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The maths behind the madness

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via The maths behind the madness.

Our interactive guide to reducing government debt

AFTER a brief hiatus, the euro crisis is back. Spain’s ten-year borrowing costs have been hitting highs not seen since the European Central Bank began injecting €1 trillion ($1.3 trillion) of cheap money into the banking system in December. The country managed to auction €2.5 billion worth of bonds on April 19th, but paid 5.7% for ten-year money, 30 basis points higher than the last auction in January.

At what point does the debt of a nation become unsustainable? Our interactive graphic above shows the IMF’s latest forecasts for government gross debt as a percentage of GDP through to 2017. It also allows you to input your own long-term assumptions to project the likely path of debt out to 2020.

There are two things that matter in government-debt dynamics: the difference between real interest rates and GDP growth (r-g), and the primary budget balance as a % of GDP (ie, before interest payments). In any given period the debt stock grows by the existing debt stock (d) multiplied by r-g, less the primary budget balance (p).

The simple r-g assumption is one of the most important in debt dynamics: an r-g of greater than zero (when interest rates are greater than GDP growth) means that the debt stock increases over time. An r-g of less than zero causes it to fall.

Our interactive model uses the nominal interest rate (i)—approximately equivalent to the ten-year bond yield—and allows you to input your own inflation rate, ∏. Inflation helps reduce the total debt stock over time, by reducing the real value of debt. In our model and using approximations, r-g becomes i – ∏ – g. The greater the inflation rate, the lower r-g becomes.

The second consideration is the primary budget balance. A primary budget surplus causes the debt stock to fall, by allowing the government to pay off some of the existing debt. A primary deficit needs to be financed by further borrowing. As European peripheral countries have found out to their cost, interest rates increase when governments run large budget deficits, and as they do it becomes increasingly difficult to reduce r-g to a sustainable level.

In reality, these variables are all related. When inflation rises, for instance, bondholders will expect a higher nominal interest rate on new debt. If a country runs a larger primary surplus, the interest rate it is forced to pay may fall. Adjustments in countries’ deficits will also affect their growth rates. To keep matters simple, we have ignored these interactions. Our calculator shows the evolution of a government’s debt stock based directly on the values for inflation, growth, interest rates and the primary deficit that you determine.

Just small tweaks in these figures can have striking results. Using the average IMF forecasts for 2012-17, Spain’s debt is expected to rise from 69% of GDP in 2011 to 107% in 2020. If you increase the interest rate by just two percentage points to 7.1%, the debt level rises to 124%.

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