**BIS Working Papers **

**No 660**

**Informal one-sided target zone model and the Swiss Franc**

by Yu-Fu Chen, Michael Funke and Richhild Moessner

**Monetary and Economic Department **

August 2017

JEL classification: F31, E42, C61

Keywords: Swiss Franc, target zone model, exchange rate interventions

BIS Working Papers are written by members of the Monetary and Economic Department of the Bank for International Settlements, and from time to time by other economists, and are published by the Bank. The papers are on subjects of topical interest and are technical in character. The views expressed in them are those of their authors and not necessarily the views of the BIS.

This publication is available on the BIS website (www.bis.org).

*© Bank for International Settlements 2017. All rights reserved. Brief excerpts may be reproduced or translated provided the source is stated.*

ISSN 1020-0959 (print)

ISSN 1682-7678 (online)

Informal One-Sided Target Zone Model and the Swiss Franc

Yu-Fu Chen

Business and Management Studies, University of Dundee

y.f.chen@dundee.ac.uk

Michael Funke

Department of Economics, Hamburg University and CESifo Munich

michael.funke@uni-hamburg.de

Richhild Moessner

Bank for International Settlements and National Institute of Economic and Social Research

richhild.moessner@bis.org

**Abstract**

This paper develops a new theoretical model with an asymmetric informal one-sided exchange rate target zone, with an application to the Swiss franc following the removal of the minimum exchange rate of CHF 1.20 per euro in January 2015. We extend and generalize the standard target zone model of Krugman (1991) by introducing perceived uncertainty about the lower edge of the band. We find that informal soft edge target zone bands lead to weaker honeymoon effects, wider target zone ranges and higher exchange rate volatility than formal target zone bands. These results suggest that it would be beneficial for exchange rate policy intentions to be stated clearly in order to anchor exchange rate expectations and reduce exchange rate volatility. We also study how exchange rate dynamics can be characterized in models in which financial markets are aware of occasional changes in the policy regime. We show that expected changes in the central bank’s exchange rate policy may lead to exchange rate oscillations, providing an additional source of exchange rate volatility, and to capture this it is important to take into account the possibility of regime changes in exchange rate policy.

Keywords: Swiss franc, target zone model, exchange rate interventions

JEL-Classification : F31, E42, C61

The views expressed are those of the authors and should not be taken to reflect those of the Bank for International Settlements. We would like to thank Bill Allen, Raphael Auer, Emanuel Kohlscheen, Dubravko Mihaljek, an anonymous referee, and participants at the 25th Annual Symposium of the Society of Nonlinear Dynamics and Econometrics 2017 in Paris and at a seminar at the Bank for International Settlements for helpful comments.

**1. Introduction**

Since the outbreak of the euro area crisis, Swiss exchange rate policy has been a hotly debated topic. From September 2011 to January 2015 the official exchange rate policy of the Swiss National Bank (SNB) was to maintain a minimum exchange rate of CHF 1.20 per euro. To achieve this objective, the central bank stated that it would be willing to buy foreign currency in unlimited quantities. On 15 January 2015, the SNB took markets by surprise with its decision to discontinue Switzerland’s currency floor (SNB, 2015; Jerman, 2016).

Countering an appreciating currency ought to be easier than countering a depreciating currency. A central bank trying to prop up the exchange rate can run out of foreign exchange reserves. A central bank aiming for depreciation can credibly promise to buy as much foreign currency as it takes, since it can print its own currency in unlimited amounts. By discontinuing the so-called “minimum exchange rate” in January 2015, the SNB showed that a theoretically unlimited strategy to defend a currency floor may have practical bounds. In particular, there may be an upper limit to how far a central bank balance sheet can grow, including due to concerns about potential future losses on foreign exchange reserves, as suggested for example by Amador et al. (2016). Amador et al. (2016) provide a theoretical model in which a central bank wants to maintain an exchange rate peg, and responds to increases in demand for domestic currency by expanding its balance sheet. They model so-called ‘reverse speculative attacks’ triggered by the concern of future balance sheet losses of the central bank. They find that the interaction between the desire to maintain an exchange rate peg and the concern about future balance sheet losses can lead the central bank to first accumulate a large amount of reserves, and then to abandon the peg, and conclude that this was just as had been observed in the Swiss case. The fact that a Swiss referendum was held on 30 November 2014 over the proposition that the SNB should hold at least 20% of its reserves in gold (The Economist, 2014), suggests that there was some political pressure to limit losses on the central bank’s balance sheet, even though the referendum was defeated. Thus there may be a limit to conducting central bank policy by steering the exchange rate. As a consequence, the SNB resorted, like many other central banks, to cutting interest rates with all the risks of feeding market distortions that such a policy may entail (see BIS, 2010).

The model presented in this paper applies to the situation where there is no formal one-sided target zone for the currency, but where there are market perceptions of an informal one-sided target zone. This may have been the case following the discontinuation of the SNB’s currency floor in January 2015, since proxy measures for the SNB’s foreign exchange interventions presented below suggest that the SNB likely continued to intervene in the foreign exchange market.

In light of the fact that exchange rate regimes have been at the centre of academic debate and have been a major concern for policymakers in recent years, this paper develops a new theoretical model with an asymmetric informal (unofficial, not publicly announced) one-sided target zone. Such a model

incorporates the notion that the target zone can be seen as a partially credible commitment device. It can be regarded as a generalisation of Krugman’s (1991) simple target zone model of exchange rates. In the standard model, the peg and the hard edge boundaries are publicly announced and credible, and market arbitrage mechanisms and interventions take place in whatever amounts are necessary to prevent the target zone from being violated. Our paper is not a normative analysis of the best exchange rate policy or exchange rate target, especially given uncertainty about fundamental drivers of exchange rates.

