A. No hedging

We consider a small open economy producing an exportable good (the commodity) and populated by a representative infinitely-lived consumer. The consumer derives utility from consuming another good that we will call the consumption good. The consumption good will be used as numeraire. The representative consumer maximizes his utility

subject to the budget constraint

where B_t is the consumer’s holding of foreign assets at the beginning of period t, r is the risk-free world interest factor, Y_t is the domestic output of consumption good (non-commodity GDP), and X_t is the level of commodity production (and exports) expressed in terms of consumption good.

In order to better focus on the consequences of uncertain export income, we assume that the domestic output of consumption good is deterministic, and grows by a factor G in every period. Thus, the budget constraint can be rewritten in units of non-exportable income as

where lower-case variables are normalized by Y_t, for example b_t = B_t/Y_t.

The stochastic dynamics of the economy are driven by the normalized export income, x_t. Since we are interested primarily in the insurance of the price risk, we abstract from the quantity risk and assume that the country exports a constant normalized volume q of commodity. Without hedging, thus, one has

where p_t is the spot price of the commodity. We assume that the spot price follows a multiplicative error process:

where p without time subscript is the average price, ρ ≤ 1 is the persistence of the price process, and ε_t is an i.i.d. positive random variable of mean one.⁹

We assume that the domestic consumer can borrow only against future export income and we rule out the possibility of default.¹⁰ Therefore, to ensure solvency the debt level can never be larger than the minimum present discounted value of future export income. More formally the country is subject to the no-default constraint:

where the minimum is taken at time t over all the possible paths (x_t+i)_i≥1 Let us define the operator ${\underset{¯}{E}}_t{p}_{t+i}$ which given the price level at t returns the lowest possible price at t + i. This can be computed from equation (4) by assuming that the shock ε is equal to its worst realization $\underset{¯}{\epsilon }$ in every period between t and t + i

where $\underset{¯}{p}=[1-(1-\underset{¯}{\epsilon })/(1-\rho \underset{¯}{\epsilon })]p$ is the lower bound to which p_t converges if the shock is always equal to $\underset{¯}{\epsilon }$ In the absence of hedging, the maximum borrowing ability of the country can therefore be written as:

which, using (6), can be written as a linear function of the current price p_t.

In order to reduce the number of state variables in the consumer’s optimization problem with hedging, it is useful to define the country’s “pledgeable wealth” as

This is the sum of the country’s external wealth and of its pledgeable intertemporal export income (the minimum present discounted value of export income that can be pledged in repayment to foreigners). As can be seen using (3) and (5), period-t consumption must be lower than the country’s current non-export income plus its pledgeable wealth,

The country’s consumption/saving problem can thus be expressed in Bellman form as

subject to

This yields the Euler equation

We assume the impatience condition

which, as discussed in Carroll (2008) and Jeanne and Sandri (2009), prevents degenerate solutions with infinite accumulation of foreign assets. ¹¹ We solve the problem numerically by policy function iteration as explained in appendix III. The dynamics of w_t are then obtained from the intertemporal budget constraint (9) with c_t = c(w_t, p_t). The dynamics of the net foreign asset position b_t are derived by netting out current export income x_t and Φ (p_t). As discussed in detail by Jeanne and Sandri (2009), the model features a target net foreign asset position, $\widehat{b}$ towards which the economy tends to converge over time.¹² The target could be positive or negative. It is increasing with the variance in export income and with the domestic consumer’s patience.

We have not allowed the country to have both foreign assets and foreign liabilities at the same time, but a simple extension of the model makes this possible. Let us assume that the country has gross foreign assets $B_t^{\prime }$ and gross external debt D_t. Assets and liabilities bear the same interest rate r. There is furthermore an external credit constraint $D_t\le {\overline{D}}_t$. It is generally assumed in the literature that the maximum level of gross debt ${\overline{D}}_t$ is proportional to the level of domestic output or of domestic collateral. The constraint involves gross debt because the foreign assets cannot be pledged in repayment to foreign creditors (they are protected by sovereign immunity).

