2 -4 -6 -6 -8 -8 0.1 0.2 0.3 0.4 0.5 0.6 u(ζ) -2 -4 0 w(ζ) 4 2 -10 x 10 0.7 ζ∗ 0.8 0.9 1 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ζ 1 ζ Dotted : a = 1 as in Figure 1. Plain : a = 0.94. Figure 1 Figure 2 The base case 12 Higher supply elasticity u(ζ) and w(ζ) are computed on a grid for ζ ∈ [0, 1] of width 0.02. The first step is the computation of U2 (λ, µ) and V2 (λ, µ) on a grid of values (λ, µ) ∈ [0, 1] × [0, 1]. 24 When the value of ζ, is sufficiently large, the price conveys an informative signal on the high state if that is the true state. The anticipation that such an information will be provided at the end of the first period and will generate a successful continuation of the attack in period 2 raises the value of a purchase order above the value of the option of delay. The gain of buying early compensates for the risk of finding out at the end of period 1 that the state is bad and enduring a capital loss in period 2. 5.2 Impact of parameters The elasticity of supply by market makers Suppose that in the first period only, a perturbation increases the elasticity of the supply S(p) = (p − 1)/a. The value of a is lowered in the first period and unchanged in period 2 for the computation of U2 and V2 , (for simplicity). The new graphs of u and w are presented in Figure 2 by the solid curves. The dotted curves are the same as in Figure 1. To interpret the effect of a higher elasticity, recall that in the first period, either the mass of orders y1 is greater than the critical mass X̄ and a devaluation takes place, or y1 < X̄ in which case no devaluation takes place and agents learn from the observation of y1 (which is equivalent to the observation of the price p1 ). An increase in the supply elasticity of the asset has two effects: it raises the critical mass X̄, and it lowers the price p1 for given y1 . The first of these two effects is significant only if the strategy to buy, ζ1 , is sufficiently large to trigger a devaluation in the first period. If ζ1 is relatively small, the probability of a 25 devaluation in period 1 is small and the first effect can be neglected. The second effect lowers the cost of an order for a given amount of information. Hence, the payoff of an order is increased while the payoff of delay is not changed significantly. The combination raises the equilibrium value of ζ ∗ as shown in Figure 2. When the value of ζ is not small, the first effect may not be neglected. The higher value of the critical mass X̄ generates a lower probability of devaluation in the first period. Hence the risk of missing the capital gain from the devaluation is lower and the payoff of delay is higher, as shown in Figure 2. (When ζ1 is near one, the payoff of delay is always near zero). Suppose now that the elasticity of supply is moderately lower in the first period (higher a). The previous arguments apply a contrario. The payoffs are represented in Figure 3. In the example of the figure, the equilibrium with a speculative attack and arbitrage disappears because of the higher cost of an order. If the elasticity of supply is decreased from the base case by a larger amount, there is an equilibrium with a speculative attack in which all speculators buy without delay with ζ1 = 1 (Figure 4). Here the first effect which was described previously operates. A lower elasticity entails a lower critical mass and therefore a higher probability of devaluation in the first period13 . Partially informed market makers The previous discussion shows that the supply elasticity and the liquidity of the market have an ambiguous impact on the likelihood of a speculative attack. 13 In the example, the mass of speculators in any state is higher than the critical mass if 1/a < 0.7 or a > 1.43. 