We find that informal soft edge target zone bands lead to weaker honeymoon effects, wider target zone ranges and higher exchange rate volatility than formal target zone bands. The honeymoon effect stabilises the exchange rate relative to its fundamentals. It refers to the phenomenon that, if the Swiss franc is close to the informal floor, the probability increases that the Swiss franc will hit the floor, which leads to perceived interventions by the central bank. These results suggest that it would be beneficial for exchange rate policy intentions to be stated clearly in order to anchor exchange rate expectations and reduce exchange rate volatility. We also study how exchange rate dynamics can be characterized in models in which financial markets are aware of occasional changes in the policy regime. We show that expected changes in the central bank’s exchange rate policy may in addition be a source of exchange rate oscillations, providing an additional source of exchange rate volatility, and to capture this it is important to take into account the possibility of regime changes in exchange rate policy.

The paper proceeds as follows. Section 2 presents the evolution of the Swiss franc-euro exchange rate, and Section 3 presents proxy measures for the SNB’s foreign exchange interventions as a function of the EUR/CHF exchange rate, at both a weekly and monthly frequency. Section 4 presents the model and its solution using Switzerland as a motivating example. Analytically, the modelling approach lies at the crossroads of the literature on exchange rate target zones originated by Krugman (1991) and the literature on intervention in the foreign exchange market. With the aim of parsimony in mind, but also wanting to ensure a fair degree of reality, we extend and generalize the standard target zone model by introducing perceived uncertainty about the lower edge, as following the policy change in January 2015. Section 5 concludes, draws some policy implications and hints at future research.

**2. Swiss franc-euro exchange rate and currency floor**

Within minutes of the announcement to discontinue the minimum exchange rate on 15 January 2015, the Swiss franc rose by 40 percent against the euro (see Figure 1). Following the announcement, the Swiss franc even reached parity against the euro. Following this initial overshooting, the Swiss franc subsequently depreciated from this low level. The model of this paper for an informal one-sided target zone for the Swiss franc presented in Section 4 applies to this latter period, after the initial overshooting. That is, our model applies to a situation where there was no longer a formal one-sided target zone, but when there may have been market perceptions of an informal, not publicly announced, one-sided target zone, since the SNB likely continued to intervene in the foreign exchange market, as suggested by the proxy measures for the SNB’s foreign exchange interventions presented in Section

The currency floor had been introduced in 2011 to halt the appreciation of the Swiss franc, and it protected the country’s many exporters. It had been effective in preventing an even larger erosion of the competitive position of Swiss exporters. According to the SNB’s assessment on 15 January 2015, “The minimum exchange rate was introduced during a period of exceptional overvaluation of the Swiss franc and an extremely high level of uncertainty on the financial markets. This exceptional and temporary measure protected the Swiss economy from serious harm. While the Swiss franc is still high, the overvaluation has decreased as a whole since the introduction of the minimum exchange rate. The economy was able to take advantage of this phase to adjust to the new situation” (SNB, 2015). Streit (2016) finds evidence for Swiss non-financial firms of significant reductions in total stock return volatility as well as market risk following the introduction of the currency floor. In order to counter the appreciation of the Swiss franc, the SNB also announced on 18 December 2014 that it would introduce a negative interest rate of -0.25% on sight deposit account balances at the SNB starting on 22 January 2015, which was reduced further to -0.75% on 15 January 2015, in order to “ensure that the discontinuation of the minimum exchange rate does not lead to an inappropriate tightening of monetary conditions” (SNB, 2015).

**3. SNB Foreign Exchange Interventions**

The model presented in Section 4 applies to the situation where there is no formal one-sided target zone for the Swiss franc, but where there are market perceptions of an informal, not publicly announced, one-sided target zone. This may have been the case following the discontinuation of the SNB’s currency floor in January 2015, since proxy measures for the SNB’s foreign exchange interventions presented in this Section suggest that the SNB likely continued to intervene in the foreign exchange market.

Between the dates of the introduction of the CHF 1.20 per euro floor and its discontinuation, the SNB ’ s foreign currency reserves increased significantly. At the end of December 2014 they reached CHF 495 billion or almost 80 percent of Swiss GDP - up from CHF 254 billion or almost 45 percent of Swiss GDP at the end of 2011. And with the ECBs quantitative easing programme, the required amounts of foreign exchange intervention to maintain the floor were about to become far higher, increasing the risk of larger losses on foreign exchange reserves in future. For example, SNB governing board member Zurbrugg mentioned that maintaining the minimum exchange rate would have required increasingly higher foreign exchange interventions: “According to SNB Governing Board member Fritz Zurbrugg, the [Swiss] National Bank would have had to intervene with increasingly large amounts in the foreign exchange market in order to maintain the minimum exchange rate. [...] "Extrapolated to one month, we would have had to intervene with around 100 billion francs in January alone", Zurbrugg said in an interview last week [...].” Following the discontinuation of the CHF 1.20 per euro floor in January 2015, the SNB’s foreign exchange reserves continued to increase. In the announcement to discontinue the minimum exchange rate on 15 January 2015, the SNB mentioned that it will “continue to take account of the exchange rate situation in formulating its monetary policy in future. If necessary, it will therefore remain active in the foreign exchange market to influence monetary conditions” (SNB, 2015). Since then the SNB has remained vague about exchange rate interventions ex-post, and generally does not confirm speculations about interventions, with a few exceptions. The SNB also reduced the negative interest rate to -0.75% on 15 January 2015, in the words of former vice-chair Danthine, to “reinstate an interest rate differential with respect to the eurozone, in particular, to limit the appreciation of the CHF that was viewed as the inevitable result of abandoning the floor” (Danthine, 2016).

The most recent SNB interventions have mostly not been publicly announced by the SNB. Ahead of the Brexit referendum, for instance, the SNB did not announce that it would intervene, but referred more generally to taking measures if necessary: “As a small open economy Switzerland is strongly exposed to developments abroad. When the United Kingdom votes next week on whether to remain in the European Union, it can lead to increased uncertainty and turbulences. In this context we will monitor the situation closely and will take measures if needed.” (Jordan, 2016).