Under these assumptions, the model endogenizes the difference $B_t=B_t^{\prime }-D_t$ but the levels of $B_t^{\prime }$ and D_t are indeterminate. If we assume that the external credit constraint is binding $(D_t={\overline{D}}_t)$, the model gives us a unique level of gross foreign assets $B_t^{\prime }$. We can think of $B_t^{\prime }$ as a commodity fund coexisting with external debt.

B. Futures

We now assume that export income can be hedged using futures or forward contracts.¹³ Futures allow the domestic consumer to secure a price θ periods ahead at a level given by the operator

where τ_θ is the premium on the future price, which could be positive because of investors’ risk aversion or a liquidity premium.

One can use the premium τ to capture different assumptions about the availability of hedging instruments. For example, the absence of hedging instruments corresponds to τ_θ = +∞ for θ ≥ 1. Hedging instruments whose liquidity decreases with the horizon can be modeled as an increasing path for τ_θ. Financial innovation can be defined as a reduction in τ at certain horizons.

Determining the optimal hedging policy for an arbitrary cost structure {τ_t}_t=0,..,_+∞ is a computationally demanding problem with a very large state and choice space.¹⁴ To simplify, we will assume that hedging is costless up to horizon θ, and infinitely costly at longer horizons:

It follows that the consumer hedges all the export income θ periods ahead. In the absence of risk premium, the futures are priced at their actuarially fair value, so that a risk-averse representative agent takes full advantage of them by hedging all his export income. Note that in this frictionless setting the country could achieve full insurance by rolling over one-year forwards.¹⁵ However, various costs associated with rolling over short-term contracts may limit the effectiveness of this strategy in practice. Alternatively, we could allow for rolling over the hedge forward at no cost and re-interpret θ as the horizon of the hedging strategy rather than the maturity of the underlying contracts.

The introduction of futures improves the welfare of the representative consumer through two channels. The first channel involves smoothing export income and consequently consumption. The second, somewhat less obvious mechanism works through the country’s “external balance sheet”.

Regarding income smoothing, hedging allows the country to sell its commodity exports at the conditional expected price θ periods ahead, (p + ρ^θ (p_t−θ − p)). Income volatility, thus, is reduced by the factor ρ^θ. The reduction in income and consumption volatility increases with the horizon of hedging and decreases with the persistence of the price process. In particular, the volatility is reduced to zero by hedging at any horizon θ ≥ 1 if there is no persistence in the price process (ρ = 0). The associated benefits are conceptually similar to the welfare gains from eliminating the business cycle, as measured by Lucas (1987).

The balance-sheet benefits of futures are twofold. First, decreasing the volatility of income reduces the target level of net foreign assets, $\widehat{b}$. The country can thus “save on precautionary savings”, and increase the present discounted value of domestic consumption without making any sacrifice in terms of consumption volatility. Second, the introduction of futures relaxes the external borrowing constraint. This effect is theoretically more subtle than the other ones but—as we will see—may have a larger impact on domestic welfare. We spend the rest of this section to explain it.

Let us assume that futures are introduced at time t* (starting from a situation without hedging instruments). The maximum borrowing capacity of the country increases from Φ_t* given by (7) to

where the superscript f is used to identify variables with a different definition under futures. The first term is the present discounted value of the export income that is sold forward in t*, whereas the second term is the minimum present discounted value of the export income that will be sold after period t* + 1. Comparing (7) and (14) shows that hedging increases the country maximum borrowing capacity $({{\Phi }}_{t*}^f>{{\Phi }}_{t*})$.¹⁶ Hedging allows the country to secure a higher minimum level of export income, and so to issue more default-free debt. The minimum level of export income is increased by hedging because the exports that are sold forward are valued at the expected price rather than at the worst possible spot price that can be later obtained.¹⁷

The equilibrium can be solved for in the same way as in the model without hedging, as explained in more detail in Appendix II. The introduction of hedging increases pledgeable wealth to

The transition equation for pledgeable wealth becomes,

In each year pledgeable wealth increases by the present discounted value of the commodity export θ periods ahead sold forward R^−θqF(p_t+1, θ), rather than by current export valued at the more volatile spot price qp_t+1. The maximization problem is equivalent to the one under no hedging, but with transition equation (16) for pledgeable wealth which at time t* is newly defined according to (15).