26 Thoughout the paper, market makers have ex ante a vanishingly small probability that there is a large mass of speculators. Suppose now for a moment that this ex ante belief is not negligible any more. When market makers see a price rise in the period, they revise upwards their probability of a high mass of speculators. This effect is equivalent to a downward shift of their supply. The supply schedule is therefore less elastic. In the examples of Figure 4, a small increase in the belief of market makers (probability of state θ1 ) may prevent the occurence of a speculative attack because it raises the cost of communication between speculators through the market. A large increase in the belief facilitates a speculative attack which succeeds immediately with no communication between speculators. -3 8 x 10 0.035 6 0.03 w(ζ) 4 0.025 2 0.02 0 0.015 u(ζ) -2 0.01 -4 0.005 -6 0 -8 -0.005 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.01 ζ w(ζ) u(ζ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Dotted : a = 1 as in Figure 1. Dotted : a = 1 as in Figure 1. Plain : a = 1.2. Plain : a = 1.3. Figure 3 0.9 1 ζ Figure 4 Low supply elasticity Lower supply elasticity Noise traders The noise traders, a standard feature of the model, affect the information content of the first period price. If the variance of noise trading is higher, the price in the first period is less informative (see equations (4) and (12)). When the state 27 is high, agents are on average less informed about it at the end of the first period (for given ζ). This lowers the probability of a speculative attack in period 2, and therefore the incentive to attack in period 1. In Figure 5, the variance of the noise is higher than in the base case (Figure 1), for the first period only. There is no equilibrium with a speculative attack. -3 8 x 10 -3 8 6 6 w(ζ) 4 2 0 0 u(ζ) -2 -4 -6 -6 -8 -8 0.1 0.2 0.3 0.4 0.5 u(ζ) -2 -4 0 w(ζ) 4 2 -10 x 10 0.6 0.7 0.8 0.9 1 ζ -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Dotted : ση /X̄ = 0.08 as in Figure 1. Dotted : r = 1 − δ = 0.01 percent. Plain : ση /X̄ = 0.1. Plain : r = 0.25 percent. Figure 5 ζ Figure 6 Higher trade noise Low and high interest rate The discount rate The discount rate is defined as r = 1 − δ. Suppose an attack takes place in the first period with an arbitrage between delay and no delay. The probability of a devaluation in the first period is very low. An order in the first period pays off because of the relatively low price in the first period and the expectation of a continuation of the attack with a devaluation in the next period. An important difference between the payoffs of an order and of delay is the foregone interest when the agent buys the asset in the first period. This effect is illustrated in 28 Figure 6 where the discount rate takes two values, r = 0.01% and r = 0.025%. A higher discount rate lowers both the payoff of an order and that of delay, but the second effect is negligible14 . 6 Defense Policies A defense policy is successful if it abolishes the coordination equilibrium with a speculative attack. Three types of policies are considered here: (i) widening the band of fluctuations, (ii) stabilizing the exchange rate through trading, and (iii) random interventions. For each policy it is assumed that the central bank cannot observe the state θ. Widening the band of fluctuations The widening of the band with a constant rate of devaluation A is equivalent to an increase in γ. In the present model, if γ = A there is no capital gain if a devaluation takes place, hence no expected profit. If the attack fails however, there is a capital loss. Hence, there is no speculative attack in an equilibrium. By continuity, that property holds if γ ∗ < γ ≤ A for some γ ∗ (0 < γ ∗ < A). A speculative attack can be prevented by a suitable widening of the band of fluctuations. A relevant episode occurred at the end of July 1993 with the speculative attack against the French Franc which was part of the ERM. The regime had margins of 2.25 percent on each side of a reference level. After trying unsuccessfully to 14 This is particularly clear for ζ = 0: the payoff of delay is obviously 0. An agent who buys now pays 1 and gets roughly a present value of δ = 1 − r in the next period, as can be verified in Figures 5 and 6. 