Volume 46 Number 181,

April-June 2015

April-June 2015

The Limits of Exchange Rate Stability in Mexico

Date received: July 2, 2014. Date accepted: December 10, 2014

Abstract

This text analyzes conditions for stability in a dynamic simulation model of Banxico’s sterilized interventions in the time period 1976-2013. The analysis reveals that two of the most striking vulnerabilities of this strategy were: *a)* the potential for the external reference interest rate for Mexico to rise above the domestic rate and *b)* the risk that oil revenue in the country would fall. In light of this change, Banxico is faced with the policy dilemma of choosing between tightening monetary restrictions to increase the domestic interest rate above the external rate or doing the opposite, because both options could counterbalance these vulnerabilities. The former would come at the cost of shifting the burden of the losses from the adjustment to non-financial sectors in the country, while the second would distribute these losses progressively.

Keywords: Exchange rate stability, exchange rate policy, dynamic simulation model, interest rate, exchange rate.

** BACKGROUND**

Since 1996, the government has implemented an explicit intervention policy for the accumulation and decumulation of foreign currency in the International Reserve to regulate the exchange rate (Banxico, 1996) through temporary mechanisms to buy and sell dollars (Banxico, 2008 and 2014a). Moreover, since 2009, the government has exercised the monetary policy of inflation targeting (Banxico, 2014b), whereby it sets an operating target for the daily interbank interest rate, in which the most important instrument has been the buying and selling of public debt bonds (Banxico, 2007). The government, however, has not publicly recognized that it is managing the two strategies in a coordinated fashion, despite having the necessary legal authority (Banxico, 2014c, Art. 23), likely because in doing so, it has been able to avoid adhering to any exchange rate target, in keeping with recommendations of the International Monetary Fund (IMF) on the subject (Capraro and Perrotini, 2011, as well as Calvo, 1991).

The hypothesis of this paper is that, despite the reticence of the IMF and even Banxico itself, in practice, monetary and exchange rate policies in Mexico are now combined and constitute a sterilized interventions strategy, which has been in effect since the fixed exchange rate regime was abandoned and therefore was in place even before the advent of the foreign currency purchase and sale mechanism and anti-inflation policy based on the inflation-targeting model. More and more studies that analyze this model have documented the link between the aforementioned policies when this model is applied in open economies (Ball, 1999; Svensson, 2000; Taylor, 1995 and 1999), corroborating the validity of the hypothesis proposed, at least during the period in which this model has been in force in Mexico. An additional argument in support of this supposition is the epidemic case of the fear of floating (Calvo and Reinhart, 2002), caused, among other factors, by the “dollarization of external liabilities (or ‘original sin’)” (Mantey, 2013a).

Sterilized interventions are also evident in the data: the increase of 1.9 trillion pesos recorded in the foreign exchange reserve between 1976 and 2013 and the issuance of 1.5 trillion in public debt instruments by Banco de México, in excess of the debt used to finance the broader economic deficit and monetary issuance, which imply that this institution hoarded in its coffers, principally as liabilities for monetary regulation, practically all of the current liabilities generated by exchange rate interventions.

Although the net accumulation of foreign currency implied by the above figures tends to be considered as the aspect that best explains the exchange rate stability that has prevailed in Mexico in recent years, there is unquestionably a relationship between interventions and the evolution of the exchange rate, as even authors prone to the inflation-targeting strategy have noted the increase in exchange rate volatility that can be attributed to this anti-inflation policy, in spite of interventions (Svensson, 2000). Other authors go even further, doubting that these interventions have any influence at all on levels and the volatility of exchange rates (Disyatat and Galati, 2005; Gimaraes and Karacadag, 2004). The empirical evidence for Mexico, however, is not far removed from this finding if we consider that the country has experienced some of the most severe inflation, exchange rate and capital flight crises, even when there was a sterilized intervention strategy in place.

Nor can there be any doubt as to the anti-inflationary effect of monetary restriction implied in sterilized interventions. Even authors inclined towards the inflation-targeting policy question its effectiveness when not correctly calibrated (Ball, 1999; Ball and Mankiw, 2002; Svensson, 1997 and 1999; Svensson and Woodford, 2005; Taylor, 1979 and 1980) and, when applied in open economies, when it does not incorporate some rule for the exchange rate as an instrument (Ball, 1999; Svensson, 1998). In turn, the data reveals that the government of Mexico has not abandoned direct control of leading prices, despite having implemented the inflation-targeting strategy, as is the case for wages, gasoline, electricity and urban transportation, as well as the exchange rate itself.

Although the model described in this document does not deny that sterilized interventions influence the evolution of the exchange rate and inflation, it does assume that precautionary or stabilizing measures have not been the only, much less principal, motivation. The hypothesis of this model is that the purpose of these measures was to satisfy the requirements demanded of the country by private deregulated investors and international financial bodies that have provided essential emergency funds. By acting to maximize their gains and guarantee that the latter are paid, these entities have made the requirement that the Mexican macroeconomic strategy adopt this same objective as its main motivation a condition for financing, and because the amount and capacity to pay these gains are defined by the results of the sterilized intervention strategy, this strategy has become one of the mechanisms that most contributes to this objective of maximization and, in that sense, to entrenching financialization in the country.

In practice, these requirements have implied the imposition of a variety of specific results on Mexico’s economic policy, such as sustaining balances at all costs and controlling inflation. In the realm of capital exchange abroad, achieving the former of these two outcomes has meant the country has had to sustain equivalence between the costs and benefits of foreign currency accumulation, and to find this balance, in light of the ease with which foreign capital enters and exits the country and the unilateral creation or destruction of liquidity in reserve currencies carried out by international financial markets (*op. cit.*), Mexico has been obliged to intervene in its exchange rate market. By combining this demand with the requirement to control inflation, the country has also been forced to sterilize the monetary impact of its exchange rate interventions. With these interventions and sterilizations, Banxico has effectively exercised combined control over inflation, the interest rate and the exchange rate, ensuring that their values guarantee the concentration of the majority of the domestic stock in assets of the financial sector and liabilities of the non-financial sector, making it clear that sterilized interventions are a financialization policy.

