**Note 3:** The last term on the RHS of (1’) is the capital gain or loss on $ debt due to nominal exchange rate changes. This gets added on to the nominal debt stock, D, which is expressed in local currency. Using the definition of in Note 1 and dividing and multiplying by , this term can be replaced by α , where, once again using IMF notation, is nominal exchange rate depreciation against the $. tt−1 tt−1 αα eett−eett−1 ee DD Using the information in Notes 1-3, (1’) can be rewritten as follows: αα≡ eett−1

Using the information in Notes 1-3, (1’) can be rewritten as follows: (2) . Comparing equation (2) with (1), we see that the only difference is the new term tt tt−1 tt tt−1 α because we are now DD −DD = PPDD + [ii+ α ]DD in a two-currency environment—but note that i is now a composite of nominal interest rates on local currency and $ debt. It is worth noting that, after dividing and multiplying the terms on the RHS of equation (1’) above by and where needed, equation (1’) can also be written: tt−1 DD (2’) tt−1 . ee Comparing (2) and (2’) gives us the formula for i, information that we shall use later, with tt tt−1 tt tt−1 translated DD −DD = PPDD + [(1 −αα)ii<sup>LL</sup> + ααii<sup>$</sup> (1 + αα) + α ]DD into local-currency equivalent terms: ii<sup>$</sup> (3) . The next step is to express nominal debt as a ratio of GDP=Y. Let d=D/Y denote this ratio. Note that ii= (1 −αα)ii<sup>LL</sup> + ααii<sup>$</sup> (1 + αα) = YYtt , where G is nominal GDP growth. Then dividing both sides of (2) by and multiplying and YYtt−1 dividing by where needed and collecting terms, we get the expression: tt (1 + GG) YY (4) YYtt−1+ <u>( )</u> , where is the ratio of the primary deficit to GDP. 1+ii+α tt tt tt−1 tt Subtracting dd = ppdd (1+GG) dd from both sides of (4) and rearranging gives: ppdd (5) ddtt−1 ≡DD+ [tt−1/YYtt−1 ) ~~]~~ . (ii−GG+α tt tt−1 tt tt−1 The second term on the RHS of (5) is the formula for “Automatic debt dynamics” in the DSA template, dd −dd = ppdd (1+GG) dd The second term on the RHS of (5) is the formula for “Automatic debt dynamics” in the DSA template, capturing the combined effects of interest rates, growth rates and exchange rates. Before proceeding, three points on notation need be made:

1. <mark>Box 3 in IMF (2022) uses ρ to denote (1+G), where G is the nominal growth rate of GDP. For expositional ease, we shall simply use (1+G).</mark>

2. <mark>z (see ‘Step 1’ in Box 3, IMF 2022) denotes the depreciation of the bilateral real exchange rate, R,</mark> e ∗ relative to the US. By definition, ≡ ~~,~~ such that dP*/P and dP/P are inflation in the US and in $ <mark>the EM measured by the GDP deflator in the US and in the EM, denoted π and π respectively.</mark> In this case, is given implicitly by the equation RR ee (1 = <u>(1 )(1+</u> $) ~~.~~<sup>3</sup> LL (1+ ) +αα ππ 3. <mark>Recall that α represents the share of $ debt at the end of the previous year. Box 3 incorrectly</mark> LL zz ≡ddRR/RR + zz) ππ <mark>gives it the time subscript t instead of (t-1) and adds a superfluous superscript, which we ignore.</mark> Noting that (1+G) equals ) , where g is real GDP growth and is inflation measured by the GDP deflator and that ) , where r is the real interest rate, equation (5) can be manipulated to get: (1 + gg)(1 + ππ<sup>LL</sup> ππ<sup>LL</sup> (1 + ii) = (1 + rr)(1 + ππ<sup>LL</sup>

> 3 Box 3 (IMF 2022) inadvertently defines z as the real exchange rate instead of as its depreciation. It also needs to specify that inflation should be measured on an end-year basis for consistency with the assumptions underpinning the DSF. We use $ and L to denote US variables and EM variables respectively instead of the f (foreign) and d (domestic) used in Box 3 of IMF (2022) because (i) d also denotes the public debt-to-GDP ratio and (ii) all foreign currency borrowing is assumed to be in US dollars.

5