Following Brexit, the SNB publicly announced that it had intervened: “"Following the United Kingdom’s vote to leave the European Union, the Swiss franc came under upward pressure," the central bank said via e-mail. "The Swiss National Bank has intervened in the foreign exchange market to stabilize the situation and will remain active in that market"” (Bloomberg, 2016). SNB president Jordan confirmed the intervention over the weekend following Brexit: “"We have always said that we are active in the foreign exchange market if necessary," SNB Chairman Thomas Jordan said at a finance event in Bern, Switzerland. "A situation like we experienced over the weekend is a situation which warranted this need and we went in to stabilize the market."” (Reuters, 2016). Another exception was at the height of the euro area sovereign debt crisis (Bloomberg, 2015, as cited below).

Some evidence suggests that foreign exchange intervention is often geared to limiting what policymakers may see as unwarranted, and possibly harmful, deviations from equilibrium levels (Daude and Yeyati 2014, Bank for International Settlements 2005, 2013). Daude et al (2014) present evidence for the prevalence of such exchange rate intervention in emerging economies, and show that it mitigates real exchange rate volatility. An overvalued exchange rate may be generated in the presence of safe haven effects arising in situations of pronounced uncertainty and/or if the exchange rate overreacts to news. In this case, exchange rate intervention against the direction that the exchange rate is moving may be a useful instrument to reduce the distortion.

The theoretical literature on optimal exchange rate interventions is enormous. For a seminal paper, see Boyer (1978). Reeves (1997) constructed a game-theoretic framework with partial credibility and nonrational expectations to examine the mode of action of the signalling channel. In this set-up exchange rate interventions represent signals of future monetary policy and hence affect exchange rate expectations. Note that it is not at all clear that exchange rate interventions will always be beneficial. In particular, the central bank may not know the true economic fundamentals but face uncertainty and misperceptions about them. Moreover, the central bank may not be capable of correctly identifying the equilibrium exchange rate implied by the fundamentals or, even if it would be able to do so, the costs of the exchange rate interventions may be too high relative to the benefits. Furthermore, persistent changes in investor preferences can have persistent effects on exchange rates.

Another argument supportive of a case for exchange rate interventions is that the authorities’ information set may be superior to the information set of market participants. On this argument the central bank could make the intervention data set publicly available, since this would eliminate the informational asymmetry.

Archer (2005) argues that transparent foreign exchange intervention is preferable to secret intervention because it increases the power of the signalling and coordination channels, which increases the effectiveness of intervention. A number of explanations have been put forward to explain why foreign exchange interventions may not be made public. In a situation of high uncertainty and volatility of foreign exchange markets, it may be risky for a central bank to announce publicly the intention of intervening, especially if the effectiveness may not be believed by market participants (see Cukierman and Meltzer, 1986).

Transparency about foreign exchange intervention and exchange rate policy may have adverse effects, which need to be weighed against the potential benefits of transparency about exchange rate policy and intervention implied by the signalling channel. Possible adverse effects include increased pressure on the exchange rate from possible speculative attacks, for which communication about foreign exchange intervention or exchange rate policy could function as a coordination device, and increases in the size of speculative flows and in the costs of carrying reserves (Chutasripanich and Yetman 2015). Ambiguity about past and planned amounts of foreign exchange intervention and desired exchange rates may help a central bank counter or prevent coordinated speculative attacks. This is in contrast to communication about the monetary policy framework in the form of an inflation target, and in contrast to communication about the policy rate (for surveys see Blinder et al., 2008 and Moessner et al., 2016), where no such adverse effects from speculative attacks are present.

For example, in Mexico a transparent rule-based foreign exchange intervention mechanism introduced in December 2014 was discontinued in February 2016 because of pressures on the exchange rate, including due to apparent speculation by foreign exchange traders against the rule-based mechanism, while the possibility of future discretionary intervention was kept open (Domanski et al, 2016). Due to an increase of electronic trading volumes (accounting for around two thirds of foreign cash market turnover in 2015, an increase from less than 50% before the global financial crisis), the effects of foreign exchange intervention on the exchange rate may depend more and more on the central bank’s ability to surprise markets (Domanski et al, 2016). Some papers find that central banks use secret foreign exchange intervention to maximise the impact on the exchange rate (Dominguez and Frankel, 1993, Neely, 2001, Sarno and Taylor, 2001).

Weekly FX intervention proxy

Sight deposits are currently the most important means of financing for SNB currency purchases. Thus changes in total sight deposits in Swiss francs at the SNB are a proxy for the SNB’s FX interventions (Auer, 2015). But sight deposits can also change for reasons other than interventions, for example due to flight-to-safety by banks, and they are therefore only an imperfect proxy measure for foreign exchange interventions.

This proxy measure of changes in total sight deposits in Swiss francs at the SNB is available at weekly frequency from the SNB, which is at higher frequency than the monthly foreign exchange reserves data, and therefore gives a more time-exact picture of the associated exchange rate levels. Changes in total sight deposits in Swiss francs at the SNB, from the week starting 19 January 2015, i.e. after removal of minimum exchange rate, to mid-July 2016, against weekly averages of the EUR/CHF exchange rate, are shown in Figure 2.

We can see strong evidence of intervention around parity of the euro against the CHF - total sight deposits in Swiss franc at the SNB increased by CHF 26.2 billion in the week starting 19 January 2015. We can also see some evidence of intervention at levels of the EUR/CHF exchange rate between around 1.03 and 1.11, with a possible concentration at a level of around 1.08. This is close to the level of 1.07, which economists expected to be defended by the SNB, as mentioned above (Neue Züricher Zeitung, 19 June 2016). The measure post-Brexit only, i.e. from the week starting 20 June 2016, is marked as filled triangles in Figure 2.

Monthly FX intervention proxies

Another proxy measure for the SNB’s FX intervention is changes in foreign currency reserves. We use data on total foreign currency reserves in convertible foreign currencies. A further proxy measure is changes in foreign currency reserves adjusted for valuation changes.These two proxy measures are shown in Figure 3 against monthly averages of the EUR/CHF exchange rate, and suggest that the SNB may have intervened at levels of the EUR/CHF exchange rate of around 1.04 to 1.10. But foreign

currency reserve changes may occur for reasons unrelated to central bank intervention. Moreover, the valuation adjustment is imperfect. These two proxy measures are therefore only imperfect measures. Furthermore, since these are monthly measures, we cannot relate them to the exact exchange rate levels, but only to monthly averages.