In the midst of these sterilized intervention strategies that aim to sustain and deepen the process of financialization, the most visible effect of the monetary restriction implied in these sterilizations has been the upward pressure on the interest rate, a feature in countries similar to Mexico that have also adopted the inflation-targeting model as an anti-inflation policy, because in these cases, the central banks have tended to raise domestic interest rates in response to capital flight, and reduce these rates in response to capital inflows, whether they are intervening in the exchange rate market or not (Ostry, Ghosh and Chamon, 2012). Because financial gains are positively indexed to the domestic interest rate, this explains the tendency of sterilized interventions to concentrate domestic assets in favor of the financial sector (Rodríguez, 2014).

Interventions, however, have provided governments with the currencies that they will sooner or later use to pay these gains to investors in public bonds with access to international capital markets, which is evident due to the synchronization that has existed since 1976 between the evolution of outflows of the International Reserve and deficits due to interest and the profits of the factorial services balance (Rodríguez, 2014). The majority of these investors are pension fund managers, and the funds are owned by the major international financial corporations maintained by practically all banks that operate in Mexico. They also include foreign investors not related to pension funds, but in any case associated with foreign corporations. The transfer of this foreign currency from the reserve to corporations, meanwhile, operates through a mechanism (described in Rodríguez, 2014) that allocates the cost of the strategy to the workers, thereby linking two of the antagonists of financialist capitalism.

It is precisely in periods of capital flight when redistribution in favor of the financial sector is made effective and exchange rate volatility emerges or increases (Calvo and Mishkin, 2003). This redistribution to the benefit of the financial sector is also greater when withdrawals deplete the foreign exchange reserves of a country and are accompanied by chaotic spikes in the exchange rate (*op. cit.*). During these periods, the number of interventions has tended to fall, but sterilization efforts expand, increasing the exchange rate and the risk premium for investments in domestic currencies. The reduced number of interventions means that during these time periods, Mexico has experienced greater exchange rate flexibility, although without abandoning the prevailing exchange rate regime, implying that more than reflecting a change in the regime (Céspedes, Chang and Velasco, 2004; Eichengreen and Hausmann, 1999), variations in the valuation of risk reflect, essentially, uncertainty regarding the capacity of the country to repay its debt, whether valuated in foreign currency or not, because the beneficiaries of the same are international investors, which takes us back to the problem of “original sin” (Eichengreen and Hausmann, 2003; Eichengreen, Hausmann and Panizza, 2007).

This means that Mexico can reduce the regressive concentration of its foreign currency stock if it avoids the instability of external variables involved in the sterilized intervention strategy, making it key to establish, in this context, the limits of that instability. With that in mind, this article aims to elucidate these stability limits,^{1} analyze their principal determinants and evaluate the potential consequences of these variables reaching these thresholds. These limits will be defined based on a segmented simulation model of difference equations with the variable coefficients presented in this paper.

With that said, the next section provides theoretical context to the model introduced and solved, respectively, in Sections 2 and 3. Section 4 presents the stability conditions derived from the solution. Finally, by way of conclusion, this article compares the current stability conditions with those that prevailed in the past. This paper has three appendices that can be requested from the author,^{2} one of which addresses mathematical procedures, a second on data and the third containing the full list of bibliographical references.

For space reasons, there is no section proposing policy alternatives to sterilized interventions. In this regard, the debate has focused on the possibility of abandoning this strategy to make the exchange rate more flexible. However, this paper would not propose this policy, because it considers it to be unfeasible and recognizes the perverse impact of exchange rate fluctuations on countries that cannot infinitely increase their availability of foreign currency since they do not issue the reference currency. On the contrary, it assumes and demonstrates that the exchange rate has faced stricter stability limits and therefore has a higher potential for chaotic behavior in time periods in which the country has had to confront foreign currency limitations to moderate its volatility, obliging the nation to implement regimes with greater flexibility. This flexibility, meanwhile, has made the redistribution of reserves more regressive and concentrated assets in the hands of the financial sector. In this environment, full exchange rate flexibility would bring with it even greater risks than the sterilized interventions strategy (Mantey, 2013b). This means that a strategy that reverses the perverse effects of sterilized interventions does not necessarily favor the progressive redistribution of income and reserves and only then support financial gains; nor does it prevent these gains from leaving the country. However, implementing this without exchange rate risks requires the consensus of the bodies charged with managing international reserves. In this context, it is more useful to define targets for sterilization policies, public spending, taxes and trade exchange, rather than the exchange rate regime (Calvo, 2003).

** 1. THEORETICAL FRAMEWORK
**

Lavoie and Seccareccia (2004) demonstrated that double-target strategies, like the sterilized interventions model, can be sustained as long as the domestic currency is pressured to appreciate, the central bank has in its toolbox more than one sterilization instrument and the costs of sterilization are null. Solorza (2013) argues that the first two conditions are true of Mexico, but not the third.

Bofinger and Wollmershäuser (2001), meanwhile, signal that one of the conditions for sustainability is defined by the cost of the sterilization (C_{t}^{s}), which should be null (C_{t}^{s} =0), and has two components: the cost due to interest (C_{t}^{i}) and the loss or return due to the valuation of the monetary reserve (C_{t}^{v}). The first, determined by the difference between the domestic interest rate (i_{t-1}) and external interest rate (i^{*}_{t-1}), and the second by the percentage variation of the exchange rate (s_{t}). By combining both components, we obtain the condition for the policy to be free of charge, which is: C_{t}^{s} =0=i_{t-1} - i^{*}_{t-1} - ( s_{t} - s _{t-1}), which can be expressed as an * ex post * formulation of the condition of interest rate parity, given by: s_{t} - s_{t-1} = i_{t-1} - i^{*}_{t-1}. This parity indicates that the floating administered is viable, as long as the trajectory of the exchange rate follows the interest rate differential.