29 ward off the speculators through trading, the central banks of the monetary union raised the bands of fluctuations to 15 percent. The change of regime stopped the attack15 , as illustrated in Figure 1 of Obstfeld (1996). Stabilizing the exchange rate through trade intervention There are two types of trade interventions by the central bank, those which are deterministic and predictable, and those which are random and surprise the speculators. A trade policy which is determined by the exchange rate is predictable by rational agents. As an example, assume that the central bank supplies a quantity of foreign currency according to the linear rule SB = b(p − 1), with b > 0. (13) With b > 0, the central bank attempts to reduce the fluctuations of the exchange rate. Such a policy requires a positive level of reserves R. The problem of defense is interesting only if the reserves cannot prevent a speculative attack under perfect information, i.e. if R < θ1 − X̄ (as in Obstfeld, 1996). Such a constraint imposes 15 The last exchange rate between the DM and the FF before the change of regime was at the top of the band at 3.4304. The day after the change of regime (August 2), the rate increased to 3.5080 then fell to 3.46040 two days later. It then began to increase again and to hover around 3.50. However, by that time the information had probably changed. Agents expected the interest rate to be lowered in France to take advantage of the greater exchange rate flexibility and reduce unemployment. Eventually, expectations were mistaken: such a policy was not conducted by the central bank. After hovering between 3.48 and 3.55 until the beginning of December, the exchange rate decreased steadily during the last month of the year to end at 3.40. In agreement with the policy interpretation in this paper, after the exchange rate returned to its mid-band level, the central banks felt no need to reduce the bands back to the original value of 2.25 percent. The wider band contributed to the stabilization of the exchange rate. (For a discussion of the events see Buiter et al., 1998). 30 a restriction on the stabilization policy. Since SB ≤ R for all values of p < 1 + γ, we must have b ≤ R/γ. Speculators with rational expectations anticipate the policy. We assume accordingly that they know the value of the policy parameter b and know that the total supply of foreign exchange is equal to (p − 1)/a with a = 1/(b + 1/a) < a. The impact of the policy is the same as that of an increase in the supply elasticity of foreign exchange, or the liquidity of the market. We have seen in the previous section (Figure 2), that a higher elasticity of supply enlarges the domain of beliefs in which a speculative attack may take place. In the present model, a central bank which reduces the fluctuations of the exchange rate does not alter the function of the exchange rate as a coordination device, but it reduces the risk taken by speculators. Such a policy facilitates speculative attacks. When the price of the foreign currency rises (because of a noise shock or an attack), a central bank which conducts a deterministic (and predictable) policy should not sell the foreign currency but it should buy. Random intervention Trades by the central bank which cannot be predicted by speculators have to be random. Assume that the central bank supplies a random amount R which is normally distributed N (0, σR ) and set before the opening of the market. Rational speculators know the parameter σR but cannot observe R. The random trading by the central bank adds noise and thus reduces the information content of the price in the first period. The intuition that the smaller information reduces the possibility of coordination between speculators is verified by numerical simulations. 