Despite the drawbacks of the three proxy measures, all provide some indication of possible foreign exchange intervention by the SNB for levels of the EUR/CHF exchange rate between around 1.04 and 1.10, which we use to motivate the calibration of our informal edge target zone model presented in the next section. However, the evidence from these three proxy measures is also consistent with the SNB having tried to prevent a CHF appreciation to varying degrees at all exchange rate levels observed since 2015, rather than having a clear intervention threshold to enforce a particular lower bound.

**4. An Informal Single-Edged Target Zone Model**

The model presented in this Section applies to a situation where there is a formal one-sided target zone which is only partially credible. It also applies to a situation where there are market perceptions of an

informal, not publicly announced, one-sided target zone, as may have been the case following the discontinuation of the SNB’s currency floor in January 2015, as discussed in Section 3.

Our model can be regarded as a generalisation of Krugman's (1991) simple target zone model of exchange rates. In the standard model, the peg and the hard edge boundaries are publicly announced and credible, and market arbitrage mechanisms and interventions take place in whatever amounts are necessary to prevent the band from being violated. A number of papers have modified and relaxed the assumption of perfectly known bands. Klein (1992) and Chen et al. (2013) have presented models in which the width of the band is unknown to the public. Thus, the dynamics of the exchange rate is not only driven by fundamentals but also by expectations with respect to the width of the band. In this context, interventions reveal the true edge of the band. This mechanism stresses the information role of interventions.

In contrast, our model develops the notion that the exchange rate band can be seen as a partially credible commitment device, within a new theoretical framework with an asymmetric informal onesided target zone. Moreover, in the absence of an announced target zone, a central bank’s policies, including foreign exchange interventions, may generate perceptions in the market that there is an informal one-sided target zone. The contribution of the paper comprises theoretical exchange rate modelling. We do not claim to have developed a way to model the day-to-day movements of the EUR/CHF exchange rate since 15 January 2015.

Standard Krugman model

It is natural to consider the basic Krugman (1991) framework with a perfectly known and credible strong band as a benchmark. The model is based on the flexible-price monetary model and thus purchasing power parity is assumed to hold continuously. We do this, despite the fact that predictions derived from the standard Krugman model have not stood up well in empirical tests (see, for example, Svensson (1992)). We feel that this choice is justified, because our main focus in this paper is on theoretical modelling issues. Moreover, Lera and Sornette (2016) find that during the period when the SNB enforced the minimum exchange rate of 1.20 Swiss francs per euro, the EUR/CHF exchange rate was well explained by Krugman's target zone model, but seemingly only under strong pressure pushing the exchange rate close to the boundary of the target zone.

In the standard target zone framework, the dynamics of the exchange rate results from the intervention obligations. This gives rise to a reflected or regulated Brownian motion and the exchange rate exhibits mean reversion towards the band. After 15 January 2015, market participants are likely to have formed expectations of the informal band based on the actual exchange rate and suspected interventions. Thus, the log of the exchange rate *S* can be written as the sum of the log of the fundamentals *F* and its expected change:

where £■[•] denotes the expectations operator for exchange rate changes with *A* >0, lnS = s and lnF = /. Following the usual assumption of the target zone framework, we assume that the log of the fundamentals follows a simple Brownian motion without a drift,

where *o* represents the constant risk parameter of the log of the fundamentals and *W _{t}* denotes the standard Wiener process. Applying Ito’s lemma to the expectations term yields.

it is obvious for a one-sided target zone with a credible lower band that the analytical solutions for the steady-state observations for financial market participants are as follows,

where *A *>0 is a positive parameter and *r* = + *lj^.* The meaning of equation (4) is straightforward.

Without central bank interventions, financial market participants would anticipate a one-to-one relationship between the exchange rate and the fundamentals. In other words, *s = f* shows the unregulated exchange rate dynamics. The exponential term *Ae~ ^{r}?* causes the bending of the exchange rate function and thus generates the target zone nonlinearity. We need to determine the parameter

The term smooth pasting denotes the phenomenon that the path of the exchange rate smoothes out on its way to *s ^{l}* and that its slope becomes zero when it actually hits

which gives us the constant *A* as:

The associated relationship for the fundamental value f^{l} and lower band s^{l} thus is

Equations (5) and (6) imply that the intervention-value parameter *A* is a function of f^{l} and hence a function of *s*^{l}: Rearranging and solving for s finally gives us the reduced form relationship between the exchange rate and the fundamentals *f* driven by the lower band *s*^{l}:

The linear part, *f,* represents the solution for a free float. The nonlinear part, —-— *e *^{rf}, is known as the honeymoon effect and refers to the phenomenon that if the Swiss franc is close to the informal floor, the probability increases that the Swiss franc will hit the floor, which leads to perceived interventions by the central bank. As a consequence, the probability that the Swiss franc will depreciate is higher than the probability that it will appreciate further. From this it follows that the Swiss franc will appreciate less than given by the fundamentals *f* alone. In other words, the honeymoon effect stabilises the exchange rate relative to its fundamentals. Intuitively, equation (9) is characterized by the following features:

- For very small values of
*s*^{l}, such as the limiting case*s*co, the interventions go to zero as^{l}^ —^ 0. Thus, the exchange rate becomes almost free floating._{e}^{rsl}~^{1} - For a very small adjustment parameter or very small risk parameter in equations (2) and (3), i.e. 2^0 and/or
*a ^*0, we have*r*^o. This means that interventions are ineffective for 2^0 and unnecessary for*a ^*0. - For
*f*the intervention term disappears and the exchange rate is free floating as only a lower bound*s*exists.^{l}

Normal distribution for s^{l}

In the original Krugman (1991) model, the reduced form relationship between exchange rates and fundamentals is driven by the perfectly known and credible floor. Conversely, what happens when the relationship is driven not by the floor itself but by expectations regarding the band? How can we model perceived uncertainty of market participants as to *s ^{l}* and the central bank’s determination to defend the informal floor?