Frenkel (2007) showed that the above condition is equivalent to the requirement that the cost of sterilization be null or negative, satisfying i_{t} ≤ i^{*}_{t} + e_{t} for e = s_{t} - s_{t-1}. Frenkel, however, asserts that this condition is not complete, as it omits the negative impact of the relationship between the stock of liabilities issued for purposes of sterilization (L_{t}) and the value in the domestic currency of the stock of the foreign exchange reserve (RE_{t}) on the interest rate, better expressed as 1) i_{t} ≤ (i_{t}^{*} + e_{t}) / (L/RE)_{t}. The consequence is that the limit imposed on the interest rate, by the sum (i_{t}^{*} + e_{t}), is not impossible to overcome, because the interest rate can be higher than this sum if the ratio (L / RE)_{t} is less than one. One case that Frenkel analyzes is that of an economy whose government implements a truncated inflation-targeting policy that responds exclusively to the inflationary gap, demonstrating that sterilized interventions are guaranteed to be sustainable in a framework in which the interest rate rises more than the inflationary gap, when(L / RE)_{t} is less than unity.

Unlike the Frenkel model, the system described in Section 2 does not study the implications of the sustainability condition, defined by equation 1, for the interest rate, but rather the impact of that same condition on exchange rate variations. In the Frenkel model, the final impact is defined by isolating from equation 1 the variable e_{t} : e_{t} ≥ i_{t} (L / RE)_{t-1} - i_{t}^{*}. In the model introduced in Section 2, this final condition is studied, but in the * ex post * condition of interest rate parity, defined by *2)* e_{t} ≥ i_{t-1} (L / RE)_{t-1} - i_{t-1}^{*}. Then, equation 2 is re-expressed in terms of a risk premium on the exchange rate φ_{t}, as: e_{t} ≥ i_{t-1} - i_{t-1}^{*} + φ_{t}, which, for Frenkel, takes the form of equation *4)* φ_{t} = i_{t-1} [(L / RE)_{t-1} - 1], which becomes evident because by substituting 4 into 3, we obtain condition 2, and when we isolate from it i_{t-1}, we obtain sustainability condition 1. Equation 2, meanwhile, clarifies that Frenkel’s sustainability condition, which proposes a value less than unity for the ratio (L / RE), is equivalent to the condition that this same strategy involve a non-positive risk premium on the exchange rate, because with (L/RE)_{t-1} ≤ 1, φ_{t} ≤ 0 and, in this case, both 1 and 2 are satisfied. This means that Frenkel’s condition can also be overcome, because if (L / RE)_{t-1} is greater than unity, the sterilized interventions strategy and the inflation-targeting strategy are sustainable for a domestic interest rate that is growing faster than the inflationary gap, if the cost of the policy is transferred to the exchange rate in the form of a positive risk premium, which implies exchange rate volatility.

The model in Section 2 also coincides with the Frenkel model in the aspect of accounting for the sustainability condition. For Frenkel, the sustainability condition emerges from the need to maintain the variation of liabilities (P_{t}) below the variation of assets (RE)_{t} of the government; that is, the need to satisfy the inequality: ∆P_{t} ≤ ∆ (RE)_{t}. The model in Section 2 uses this same condition but interprets the variation of liabilities as the cost of the strategy and asset variation as the benefit of the same. In addition to that, the valuation of the cost and benefit is associated to the specific financing sources available to Banxico, to adjust the simulation exercise of the aforementioned model to the data available for the case studied and to obtain specific stability limits for this case.

The other difference between Frenkel’s work and the model introduced in the next section is related to analysis over time. The Frenkel model defines and analyzes the sustainability limit of the strategy; the Section 2 model, meanwhile, assumes that sustainability is guaranteed in all periods, because the equivalence between asset and liability variation is also guaranteed in all periods. If the former of these variations, identified by Frenkel as ∆P_{t}, is denoted as ∆SD_{t}^{e} and, the latter, identified as ∆SE_{t}, is denoted as ∆SR_{t} , its equivalence can be expressed as: ∆SD_{t}^{e} = ∆SR_{t}, which is assumed to be met for the entire t in this work, as it is interpreted as the equilibrium condition of the model. This artifice allows us to concentrate the analysis on the stability of the strategy over time, by proposing equations for SD_{t}^{e} and SR_{t}. With that, we obtain a system of three equations, sufficient to solve the variations in the reserve (∆SR_{t}), the liabilities (∆SD_{t}^{e}) and the exchange rate (∆τ_{t}). In the case of the exchange rate, the solution is re-expressed in terms of the * ex post * condition of interest rate parity, to discover the risk premium included in its definition, producing expressions for the three variables in the form of a set of difference equations with variable coefficients. The equations that express the time paths of these difference equations, and are solutions to them, define the conditions of temporal stability of these paths, analyzed in Section 4.

The overwhelming implication of this analysis is that Mexico was able to maintain its sterilized interventions policy and, in fact, it was always sustainable for various degrees of exchange rate control, even when interest rates were higher than the sum of the external interest rate and the exchange rate, and did not satisfy the condition that the L / RE ratio be less than unity, because the country accepted charging positive risk premiums on its exchange rate. By generating strong exchange rate volatility, this charge also reveals that the sterilized interventions strategy has not always been a stabilizing policy, which challenges the legitimacy of implementing it from the perspective of distributive fairness and the stated objective of the strategy. It also makes evident the need for the rest of economic policy to compensate for the impact of that cost on the balance sheets of economic agents not benefitted by exchange rate volatility or the increase in the domestic interest rate greater than inflation.

** 2. PROPOSAL OF THE MODEL**

In addition to what was discussed in the above section, the model presented here is based on the hypothesis that the sterilized intervention strategy implemented in Mexico did not always have the same availability of resources to be financed, and the sources of financing changed over time, segmenting the evolution of the external variables involved in this strategy into four major periods. These changes in financing sources, meanwhile, generated the costs and benefits of the strategy, which varied for each of these four time periods, with the market response reflected in the risk premium charged to the exchange rate. These periods were defined based on the empirical data available, which indicates that three of the four main time periods coincide, in turn, with the stages through which this paper assumes that the financialization process in Mexico has passed (proposed in Rodríguez, 2012).