A. Calibration

Our benchmark calibration is reported in Table 3. The values of r, γ, and β are standard in the literature. We choose a deterministic growth rate of 1 percent and will discuss the sensitivity to this parameter in the next section.

Table 3.

Benchmark calibration

Table 3.

Benchmark calibration

r	γ	β	G	ρ	σ	qp
0.04	2	1.04−1	1.01	0.9	0.2	0.2

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Table 3.

Benchmark calibration

r	γ	β	G	ρ	σ	qp
0.04	2	1.04−1	1.01	0.9	0.2	0.2

The persistency and volatility of price shocks are calibrated by reference to the time series properties of commodity prices. We estimate the AR1 process with multiplicative error term in equation (4) using annual commodity price data taken from the International Financial Statistics and deflated by the export unit value for industrial countries. We assume that the error term is lognormally distributed with mean one and standard deviation σ, and perform the estimation using Maximum Likelihood as described in Appendix IV. The results are reported in Table 4 together with the ratios of commodity exports to non-export GDP, qp, obtained as the simple average of the data in Table 1.

Table 4.

Calibration by commodity

Table 4.

Calibration by commodity

Commodity	ρ	σ	qp(%)
Aluminum	0.71	0.14	15.0
Cocoa	0.98	0.42	13.6
Copper	0.87	0.20	18.2
Fish	0.88	0.10	10.3
Gold	0.96	0.16	13.6
Iron	0 98	0.14	22.2
Natural Gas	0.97	0.27	21.4
Petroleum	0.98	0.21	38.0
Sugar	0.65	0.34	18.2

View Table

Table 4.

Calibration by commodity

Commodity	ρ	σ	qp(%)
Aluminum	0.71	0.14	15.0
Cocoa	0.98	0.42	13.6
Copper	0.87	0.20	18.2
Fish	0.88	0.10	10.3
Gold	0.96	0.16	13.6
Iron	0 98	0.14	22.2
Natural Gas	0.97	0.27	21.4
Petroleum	0.98	0.21	38.0
Sugar	0.65	0.34	18.2

As documented in the literature (Cashin, Liang and McDermott (2000)), we find that shocks to commodity prices are very persistent, with autocorrelation coefficients usually above 0.85. Our benchmark value for the persistence parameter, 0.9, is somewhat higher than the average level of ρ across commodities (0.86), but lower than the estimated persistence for oil, which is the highest across all commodities in our sample and close to 1. We find that the year-to-year volatility in the price varies from 10 to 40 percent depending on the commodity: in the calibration we choose σ = 0.2, close to the cross-commodity average as well as the volatility of oil prices. Finally, we set the ratio of commodity export to non-commodity GDP to 20 percent, close to the average across commodities.

B. Benchmark results

By numerically solving the model, we find that under the benchmark calibration the country’s target level of net foreign assets is equal to zero, $\widehat{b}=0$. In other terms, the country’s gross foreign assets fully cover its gross external debt. The impatience condition (12) is satisfied because of the positive growth in domestic income, against which the domestic consumer would like to borrow. Although impatience is generated by a relatively moderate growth rate in income (1 percent), the borrowing motive is sufficiently strong to dominate the precautionary savings motive, so that the country would like to be a net debtor. However, the country cannot be a net debtor because the process (4) implies that future export income could fall to zero with a nonzero probability. In the absence of hedging, thus, the country simply converges to a zero level of net foreign assets.

As common in the literature, we express the welfare gains from hedging in terms of the permanent percentage increase in annual consumption that yields the same welfare gain for the representative consumer in the economy without hedging (at the target level of net foreign assets, $b_{t*}=\widehat{b}$ and at the equilibrium price p_t* = p).¹⁸ More formally, by exploiting the homotheticity of the CRRA utility function, the welfare gain from the introduction of hedging is defined as the α^f which solves:

so that

As already discussed in section III.B, hedging improves welfare through two channels: first, by reducing the volatility of income; and second, by allowing the country to reduce its net foreign asset position. In order to measure the welfare gain from the first channel, we use the smoother income process allowed by hedging but keep the same consumption function (and so the same target level of assets) as in the absence of hedging. We are therefore exclusively assessing the gains from reducing the income and consumption variance, without allowing for the optimal depletion of the net foreign asset position.