31 In the example of Figure 5, the speculative attack is eliminated when σR ≥ 0.06X̄. For a policy of random interventions, some reserves are required (since the foreign currency has to be sold at times), contrary to the policy of deterministic trade. However, these reserves may be significantly smaller than what would be required under perfect information. This is an important implication of the present model for policy. By trading in a non predictable manner, the central bank can prevent speculators from coordinating an attack. The amount of reserves which are required for this policy can be smaller than what would be required if speculators had complete information on their total resources16 . The interest rate A standard defense policy is to raise the interest rate. This policy is illustrated in Figure 6. It is particularly powerful in the context of the model. For the parameters of the figure, an interest rate of 0.25 percent defends the regime. The policy is effective because it raises the cost of “communication” through the price at the beginning of an attack when no devaluation takes place yet. If the parameters of the model and the beliefs are such that there is an attack by all speculators which must succeed or fail in the first period, raising the interest rate is significantly less effective. This case is illustrated by Figure 4. Given the elasticity of supply in that figure, the interest rate has to be raised from 0.15 percent to 4 percent to prevent the speculative attack. 16 In Figure 5, the central bank needs a level of reserves equal to R = 0.12X̄. (The normal distribution can be considered as an approximation of a distribution where the trade by the central bank is bounded). If all speculators have knowledge of the high state and attack, the central bank’s reserves R are too low to fend off the attack: θ1 − X̄ = 0.3X̄ > R. 32 APPENDIX Lemma 1 Assume that in period t, all asset holders sell and no agent buys the asset. If a devaluation occurs, it is triggered by the noise traders, with ηt > X̄ (equation (1)). In that event, an asset holder is on the long side of the market and gets his order to be executed at the highest price in the band, 1 + γ. If no devaluation occurs, the sell order is executed at the market price p = 1 + aηt . The payoff of selling is therefore V̄ with   dF (η) + a V̄ = 1 + γ η>X̄ Since  ηdF (η). ηX̄ (η − X̄)dF (η) = 1 − β. Lemma 2 We first prove that if all asset holders sell in period t and no agent buys in period t or after, no asset holder wants to delay his sale in period t. Assume that one asset holder delays his sale until period t + k, 1 ≤ k ≤ T + 1 − t. If a devaluation occurs in period t+i with i < k, he sells in period t+i+1 because of the discount factor. Hence, the delay until period t + k is contingent on no devaluation before period t + k. Since no speculator holds the asset after period t, the demand for the asset is driven only by the noise traders and a devaluation occurs in period i only if no devaluation has occurred before and ηi > X̄, with a probability α ≥ 0. If the agent sells after a devaluation, he gets 1 + A. If he sells with no prior devaluation, he gets V̄ (Lemma 1). The payoff of holding until period t + k is 33 therefore vk  = δ α(1 + A) + ... + ((1 − α)δ)k−1 α(1 + A) + (1 − α)((1 − α)δ)k−1 V̄ δα(1 + A) + δ(1 − α)V̄ , 1 − (1 − α)δ <  with V̄ = 1 − β. Delaying the sale is not optimal if vk < V̄ . A sufficient condition is δα(1 + A) < (1 − δ(1 − α))2 (1 − β). For any (δ, β ∗ ) ∈ (0, 1) × (0, 1), there is α1∗ > 0 such that if α < α1∗ and β < β ∗ this condition holds. We now prove that no agent would purchase the asset in any period