The first of these time periods coincides with the transition stage from the import substitution model to the financialization model (which began in 1976 when Mexico abandoned the fixed exchange rate regime and ended in 1988, when the country experienced its most severe oil crisis). This time is characterized as the period in which Banxico financed the build-up of reserves with the sale of PETROBONOS and valued the costs of this foreign currency as the equivalent of variations in the external debt balance. It should be noted that this is still the indicator that Banxico uses to value the cost of foreign currency because it considers it to be the factor that best reflects the opportunity cost of the foreign currency, as it approximates the yield on its alternative use, as well as the scale and variability of payments abroad, both due to the export deficit as well as to the inflow of capital and a substitute, however imperfect, for the reserve itself (Banxico, 2009). With this source of financing, the benefits of the reserve or variation in assets (ΔSR_{t} / SR_{t-1}) was positively related to yields due to interest (r_{t+1} - r_{t+1}^{e}) and exchange rate yields (∆τ_{t+1} / τ_{t}) generated by oil revenue (∆Y_{t}^{p} / Y_{t-1}^{p}) invested in the very foreign exchange reserve. The costs of these assets (ΔSD^{e} / SD_{t-1}^{e}) were determined by the domestic interest rate (Δr_{t} / r_{t-1}), which was the rate paid for the majority of PETROBONOS issuances (Rodríguez, 2012).

The second period, which coincides with the stage of leveraged buying in the financialization process of Mexico, began in 1989 and ended in 1994, during the second-most severe crisis the country has experienced since 1925. The cause of the change was the collapse of oil income the year prior, as it removed the only source of financing for the foreign exchange reserve in the period before. In response to this scenario, the government had to use practically all of the additional income from the public sector budget to compensate for the drop in oil revenue and, moreover, to renegotiate the country’s external debt and seek limited exchange rate regulations. This income principally came from the privatization of companies that at that point were still state enterprises. With that done, the reserve increased in the same proportion as public oil (∆Y_{t}^{p} / SR_{t-1}) and non-oil (∆Y_{t}^{t} / SR_{t-1}) income grew, net of the cost of the domestic debt generated by the sterilization of the monetary effect of foreign currency accumulation, defined by the interest (r_{t} - r_{t}^{e})(1 - ∆τ_{t} / τ_{t-1}) of the debt in government securities (∆SD_{t}^{iv} / SR_{t-1}). The variation of liabilities (∆SD_{t}^{e} / SD_{t-1}^{e}) was determined by the difference between the external interest rate and the domestic interest rate (r_{t}^{e} - r_{t}).

The third stage, which coincides with the period of consumer debt in the Mexican financialization process, and is still ongoing, began in 1995. The change was due not only to the fact that by 1995, the product of privatizing public companies had been exhausted, but also perhaps to a lesson learned in the crisis of the year before. As such, Banxico added to the reserve, besides the resources from sources in the previous stage, including public oil (∆Y_{t}^{p} / SR_{t-1}) and non-oil (∆Y_{t}^{t} / SR_{t-1}) income, enough foreign currency resources to cover the cost of interest (r_{t}^{e} - r_{t}) and exchange rate costs (∆τ_{t} / τ_{t-1}) derived from its debt in securities (∆SD_{t}^{iv} / SR_{t-1}). This policy of adding a portion of the product of domestic debt to the sources of foreign currency acquisition linked the issuance of domestic public bonds with the accumulation of reserves available to exercise exchange rate management, giving rise to an actual sterilized interventions strategy where the intermediary was the inflation-targeting policy. With respect to liabilities, the government began to value its debt globally (*op. cit.*), managing it to maintain total indebtedness, external (∆SD_{t}^{e} / SD_{t-1}^{e}) plus domestic {[((∆SD_{t}^{i} / SD_{t-1}^{i})(1 + r_{t}^{e}) / (1 - r_{t})]}, identical to the variations in oil income (∆Y_{t}^{p} / Y_{t-1}^{p}) [(1 + r_{t}^{e}) / (1 + r_{t})], because these resources were the principal guarantee of repaying the commitments in foreign currency that the country maintained. In other words, the government managed its total debt to achieve the following equivalence (∆SD_{t}^{e} / SD_{t-1}^{e}) + [(∆SD_{t}^{i} / SD_{t-1}^{i})(1 + r_{t}^{e}) / (1 - r_{t})] = (∆Y_{t}^{p} / Y_{t-1}^{p}) [(1 + r_{t}^{e}) / (1 + r_{t})]. This equivalence positively linked the cost of liabilities used to accumulate foreign currency (∆Y_{t}^{p} / Y_{t-1}^{p}) [(1 + r_{t}^{e}) / (1 + r_{t})] and negatively linked with the cost of domestic indebtedness [(∆SD_{t}^{i} / SD_{t-1}^{i})(1 + r_{t}^{e}) / (1 - r_{t})].

This strategy for accumulation and financing foreign currency was so successful that Mexico was able to increase its reserve assets to unprecedented levels; even so, the strategy was abandoned in 2002 when Banxico adopted the objective of reducing the rate of that accumulation, perhaps because it was considered excessive, modifying the determinants of the reserve, which responded to a new model that has been in place since 2007. The foreign currency decumulation strategy has implied reducing asset variation (∆SR_{t} / SR_{t-1}) to a level that only backed the exchange rate gains (∆τ_{t} / τ_{t-1}) and gains for domestic interests (r_{t} - r_{t}^{e}) of the public internal debt (∆SD_{t}^{i}), removing other sources of financing from the reserve (∆Y_{t}^{p} + ∆Y_{t}^{t}), an amount equivalent to the internal debt (∆SD_{t}^{i}) not allocated to the payment of interest plus two times the internal debt in securities (∆SD_{t}^{iv}). The valuation of the cost of these assets was not modified.

In 2008, the government began shaping the currency exchange strategy that is still in force. This change was not derived from a lack of resources, but rather the fact that Mexico was able to rely on foreign currency from the IMF that it allowed it to usher in a new period of net reserve accumulation that combined two sources of financing, external and domestic resources. This new approach was reflected in the vigorous growth of foreign currency accumulation of Banxico in 2008, because it received the impact of the initial investment made by the government in that savings account, oriented towards opening and sustaining the market of public bonds valued in foreign currency that helped push the country into foreign markets (Banxico, 2009). In reality, the foreign exchange reserve grew at the same rate as public oil (∆Y_{t}^{p}) and non-oil (∆Y_{t}^{t}) revenue, in addition to the resources that backed the new domestic debt in securities (∆SD_{t}^{iv}) with exchange rate earnings (∆τ_{t} /τ_{t-1}) and earnings on interest (r_{t} -r_{t}^{e}) included, heralding, perhaps, the expectation of expanding the foreign market for Mexican public bonds valued in foreign currency to the same level as the public bond market valued in domestic currency. The cost of that foreign currency, however, was modified in 2009, when it rose extraordinarily because it was indexed to the stock of bonds issued abroad in 2008, which was entirely taken account in the valuation of that cost in 2009, as if it had been a flow, because it began at the practically null level of external debt in 2008 and had a value equivalent to the variation in internal debt in securities (∆SD_{t}^{iv} / SD_{t-1}^{iv}) plus oil income (∆Y_{t}^{p} / Y_{t-1}^{p}), similar to the inflow of reserve foreign currency of 2008. In this way, the 2008-2009 period can be considered as a transition period, because it combined, in 2008, a new valuation of the reserve with the valuation of the cost of foreign currency in the previous stage and, in 2009, the valuation of the 2008 reserve with a new valuation of the cost of the foreign currency, all of this to adapt the two extraordinary valuations of foreign currency that entered the reserve in 2008, which was reflected as a cost of this foreign currency in 2009, equivalent to the variation of the internal debt in securities plus the variation of oil income, reflecting the expectation of increasing to that level the issuance of domestic bonds abroad. After this one-time increase, starting in 2010, the valuation of the cost of foreign currency continued to grow at an amount indexed to the interests generated by its value the previous year.

The specific way in which all of these variables were combined in each of the previously defined periods to determine the variation of assets (∆SR_{t} / SR_{t-1}) and liabilities (∆SD^{e} /SD_{t-1}^{e}) derived from the accumulation of foreign currency, is described by the equations in Table 1, which also defines the prevailing exchange rate in the market, resulting from making the costs and benefits of the foreign currency accumulated in the reserve equivalent. In other words, it is the result of substituting each of the expressions for (∆SR_{t} /SR_{t-1}) and for (∆SD^{e} / SD_{t-1}^{e}) in the equivalence (∆SR_{t} / SR_{t-1}) = (∆SD^{e} / SD_{t-1}^{e}) and then isolating the exchange rate from the result and adding to it what is described in Table 1 as the gross risk premium. It is noted in the table that with this procedure the market exchange rate can also be expressed as the sum of * ex post *interest rate parity and the net risk premium. The details of these calculations can be reviewed in the Mathematical Procedures Appendix.

The simulation of the equations defined in Table 1 very closely follows the way in which these same indicators evolved and their growth rates during the period analyzed, with a correlation of around 86%^{3} between simulations and estimated observations, a testimony to the empirical validity of these equations. With that said, and in any event, taking into account the limitations of the model, the analysis in Section 4 explains this behavior based on the stability conditions derived from solving the difference equations shown in Table 1.

** 3. THE SOLUTION TO THE MODEL**

To resolve the temporal dynamics of the four systems of equations presented in Table 1, they were summarized or represented through three more general equations that took the following form: SR_{t} = ρ_{1t}; SR_{t-1} + ρ_{0t}; SD_{t}^{e} = α_{1t} SD_{t-1}^{e} + α_{0t} and τ_{t} = β_{1t} τ_{t-1} + β_{0t}. These are equivalent to the equations in Table 1 when their parameters are restricted to the values described in Table 2, with null values for coefficients not presented there, constituting a model of three difference equations with variable coefficients and a time path that takes the general form of: y_{t} = (Π_{k=0}^{t-1}γ_{1k})y_{0} + ∑_{m=0}^{t-1} (Π_{k=m+1}^{t-1} γ_{1k}) γ_{0m} (Shone, 2002, pp: 102-105). In this last equation, y_{t} can take the values of SR_{t}, SD_{t}^{e} and τ_{t}, associated, respectively, to values for γ_{1k} equivalent to α_{1k}, β_{1k} and ρ_{1k}, as well as to values for γ_{0m}, equivalent to α_{1k}, β_{1k} and ρ_{1k} and to values for γ_{0} equivalent to SR_{0}, SD_{0}^{e} and τ_{0}, assuming that these last are values acquired in the initial period, t = 0, for the international reserve, external debt and the exchange rate, respectively.

The equations for time paths, or the solutions to the difference equations, meanwhile, serve to analyze the dynamic stability of the variable they represent. In the simplest case, they produce stable solutions or solutions that are constant over time for these variables if the multiplicands different from SR_{0}, SD_{0}^{e} and τ_{0} are less than or equal to unity and the summands maintain a constant value, resulting in unstable solutions in the contrary case. In this scheme, with the simplest system possible, these two conditions are satisfied if the absolute value of the parameters of α_{1k}, β_{1k} and ρ_{1k} are less than or equal to unity from period to period y and for unit values of γ_{1k}, the sums ∑_{m=0}_{t-1} (Π_{k=m+1}^{t-1} γ_{1k}) γ_{0m} tend towards a constant value. The following are some conclusions of the analysis of the specific value of these parameters, as well as their stability limits.

** 4. LIMITS ON EXCHANGE RATE STABILITY**

The limits on exchange rate stability, external debt and the international reserve derived from the time paths of the variables involved in the sterilized interventions strategy are summarized in Table 2. The most significant conclusion that emerges from analyzing the evolution of these conditions for the segment between 1976 and 1988 is that the strategy implemented during that time period in terms of foreign currency accumulation, external debt and the exchange rate did not contain any stabilizing mechanism, and in fact that government applied an interest rate policy against this purpose.

Looking at the process that governed external debt, the requirement to achieve stable development is represented by the inequality: |(1 + r'_{t} | ≤ 1, which implies the need for negative growth rates for the domestic interest rate. For the exchange rate, stability would require satisfying the inequality: |{1 + r_{t} - r_{t}^{e} + φ_{t} }| ≤ 1. Because a positive and growing differential for the domestic interest rate prevailed in this time period, satisfying this inequality would require the risk premium offered by the exchange rate market, given by: φ^{t} = r^{'}_{t} / (1 + Y_{t}^{p'}) to be negative enough to compensate for the expansive effect of the positive differential over the exchange rate and the risk premium could take negative values only if oil income and the interest rate moved in opposite directions: if (1 + Y_{t}^{p'}) was positive with r^{'}_{t} negative and vice versa. Achieving this with a negative r^{'}_{t} would also ensure exchange rate stability. In reality, although the government applied a strategy of decreasing interest rates, they never went to a negative growth rate and, contrary to what is required for exchange rate stability, they moved in the same direction as oil income growth until 1985. The stability of the reserve would require satisfying the inequality: |{[(1 + Y_{t}^{p'}) (r_{t+1} - r_{t+1}^{e} + r_{t} - r_{t}^{e} )] + r'_{t} }| ≤ 1, because in this time period both the interest rate growth rate (r'_{t} > 0) as well as the interest differential [(r_{t} - r_{t}^{e}) > 0] were positive, so the increase in oil income was compatible with this condition if the differential mentioned was less than or equal to unity, satisfying: -3 ≤ (r_{t+1} - r_{t+1}^{e}) ≤ 1; as was comfortably the case before 1986, but since then, oil income has fallen continuously, an in light of this decrease, the interest differential had to exceed unity to satisfy: -2 ≥ r_{t+1} - r_{t+1}^{e}) ≥ 1. What happened in practice is that the government reversed the downward trend of interest rates in this time period, adjusting to the condition of exchange rate stability, but it did not manage to raise them enough to stabilize the reserve.

It was in 1989 that the government included in its reserve financing model a stabilizing mechanism meant to regulate reserve fluctuations, more due to a lack of resources than to any clear planning efforts. In that model, the reserve began to be influenced by the debt in securities, with the magnitude and direction of the impact governed by the domestic interest rate differential. Despite this mechanism, just as in the previous stage, the model adopted between 1989 and 1994 had incompatible conditions of stability for the three variables studied. Specifically, while the stability of the exchange rate would require that the following inequality be satisfied: |(1+r_{t} -r_{t}^{e})| ≤ 1; and as such the interest differential would be negative but greater than -2; the stability of external debt needed this same differential to be positive but less than 2, satisfying: |(1+r_{t}^{e} - r_{t})| ≤1; finally, the stability of the reserve, subject to the following sum remaining constant: {∆Y_{t}^{p} + ∆Y_{t}^{t} + (r_{t} -r_{t}^{e})[1 - (r_{t} - r_{t}^{e} )] ∆SD_{t}^{iv} }, would require that the differential not be null in response to changes in oil income that were not compensated with modifications in non-oil income.

The evident contradiction would occur between external debt and the exchange rate, and is derived from two causes. One is that the model did not link the stabilizing mechanism for the reserve with the evolution of external debt. The second is that by forming solid expectations that investors would pay interest in domestic currency, in liquid foreign currencies and in the very short term, equivalent to the interest differential, the economic policy strategy brought the net risk premium contained in the exchange rate to zero, canceling out the presence of any mechanism to compensate for the exchange rate effect of that differential.

At the beginning of the model, external debt was a stable process that tended to converge around zero, while the exchange rate was unstable and tended towards infinity. Based on this development, the government implemented a monetary policy rate oriented towards closing the gap between the domestic and external interest rates. At one point this gap was as high as 60 percentage points, but it fell to 0.83 percentage points between 1988 and 1993. With these figures, the characteristic ratio greater than unity that had been a feature of the process that governed the exchange rate before 1994 was reduced practically to unity, increasing the root less than unit to nearly the same value for the process that defined external debt. Although this situation was compatible with a unit root process with a stable trend for both variables, it depended on the stability of the reserve.

The reserve was determined by the capacity of the country to stabilize income sources and outlays from the savings account, which required the implementation of a constant fiscal policy that maintained the sum of variations of non-oil income (∆Y_{t}^{t}), oil income (∆Y_{t}^{p}) and interest on the issuance of securities +(r_{t} - r_{t}^{e} ) [1- (r_{t} - r_{t}^{e} )] ∆SD_{t}^{iv}) in a fixed amount, included in: {∆Y_{t}^{p} + ∆Y_{t}^{t} + (r_{t} - r_{t}^{e})[1 -(r_{t} - r_{t}^{e} )] ∆SD_{t}^{iv} }, which could be achieved by compensating for budgetary income fluctuations with changes in direction against the interests on the debt in securities. This mechanism, which is still place in today, gave the government greater control of its reserve in the time periods in which it was able to decide the direction and magnitude of its issuance of domestic securities; even so, as the gap between the domestic and external interest rates closed, the effect of the mechanism faded to zero. Specifically, between 1989 and 1992, interest on the debt in securities reinforced the impact of the type of revenue on the reserve with less variation in absolute value than oil income. In fact, the drop in debt in securities weighted by (r_{t} - r_{t}^{e}) [ 1-(r_{t} - r_{t}^{e} )], from 1992, was practically of the same magnitude as the increase that non-oil income experienced, while oil income essentially remained the same, allowing the reserve to fall to its lowest point between 1989 and 1994. In the last two years of the period, however, the interest differential practically fell to zero, meaning that changes in the debt in securities stopped influencing stability conditions, which were then defined by the capacity of the government to stabilize the sum (∆Y_{t}^{p} + ∆Y_{t}^{t}), which it did not manage to do. Specifically, in 1993, the two types of income both fell, meaning the sum became negative and caused, the following year, the most significant downwards adjustment of the reserve in the 1990s. To this we can add that the exchange rate and external debt reached the unit root.

By 1995, this evolution had become rather chaotic and continued to shift pursuant to a new model of behavior in effect since 2001, which linked the stabilizing mechanism of the reserve with external debt, by valuing it in conjunction with internal debt. This implied a risk premium that was not zero, equivalent to φ_{t} = -(∆τ_{t} / τ_{t-1}) r_{t}^{e}, which moderated the evolution of the exchange rate, because it was reduced with devaluations [if ∆τ_{t} / τ_{t-1} > 0, then φ_{t} = -(∆τ_{t} / τ_{t-1})r_{t}^{e} < 0], reducing the exchange rate adjustment, defined by: ∆τ_{t} / τ_{t-1} = r_{t} - r_{t}^{e} + φ_{t}. In this period, a solution was satisfied for the exchange rate given by: τ_{t} = {Π_{k=0}^{t-1}[(1 + r_{k}) / (1 + r_{k}^{e})]} τ_{0} = 3τ_{0}, under which [(1+r_{1}) / (1 +r_{1}^{e})] = 3, and as such, r_{t} = 2 + 3r_{t}^{e}, implying that the interest rate should be between two to three times the external interest rate, equivalent to 20%, to stabilize the exchange rate at three times its 1994 value, which was precisely what happened. Stability of external debt, meanwhile, was defined by the condition: |{1 + [Y_{t}^{p'} (1 + r_{t}^{e}) / (1 + r_{t})] - [SD_{t}^{i'} (1 + r_{t}^{e}) / (1 - r_{t})]}| ≤ 1, which to be fulfilled would require that the relationship between the growth of that variable and oil income growth remain below: [(1-r_{t}) / (1 + r_{t})], in absolute value, a condition that was true of practically the entire period, making external debt fall with a trend towards zero with an evolution that compensated the trend growth of the reserve. The reserve stability had the same determinants as in the previous period, requiring that the government maintain the term {∆Y_{t}^{p} + Y_{t}^{t} + {[(1 + r_{t}) / (1 + r_{t}^{e})] - 1} (r_{t} - r_{t}^{e})∆SD_{t}^{iv} } at a constant value, which differs from the value before 1995, because the effect of variations in the internal debt in securities was weighted by: {[(1+r_{t}) / (1+r_{t}^{e})] - 1} (r_{t} - r_{t}^{e}). Because in this period the interest differential, together with the ratio [(1 + r_{t}) / (1 + r_{t}^{e})] was positive, variations in the debt in securities recovered its impact on the reserve, increasing it at least proportionally, because, with the exception of 1997, {[(1 + r_{t}) / (1+r_{t}^{e})] -1}(r_{t} - r_{t}^{e}) was less than unity; in that context, internal debt was managed to compensate the variations in non-oil income but with the opposite sign.

Although external debt displayed clearly stable evolution that changed neither in its formulation nor trend, the currency exchange market modified the valuation of risk starting in 2002, through a mechanism in force until 2013, that paid for the risk of the increase in the differential between the external and domestic interest rates, by defining the risk premium as φ_{t} = [(r_{t-1}^{e} - r_{t-1}) / τ_{t-1}]. So, in addition to requiring that φ_{t} trend to a constant value, the stability condition for the exchange rate was the inequality |(1 + r_{t} - r_{t}^{e} )| ≤ 1, and because this was satisfied and (r_{t} - r_{t}^{e}) stabilized at around 0.03, the exchange rate tended to appreciate at a magnitude twice that average each year, between 2002 and 2007, with the product (1 + r_{t} - r_{t}^{e}) (r_{t-1}^{e} - r_{t-1}) correcting the fluctuations. The exchange rate stability implied by the aforementioned, as well as the new risk valuation, was reflected in the reserve strategy adopted in the time period, which caused accumulation to decrease. During those years, the valuation of the reserve was maintained, and in fact the effect of internal debt in securities on that variable became fixed and inversely proportional, as the stability condition was defined by the requirement of maintaining the following sum constant: {∆Y_{t}^{p} + ∆Y_{t}^{t} - ∆SD_{t}^{iv} + {(∆SD_{t}^{i}) (r_{t} - r_{t}^{e})[ (r_{t-1}^{e} - r_{t-1}) - (1 / τ_{t-1})]}}, to which, unlike that of the previous period, the impact of ∆SD_{t}^{iv} was pegged at -1. In addition, it included what could be considered the first version of a corrective mechanism for that fixed proportion, initially associated to variations in total internal debt (∆SD_{t}^{i}), weighted by the interest differential (r_{t} - r_{t}^{e}) and by an adaptive valuation of the exchange rate effect on that differential [(r_{t-1}^{e} - r_{t-1}) - (1 / τ_{t-1})]. The correction, however, was small, because the interest gap closed once again. With that, the reserve stability was determined by the capacity of the government to manage the fluctuations of its debt in securities to compensate for the income fluctuations, with the difference as compared to the previous period that both variables had to move in the same direction and magnitude because they carried opposite signs in the definition of the reserve.

From 2008 to 2013, the evolution of the reserve was determined by the term: {{∆Y_{t}^{p} + ∆Y_{t}^{t} + ∆SD_{t}^{iv} + φ_{t} (∆SD_{t}^{iv})}, which differs from the term in the previous stage because the impact of variations in the debt in securities became positive. In addition, because the correction mechanism of that impact was defined by the debt in securities and not by total domestic debt, and was weighted by the risk premium φ_{t} = [(r_{t-1}^{e} - r_{t-1}) / τ_{t-1}]. Because in this time period the exchange rate fluctuations were determined by ∆τ_{t} / τ_{t-1} = (r_{t} -r_{t}^{e}) + φ_{t}, it is evident that the correction mechanism transmitted to the reserve what happened with exchange rate fluctuations in response to modifications in φ_{t}; Specifically, a higher risk valuation, derived from reductions in the domestic interest rate or increases in the external interest rate, had a devaluation effect that increased the proportion of internal debt converted into the reserve and vice versa. As the interest differential has closed, the impact of this correction mechanism has faded, making the stability of the reserve dependent on the capacity to compensate fluctuations in public income with the debt in securities.

In 2009, the government increased its access to the international capital markets, increasing external debt significantly only that year, by inserting it in a process with a characteristic root defined by {1 + [(1 + r_{t}^{e}) / (1 + r_{t})] + [1 + (∆SD_{t}^{iv} / SD_{t-1}^{iv})] + [1 + (∆Y_{t}^{p} / Y_{t-1}^{p})]}, which is greater than 3 and implied, as such, an increase of over 300% of that variable. However, growth has slowed since 2010, returning to the unit root, with a stability condition given by:|{[(1 + r_{t}^{e}) / (1+r_{t}) + {[1 + (∆Y_{t}^{p} / Y_{t-1}^{p})] + [1 + (∆SD_{t}^{iv} / SD_{t-1}^{iv})]} r_{t}^{e} }| ≤ 1, in which the effects of the debt in securities and oil income were weighted by the external interest rate, which was decreasing and less than unity.

