# How can a sovereign debt ratio be high, but debt-servicing ratio be low?

I'm quoting this May 22 2020 interview with NYU Stern Prof. Nouriel Roubini.

### In other words, you’re saying that because of structural weaknesses in the economy, even modest inflation would be crisis-inducing because key economic actors are dependent on near-zero interest rates?

For the last decade, debt-to-GDP ratios in the U.S. and globally have been rising. And debts were rising for corporations and households as well. But we survived this, because, while debt ratios were high, debt-servicing ratios were low, since we had zero percent policy rates and long rates close to zero — or, in Europe and Japan, negative. But the second the Fed started to hike rates, there was panic.

I never studied accounting and don't understand the embolding. Investopedia defines Debt to Asset ratio = Total debt/Total assets. I can't find the definition of sovereign debt-servicing ratio on Google! The one for family on Investopedia appears irrelevant.

Debt-Service Coverage Ratio $$= \dfrac{\text{Revenue - Certain Operating Expenses}}{\text{Total Debt Service also known as Current debt obligations}}$$

Your definition is not applicable to governments. The definitions used depend on governments.

For governments with external debt (Foreign currency debt), there are standard definitions: link to IMF external debt document.

But for a country like the United States without external debt, there does not appear to be a standard definition. People generally refer to the interest payments/GDP ratio. Since the concept is not particularly meaningful for a floating currency sovereign, there is no pressure to develop a standard definition.

This is because the cost of servicing debt depends critically on the interest rate on that debt. When interest payments are low the cost of servicing debt will be lower as well.

A general debt servicing ratio can be expressed following this Fed study as:

$$DSR = \frac{i_t}{1- (1+i_t)^{-s}} \frac{D_t}{Y_t}$$

While the authors from Fed apply this debt service ratio for households - the formula is generally valid and can be easily repurposed and applied to government or any other organization. It is not necessary the best way of measuring servicing ratio for government (as explained below), but its a good way to understand the intuition.

You can think of the $$D_t/Y_t$$ part as debt to GDP ratio. The $$i$$ is the interest on debt, and $$s$$ is the average remaining maturity. To see how even with high debt ratios the debt servicing cost ratios can be low lets consider few examples. Lets suppose for the sake of argument the $$s$$ is always $$1$$ to simplify calculations. Moreover, lets assume that country has debt-to-GDP ratio of $$200\%$$ (thats quite high - according to the World Bank data it would be higher than the most recent debt-to-GDP of any country they report - although the WB data are not very recent). Now lets consider 4 scenarios, scenario 1 where interest rate is $$5\%$$, scenario 2 where interest is $$1\%$$, scenario 3 where $$i$$ is $$0\%$$ and 5 where it is $$-0.5\%$$.

In scenario 1 the $$DSR$$ will be: $$\frac{0.05}{1- (1+0.05)^{-1}} 2 = 2.1$$

in scenario 2 the $$DSR$$ will be $$\frac{0.01}{1- (1+0.01)^{-1}} 2 = 2.01$$

In scenario 3 the $$DSR$$ will be $$\frac{0}{1-(1)^{-1}} 2$$ which is undefined as it would require you to divide but zero, but the limit of this function from both sides approaches 2 which is the debt to GDP ratio - implying there is no extra cost of servicing the debt beside paying down the principal

In scenario 4 the $$DSR$$ will be $$\frac{-0.005}{1- (1-0.005)^{-1}} 2= 1.99$$ implying that the servicing debt is now even cheaper than just paying down the principal - you get paid for borrowing.

So as you can see that low interest rates make debt servicing progressively cheaper and the rate is non linear - moreover, the above examples all include average debt to maturity 1 for simplicity but actually allowing for it to be higher the effect would be even stronger due to compounding.

Hence while the debt-to-GDP ratio can be very high, borrowing at low (or even negative) interest rates results in progressively lower debt servicing ratios.

As mentioned above the first formula is not necessary the most appropriate for measuring the servicing costs of government. A common debt servicing ratios applied to sovereigns reported by institutions such as World Bank is servicing cost over export, defined as:

$$X_t = S_{t}/E_{t}$$

And this ratio is more relevant when it comes to the government since as for example Frank and Cline point out:

the rationale for the use of the debt service ratio [authors reference the above service costs to export ratio] as an indicator of a country’s debt servicing capacity is that an increase in the debt service ratio indicates increased vulnerability to foreign exchange crises. Any shortfall in foreign exchange earnings or capital imports which is not covered by exchange reserves must be met by reducing imports: since aebt service is a fixed obligation, the higher the debt service ratio, the greater is the relative burden on import reduction for a given shortfall in foreign exchange.

This being said, I think that the intuition behind the quote can be better understood from that repurposed household debt service cost ratios and caries to this. Moreover, as correctly pointed by @BrianRomanchuck in his +1 answer there is no standard definition for countries with no external debt which is important thing to keep in mind.

However, all above being said from the context of the article I dont think the author had in mind any accounting debt servicing ratio. Based on the context of the article "debt-servicing ratios were low" it seems to me that the author just wanted to say the relative costs of servicing debts are low thanks to the low interest rate which logically also implies that any servicing cost ratio (which will presumably have servicing cost in numerator) will be low. The article is an interview, I think the author just could not think of better way to say it on the spot, I would not read too much into it.

• Fed is not an acronym, it is a shorthand for “Federal Reserve”, and so should not be capitalised as FED. May 26 '20 at 12:13
• @BrianRomanchuk you are of course completely right - my bad I corrected the grammar
– 1muflon1
May 26 '20 at 12:15