i, t ≤ i ≤ T + 1. From the first part of the proof, if an agent purchases the asset, he sells it the next period. His order is executed only if there is no devaluation in the same period. The payoff of such a strategy is  u= ηX̄ ηdF (η) = β + aαX̄ = β + γα, then u = −(1 − α)(1 − δ)(1 − β) + α(γ + β). For any (δ, β ∗ ) ∈ (0, 1) × (0, 1), there is α2∗ > 0 such that if α < α2∗ and β < β ∗ , u < 0. The value of α∗ in the Lemma is α∗ = M in(α1∗ , α2∗ ). Lemma 3 Let V S be the payoff of an asset holder who sells without delay:  VS =  yt > X̄ (1 + γ)dFt + yt < X̄ (1 + ayt )dFt , with dFt = dF (θ, η; µt−1 ). Let Vt be the value of holding the asset until the end of the period:  Vt = yt > X̄  (1 + A)dFt + δ  yt < X̄ 34  Vt+1 λ, µt (yt , µt−1 ) dFt . Vt − V S = (A − γ)   dFt + yt > X̄ yt < X̄ (δVt+1 − 1 − ayt )dFt . Using the expression of Ut in (10) and (6), (A − γ)   dFt + yt > X̄ yt < X̄ (δVt+1 − (1 + ayt ))dFt > Ut ≥ 0. Lemma 4 First, if there is a speculative attack (φt > 0), using the definition of Ut in (10) and (6), Ut = (A − γ)   φt (λt−1 , µt−1 )θ yt > X̄  +  M ax (X̄ − λt−1 )θ − η, 0  yt < X̄ dFt  δ Vt+1 − (1 + ayt ) dFt , Using (11), Vt = 1 + Ut + G(φt (λt−1 , µt−1 ), λt−1 , µt−1 ), where the function G is independent of the period t:  G(ζ, λ, µ) = A  y > X̄ 1−(  +a A − γ M ax(X̄ − λθ − η, 0)  dF (θ, η; µ) ) A ζθ ydF (θ, η; µ), y > X̄ with y = ζθ + η. Second, if φt (λ) = 0, from Assumption 1, all asset holders sell: Vt = V̄ and Ut = 0. We define G(0, λ, µ) = −β. If ζ > 0, G(ζ, λ, µ) is differentiable in ζ at φ(λ, µ). Recall that y = (λ+ζ)θ+η. In the expression of Gζ , the sum of the terms generated by the derivatives of the bounds of integration is equal to zero and Gζ = (A − γ)  y>X̄ M ax(X̄ − λθ − η, 0) dF > 0. ζ 2θ Proposition 1 We proceed by backward induction. In period T +1, φT +1 ≡ 0. Hence φT ≥ φT +1 . Assume that φt+1 ≥ φt+2 . 35 Consider first the case where φt+1 (λ, µ) = 1 − λ. Since all currency holders in period t + 1 place an order, there cannot be new orders in period t + 2, and by Assumption 1, the game ends in period t + 2. This equilibrium strategy in period t + 1 is also an equilibrium strategy in period t (with the same (λ, µ)) and it is the highest equilibrium strategy. We have φt (λ, µ) = φt+1 (λ, µ) = 1 − λ. Assume now that 0 < φt+1 (λ, µ) < 1 − λ: there is an arbitrage in period t + 1 between purchase and delay. We show that in period t, if (λt−1 , µt−1 ) is identical to (λ, µ), then delay to order yields a lower payoff than no delay. Recall the expression of ut in (6), and of wt in (7): ut (ζ; λt−1 , µt−1 ) (A − γ) =    + yt < X̄    δ Vt+1 ζ + λt−1 , µt − (1 + ayt ) dF (θ, η; µt−1 ).   wt (ζ; λt−1 , µt−1 ) = δ M ax(X̄ − λt−1 θ − η, 0) dF (θ, η; µt−1 ) ζθ yt > X̄    M ax Ut+1 ζ+λt−1 , µt (yt , µt−1 ) , 0 dF (θ, η; µt−1 ), yt < X̄ Applying Lemma 4 to Vt+1 and Ut+1 , we have  ut (ζ, λ, µ)−wt (ζ, λ, µ) = H(ζ, λ, µ)+δ y≤X̄   G φt+1 (ζ+λ, µ ), ζ+λ, µ dFµ , (14) where dFµ = dF (θ, η; µ), y = (λ + ζ)θ + η, µ is the updated belief which depends on (µ, ζ + λ, η) as specified in (5), and the function H is defined by H(ζ, λ, µ) = (A − γ)   M ax(X̄ − λθ − η, 0) (1 − δ + ay)dF, dF − ζθ y > X̄ y < X̄ which is independent of t. If 0 < φt+1 (λ, µ) < 1 − λ, the payoff of an order and of delay are equal in period t + 1. Hence,  0 = H(φt+1 (λ, µ), λ, µ) + δ  (φt+1 (λ,µ)+λ)θ+η≤X̄ 36  G φt+2 (ζ + λ, µ ), ζ + λ, µ dFµ . Consider the difference between the payoff of an order and that of delay in period t, for the same value (λ, µ) and the same strategy φt+1 (λ, µ). Substituting in (14), ut (φt+1 (λ, µ); λ, µ) − wt (φt+1 (λ, µ); λ, µ) =  H(φt+1 (λ, µ), λ, µ) +  (φt+1 (λ,µ)+λ)θ+η≤X̄  G φt+1 (ζ + λ, µ ), ζ + λ, µ dFµ . Since by assumption φt+1 ≥ φt+2 and G is increasing in its first argument (Lemma 4), ut (φt+1 (λ, µ); λ, µ) − wt (φt+1 (λ, µ); λ, µ) ≥ 0. By definition of φt in (9), φt (λ, µ) ≥ φt+1 (λ, µ). 37