** CONCLUSIONS **

The conditions that define the stability of the processes that have governed the evolution of external debt, the exchange rate and international reserves in recent years now include stabilizing mechanisms that did not exist in the past and which have allowed these variables to be unit root processes with stable trends. Even so, the first two variables continue to face incompatible conditions of stability, harkening back to the situation between 1989 and 1994. The stability condition for external debt is: |{[(1 + r_{t}^{e}) / (1 + r_{t}) + {[1 + (∆Y_{t}^{p} / Y_{t-1}^{p})] + [1 + (∆SD_{t}^{iv} / SD_{t-1}^{iv})]}r_{t}^{e} }| ≤ 1, whose second term trends to zero with r_{t}^{e} < 1, leaving the condition of satisfying: [(1+r_{t}^{e}) / (1 + r_{t})] ≤ 1, which implies: 2 ≥ r_{t} - r_{t}^{e} ≥ 0, which is identical to what was in force between 1989 and 1994, but opposite to the exchange rate, because for the exchange rate the stability conditions are, on one side: |(1+ r_{t} - r_{t}^{e})| ≤ 1, which requires that -2 ≤ (r_{t} - r_{t}^{e}) ≤ 0, to which we must also maintain constant φ_{t} = [(r_{t-1}^{e} - r_{t-1}) / τ_{t-1}]. This last, meanwhile, becomes null if the domestic and external interest rate differential approaches zero, in which case the exchange rate is in a unit root process.

Now, with positive variations in oil income and the debt in securities, reducing the interest differential to zero, when the external rate increases to the level of the domestic, generates a more violent effect on external indebtedness than in the case of the differential closing because the domestic interest rate falls to the level of the external rate, because in the latter situation the impact is unitary, equivalent to [(1 + r_{t}^{e}) / (1 + r_{t})], and in the former, the effect would include this same unitary impact, but coming from the term {[1 + (∆Y_{t}^{p} / Y_{t-1}^{p}) + [1 + (∆SD_{t}^{iv} / SD_{t-1}^{iv})]} r_{t}^{e} }, which is positive if variations in oil income and internal debt are also positive, resulting in roots greater than unity. This can be avoided if the increase in the external interest rate is compensated for by reductions in the internal debt in securities, which reduces the domestic interest rate, because it would also make the exchange rate less volatile, by turning it into a root process less than unity. It would also reduce the external debt capacity, but this drop would be compensated for by the lower exchange rate volatility. In the case of the reserve, the effect of internal debt weighted by φ_{t} would become null in the stability condition that it currently enjoys, given the need of maintaining the following sum constant, {∆Y_{t}^{p} + ∆Y_{t}^{t} + ∆SD_{t}^{iv} + φ_{t} (∆SD_{t}^{iv})}, implying that stability would depend only on the capacity of the government to compensate fluctuations in the budgetary income with domestic debt. Even so, as was made clear earlier, increases in domestic debt tend to destabilize foreign debt, because although they increase the reserve by a proportion given by: 1 – φ = (τ_{t-1} - r_{t-1}^{e} + r_{t-1}) / τ_{t-1}, they also increase external debt by the proportion r_{t}^{e}, implying that the effect on the reserve tends to exceed the effect on the increase of external debt, when the following is satisfied: (τ_{t-1} + r_{t-1} - r_{t-1}^{e} ) > τ_{t-1} r_{t}^{e}. But if the interest differential is canceled out (τ_{t-1} + r_{t-1} - r_{t-1}^{e}), it tends toward τ_{t-1}, which, with r_{t}^{e} < 1, is greater than τ_{t-1} r_{t}^{e}, meaning that closing the gap between domestic and foreign interest rates, because the domestic interest rate falls to the level of the external rate, will leave an accumulated reserve higher than external debt, but if this happens because the external interest rate goes up, there is a greater chance that the opposite will occur and the country will become insolvent. This is one of the clear vulnerabilities that Mexico faces. The other is the drop in oil income.

Falling oil income tends to proportionally reduce the international reserve thanks to its impact on the first term of the stability condition for that variable. It also tends to reduce the government's capacity to take on foreign debt, due to its presence in the second term of the stability condition for this indebtedness, with a less-than-proportional impact for r_{t}^{e}, sufficiently below unity, implying that this reduction would be less than that of the reserve, which would also bring the country to insolvency. The degree to which these impacts are transmitted to the exchange rate depends on the monetary policy response. If the monetary policy response increases the issuance of securities at the same magnitude as the drop in oil income and the domestic interest rate goes up, this increases the capacity of the country to take on foreign debt, due to the last two terms of the stability condition for that variable. However, it is unclear if there is a positive effect on the reserve, because the second term of its stability condition would remain constant and the first term would be lower. What is clear is that exchange rate volatility would increase, because the greater interest differential derived would not be entirely compensated for by the reduction in the risk premium that would also be implied. The alternative of inducing a drop in the domestic interest rate, reducing the debt in securities, would lower the external debt capacity and the reserve, but also exchange rate fluctuations. For either of these two vulnerabilities, we must consider that when the variables analyzed have reached their thresholds of instability, the way in which they will evolve becomes more than uncertain, especially when the changes respond to variables that Mexico cannot control, insofar as agents tend to react by modifying their response to risk, and with that, the entire model that governs the evolution of the mentioned variables, as well as the effect of already proven strategies. But if agents do not change this valuation, the destabilizing impact of the increase in the external interest rate and/or the decrease in oil income could be compensated for by reducing monetary restrictions, which would prevent exchange rate volatility and progressively distribute the cost of the adjustment.

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^{*} Institute of Economic Research, UNAM, Mexico. E-mail address: violetta@unam.mx.

^{1} Referring to the stability of the theory of difference equations, specifically to the term of Shone, R. (2002), pp. 85-105 and Elaydi, S. (2000), pp. 173-243).

^{2} Through the following e-mail address: violetta@unam.mx.

^{3} Due to space reasons, this paper does not present the estimates, but they are currently being updated and can be requested from the author at: violetta@unam.mx.

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