This paper provides a modelling framework for evaluating the exchange rate dynamics of a target zone regime with undisclosed bands. We generalize the literature to allow for asymmetric one-sided regimes. Market participants’ beliefs concerning an undisclosed band change as they learn more about central bank intervention policy. We apply the model to Hong Kong’s one-sided currency board mechanism. In autumn 2003, the Hong Kong dollar appreciated from close to 7.80 per US dollar to 7.70, as investors feared that the currency board would be abandoned. In the wake of this appreciation, the monetary authorities finally revamped the regime as a symmetric two-sided system with a narrow exchange rate band.
An in-depth discussion of Hong Kong’s currency board including technical details is available at http://www.hkma.gov.hk/eng/publications-and-research/reference-materials/monetary/review-of-currency-board-arrangements.shtml. For a perceptive and thorough discussion, see Latter (2007).
In the empirical exchange rate literature, the performance of currency board systems has been discussed ad nauseam. For example, Ghosh, Gulde and Wolf (2000) have found that currency boards exhibit better inflation performance than soft pegs, mostly due to a credibility effect. Oliva, Rivera-Batiz and Sy (2001) have developed a simple model of a currency board and its credibility allowing a comparison between a currency board and other exchange rate regimes in terms of the inflation – unemployment trade-off and credibility. Regarding growth, currency boards also do better than soft pegs. An explanation for this fact is provided by the growing body of macroeconomic evidence suggesting that volatility is detrimental for economic growth, especially when financial opportunities are limited. See, for example, Aghion et al. (2006) and Aghion and Howitt (2009), pp. 329–339.
The result of this surprise move was that interbank interest rates jumped and the overnight rate hit 280%. This successfully stemmed the speculative outflow of US dollars. Overnight rates dropped back to about 5% within a few days.
As a historical note, while no formal strong-side intervention point was introduced, the Subcommittee on Currency Board Operations already considered the options in this area in meetings in October 1999 and July 2000 and “agreed that there would be scope to review the arrangement again, should the need arise” [Hong Kong Monetary Authority, Research Department (2000)].
For an analysis of the strong-side pressure on the HKD and particularly the HKMA’s response, see IMF (2005). This paper also offers a simple second-generation currency crisis framework for modelling trade-offs faced by the HKMA.
Rodríuez-Lóez and Mendizabal (2006) have presented a model in which the width of the band, the credibility of the target zone regime, and the volatility of the exchange rate is made explicit. Balancing risks and benefits of fluctuating exchange rates leads to the conclusion that self-declared free-floaters find it optimal to target a narrower implicit (unofficial) band inside an officially announced broad band. This modelling result is consistent with ample indications for the presence of such narrower implicit bands.
An implicit strong-side band is likely because history has shown that Hong Kong’s policymakers place a heavy emphasis on exchange rate stability (vis-à-vis the USD). The HKMA views the currency board system as an important barometer of Hong Kong’s economic and political conditions. The associated strong external position, characterised by sizable foreign exchange reserves, provides a buffer against short-term external shocks. However, the long-term sustainability of the currency board system will depend crucially on prudent fiscal policies, and high flexibility of goods and factor markets.
For a thorough review of the theoretical and empirical target zone literature see Duarte, Andrade and Duarte (2013).
Unfortunately, Klein’s (1992) modelling approach does not lend itself naturally to the asymmetric one-sided band exchange rate regime case. It explains why we depart from Klein’s (1992) approach on technical grounds. Alternative one-sided target zone modelling frameworks without learning from interventions are available in Krugman and Rotemberg (1990) and Veestraeten (2001). With different methodology, Klein and Lewis (1993) have presented a symmetric target zone model with probability-of-intervention learning and applied the framework to the European Exchange Rate Mechanism (ERM).
A thorough description of the approach is provided by Sarno and Taylor (2003), pp. 177–184.
The interested reader is referred to Appendix A.1 for the derivation. Note that the closed-form solution in Klein (1992) is more appealing because it is derived for the symmetric case where the integrals are characteristic.
In dynamic economic models backward-looking expectations with systematic forecasting errors are inconsistent with rational behaviour. In nonlinear dynamic models, exhibiting seemingly unpredictable breaks due to the sporadic nature of the interventions, however, simple “rule of thumb” backward-looking expectation rules may yield non-systematic forecasting errors. Furthermore, numerous survey studies, such as Cheung and Chinn (2001) and Menkhoff (1998), uniformly confirm that speculators in foreign exchange markets generally do not rely on mathematically well-defined econometric or economic models, but instead follow simple backward-looking trading rules.
In Klein’s (1992) model, the first intervention is such a landmark decision for the future. This implies that after the first intervention the model with undisclosed band width collapses to the standard Krugman (1991) model with full faith in the target zone.
Whetheris determined to belong to the upper or lower interval, which influences the conditions in the conditional probability functions, is negligible for the exchange rate movements, as only a null set is integrated.
Convex functions are typically used in macroeconomic models with adjustment costs to penalise swift changes in variables and thereby to induce gradual movements over time. Among the many models with convex adjustment costs, quadratic functions have been by far the most common specification, essentially for tractability reasons. Without loss of generality and for mathematical convenience, we also assume a quadratic specification.
In our framework, we approximate the relevant considerations with the simplest functional forms to keep the model tractable and the conclusions less susceptible to certain twists in the functions. The derivation of λ is shown in Appendix A.2.