Figure 2 shows the welfare gains from futures as a function of their maturity (the case where the policy functions are held constant is denoted with an upper bar on the variable names). We observe that 1-year forward contracts yield relatively small welfare gains, equivalent to a permanent increase in annual consumption of less than one-tenth of a percent, similar to the benefits from eliminating the business cycle as quantified by Lucas (1987).¹⁹ Even extending the maturity to twenty years provides the country with limited gains, less than one half of a percent.

Welfare gains from consumption smoothing only (holding constant the target level $\widehat{b}$)

Citation: IMF Working Papers 2009, 229; 10.5089/9781451873764.001.A001

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Welfare gains from consumption smoothing only (holding constant the target level $\widehat{b}$)

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Welfare gains from consumption smoothing only (holding constant the target level $\widehat{b}$)

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Figure 3 reports the welfare gains from futures contracts once we allow the country to optimally adjust its net foreign asset position. The welfare gains are about ten times larger than in the previous case. One-year forward contracts with endogenous net foreign asset position leads to welfare gains equivalent to a permanent increase in consumption of more than one percent. Forward contracts with maturity of five years multiply these gains by more than three. Furthermore, the gains are marginally decreasing with the maturity, so that 10-year contracts provides most of the benefits.

Full welfare gains (allowing for the optimal reduction in b_t)

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Full welfare gains (allowing for the optimal reduction in b_t)

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Full welfare gains (allowing for the optimal reduction in b_t)

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The reason behind the much larger welfare gains from hedging with endogenous net foreign assets is that the introduction of futures allows the country to boost consumption by consuming a fraction of precautionary assets or by issuing debt. This can be seen from the left panel of Figure 4, which shows the variations of the consumption function with the normalized net foreign assets, b.²⁰ The curve labeled with θ = 0 corresponds to the case without hedging, whereas the other curves correspond to futures of maturities θ equal to 1, 3 and 10 years and infinity. We observe that the introduction of futures boosts consumption, which is financed by depleting the net foreign asset position. The right-hand side panel shows how the target for net foreign assets decreases with the maturity of hedging instruments, from a debt of about 80 percent of income for one-year contracts to a debt amounting to five years of income for twenty-year contracts.

Consumption functions and target net foreign asset position

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Consumption functions and target net foreign asset position

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Consumption functions and target net foreign asset position

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The fact that the introduction of hedging leads to the reduction of foreign assets raises interesting questions about the distribution of the welfare gains over time. Figure 5 shows, on the left-hand side, the dynamics of the net foreign asset position in the absence of price shocks following the introduction of futures with maturities of respectively one, five and ten years. On the right-hand side, we plot the change in consumption induced by hedging computed j years after the introduction of futures.

Dynamics of net foreign assets and consumption following the introduction of hedging

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Dynamics of net foreign assets and consumption following the introduction of hedging

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Dynamics of net foreign assets and consumption following the introduction of hedging

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The impact of hedging on domestic consumption is positive over the first ten years, but turn negative after that due to the service of foreign debt. Consumption is reduced by more than 5 percent forty years after the introduction of 5-year forward contracts. This suggests that in an overlapping-generations model, expanding the hedging possibilities would increase the welfare of current generations at the cost of future generations.²¹

C. Sensitivity analysis

Under the benchmark calibration, the model leads to three main results. First, the welfare gains from forward contracts can be large. Second, most of the welfare gains come from the relaxation of the borrowing constraint. Third, contracts with medium-term maturities already provide most of the gains. We now discuss the sensitivity of these results to alternative parameter values.

We investigate first the role of the intertemporal discount factor and of the growth rate, with and without allowing for changes in foreign assets. These are important parameters since they alter the consumer’s level of impatience and thus his desire to borrow against future income. Figure 6 shows that as we reduce the discount rate by increasing the value of β, the country targets a larger net foreign asset position as a percentage of GDP. As β approaches G^γ/R the impatience condition (12) becomes an equality and the foreign asset target goes to infinity.