To keep the model tractable, and consistent with the empirical evidence in Section 3, we assume that the interventions have a normal distribution for *s*^{l}, and equation (8), which shows a linear relationship between *s ^{l}* and

Market participants can only observe the interventions with a mean value s^{le} for exchange rates and observations have a normal distribution around s^{l}. It is obvious that f[s^{(}] = s^{le} and E[f^{l}]* = *f^{le}. With the above distribution for *s*^{l}, we can analyse the response from financial market participants by taking expectations of value-matching condition equation (5) conditional upon the observed normal distributions to determine how the uncertainty about the SNB’s preferences and thus the informal band feeds back into the dynamics of the exchange rate. The exponential term e rr of equation (5) has the properties

The corresponding smooth-pasting condition around the observed f^{le} is denoted by

Substituting equation (13) back into equation (4) we finally get the law of motion of the exchange rate in the informal band system

Compared with equation (9), the impact of interventions is reduced by the factor *e~A ^{r }*

Several properties of the solution are apparent from Figure 4. First, as expected, the resulting dynamics of the exchange rate is ./-shaped even in the absence of a formal lower floor. Second, an important difference between the solution for a fully credible formal lower floor and a perceived informal lower floor is that a unique exchange rate is defined as a function of the fundamentals for the traditional (credible) one-sided target zone, while a family of such curves is defined for the perceived informal one-sided target zone. Third, for perceived uncertainty about the informal lower floor (ct_{s} > 0), the resulting /-shaped curve is a monotonically increasing function of the level of *o*_{s}. Put the other way round, the moderating effect is the stronger, the greater the credibility of the policymaker and the lower the uncertainty about the lower band. Fourth, while all curves are almost identical near s = ln(1.07), the informal band curves become steeper than the hard band curve as the exchange rate approaches the edges. Hence, informal soft edge bands lead to weaker honeymoon effects, wider target zone ranges and higher exchange rate volatility.

Since the honeymoon effects become weaker as uncertainty about the informal lower band increases, as shown in Figure 4, a policy conclusion from the numerical simulations is that central bank communication about exchange rate policy would help to anchor exchange rate expectations and reduce exchange rate volatility. A central bank’s exchange rate policy intentions regarding a lower band could be stated clearly to reduce misinterpretation. Furthermore, misreading of a policy could be corrected in a timely manner. In line with this conclusion, Archer (2005) argues that transparent FX intervention is preferable to secret intervention (see Section 3).

However, transparency about foreign exchange intervention and exchange rate policy may also have adverse effects, which would need to be weighed against the potential benefits of transparency about exchange rate policy and intervention implied by our model, including due to the effects of possible speculative attacks, which are not modelled here (see the discussion in Section 3). Moreover, uncertainty and misperceptions about fundamentals, which are not modelled here, could affect the desirability of transparency.

Markov switching framework

Next, we allow for exchange rate dynamics with discrete regime shifts. The pivotal requirement for the honeymoon effect to function properly is that the public believes that the informal target zone will persist. So far a point estimate for s augmented with a normal distribution has been assumed. Given that Markov-switching models have become a popular tool to allow for parameter instability and parameter uncertainty, we go beyond the previous exercises and pursue the analysis further by augmenting the framework with a bivariate Markov-switching model.In other words, we characterize financial market’s uncertainty about the informal floor via the possibility of regime changes. The main advantages of regime-switching models are that they (i) provide a high goodness of fit in face of policy switches; (ii) can accommodate a wide variety of macroeconomic shocks and nonlinearities; (iii) can visualize the impact of unobservable variables such as credibility of announcements or anticipation of forthcoming events; and (iv) can take the prospect of policy shifts and exchange rate dynamics alternating between regimes changes into account.

How do we introduce such dynamics into the informal target zone model?To fix ideas, we assume that the exchange rate dynamics is subject to discrete regime shifts governed by a two-regime Markov process with constant transition probabilities. Formally, 0 is the probability of state 1 jumping to state 0, and 1 — 0 is the probability of state 1 remaining in the same state. Similarly, *9* denotes the probability of state 0 jumping to state 1, and *1 — 9* is for state 0 remaining in the same state. Thus,

the probability of shifting from state 0 to state 1 in an interval of time of length *dt* is 6dt, and the probability of switching from state 1 to state 0 is 0dt. Note that the jump probabilities need not be identical because the central bank may have asymmetric preferences. A perfectly credible lower bound is obtained by setting the switching probabilities to zero. Regime switches might occur in an attack- type crisis when the Swiss franc suddenly comes under pressure or appreciates strongly in ways inconsistent with the present policy regime. Therefore it is reasonable to assume s/, > in the recurrent two-state process. In other words, we assume that the exchange rate triggering interventions in state 0 is higher than in state 1. This assumption is made largely for convenience.

In the following we analyse some of the channels through which cross-regime effects come about and assess their policy relevance. In the following subsection we assume that market participants know the current state of the exchange rate regime and are able to conjecture the switching probabilities. Under these two assumptions, the two regimes can be characterised as follows:

where sl0 (s{e) is the expected lower band for state 0 (1). The above equations are obtained by taking the expected value of the lower bands for state 0 and state 1, respectively. The next task is to solve for the Markov-switching interventions. The state-dependent law of motion of the exchange rate in the Markov switching version of the model is given by

where o0 and o1 are the risk parameters for the fundamentals in state 0 and state 1, respectively. What is the implied momentum of this assumption? The existence of two states introduces a complication, in that the exchange rate will jump to the new edge of the informal band as appropriate

by means of an instantaneous discrete intervention. There are two cases to be distinguished. When a policy change from state 0 to state 1 occurs, and the actual exchange rate at this juncture satisfies the condition s_{0} > , then no interventions will take place. The reason is that the central bank tolerates a further appreciation of the exchange rate beyond Sq up to . On the contrary, in the case of a reverse jump from state 1 to state 0, an actual exchange rate in the range Sq > s-l > will trigger an immediate intervention to sustain the new regime 0. This implies that over the range Sq > s-l >,we need to change equation (18) as follows so as to satisfy the boundary conditions:

In the next step, we modify the standard method for solving for the exchange rate in terms of the fundamentals, to allow for there being two states of the world. There are two cases to be distinguished. In the first case the new regime satisfies the range Sq^{6} > s-l > s{^{e}.

