Items 2 to 4 in the list capture the impact of interest rates, growth rates and exchange rate changes on the debt ratio. They add up to “automatic debt dynamics”, or ADD—presumably because these cannot be easily influenced by policy in the short run—while the first item, the primary deficit, captures policy effort.

In gauging the sustainability of public debt dynamics, the two key variables are the ratio of the primary deficit to GDP, pd, and ADD, which encapsulates the effect of the term (r-g), that is, the difference between the real interest rate on public debt and the real growth rate of GDP.<sup>2</sup> While AEs tend to operate in a single-currency environment, that is, the public sector tends to borrow exclusively in the local currency, most EM public sectors borrow both in the local currency and in foreign currency, such as the US dollar ($), euro or Japanese yen. This requires modifications in the DSA template, as we shall see.

We now derive the difference equation for d starting with a single-currency environment and then adding on the $ as an additional currency. For simplicity, and since our focus is on underlying public debt dynamics, we shall assume items 5 and 6 in the list above, namely, privatization proceeds and the SFA, are both zero. We shall also narrow the discussion to the government.

The change in nominal government debt is equal to the fiscal deficit. In other words, the gap between government spending and revenues (including central bank profits transferred to the Ministry of Finance) is filled by borrowing: (1) , where: = Nominal debt in local currency tt tt−1 tt tt tt−1 _at the end_ of year t DD −DD = FFDD = PPDD + iiDD tt = Nominal fiscal deficit _during_ year t DD tt = Primary deficit = non-interest expenditure minus total revenue FFDD tt = Interest payments during year t, with i denoting the nominal interest rate for year t. PPDD Two assumptions are embedded here. First, interest payments are incurred on the stock of inherited tt−1 iiDD Two assumptions are embedded here. First, interest payments are incurred on the stock of inherited debt, that is, debt at the end of the previous year. Second, interest is paid at the end of the year. These assumptions are consistent with the IMF’s computation of the “effective interest rate” as interest paid during the year divided by the total nominal debt stock at the end of the previous year. This is, by construction, a nominal interest rate.

Now split D into its local currency (L) and foreign currency ($) components and rewrite equation (1) as $ $ (1’) ) , LL where DDtt −DDtt−1 and = {PPDD$ tt are local currency and $-denominated debt, respectively, at the end of the previous + �ii<sup>LL</sup> DDtt−1 + ii<sup>$</sup> DDtt−1eett�} + DDtt−1(eett −eett−1 year, and LL are the nominal interest rates on local currency debt and foreign currency ($) debt, and e is tt−1 tt−1 the end-of-year exchange rate (local currency price of $, so a rise in e means a depreciation). DD DD ii<sup>LL</sup> ii<sup>$</sup> $ **Note 1:** Debt in local currency at the end of year t-1 is given by . Let denote $ the share of foreign currency debt at the end of year (t-1), that is, LL . tt−1 tt−1 tt−1 tt−1 DD = DD + DD . ee αα **Note 2:** The expression in curly brackets on the RHS of (1’) is the fiscal deficit (primary deficit plus interest tt−1 tt−1 tt−1 αα≡DD . ee /DD payments). The expression in square brackets represents total interest payments (sum of interest on local currency and $ debt in local currency terms) for year t. By the IMF’s definition of the effective nominal interest rate, namely, interest payments divided by the debt stock at the end of the previous year, the two terms in square brackets can be replaced by . tt−1 iiDD

> 2 <mark>For example, in continuous time, the trajectory of the government or public debt-to-GDP ratio d is given by the differential equation d = pd + (r −g)d . Suppose d exceeded some market-determined threshold (as signaled by interest rates and bond spreads) while pd>0 and r>g. The debt trajectory would be considered explosive absent decisive reform to raise primary surpluses. Bringing r down or raising g may be harder in the short run.</mark>

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