Welfare gains as a function of discount factor and growth rate

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Welfare gains as a function of discount factor and growth rate

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Welfare gains as a function of discount factor and growth rate

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Intuitively, a higher degree of patience should reduce the utility cost of holding precautionary assets and so should also reduce the welfare gains from substituting those assets by hedging. The top-right panel of Figure 6 confirms that this is indeed the case. The figure shows how the welfare gains from hedging at an infinite horizon vary with β under two assumptions: if the country does not adjust its foreign assets target $(\overline{\theta }=+\infty )$ and if it does (θ = +∞). As β increases, the welfare gains coming from the balance sheet channel, measured by the gap between the two curves, shrinks and converges to zero as β comes close to its upper bound. This is so even though hedging reduces the level of precautionary foreign assets by a very large amount when β is large. The large reduction in foreign assets does not lead to large welfare gains because the consumer is patient and does not suffer much from postponing consumption. The main lesson from Figure 6 is that hedging yields large welfare gains if the domestic consumer is impatient, and it does so by relaxing his external borrowing constraint.

The bottom panels of Figure 6 show the variations of the target level of foreign assets and of the welfare gains from hedging with the growth rate. Increasing the growth rate has the same effect as decreasing patience since this makes the consumer more willing to borrow against future income. Thus the welfare gains from relaxing the borrowing constraint increase with the growth rate.

We now consider the sensitivity of the results to the properties of the price process. The left-hand side plot of Figure 7 shows that under the benchmark calibration, changing the persistence of the price process, ρ, does not affect the target level for foreign assets, which remains equal to zero. The middle plot displays the welfare gains from moving to full insurance (i.e., with forward contracts of infinity maturity) respectively with and without allowing the country to borrow. We observe that while the welfare gains from a smoother income are relative small, they become significant if the income shock is very persistent. This is simply due to the fact that the unconditional variance of the price process increases with its persistence.

Welfare gains as a function of the shock persistency, ρ

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Welfare gains as a function of the shock persistency, ρ

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Welfare gains as a function of the shock persistency, ρ

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Finally, the right-hand side plot shows the fraction of the welfare gains from full insurance that can be obtained with futures of finite maturities. The figure reveals that if the shocks are fully transitory (ρ = 0), even a one-year forward contract is able to provide the country with complete insurance. This is easily understood by considering that with transitory shocks the expectation of tomorrow’s price is always equal to its average. As the persistence of the shock increases, short-horizon contracts yield a lower share of the maximum welfare gains. At very high persistence we observe that even ten-year forward contracts provide the country with only around fifty percent of the welfare gains from full insurance.

The sensitivity of the welfare gains to the standard deviation of the shock in export income is shown in Figure 8. In the absence of hedging, higher volatility leads to a larger stock of precautionary savings. The welfare gains from moving to full insurance are also increasing with the volatility of the price innovation, and the gains from income smoothing become significant for high levels of income volatility. We also see that the welfare gains from increasing the horizon of hedging are not sensitive to the level of income volatility.

Welfare gains as a function of the shock variance, σ

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Welfare gains as a function of the shock variance, σ

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Welfare gains as a function of the shock variance, σ

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Summing up, the welfare gains from hedging export income may be much larger than the welfare costs of the business cycle. This is because hedging does more than simply smoothing the consumer’s income: it also relaxes the country’s external borrowing constraint, which yields a large welfare gain if the consumer is impatient.

D. Welfare gains by commodities

Having quantified the welfare gains under our benchmark calibration and discussed their sensitivity, we now calibrate the model using data on key commodities. Table 5 reports the welfare results, computed using the parameters from Table 4. The first two columns report the benefits of full insurance, respectively with and without adjustment of the target level for foreign assets. The subsequent columns report the gains reaped with forward contracts of maturities of 1, 3, 5, 10 and 20 years.

Table 5.

Welfare gains from futures by commodity

Table 5.