The interested reader might look at the derivation in Appendix A.3. A hypergeometric function can be defined in the form of a convergent hypergeometric series. Many functions can be expressed as special cases of a hypergeometric function (eg exponential, gamma, trigometrical and the Bessel functions).
The HKMA’s foreign currency market interventions are carried out in an open and transparent manner and so are public knowledge. In all cases, the interventions are announced on the day they occur. Agents can therefore distinguish between movements in f due to interventions and fluctuations due to equation (3).
Technically expressed, the coefficients ai in equation (32) show the importance of the last interventions. However, expectations are also influenced by V1 (compare the effects of different V1 by means of Figure 3). Therefore, there is also room for expectations, which are not primarily anchored by past interventions.
The 1-year forward rate of the HKD was consistently outside the convertibility zone between October 2005 and the start of 2007. This is known as the 100% credibility test developed by Svensson (1992, 1994) and indicates that financial market participants have initially revealed skepticism about the ability of the new strong side CU to limit exchange rate fluctuations. Intermittent upward pressure on the HKD occurred again in autumn 2007 when HKMA interventions again were aimed at anchoring market expectations.
The HKMA would not be the first central bank to do this. For example, the ERM currencies were normally allowed to fluctuate no more than 2.25% above or below their fixed bilateral rates. The Netherlands and Austria had narrow bands of ±0.6%, while Portugal and Spain had wider bands of ±6%. Italy and the UK were forced to leave the ERM on 17 September 1992 when both currencies came under speculative pressure. Fluctuation bands were then widened to ±15% in 1993 to avoid defending the indefensible.
Interventions refer to net injections or withdrawals of funds by the HKMA in the interbank money market. For the daily market operations data, see http://www.info.gov.hk/hkma/eng/statistics/msb/index.htm.
Interventions were necessary because markets believed that the HKD would appreciate alongside the RMB made the automatic adjustment mechanisms of the currency board system ineffective. For an econometric logit analysis of monetary operations conducted by the HKMA, using daily data for the one-sided regime between September 1998 and December 2001, see Gerlach (2005). For an analysis of intra-marginal interventions, also see Svensson (1992).
In the theoretical modelling framework, V1 is assumed to be exogenous, neglecting central bank incentives to influence expectations with announcements. Rational central banks choose “verbal intervention” as a toolkit since it has the ability to enhance the predictability of monetary policy decisions and potentially to help achieve central banks’ macroeconomic objectives. On the other hand, when optimal policies are dynamically inconsistent, announcements will only be considered cheap talk. For a survey of this partially credible commitment device, see Blinder et al. (2008).
Further points after the next interventions yielded qualitatively similar, although quantitatively different, results. For brevity of exposition, only the exchange rate dynamics for SII and SIII is presented here. Interested readers may obtain further calibrations of the model dynamics from the authors upon request.
Please note that the upper band of the target zone has not been tested to the limit during the sample period.
See, for example, Chen, Funke, and Glanemann (2011).
A.1 Derivation of equation (19)
At first we solve the expectation values E(A1(Vl)) and E(A2(Vl)). Manipulations of the integrand provide the primitive
Now we derive the closed form expression (19) using both expectation values
A.2 Derivation of λ
An easy way to choose λ properly is to derive it from condition (25), which claims that
A.3 Derivation of equation (29)
Before we prove equation (29), we provide a short introduction to the hypergeometric function. The hypergeometric function 2F1 is the convergent Gauss hypergeometric series
where the circle of convergence is the unit circle |z|=1 and Γ(‧) denotes the gamma function.
The relationship between the factorial and the gamma function is defined as Γ(n+1)=n! for all n∈ℕ. The functional equation of the gamma function is xΓ(x)=Γ(x+1) for all x∈ℝ+.
An important property of hypergeometric functions is that the six functions 2F1[a±1, b; c; z], 2F1[a, b±1; c; z] and 2F1[a, b; c±1; z] are contiguous to 2F1[a, b;c;z]. They are used to express one of them as a linear combination of any two of the other contiguous functions and are derived by Gauss. The two relations that are applied in the following are
Another important property is
Property 3:2F1[a, b; b; z]=(1−z)–a
A useful overview of the linear combinations and other interesting relations is given in Abramowitz and Stegun (1972).
Being equipped with this short introduction to hypergeometric functions 2F1, we turn to the calculation of the expectation E(A1(Vl)).
In order to prove the third equality, we show that the derivative of
The third equality holds because the first summand in the line above is zero. For the fourth equality we use the functional equation of the gamma function, which results in a hypergeometric function with new parameters.
According to the order of the equal signs the properties 1–3 are applied:
Now, we have obtained all the ingredients for differentiating the above mentioned primitive.
The other expectation is given without proof of the primitive, as it is derived in a like manner.
Hence the closed form solution results in
A.4 Calculation of the coefficient aA
Suppose that market participants assign the same weight to the last N=8 interventions in their expectations formation process. In the case where SIV=ln(7.77) and SV=ln(7.754), these market operation dates and the corresponding HKD spot exchange rates are:
As a start, this enables us to calculate the logarithmised exchange rates
This equation is derived from equation (33). However, the question arises as to why the left hand side is equal to SIV. One has to take into consideration that whenever a smaller fundamental f is observed, the interval of possible intervention triggering exchange rates is truncated. Where does that leave us? In case of SIV, the original interval
For the simulation of the exchange rate dynamics in SVI=ln(7.76) three further interventions are to be included in the computation:
The corresponding value of aA thus evolves as aA≈0.546.
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