The Math Of European Debt

If you have 1000 Euros you want to invest, what do you do?

One thing you could do is buy government bonds (government debt). Of course, you have a bunch of different governments to choose from. For each country, there’s an effective interest rate you earn on their bonds. If the rate on 1 year bonds is 5%, and the end of 1 year, you have €1050 in your hand.

So, which country do you choose?

Let’s suppose country A has bonds that yield 5% and country B has bonds that yield 10%. If you buy A’s bonds, you end up with €1050. If you buy B’s, you end up with €1100. The exchange rate between Euros and Euros is fixed at 1:1, so B’s bonds are, it seems, a clearly a better choice. As investors buy B’s bonds, the price goes up, which has the effect of bringing the yield down. Eventually, it should not matter what country’s bonds you buy, the yields should all be the same.

Sure enough, in September 2008,

  • The yield on German 1 year bonds was about 4.1%
  • The yield on Irish 1 year bonds was about 4.1%
  • The yield on Greek 1 year bonds was about 4.2%

Now, things have changed. Today,

  • The yield on German 1 year bonds is about 0.07%
  • The yield on Irish 1 year bonds is about 4.9%
  • The yield on Greek 1 year bonds is about 1143%

Why the difference? Why aren’t the bond yields the same? Why would anyone buy the low-yielding bonds over the higher-yielding options?

The reason is simple : would you put your money in Greek bonds now?

If you put your €1000 money in German 1 year bonds, at the end of a year you’ll have €1000.70. The same €1000 in Greek bonds will leave you with €12430 in a year’s time, or €0 if Greece defaults on its debt.

Apparently, this slim chance at €11430 profit in Greece, combined with the greater chance of a €1000 loss, is worth the same as the relative certainty of the €0.70 profit from German bonds. If the probability of Greece defaulting is p, that means (1-p) x 11430+p x (-1000) = 0.7, so p=91.95%.

You could say that “the market” reckons there’s only a 1 in 12 chance that Greece will pay out on its 1 year debt.

You can do the same calculation on any other country. If you put €1000 into Irish bonds, you get €49 profit with probability 1-p, and lose €1000 with probability p. This gamble is worth €0.70, so (1-p) x 49 + p x (-1000) = 0.7, and p=4.6%. Apparently investors are confident Ireland will pay up, but still think there’s a 1 in 21 chance they won’t.

You can do this calculation with other Eurozone countries too. You can even do it with 10-year bond yields (which are easier to find online). For example, the German 10-year yield is 1.4%, the Spanish yield is about 5%. If you bought €1000 of these bonds and held them for a year, you’d get €14 from Germany, or a chance at €50 from Spain. Why don’t you try to calculate the risk of a Spanish default yourself? (Hint : the answer is about 3.4%)

These probabilities have to be taken with a pinch of salt, but the difference in bond yields in Europe arises because investors aren’t really sure if European countries will honor their debts. After all, they can’t print their own money, they depend on the European Central Bank. And the ECB has looked extremely unwilling to print extra Euros on demand to cover the debt of member countries.

Hence, because the ECB is unwilling to promise to back their member countries, there’s a non-zero probability of these countries defaulting, which in turn affects the bond yields in a simple, predictable, mathematical way.