Welfare gains from futures by commodity

Commodity	${\alpha }_{\infty }^f$	${\overline{\alpha }}_{\infty }^f$	${\alpha }_1^f$	${\alpha }_3^f$	${\alpha }_5^f$	${\alpha }_{10}^f$	${\alpha }_{20}^f$
Aluminum	3.8	0.0	1.7	3.0	3.4	3.8	3.8
Cocoa	6.3	2.5	0.5	1.3	2.0	3.3	4.8
Copper	4.5	0.3	1.2	2.5	3.2	4.0	4.4
Fish	3.1	0.0	0.7	1.5	2.0	2.6	3.0
Gold	3.9	0.2	0.5	1.2	1.7	2.5	3.3
Iron	5.4	0.7	0.5	1.2	1.8	2.8	4.0
Natural Gas	6.7	2.0	0.7	1.7	2.5	4.0	5.4
Petroleum	8.7	2.8	0.9	2.3	3.3	5.2	7.1
Sugar	4.5	0.3	2.4	3.8	4.2	4.5	4.5

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Table 5.

Welfare gains from futures by commodity

Commodity	${\alpha }_{\infty }^f$	${\overline{\alpha }}_{\infty }^f$	${\alpha }_1^f$	${\alpha }_3^f$	${\alpha }_5^f$	${\alpha }_{10}^f$	${\alpha }_{20}^f$
Aluminum	3.8	0.0	1.7	3.0	3.4	3.8	3.8
Cocoa	6.3	2.5	0.5	1.3	2.0	3.3	4.8
Copper	4.5	0.3	1.2	2.5	3.2	4.0	4.4
Fish	3.1	0.0	0.7	1.5	2.0	2.6	3.0
Gold	3.9	0.2	0.5	1.2	1.7	2.5	3.3
Iron	5.4	0.7	0.5	1.2	1.8	2.8	4.0
Natural Gas	6.7	2.0	0.7	1.7	2.5	4.0	5.4
Petroleum	8.7	2.8	0.9	2.3	3.3	5.2	7.1
Sugar	4.5	0.3	2.4	3.8	4.2	4.5	4.5

We observe that the welfare gains from hedging are potentially very large, ranging from 3 percent to more than 8 percent of consumption in the case of Petroleum. As discussed in section C, the benefits are increasing with the persistence and variance of the price shocks and are obviously greater for countries whose commodity export income accounts for a larger share of total GDP.

Countries exporting commodities with low shock persistence, such as Aluminium, obtain most of the gains from hedging at relatively short maturities, while exporters of commodities subject to more persistent shocks, like Petroleum, derive large welfare benefits from extending the horizon of hedging.

A. Options

Forward contracts help the country to avoid negative shocks to commodity prices, but at the same time prevent it from reaping the gains of positive fluctuations. Governments can therefore be reluctant to adopt this hedging strategy since strong political opposition can arise during periods of increasing commodity prices. A more politically viable alternative may be to insure exclusively against the downside risk by purchasing put options that pay off only if the price of the commodity falls below the strike price. This section looks at the welfare gains from hedging price risk with options of different maturities and with different strike prices.

Assume that options of maturity θ are unexpectedly introduced at time t*. The strike price is constant and equal to a fraction ζ of the equilibrium price p. Consistent with our assumptions about futures, we assume that options are sold at the fair risk-neutral price, so that the country will buy enough options to fully cover its export income.

At the time when options are introduced, the maximum amount of default free borrowing for the representative consumer is given by:

where we use superscript o to distinguish variables that refer to options, and ξ_i(p_t) is the price of an option with maturity i and strike price pζ when the export price is p_t The first term corresponds to the cost of the options purchased at time t*. The second term reflects the minimum present value of future export income net of the cost of buying the options. We use the fact that max(p_t+i, ζp) − ξ_θ(p_t+i) is increasing with p_t+i and so is minimized for $p_{t+i}={\underset{¯}{E}}_t{p}_{t+i}$.