In the second case, > *s ^{l}_{Q}^{e}* and s

where *r _{0}* —

The Markov-switching version of the model is characterised by the fundamentals /0ie and fle perceived by market participants and the associated exchange rates sl0e and s{e. The expected values of the lower bands, sl0e and s{e, then reflect the corresponding values of /Je and f{e. The regime- dependent value-matching and smooth-pasting conditions at the Swiss franc’s floor are given by:

Numerical examples

In consideration of equations (23) and (24), we can numerically determine the fundamentals /_{0} and *f-jf* at the perceived regime-dependent intervention bands. After having obtained the values of *A _{00}, A_{01},* and

In Figure 5 we show numerical simulation results of how exchange rate dynamics can be characterized in models in which financial markets are aware of occasional changes in the policy regime. Figure 5 shows representations of the ./-shaped dynamics of the exchange rate for different values of Sq and for given jump probabilities 0 — 9 — 0.2 on a case-by-case basis. Thus, the discontinuity is represented by a shift in the perceived mean. In the upper left (right) graph a policy regime change from Sq — ln1.07to — ln1.02(s{ — ln1.00) is simulated. In the lower left graph Sq — ln1.20and — ln 1.07 are simulated. In other words, this panel presents the case of a perceived switch back to the regime prior to 15 January 2015. Finally, the lower right panel gives the dynamics of the exchange rate for the two regimes Sq — ln 1.02 and — ln 1.00.

Four interesting results emerge. First, the exchange rate dynamics alternates between regimes (see the two J-curves in each of the panels). Second, there is mutual interdependence of the /-curves across regimes. The mere fact that a regime change might happen has effects even when no regime change has taken place. Third, despite this fact it is notable that the lower floor remains a stabilizing instrument even under regime-switching expectations, as evidenced by the /-shaped dynamics in each of the panels in Figure 5. Put differently, even in the presence of Markov-switching and boundaries that are not publicly announced, the exchange rate is stabilized relative to the case of a free float. Fourth, the story is more subtle than has been acknowledged so far. Figure 5 indicates that in the Markov- switching version of the model the exchange rate will jump to a new /-curve as soon as the perceived lower floor has changed. Put differently, rightly or wrongly expected changes in the central bank’s exchange rate policy cause shifts in the /-curves and may lead to exchange rate oscillations. This additional source of volatility cannot be detected if uncertainty is computed conditioning on a sole regime. Therefore, it is important to take into account the possibility of regime changes.

Next, we also present numerical simulation results which show that the contribution of the different regimes to the overall volatility does not depend only on the size of the policy shocks, but also on the shocks’ frequency and persistence. In Figure 6 we relax the assumption of given jump probabilities 0 — *9* — 0.2. To repeat, *9* denotes the probability of state 0 jumping to state 1 and 0 is the probability of state 1 jumping to state 0. By contrast, Sq — ln 1.07 and — ln 1.02 is assumed throughout. Under these assumptions, the four graphs summarize the model simulation results for different jump probabilities 0 and *9*. The comparison indicates how different jump probabilities change the dynamics of the exchange rate.

Again three interesting results emerge. First, the jump probabilities and thus policy shifts play a big role, as can be seen for example by comparing the /-curves for the jump probabilities of (0.1,0.1) in the top left panel with those for the higher jump probabilities of (0.5,0.5) in the bottom right panel. Second, the comparison of the /-curves in the different panels reveals that a less (more) likely regime change leads to smaller (larger) cross-impacts. As a result, the importance of the regime that is in place at a particular point in time is substantially reduced when taking into account the possibility of future regime changes. Thus, accounting for potential cross-regime effects appears to be an important element of a good policy design. Based on this model with regime changes, effective central bank communication about the target zone bands may be helpful in reducing exchange rate volatility from such cross-regime effects. Third, occurrences which look like speculative attacks on the Swiss franc may be associated with market perceptions of a policy regime switch having taken place. The numerical results indicate that perceived possible regime switches and speculative attacks may put the central bank under strain, even though it has not changed its foreign exchange market policies yet. Effective central bank communication may be helpful in order to anchor exchange rate expectations and reduce exchange rate volatility.

This ability of the univariate Markov-switching model to generate non-trivial connections between the dynamics of the exchange rate in both regimes represents a promising feature and opens the door to study the link between exchange rate uncertainty and real activity in a multivariate setting. This might be useful in future research in light of the attention that uncertainty has recently received in the profession following the seminal contribution of Bloom (2009).

**5. Summary and Conclusions**

This paper develops a new theoretical model with an asymmetric informal (unofficial) one-sided target zone, using Switzerland as an example. The model incorporates the notion that a target zone can be seen as a partially credible commitment device. Analytically, the modelling approach lies at the crossroads of the literature on exchange rate target zones originated by Krugman (1991) and that on intervention in the foreign exchange market. With the aim of parsimony in mind, but also with the aim of ensuring a fair degree of reality, we extend and generalize the standard target zone model by introducing perceived uncertainty about the lower band following the removal of the minimum exchange rate of CHF 1.20 per euro in January 2015.

