Macro-hedging and sovereign default for commodity exporters Chang Ma Johns Hopkins University Fabián Valencia International Monetary Fund This draft: January 31, 2017 Abstract Motivated by Mexico s oil hedging program, this paper studies the benefits of macrohedging in a model with sovereign default. We show that hedging against commodity price risk enhances domestic welfare through two channels: first, by reducing income volatility; and second, by reducing countries incentive to default and thus relaxing the endogenous borrowing limit. The majority of welfare gains come from the second channel since the default spreads on sovereign bond are reduced. Quantitative results suggest that the size of welfare gains is at the magnitude of 0.5% of annual consumption increase if the country pays an actuarially fair price on the hedging instruments. However, the benefit vanishes if the country has to pay a higher enough premium. Furthermore, considerable welfare gains can be also found using different macro-hedging instruments such as forwards and GDP-indexed bonds. Keywords: Hedging, Commodity exporters, Sovereign debt, Default JEL Classification: F3; F4; G1 The authors thank Ali Alichi, Michal Andrle, Ravi Balakrishnan, Paolo Cavallino, Stephan Danninger, Olivier Jeanne, Laura Kodres, Anton Korinek, Leonardo Martinez, Dora Iakova, Andrea Pescatori, Francisco Roch, Damiano Sandri, Alejandro Werner, Jinfan Zhang and and seminar participants at IMF brown bag seminar for helpful comments. All errors are our own. Address: Wyman Park Building 544E, 3400 N. Charles St, Baltimore, MD 21218. Email: cma18@jhu.edu. Address: 700 19th St NW, Washington, DC 20431. Email: FValencia@imf.org.
1 Introduction Commodity exporters can insure against fluctuations in commodity prices by accumulating assets in commodity stablization funds, issuing defaultable debt or hedging with financial instruments. The first strategy is dominated by the latter two from a theoretical perspective since more insurances are provided to the country from either defaultable debt or macro-hedging 1 (See Borensztein et al. (2013) and Borensztein et al. (2017) for quantitative calculations of welfare gains from using defaultable debt or hedging as opposed to stabilization funds). The relationship between defaultable debt and macro-hedging, however, is less clear, even from the theoretical perspective. Clearly, both methods provide some insurances to the country at certain costs: default might lead to an output loss or an exclusion from financial market for certain periods while buying hedging instruments might be too costly and risky especially if the country has no corresponding expertise in managing financial risk. In reality, many countries, especially less developed countries, default frequently in their external liability but few of them actually conduct macro-hedging programs to smooth their income. Interestingly, Mexico is the only country, to our knowledge, who is involved in a macro-hedging program since 2000. 2 Meanwhile, Mexico has experienced 8 episodes of default in its sovereign debt since 1800 and the last default episode happens in 1982. 3 Motivated by Mexico s experiences, we want to explore the following questions: what is the relationship between defaultable debt and macro-hedging? How much are the welfare gains from applying macro-hedging strategies in the presence of defaultable debt? Will the cost of hedging affect the welfare gains? In this paper, we address these questions using a willingness-to-pay framework á la Eaton- Gersovitz model. In this framework, the country is exposed to a risk in the price of its commodity exports. The country can insure against this risk by issuing defaultable debt or using hedging instruments. Defautable debt allows the country to default whenever it is optimal to do so. Meanwhile, hedging instrument helps the country to smooth income fluctuations in its export 1 Daniel (2001) argues that stabilization funds are inherently flawed since the international oil spot price does not have a well-defined time-invariant equilirbium value. 2 Other countries or regions have participated in macro-hedging programs as well but only Mexico has a largescale annual program over a long period. For example, Ecuador terminated its oil hedging program in 1993 after a loss of 20 million dollars to Goldman Sachs followed by a political storm. 3 The data on Mexico s default is from Carmen M. Reinhart s website: http://www.carmenreinhart.com/ data/browse-by-topic/topics/7/. 1
income. In our baseline framework, we assume that the commodity is oil and the hedging instrument is put options since Mexico is an oil exporter and it has adopted put options as macrohedging instruments for nearly 20 years. We firstly calibrate the model with both defaultable debt and put options to Mexico s data from 1996-2015 since it is the relevant period when Mexico participates in oil hedging program. We then compare the benchmark model with an otherwise equal defaultable debt model without put options. The difference between these two economies is considered to be the welfare gains from using put options and the we find the gains are at the order of 0.5% permanent increase in annual consumption. The welfare gains come from two channels. The first channel is the income smoothing channel as in Lucas (1987): lower income fluctuations transfer into a smoother consumption path and thus a higher welfare gains since representative agents are risk averse to fluctuations in consumption. In his calculation, Lucas (2003) argues that the welfare gains from smoothing business calculation are very small for the United States. We conduct the same calculation using data from Mexico and find that the welfare gains are at the same magnitude as that in U.S., around 0.05% of permanent annual consumption increase. In our model, we provide a way to isolate the welfare gains from the income smoothing and the relaxation of endogenous borrowing limit which will be explained further later. We find that the welfare gains from the income smoothing are around 0.08% of permanent annual consumption increase, the same magnitude as that in the data. Apparently, the majority of welfare gains in macro-hedging is not from the income smoothing channel but the other source. The second channel is from the relaxation of endogenous borrowing limit. In our framework, countries might choose to default whenever it is in their best interests to do so. Investors understand the countries incentive and thus demand a higher default spread when the probability of future default is higher. The default spread imposes an endogenous borrowing limit on the countries borrowing capacity in each period since the default spreads above a certain threshold are too high and it is not optimal for the country to borrow at that price. In the presence of hedging, future income is higher and the default incentive is thus reduced, which lowers the default spread and thus relaxes the endogenous borrowing limit. This channel is the majority of total welfare gains since the country is allowed to borrow more and gets a better risk sharing from the defaultable debt contract. From this perspective, we shall view hedging as complementary to defaultable debt. Indeed, we quantify the welfare gains from this channel by providing 2
the country in a no-hedging world with the same default spreads as that in a hedging world and find that the gains are at the magnitude of 0.40% permanent increase in annual consumption, which is 83% of the total welfare gains. We also find considerable welfare gains using other types of hedging instruments, such as selling forwards and GDP-indexed bonds. In our quantitative results, selling forwards can generate a higher welfare gains: at the order of 1.43 %, which is at the same order of gains as in Borensztein et al. (2013). The gains are larger than our benchmark model since selling forwards does not require any cost ex-ante while purchasing put options requires sacrificing current consumption. Depending on the degree of indexation, the gains from selling GDPindexed bonds differ. Intuitively, when the country can sell fully indexed bonds to investors, the country is effectively access to Arrow-Debreu securities and thus has perfect risk-sharing with foreign investors. As a result, the gains should be very large and we find the gains are at the order of 8 % of permanent consumption increase, much larger than the gains from put options and forwards. Also, this number is larger than the quantitative results of GDP-indexed bonds in Hatchondo and Martinez (2012) since we use a much lower discount rate than theirs in order to match the data. The gains become smaller when the discount rate increases and we find the similar size of welfare gains when using the same discount rate as in Hatchondo and Martinez (2012). Furthermore, we find that when the degree of indexation decreases, the gains decrease since the degree of hedging becomes smaller. Literature review: This paper is at the intersection of several lines of literature. First, our paper belongs to the literature on quantitative models of sovereign default. Essentially, this literature focuses on a willingness-to-pay problem where a country decides to honor its external debt or not. In a seminal paper, Eaton and Gersovitz (1981) argues that international borrowing can be rationalized in a model with reputation concern even if there is no explicit mechanism deterring a government from repudiating its external debt and the punishment of no repayment includes exclusion from future borrowing. 4 Aguiar and Gopinath (2006) and Arellano (2008) generalize Eaton-Gersovitz framework in quantitative models and find this framework can closely match the data in terms of business cycle feature. Mendoza and Yue (2012), on the other hand, provide a general equilibrium framework in order to endogenize the cost of default and shows 4 Indeed, this punishment could be very persistent. Reinhart et al. (2003) find that the history of default matters for the sovereign borrowing, which they called debt intolerance. 3
that the model can match both sovereign default episodes and business fluctuation dynamics. Moreover, Hatchondo and Martinez (2009) and Chatterjee and Eyigungor (2012) extend the framework into a longer duration of debt and find that models with longer maturity debt can better match the data. The Eaton-Gersovitz model has been proved to be successful in providing a useful benchmark to investigate various of interesting questions. 5 For example, Cuadra and Sapriza (2008) introduce political uncertainty into a sovereign default model and show that politically unstable and more polarized economies are associated with higher default rates and interest spreads. Na et al. (2014) characterize the jointly optimal default and exchange-rate policy in a sovereign default model and nominal wage rigidity. Unlike the previous literature, we use the sovereign default framework to investigate the benefits of macro-hedging. In particular, we focus on the channel through which hedging increases social welfare. Second, our paper belongs to the branch of literature that calculates the welfare gains of macro-hedging. For example, Caballero and Panageas (2008) show that optimal hedging strategies can reduce the precautionary savings for a country facing against sudden stop risks in capital flows. Borensztein et al. (2013) find that the welfare gains from macro-hedging against commodity price fluctuations are at the magnitude of several points of permanent consumption increase. In their framework, the insurance that country would adopt other than financial innovations is the risk-free bond. The channel is through either the reduction of precautionary savings as in Caballero and Panageas (2008) or the relaxation of external borrowing limit as in Borensztein et al. (2013). In our paper, however, the country cannot issue risk-free bond. Instead, it chooses to strategically default whenever it is optimal to do so. Therefore, the borrowing limit is endogenous rather than exogenous. Our paper is closely related to Hatchondo and Martinez (2012), Borensztein et al. (2017) and Lopez-Martin et al. (2016), where Hatchondo and Martinez (2012) investigates the role of GDP-indexed bond in an otherwise equal sovereign defaultable bond model and Borensztein et al. (2017) studies the role of catastrophic (CAT) bond in a framework with either risk-free or defaultable bond. There are two main differences between our work and these two papers. First, both GDP-index bonds and CAT bonds provide a complete insurance to the country and thus complete the market. In our framework, however, hedging instrument such as put options only provides partial insurance to the country. As a result, one should 5 To see more works in sovereign default models, we suggest the readers to read the survey papers such as Eaton and Fernandez (1995) and Aguiar et al. (2014). 4
expect relatively lower welfare gains from our calculation. Secondly, our exercise motivates from Mexico s oil hedging program and we can calibrate our model to Mexico s data. Hatchondo and Martinez (2012) investigate the GDP-indexed bond mainly through a theoretical exercise even if the author motivates the paper from many other countries experience in issuance of GDPindexed bond. In reality, however, the data on GDP-indexed bond is sparse, which does not allow for a rigorous calibration. 6 Borensztein et al. (2017), however, focus mainly on the welfare gains from CAT bond with respect to default-free bond. In the end of their paper, the author provides a generalization for the welfare gains with respect to defaultable debt but their cost of default is very artificial in the sense that they assume a welfare loss under default. 7 Moreover, they do not calibrate their model with defaultable debt to the data since their main focus is the welfare gains with respect to default-free debt. Lopez-Martin et al. (2016) also calibrate their sovereign default model using Mexican data. However, there are several differences. Firstly, they investigate the effectiveness of financial instruments in reducing the volatility of macroeconomic variables where we focus on the welfare gains from adopting financial instruments. Secondly, they assume that government pays zero cost in access to financial instrument and ask the question that how much government would like to pay for financial instrument where we assume that government needs to pay an actuarially fair price for the financial instrument and calculate the benefit of macro-hedging. Moreover, we focus more on the channel through which financial instruments enhance social welfare. In particular, we find that the main channel is through the relaxation of endogenous borrowing limit. The organization of this paper is as follows: section 2 provides institutional details about Mexico s oil hedging program; section 3 presents a benchmark model; section 4 presents the calibration procedure; section 5 presents the quantitative results and section 7 concludes. 6 Furthermore, some countries issue GDP-indexed bond in default episodes, such as Argentina in 2002 and Greece in 2012. It is hard to make the case that the country uses GDP-indexed bond for hedging purpose. It is more likely that country uses GDP-index bond for debt-restructuring purpose. 7 Unlike most literature in sovereign default literature, the cost of default comes from either output loss or exclusion of international market. 5
2 Mexico s oil hedging program We focus on the experience of Mexico since it is involved in the macro-hedging program for nearly 16 years. To be specific, Mexico government purchases put options in each fiscal year from Wall Street banks to hedge against its oil export price risk in the next fiscal year. The put options entitle the Mexican government to sell its oil at a predetermined price (strike price), which provides a floor to its downside risk. The underlying asset of the put options is the crude oil which Mexico government sells to the foreigners and the strike price is determined using a complicated formula that tries to estimate the long run equilibrium level of oil prices. The maturity of the put options is one year since the government wants to hedge against its annual fiscal budget. On the maturity day, the payoff from the put options is the difference between the strike price and the average oil price in the past year. If the former is larger than the latter, the government chooses to exercise the put options and its downside risk from oil price is hedged. Otherwise, the put options are out of money. Figure 1 plots the oil price (in black line) and the strike price (in red line) from the Mexico s oil hedging program. One can see that Mexico has exercised its put options twice in history: 2009 and 2015. Given the current oil price and the strike price set in 2015, it is expected that Mexico will exercise again in 2016. The payoffs from exercising put options are large compared with the cost of purchasing. Historically, the cost is around 0.1% of its GDP but the benefit is around 0.6%. The net benefits up until now are positive, which makes the hedging program attractive to other countries facing the similar shocks. However, all this calculation is from an ex-post perspective. A more relevant question is the benefit of hedging ex-ante. To formally answer that question, we need to establish an analytical framework. Figure 2 plots the oil price and the sovereign spreads in Mexican sovereign bonds. These two series are negatively correlated and the correlation is -0.59, which implies that oil price is associated with sovereign spreads. Interestingly, the standard deviation of sovereign spreads has declined from 300 basis point before 2000 to 82 basis point after 2000. The average spreads has dropped from 603 basis point to 222 basis point. Yet the standard deviation of oil price does not change so much: 1.03 before 2000 and 1.06 after 2000. Indeed, it is hard to attribute the change of spreads entirely to the oil hedging program. However, hedging could potentially reduce the sovereign spreads since it facilitates a better risk management. 6
3 The Model Economy We start from a standard sovereign default model as in Aguiar and Gopinath (2006) and Arellano (2008) to investigate the benefit of hedging. To quantify the benefit of hedging, we setup a counterfactual economy where no macro-hedging tools are allowed. The difference between these two economies is considered as the benefit/cost of hedging. In our benchmark model, we allow the economy to sell put options since Mexico has been following this strategy over a decade. For robustness, we also investigate the benefit of other potential macro-hedging tools. 3.1 Benchmark Model Consider a small open economy populated with representative infinitely-lived consumers. The consumer derives utility from consuming consumption good which we use as numeraire. The representative agent has CRRA preference over consumption stream {C t } t=0 as follows: E 0 t=0 β t C1 γ t 1 γ (1) where β is the discount factor and γ is the coefficient of risk aversion. The total income Y t has two components: the non-oil income (f t ) and oil-component income (x t ) as follows: where p t and Q t are the price and quantity of oil respectively. Y t = f t + x t f t + p t Q t (2) Source of risk: We assume that the shock to this country is from the price of oil: p t, where its process follows an AR(1): [ log p t = (1 ρ) log(p) 1 2 σ 2 1 ρ 2 ] + ρ log p t 1 + ε t (3) where p is the unconditional mean for commodity price, ρ < 1 is the persistent parameter and ε t N(0, σ 2 ). In order to better focus on the consequence of uncertain oil income, we assume that nonoil income f t is deterministic, and grows at the rate of G in every period. We normalize all the variables in the economy by f t and present the normalized economy from now on. The 7
normalized income and utility function can be written as y t = Y t Q t = 1 + p t = 1 + p t Q f t f t (4) E 0 (βg 1 γ ) t c1 γ t 1 γ (5) t=0 where we assume that the normalized quantity Q = Qt f t is constant since our primary interest lies in the risk from price of oil, and c t = Ct f t is the normalized consumption at period t. The government is benevolent and its objective is to maximize the utility of all the households. 8 The government can buy one-period discount bond b t+1 at price q t in each period, where q t is determined in equilibrium and thus depends on the state of economy. In each period t, government can choose whether to default or not. When the government chooses to default, the economy is excluded from international bond market for current period. It stays in the state of default for several periods and might re-enter the market with probability of λ (0, 1). Moreover, there will be a one-period output loss h (y t ) in the state of default. Macro-hedging: We allow the government to purchase put options as macro-hedging for a fraction α of its oil income. Put options provide a mechanism for the government to hedge against the downside risk of export income. Specifically, government can purchase put option that allows the country to sell the commodity at the price of p t at t + 1 and the unit cost of options is ξ( p t ). Here the strike price p t is time-variant and set at the period t. This is to mimic the practice of Mexican government in its hedging strategy: in the data the strike price is time-variant and supposed to hedge the downside risk in one-year ahead. In the state of default, we assume that the country has no access to the put options. It is convenient to denote the beginning-of-period wealth by w t = y t + b t. The state of this economy can be summarized by two state variables: (w t, p t ). Let V (w t, p t ), V c (w t, p t ) and V d t (p t ) denote the value function at the beginning of period t, the value function of not default and the value function of default respectively. default if and only if V d (w t, p t ) > V c (p t ). Therefore, ( ) V (w t, p t ) = max V c (w t, p t ), V d (p t ) At each period t, the government chooses to 8 It is standard in sovereign default literature that government makes decision on behalf of domestic agents. The rationale is that government can always decentralize its allocation through taxes and transfers. 8
If the economy does not default, we have: V c c 1 γ t (w t, p t ) = max c t,b t+1 1 γ + βg1 γ E t V (w t+1, p t+1 ) (6) s.t. c t + q t Gb t+1 + α QGξ( p t ) = w t w t+1 = y t+1 + α Q max{ p t p t+1, 0} + b t+1 Here the bond holding b t+1 and cost of options ξ( p t ) are multiplied by a growth factor G since the economy grows at the rate of G. In period t + 1, the wealth w t+1 is hedged by the amount α Q max{ p t p t+1, 0} whenever the country exercises put options, i.e. p t > p t+1. If the economy defaults, in next period there is a probability λ that the economy will be redeemed and re-enter the international market. If redeemed, all the past debt is forgiven and the economy starts with zero net assets. Furthermore, a country in default status cannot purchase put options. In a recursive form, we have; V d (p t ) = c 1 γ [ ] t 1 γ + βg1 γ λe t V (w t+1, p t+1 ) + (1 λ)e t V d (p t+1 ) s.t. c t = y t h (y t ) (7) w t+1 = y t+1 Risk-Neutral Investors: We assume that both the sovereign bonds and put options are purchased/sold by a continuum of risk-neutral investors. Denote the world risk-free rate by r, and the default function D(w t, p t ) = 1 if government defaults and zero otherwise. Then no-arbitrage condition requires that q t (b t+1, p t ) = E t [1 D(y t+1 + b t+1, p t+1 )] 1 + r ξ( p t ) = E t[max{ p t p t+1, 0}] 1 + r Note In our benchmark model, we do not allow countries to choose whether to purchase put options or not and how much share of export to hedge for the sake of simplicity. That said, the welfare gains should be considered as a lower bound of hedging. 3.2 Benefit of Macro-Hedging To quantify the gains from hedging, we setup a counterfactual economy where no hedging is allowed. In order to differentiate these two scenarios, we use variables with tilde to denote value 9
functions in the no macro-hedging economy. The state variables (w t, p t ) are defined the same as before. So, in each period, the value function is defined as follows. ) c Ṽ t (w t, p t ) = max (Ṽt (w t, p t ), Ṽ t d (p t ) If the economy does not default, we have: Ṽt c c 1 γ t (w t, p t ) = max c t,b t+1 1 γ + βg1 γ E t Ṽ t (w t+1, p t+1 ) s.t. c t + q t Gb t+1 = w t, w t+1 = y t+1 + b t+1. If the economy defaults, Ṽ d t (p t ) = [y t h(y t )] 1 γ 1 γ [ ] + βg 1 γ λe t Ṽ t (w t+1, p t+1 ) + (1 λ)e t Ṽt d (p t+1 ) The bond price is pined down by q(b t+1, p t ) = E t [1 D(y ] t+1 + b t+1, p t+1 ) 1 + r 4 Calibration The model can be solved using value function iteration and the procedure is described in Appendix C. Benchmark model is calibrated to match Mexico s data from 1996-2015 where it includes the periods when Mexico s government starts to use put options for hedging. Each period in our model refers to a year and the parameter values are reported in Table 1. Output cost function As is known to the literature, the output cost function is very crucial for quantitative results in the type of models following Eaton and Gersovitz (1981). As Arellano (2008) points out in her paper, the asymmetric feature of output cost function is the key to generate reasonable default rate and thus default spreads in the model. Since our purpose is to investigate the role of macro-hedging in this framework, we just follow her assumption for output loss function. h(y t ) = y t y, if y t y 0, if y t < y 10
In the model, there are 13 parameters that we need to assign values and we proceed by grouping them into three categories. The first category of parameter values can be directly obtained from the data, such as {r, γ, λ, p, ρ, σ, Q, α, G}. The second category of parameters is chosen to match relevant moments in the data, such as {β, y }. The last category is approximated under reasonable assumptions, such as {ξ(p t ), p t }. The real risk free interest rate r is calibrated to be 0.71 % using data of U.S. 1-year treasury bill. Risk aversion parameter γ is set to be 2, which is a standard value in the literature. The probability of redemption parameter λ is calibrated to be 0.11, which implies that the country is expected to stay in financial autarky for 9.38 years. The number matches the historical experience for Mexico. For oil income process, we have data for Mexico s oil price and its relevant parameters such as unconditional mean p, persistence ρ and volatility σ can be easily estimated using MLE method described in Appendix D. We also have data for GDP and oil revenue, and the difference between these two series is used as f t in the model for normalization. The growth rate G in the data is 3.75%, which is reasonable in small open economies. For macro-hedging programs, we have a relative short sample series. The information on hedging volumes helps pin down the share of hedging parameter α. We have time-variant strike prices and cost of hedging for some years, from which we need to get an approximation for p t and ξ(p t ). We proceed by assuming that p t = µe t [p t+1 p t ] since Mexico government applies a complicated formula to estimate the long run equilibrium value of oil prices. We choose µ such that the long run probability of exercising options is 18.75% since Mexican government exercises its put options three times out of 16 years. For the option price function, we use an actuarial fair price implied by no-arbitrage condition in the market. The calculation method is reported in Appendix E. The oil price process can be approximated using Tauchen (1986) s method to discretize an AR(1) process. We use 21 points to approximate the income process and 500 points to approximate the bond space. The model is finally calibrated by choosing {β, y } to match two important empirical moments: (1) debt to non-oil fiscal revenue ratio: 9.03%; (2) average default spreads: 3.68%. 11
5 Welfare gains from hedging 5.1 Benchmark results We measure welfare gains from macro-hedging by comparing the utility of the economy who uses put options with the utility of the economy who does not have access to put options. As common in the literature, we express the welfare gains as the permanent percentage increase in annual consumption that yields the same level of utility. Formally, the welfare gains is expressed in equation (8). We calculate this number by running a Monte Carlo simulation for the economy with and without hedging. Given the same initial condition, we can calculate the welfare as the discounted sum of utilities. The unconditional welfare gains are calculated by plugging into the utilities into equation (8). (b t, p t ) = 100 [ (V (bt, p t ) Ṽ (b t, p t ) ) 1 1 γ 1 ] (8) From the simulation, the unconditional welfare gain is around 0.4875%. This numbers are comparable to the previous study in a similar environment. For example, Borensztein et al. (2017) finds that the welfare gains from using catastrophe(cat) bonds in the presence of defaultable debt are typically small: less than 0.12 % in their calculation. They rationalize this results by claiming that the CAT bonds do not change the default threshold. However, even if the instrument does change the default threshold, the welfare gains are also small. Hatchondo and Martinez (2012) did an exercise for the GDP-indexed bond in a model with defaultable debt. They find that GDP-indexed bond could change the default threshold but the welfare gains are also small: around 0.46% in terms of permanent consumption increase. Source of welfare gains: the welfare gains from hedging mainly come from the relaxation of endogenous borrowing limit. Intuitively speaking, if the country ever borrows to the point where the investors know that it will default for sure in the future, investors will never be willing to lend. As a result, there is a borrowing limit imposed on this country. Under the hedging program, the income becomes more smoothed and the incentive to default is attenuated, which corresponds to a more favorable default spreads charged by the investors. To confirm this intuition, we compare the simulation results for economy with hedging and without hedging and report them in Table 2. We find that from hedging to non-hedging world the debt ratio is reduced from 9.77% to 8.96% but the default spreads increased from 3.21 % to 4.72%. 12