The country’s pledgeable wealth at the time of the introduction of options is thus given by:

with transition equation:

Using the same solution procedure as for futures and under the benchmark calibration in Table 3, we compute the welfare gains from introducing options with different maturities and strike prices.²² The results are shown in Figure 9. The dotted lines correspond to the case of options with strike prices at respectively 40, 60 and 80 percent of the average price. As shown by the left-hand side panel, options allow the country to issue more default-free debt by limiting the downside risk in export income. This benefit is increasing with the strike price. The welfare gains from options are increasing with their horizon and with the strike price.

Net foreign assets and welfare gains with options and futures contracts

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Net foreign assets and welfare gains with options and futures contracts

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Net foreign assets and welfare gains with options and futures contracts

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For comparison, the continuous line reports the results with futures. Futures dominate options because they hedge the risk on both sides.²³ However for maturities shorter than five years, most of the welfare gains from futures can be captured by options, including relatively inexpensive options with low strike prices. This is no longer the case at longer maturities, for which high strike prices are required to make options a good substitute to futures.

B. Default

Defaultable debt may reduce the welfare gains from hedging in two ways. First, default itself may be viewed as a substitute to hedging if it tends to occur when the price of the commodity export is low.²⁴ Second, allowing the country to default will relax the no-default constraint and allow impatient countries to issue more debt—which was an important benefit from hedging. Since default occurs in the real world, it seems important to see the extent to which defaultable debt provides a substitute for hedging.

More formally, we assume that the country defaults entirely on its foreign debt if the export price falls below a certain percentage threshold of the equilibrium price, ζp. There is no repayment following a default (the debt is completely erased). We assume that defaulting is costless: this entails no loss of output, the country maintains access to international debt market and foreigners are willing to purchase defaultable debt without requiring a pure risk premium compensating for risk aversion (although there is a default risk premium compensating for the probability of default). These assumptions obviously bias the analysis in favor of default, so that the following estimates should be viewed as an upper bound for the welfare gains from defaultable debt.

Defining π_t,i as the default probability i periods ahead from time t, the equilibrium gross interest rate on defaultable debt of maturity θ is given by:

Since default is assumed to happen only if the price falls below ζp, the country is still subject to the “natural” constraint of limiting the amount of borrowing to what it can repay in the worst possible scenario. The worst possible outcome for the country in the presence of default, is the one in which the export price drops permanently to exactly ζp, which leads to the minimum present discounted value of export income,

where the subscript d denotes the variables if debt is defaultable.

Let us assume that debt becomes defaultable at some time t*. The country’s pledgeable wealth is now given by,

with the transition equations:

where $a_t=1+w_t^d-c_t-{{\Phi }}_t^d$ is the net foreign asset position at the end of period t after consumption occurs.

As in the case of options and futures, the welfare gains from defaultable debt are closely related to the relaxation of the borrowing constraint. A higher default threshold ζ increases the level of export income the country can borrow against ζp, but also the probability of default and consequently the default risk premium. The impact of the default threshold on the borrowing ability of the country is therefore non monotonic as shown in Figure 10. We see that the maximum borrowing potential is achieved at a low levels of ζ, around 30 percent, and so are the maximum welfare gains, which can reach almost 2 percent of consumption. Somewhat unexpectedly, the possibility of default leads to the highest benefits when its occurrence is limited to very low prices. In this way, the country insures itself from the worst tail events and maximizes its borrowing capacity.²⁵

Borrowing capacity, equilibrium net foreign assets and welfare gains with defaultable debt

Citation: IMF Working Papers 2009, 229; 10.5089/9781451873764.001.A001

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Borrowing capacity, equilibrium net foreign assets and welfare gains with defaultable debt

Citation: IMF Working Papers 2009, 229; 10.5089/9781451873764.001.A001

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Borrowing capacity, equilibrium net foreign assets and welfare gains with defaultable debt

Citation: IMF Working Papers 2009, 229; 10.5089/9781451873764.001.A001

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Defaultable debt is a form of insurance and yields some welfare gains. However, it is dominated by a combination of non-defaultable debt and hedging (provided that hedging instruments are available at a sufficiently long horizon). This is so even though we made the unrealistic assumption that default did not involve any cost in terms of output or credibility. Such costs would reduce further the net value of defaultable debt as a form of insurance.