We find that informal soft edge target zone bands lead to weaker honeymoon effects, wider target zone ranges and higher exchange rate volatility than formal target zone bands. As discussed in Section 3, the honeymoon effect stabilises the exchange rate relative to its fundamentals. It refers to the phenomenon that, if the Swiss franc is close to the informal floor, the probability increases that the Swiss franc will hit the floor, which leads to perceived interventions by the central bank. Consequently, the probability that the Swiss franc will depreciate is higher than the probability that it will appreciate further, and therefore the Swiss franc will appreciate less than is given by the fundamentals alone. Our model suggests that it would be beneficial for exchange rate policy intentions to be stated clearly to reduce misinterpretation, in order to anchor exchange rate expectations and reduce exchange rate volatility. However, transparency about foreign exchange intervention and exchange rate policy may also have adverse effects, which would need to be weighed against the potential benefits of transparency about exchange rate policy and intervention implied by our model. Moreover, there may be uncertainty and misperceptions about fundamentals, which are not modelled here.

We also study how exchange rate dynamics can be characterized in models in which financial markets are aware of occasional changes in the policy regime. We show that expected changes in the central bank’s exchange rate policy may lead to exchange rate oscillations, providing an additional source of exchange rate volatility. To capture this effect, it is important to take into account the possibility of regime changes in exchange rate policy. The ability of the univariate Markov-switching model to generate non-trivial connections between the dynamics of the exchange rate in both regimes represents a promising feature and opens the door to study the link between exchange rate uncertainty and real activity in a multivariate setting.

It is clear that target zone regimes vary from one country to another and within countries over time. Our extension of the original Krugman model is, therefore, only one example of how such a regime might work. Exchange rate economics is an exceptionally exciting subject. We hope that the model presented above can contribute to it.

**References**

Adler, G. and C.E. Tovar (2011) “Foreign Exchange Intervention: A Shield Against Appreciation Winds?”, *IMF Working Paper* WP/11/165, Washington.

Allen, W. and R. Moessner (2013) “Liquidity consequences of the euro area sovereign debt crisis”, *World Economics* 14, 103-126.

Amador, M., Bianchi, J., Bocola, L. and F. Perri (2016), “Reverse speculative attacks”, NBER Working Paper No. 22298.

Archer, D. (2005) “Foreign exchange market intervention: methods and tactics“, in: Bank for International Settlements (2005) *Foreign Exchange Market Intervention in Emerging Markets: Motives, Techniques and Implications, BIS Papers No* 24, Basel.

Auer, R. (2015) “A safe haven: international demand for Swiss francs during the euro area debt crisis”, Swiss National Bank Quarterly Bulletin, 2/2015 (June), 41-53.

Bank for International Settlements (2005) “Foreign Exchange Market Intervention in Emerging Markets: Motives, Techniques, and Implications”, BIS Paper 24.

Bank for International Settlements (2010) “Low interest rates: do the risks outweigh the rewards?”, 80th Annual Report, chapter 3, 36-46.

Bank for International Settlements (2013) “Market volatility and foreign exchange intervention in EMEs: what has changed?”, BIS Paper 73.

Bianchi, F. and L. Melosi (2016) “Modeling the Evolution of Expectations and Uncertainty in General Equilibrium”, *International Economic Review* 57, 717-756.

Blinder, A., Ehrmann, M., Fratzscher, M., De Haan, J., and D. Jansen (2008) “Central Bank Communication and Monetary Policy: A Survey of Theory and Evidence”, */ournal of Economic Literature* 46, 910-945.

Bloom, N. (2009) “The Impact of Uncertainty Shocks”, *Econometrica* 77, 623-685.

Bloomberg (2015) “SNB Sight Deposits Inch Up After Franc Interventions on Greece”, 6 July, available at http://www.bloomberg.com/news/articles/2015-07-06/snb-sight-deposits-inch-up-after- franc-interventions-on-greece (accessed 11 August 2016).

Bloomberg (2016) “SNB Steps Into Currency Market Amid Brexit-Induced Stress”, 24 June, available at http://www.bloomberg.com/news/articles/2016-06-24/snb-steps-into-currency-market-amid- brexit-induced-stress (accessed 2 August 2016).

Boyer, R.S. (1978) “Optimal Foreign Exchange Market Intervention”, */ournal of Political Economy *86, 1045-1055.

Chutasripanich, N. and J. Yetman (2015) “Foreign exchange intervention: strategies and effectiveness”, BIs Working Paper No. 499.

Chen, Y.-F., Funke, M. and N. Glanemann (2013) ”Off-The-Record Target Zones: Theory with an Application to Hong Kong's Currency Board”, *Studies in Nonlinear Dynamics & Econometrics* 17, 373-393.

Cheung, Y.-W. and U.G. Erlandsson (2005) “Exchange Rates and Markov Switching Dynamics”, */ournal of Business and Economic Statistics* 23, 314-320.

Cukierman, A, Kiguel, M.A. and L. Leiderman (1994) “Choosing the Width of Exchange Rate Bands: Credibility versus Flexibility”, *CEPR Discussion Paper No.* 907, London.

Cukierman, A. and A.H. Meltzer (1986) “A Theory of Ambiguity, Credibility and Inflation under Discretion and Asymmetric Information”, *Econometrica* 54, 1099-1128.

Danthine, J.-P. (2016) “The interest rate unbound”, Hutchins Center Working Paper No. 19.

Daude, C. and E. Yeyati (2014) “‘Leaning against the wind’: exchange rate intervention in emerging markets works, VOX, 1 September, available at http://voxeu.org/article/exchange-rate-intervention- emerging-markets-works .

Daude, C., Yeyati, E. and A. Nagengast (2014) “On the effectiveness of exchange rate intervention in emerging markets”, OECD Development Centre Working Paper 324.

De Jong, F. (1994) “A Univariate Analysis of EMS Exchange Rates Using A Target Zone Model”, */ournal of Applied Econometrics* 9, 31 - 45.

Dibeh, G. (2006) “Target Zone Dynamics where the Fundamental Follows a SDE with Periodic Forcing”, *Physica A* 363, 437-445.

Dominguez, K. and J. Frankel (1993) “Does foreign-exchange intervention matter? The portfolioeffect”, *American Economic Review* 83(5), 1356-69.