This intuition can be further strengthened by looking at the price schedule under these two economy. Figure 3 shows the bond price and sovereign spreads for the economy with and without hedging. One can see that the bond is priced higher in the world with hedging, which implies a lower default spreads. To see the contribution of the reduction in default spreads, we firstly solve the model without hedging again but under the same price schedule as that in the model with hedging and then calculate the welfare gains by comparing its utility with that in the model without hedging. Presumably, the welfare gains here purely capture the benefit from relaxation of borrowing limit and reduction in default spreads. In our simulation, we find that this number is around 0.4057%. 5.2 Cost of Hedging Current calculation is based on the premise that Mexico government pays zero premium for put options to hedge. Clearly, the cost of buying put options affects the effectiveness of hedging. One should expect that the higher the cost premium, the lower the welfare gains from hedging. In order to better understand the cost of hedging, we augment the fair price of options with a premium µ. By setting µ to the different cost premiums paid by Mexico government, we want to understand how the unconditional welfare gains change with respect to µ. From Figure 4, we find that the welfare gains vanish when the cost premium is around 1.14 $ per barrel. 5.3 Feature of hedging program Before entering into a contract of put options, the government might be concerned about several questions about the feature of hedging. These questions include the strike price of the contract, the share of productions that the government is going to hedge and the productions that are exposed to commodity price risk. To better understand the sensitivity of our benchmark results with respect to the feature of hedging program, we conduct the following experiments. Strike prices: Mexico government has its own formula to calculate the strike price of put options. To simplify life, in our baseline calculation we assume that strike price is a fraction of conditional price mean. In the historical data that we have, the mean is 0.85. In the data, this number varies from year to year as in Table 3. We find from Figure 5 that the higher the strike price, the more welfare gains there are since more risk is covered by the put options. However, the cost of put options increases as the strike price increases. As a result, the welfare gains from 13
hedging cease to increase at certain point. Hedging shares: Mexico government only hedges a fraction of their oil revenue. The share of hedging α ranges from 44% to 72% as in Table 4. In the calibration, we use the average number of hedging, which is 55%. Figure 6 plots the unconditional welfare gains for different hedging share α. Clearly, one find that our results are robust to different hedging shares. Again, the higher the hedging share is, the more risk the country is covered, and thus the higher the welfare gain is. When α exceeds 100%, the country actually speculates using put options. 5.4 Robustness Check Oil price process: We also calculate welfare gains using different values for parameters in oil price process such as unconditional mean ρ, persistence ρ and volatility σ. The results are reported in Table 5. We find that the welfare gains are robust to different parameters. Other parameters: We also conduct robustness check with respect to different parameters in our model such as risk free rate (r ), risk aversion parameter (γ), probability of redemption (λ), growth rate G, output loss function y, discount rate β,and redemption parameter λ. The results are presented in Table 6. The welfare gains are robust to various of parameters. Finer grid spaces: We also apply a finer grid space for the calculation of welfare gains since Hatchondo et al. (2010) points out that a coarse grid space is subject to approximation errors. Our results are robust if we increase the number of grid points for endowment space from 21 points to 201. 6 Extensions 6.1 Selling forwards as hedging Suppose that we allow the economy to sell forward contract one year in advance. Specifically, the budget constraint changes into the following form. w t = c t + q t Gb t+1 w t+1 = 1 + Q {(1 α)p t+1 + αe t [p t+1 ]} + b t+1 14
We use the same parameters as in the benchmark model and find the welfare gains are at the order of 1.4385 % in terms of permanent consumption increase. The debt has increased from 8.96% to 13.97% and the default spread has decreased from 4.72% to 1.42%. Clearly, the gains are larger than the benchmark model since the economy does not pay any cost when selling forwards. Put options require the economy to sacrifice current consumption to hedge. 6.2 Selling GDP-indexed bonds as hedging We assume that the economy issue bonds that promise to pay in the next period a coupon s and (1 s)y t+1. Parameter s [0, 1] determines the extent to which debt is indexed to GDP. As a result, the wealth and price of GDP-indexed bonds change into the following forms: w t = c t + q t Gb t+1 w t+1 = y t+1 + b t+1 [s + (1 s)y t+1 ] q t (b t+1, p t ) = E t [1 D(y t+1 + b t+1, p t+1 )[s + (1 s)y t+1 ]] 1 + r The welfare gains are plot in Figure 7. As the degree of indexation increase, i.e. decrease in s, the welfare gains increase. The maximum gains are at the order of 8% in terms of permanent consumption increase, which is much larger than the number in Hatchondo and Martinez (2012) since they use a higher discount rate factor. Using the same discount factor, we also find the gains are at the order of 0.16%, which is at the same magnitude as in Hatchondo and Martinez (2012). 6.3 Risk averse investors We assume that international investors are risk averse. In particular, they have a time-variant pricing kernel m t, i.e. intertemporal marginal rate of substitution. Following Arellano (2008), we assume that m is an i.i.d. random variable with a constant mean equal to the inverse of the risk-free rate and with an innovation correlated with the economy s income. Here we assume that log m t+1 takes the following form to make sure that m t+1 is non-negative. log m t+1 = r νε t+1 Here ε t is the same shock to the price of oil and E[log m t+1 ] = r and var(log m t+1 ) = ν 2 σ 2. 15
The pricing of sovereign bond and options are given by the following formula. q t (b t+1, p t ) = E t [m t+1 (1 D(y t+1 + b t+1, p t+1 ))] (9) ξ t (p t ) = E t [m t+1 max( p p t+1, 0)] (10) We increase ν so as to increase the risk aversion of investors. Not surprisingly, we find that in Table 7 the welfare gains are becoming smaller as ν increases since the price of options becomes more and more expensive. We do not calibrate ν to any empirical objects but instead choose several points to calculate welfare gains. Clearly, as ν increases, the investors become more and more risk-averse and starts to charge a higher price for both put options and sovereign bonds. At certain point (ν = 0.5 in our exercise), the default spreads become negative. However, negative default spreads are inconsistent with the data so we interpret a reasonable ν should lie in the range of [0, 0.5). Given that welfare gains decrease with ν, the gains for risk-averse investors are at the range of 0.36% and 0.49%, which is roughly the same as our benchmark result. 7 Conclusion Countries can hedge against their income shocks through either issuance of defaultable debt or involvement in hedging programs. Both methods provide partial insurance to the country. In this paper, we show that these two methods are complementary in providing a better risk sharing mechanism. Macro-hedging instruments can enhance social welfare through its effect on countries incentive to default. Using Mexico s oil hedging program experience, we quantify the welfare gains are at the order of 0.5% of annual permanent consumption increase, the majority of which comes from the relaxation of endogenous borrowing limit rather than smoothing business fluctuations as in Lucas (1987). The welfare gains, however, depend on the cost of hedging instruments. If the country has to pay a higher price for the acturially fair price of the hedging instruments, the welfare gains might vanish. There are some aspects in reality that our paper does not touch and we leave them for future research agenda. Firstly, the interactions between private hedging and sovereign default. In reality, some countries are exposed to income shocks but the hedging program is involved by private firms. How should one think about the interactions between private hedging and sovereign default in a unified framework is an interesting question. Secondly, there is political pressure from using macro-hedging program. If government losses money from hedging, there 16
might be some political chaos afterward. If, however, government makes money from the hedging, there will be criticism based on the argument that public sector should not make money from financial business. All in all, the main cost of hedging is not just economical but also political. Last but not least, hedging requires expertise: Mexico has been involved in macro-hedging for nearly 10 years. It is fair to say that Mexico government has gained affluent experience in financial risk management. Other countries might not have this expertise. Before they decide to join the macro-hedging program, they should be able to evaluate their ability to manage such hedging program and corresponding risk aversion of such business on top of other economical and political concerns. 17
References Aguiar, Mark and Gita Gopinath, Defaultable debt, interest rates and the current account, Journal of international Economics, 2006, 69 (1), 64 83., Manuel Amador et al., Sovereign Debt, Handbook of International Economics, 2014, 4, 647 687. Arellano, Cristina, Default risk and income fluctuations in emerging economies, The American Economic Review, 2008, pp. 690 712. Borensztein, Eduardo, Eduardo Cavallo, and Olivier Jeanne, The welfare gains from macro-insurance against natural disasters, Journal of Development Economics, 2017, 124, 142 156., Olivier Jeanne, and Damiano Sandri, Macro-hedging for commodity exporters, Journal of Development Economics, 2013, 101, 105 116. Caballero, Ricardo J and Stavros Panageas, Hedging sudden stops and precautionary contractions, Journal of Development Economics, 2008, 85 (1), 28 57. Chatterjee, Satyajit and Burcu Eyigungor, Maturity, indebtedness, and default risk, The American Economic Review, 2012, 102 (6), 2674 2699. Cuadra, Gabriel and Horacio Sapriza, Sovereign default, interest rates and political uncertainty in emerging markets, Journal of international Economics, 2008, 76 (1), 78 88. Daniel, James, Hedging Government Oil Price Risk, IMF working papers, 2001. Eaton, Jonathan and Mark Gersovitz, Debt with potential repudiation: Theoretical and empirical analysis, The Review of Economic Studies, 1981, 48 (2), 289 309. and Raquel Fernandez, Sovereign debt, Handbook of international economics, 1995, 3, 2031 2077. Hatchondo, Juan Carlos and Leonardo Martinez, Long-duration bonds and sovereign defaults, Journal of international Economics, 2009, 79 (1), 117 125. 18
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A Tables Table 1: Parameters Parameters Value Source Standard Value in literature Risk-free interest rate r = 0.71% U.S. Real Interest Rate (1-Year Treasury Bill): 1995-2015 Risk Aversion γ = 2 Standard Value in literature Probability of redemption λ = 0.11 Average years in default for Mexico: 1800-2010 Calibrated From Mexico s Income Data Growth rate of non-oil fiscal revenue G 1.0375 Unconditional commodity price mean p 48.84 Log commodity price persistence ρ 0.8403 Log commodity price volatility σ 0.2869 Oil to non-oil GDP ratio p Q 6.46% Strike Price p t µe t[p t+1 p t], where µ = 0.74 Share of Hedging α 0.55 Cost of Hedging ξ( p t) Fair price implied by risk-neutral investors Parameter to match moments Discount rate β 0.7317 Output loss function y 1.0330 Target Moments Data Model Simulation Debt to non-oil GDP ratio 9.03% 9.77% Sovereign spread 3.68% 3.21% Table 2: Simulation results: hedging and no-hedging world Economy debt ratio default spreads default probability Hedging 9.97% 3.21% 2.38% No-Hedging 8.96% 4.72 % 3.17 % 20
Table 3: Strike Prices Time Strike price (real $) Ratio to conditional price mean 2006 40 0.73 2007 46.8 0.72 2008 70 0.72 2009 56.69 0.87 2010 63 0.90 2011 84.9 0.84 2012 86 0.88 2013 81 0.82 2014 76.4 0.86 2015 49 1.14 Table 4: Hedging Shares Time 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 Share 70% 72% 65% 52% 45% 44% 48% 50% 55% 50% Table 5: Oil Price Parameters Parameters Welfare gains ρ = 0.7 8.1076% ρ = 0.8 0.5101% ρ = 0.9 0.0164% ρ = 0.95 0.0098% ρ = 0.99 0.1005% σ = 0.3 0.4923% σ = 0.4 0.5938% σ = 0.5 0.4640% σ = 0.6 0.4962% σ = 0.7 0.5428% p = 30 0.1412 % p = 40 0.3600% p = 50 0.5100% p = 60 0.5661% p = 70 0.7195% 21
Table 6: Robustness check for different parameters Parameter r = 1% r = 2% r = 3% r = 4% r = 5% r = 6% Welfare gains 0.4394% 0.4213 % 0.3534 % 0.2776 % 0.1609 % 0.3626 % Parameter γ = 3 γ = 4 γ = 5 λ = 0.1 λ = 0.1667 λ = 0.2 Welfare gains 0.6043% 0.6368 % 4.1565% 1.3410% 0.1837% 0.0273% Parameter G = 1 G = 1.01 G = 1.02 G = 1.03 G = 1.04 Welfare gains 0.2869 % 0.2706% 0.2711 % 0.4244% 8.7213% Parameter β = 0.8 β = 0.85 β = 0.9 β = 0.95 β = 0.96 β = 0.99 Welfare gains 0.5108% 0.3438% 0.1994% 0.1467% 0.1668% 0.1018% Parameter y = 1.0194 y = 1.0323 y = 1.0647 y = 1.0776 y = 1.3 Welfare gains 0.7169% 0.4068% 0.2994% 0.1422% 0.0028% Table 7: Risk Averse Investors: hedging and no-hedging world Economy welfare gains debt ratio default spreads default probability ν = 0 Hedging 0.4875% 9.97% 3.21% 2.38% No-Hedging 8.96% 4.72 % 3.17 % ν = 0.1667 Hedging 0.4168% 9.39% 2.42% 1.84% No-Hedging 8.71% 4.32 % 2.83 % ν = 0.3333 Hedging 0.4173% 9.31% 2.22% 1.76% No-Hedging 8.35% 3.79 % 2.54 % ν = 0.5 Hedging 0.3693% 1.9979% -1.04% 0.02% No-Hedging 1.9869% -1.00 % 0.05 % 22
B Figures Figure 1: Mexico s oil hedging program 140 Mexico's Oil Price (US $ per barrel) 0.7 Flows from hedging (% of GDP) 120 0.6 100 0.5 80 0.4 60 0.3 40 20 0 Oil Price 0.2 0.1 0 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Cost Paid Payoff 23
Figure 2: Mexico s sovereign spreads and oil price 2500 Mexico Sovereign Bond Spread (bp) and oil price 160.00 140.00 2000 120.00 1500 100.00 1000 80.00 60.00 Spread Oil Price 500 40.00 20.00 0 0.00 24
Figure 3: Bond price and sovereign spreads 1 Bond Price Sovereign spreads (in percentage) 14 #108 Hedging 0.9 0.8 Hedging No-Hedging 12 No-Hedging 0.7 10 0.6 8 0.5 0.4 6 0.3 4 0.2 0.1 2 0-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 b 0-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 b Figure 4: Welfare gains for different cost premiums Welfare Gains(%) 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4 0 0.5 1 1.5 2 2.5 cost premium (Unit $) 25
Figure 5: Welfare gains for different strike prices Welfare Gains(%) 1.1 1 0.9 0.8 0.7 0.6 0.5 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 Strike Price (Ratio of unconditional price mean) Figure 6: Hedging share α 2.5 Welfare Gains(%) 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5, 26
Figure 7: Welfare gains of GDP-indexed bonds 8 Welfare Gains(%) 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Coefficient of Index 27
C Algorithm To solve the model, we need to apply value function iteration. We create grid spaces for both b t and p t and denote them by B and P respectively. Starting from an initial guess of bond price q i (b, p) for each b B and p P for iteration i = 0. Conduct the following process. 1. Staring from an initial guess of {V i (b, p), Vi d(p), V c (b, p)} for each b B and p P for iteration i = 0. 2. Update Vi+1 d (p) using equation (7), 3. Update Vi+1 c (b, p) according to equation (6). 4. Update V i+1 (b, p) using V i+1 (w, p) = max { Vi+1 c (b, p), V d i+1 (p)}. 5. Calculate the implied bond price as follows q i+1 (b, p) = E [ p p V c i+1 (b, p ) Vi+1 d (p ) ] 1 + r 6. Iterate until the endogenous objectives q j (b, p), V j (b, p), Vj c(b, p ) and Vj d (p) are close enough for j = i and j = i + 1. i D Estimation of Income Process Suppose that we find a series for price {p t } T t=1 and transform it into {log p t} T t=1. We can find an estimator for unconditional mean p = 1 T T t=1 p t. To get an estimator for the AR(1) coefficient, we assumed that For the conditional density, we have [ log p t = (1 ρ) log(ˆp) 1 σ 2 ] 2 1 ρ 2 + ρ log p t 1 +ε t }{{} µ t 1 f ( log p t p t 1, p, ρ, σ 2) = 1 (log p t µ t 1 ) 2πσ 2 e 2σ 2 2 The likelihood can be written as: L = Π T t=2 1 (log p t µ t 1 )2 2πσ 2 e 2σ 2 log L = T 1 2 log(2πσ 2 ) 1 2σ 2 T t=2 (log p t µ t 1 ) 2 28
The first order conditions require: log L ρ = 1 2σ 2 T t=2 2(log p t µ t 1 )( log p t 1 ) = 0 log L σ 2 = T 1 2σ 2 + 1 2(σ 2 ) 2 T t=2 (log p t µ t 1 ) 2 = 0 E Option Pricing The payoff of the put options is given by max{ p p t+1, 0} for strike price p and current price p t at time t. For a risk neutral investor, the put option is priced according to the following formula: ξ(p t ) = E t [max{ p p t+1, 0}] 1 + r Since we assume that log p t follows an AR(1) process, i.e. Hence, we have [ log p t+1 N (1 ρ) log(p) 1 σ 2 ] 2 1 ρ 2 + ρ log p t, σ 2 }{{} µ t ξ(p t ) = E t [max{ p p t+1, 0}] 1 + r log p = t+1 log p ( p p t+1) 1 + r = = = 1 1 + r log p ( p log p 1 + r Φ µt σ ( p log p 1 + r Φ µt σ 1 ( p p t+1 ) (log p t+1 µ t ) 2 2πσ 2 e 2σ 2 d log p t+1 ) 1 1 + r ) 1 log p 1 + r eµt+ σ 2 1 p t+1 (log p t+1 µ t ) 2 2πσ 2 e 2σ 2 d log p t+1 ( log p µt σ 2 ) 2 Φ σ 29