Domanksi, D., Kohlscheen, E. and R. Moreno (2016) “Foreign exchange market intervention in EMEs: what has changed?”, BIS Quarterly Review, September, 65-79.

Engel, C. (1994) “Can the Markov Switching Model Forecast Exchange Rates?”, */ournal of International Economics* 36, 151-165.

Hamilton, J.D. (1990) “Long Swings in the Dollar: Are They in the Data and Do Markets Know It?, *American Economic Review* 80, 689-713.

Handelsblatt (2015) “SNB musste offenbar massiv intervenieren”, available at http://www.handelszeitung.ch/konjunktur/snb-musste-offenbar-massiv-intervenieren-730123 (accessed on 2 August 2016), 26 January 2015.

Iannizzotto, M. and M. P. Taylor (1999) “The Target Zone Model, Non-Linearity and Mean Reversion: Is the Honeymoon Really Over?”, *The Economic /ournal* 109, C96 - C110.

Jermann, U.J. (2016) “Financial Markets’ Views about the Euro-Swiss Franc Floor”, *NBER Working Paper No.* 21977, Cambridge (Mass.).

Jordan, T. (2016) “Einleitende Bemerkungen von Thomas Jordan“, p. 4, Bern, 16 June, available at http://www.snb.ch/de/mmr/speeches/id/ref 20160616 tjn/source/ref 20160616 tjn.de.pdf (accessed on 12 August 2016).

Klein, M. (1992) “Big Effects of Small Interventions: The Informational Role of Intervention in Exchange Rate Policy”, *European Economic Review* 36, 915-924.

Klein, M. and K. Lewis (1993) “Learning About Intervention Target Zones”, */ournal of International Economics* 35, 275-295.

Krugman, P. (1991) “Target Zones and Exchange Rate Dynamics”, *Quarterly Journal of Economics *106, 669-682.

Lera, S. and D. Sornette (2016) “Quantitative modelling of the EUR/CHF exchange rate during the target zone regime of September 2011 to January 2015”, *Journal of International Money and Finance, *63, 28-47.

Miller, M. and A. Sutherland (1991) “Britain’s Return to Gold and Entry into the EMS: Joining Conditions and Credibility”, in: Krugman, P. and M. Miller (eds.) *Exchange Rate Targets and Currency Bands,* Cambridge (Cambridge University Press), 82-106.

Moessner, R. and W. Allen (2013) “Central bank swap line effectiveness during the euro area sovereign debt crisis”, *Journal of International Money and Finance* 35, 167-178.

Moessner, R., Jansen, D. and J. de Haan (2016) “Communication about Future Policy Rates in Theory and Practice: A Survey”, *Journal of Economic Surveys,* forthcoming.

Neely, C (2001) “The Practice of Central Bank Intervention: Looking under the Hood”, *Federal Reserve Bank of St. Louis Review,* May/June, 1-10.

Neue Züricher Zeitung (2016) “Brexit: Gravierende Folgen fuer die Schweiz befürchtet“, 19 June2016, p. 1.

Obstfeld, M. (1983) “Exchange Rates, Inflation, and the Sterilization Problem: Germany, 1975-1981”,

European Economic Review 21, 161-189.

Ranaldo, A. and E. Rossi (2010) “The reaction of asset markets to Swiss National Bank communication”, Journal of International Money and Finance 29(3), 486-503.

Reeves, S.F. (1997) “Exchange Rate Management when Sterilized Interventions Represent Signals of Monetary Policy”, *International Review of economics and Finance* 6, 339-316.

Reuters (2016) “UPDATE 1-SNB head confirms weekend intervention to weaken franc”, 29 June, available at http://www.reuters.com/article/swiss-snb-idUSL5N0ZF18F20150629 (accessed 2 August2016).

Sarno, L. and M. Taylor (2001) “Official Intervention in the Foreign Exchange Market: Is It Effective and, If So, How Does It Work?”, *Journal of Economic Literature* 39, 839-868.

Serrat, A. (2000) “Exchange Rate Dynamics in a Multilateral Target Zone”, *Review of Economic Studies* 67, 193-211.

Streit, D. (2016) “Firm-level effects of asymmetric intervention in foreign exchange markets: Evidence from the Swiss currency floor”, *Journal of International Money and Finance* 60, 289-312.

Svensson, L.E.O. (1992) “An Interpretation of Recent Research on Exchange Rate Target Zones”, *Journal of Economic Perspectives* 6, 119-144.

Swiss National Bank (2015), “Swiss National Bank discontinues minimum exchange rate and lowers interest rate to -0.75%”, press release, 15 January, available at http://www.snb.ch/en/mmr/reference/pre 20150115/source/pre 20150115.en.pdf .

Swiss National Bank (2016), Monetary policy assessment of 15 December 2016, Press release, 15 December, see https://www.snb.ch/en/mmr/reference/pre 20161215/source/pre 20161215.en.pdf (accessed on 22 May 2017).

Taylor, M. P. and M. Iannizzotto (2001) “On the Mean-Reverting Properties of Target Zone Exchange Rates: A Cautionary Note”, *Economics Letters* 71, 117 - 129.

The Economist (2014), “Setting monetary policy by popular vote: Full of holes”, 27 November, available at http://www.economist.com/news/finance-and-economics/21635014-plan-boost- switzerlands-gold-reserves-has-no-benefits-and-many-costs-full (accessed on 22 May 2017).

Tristani, O. (1994) “Variable Probability of Realignment in a Target Zone”, *Scandianvian /ournal of Economics* 96, 1-14.

Werner, A.M. (1995) “Exchange Rate Target Zones, Realignments and the Interest Rate Differential: Theory and Evidence”, */ournal of International Economics* 39, 353-367.

**Appendix A: Derivation of Equations (20) and (21) for sff > «]_ > s^{l}f and s_{0} > ^{s}o^{e}**

We need to solve equations (17) and (19) which are coupled with parameters from both states. It is obvious that the solutions to equation (19) in the text can be obtained as follows: