Compound interest – the rate of doubling

One of the key problems in understanding debt obligations is the relationship between the interest rate and the financial obligations. Compound interest is among the most commonly used phrases, yet least understood.

The concept of compound interest can be insidious. I believe the terminology “compound interest rate” is misleading; it is helpful to look at the financial obligations as the “doubling time”.

The doubling time is often estimated by the Rule of 72, and somewhat more sophisticated calculations, namely:

\mbox{doubling time} = {\log(2) \over{\log(1 + {{\mbox{rate \%} }\over100})}}

A good example is to use the latest data from CNN on the Davos 2012 conference. Here are the compound interest rates for the respective bonds (aka the “yields”), and the length of time it takes for the amount owing to double:

State Interest Rate Years to double Approximate amount owing for every $1 after 5 years (future value)
Greece 34.32% ~2.35 $5.40
Portugal 12.63% ~5.83 $1.90
Ireland 8.05% ~9 $1.50
Italy 7.02% ~10.24 $1.40
Slovakia 6.01% ~11.9 $1.30
Spain 5.35% ~13.25 $1.30
Belgium 4.34% ~16.46 $1.20
France 3.19% ~22 $1.20
Austria 3.09% ~22.65 $1.20
Finland 2.26% ~31 $1.10
Netherlands 2.19% ~31.85 $1.10
Luxembourg 2.06% ~35 $1.10
Germany 1.82% ~38.85 $1.10

One can calculate an arbitrary doubling time on Wolfram Alpha.

In other words, with some hypothetical numbers for simplicity, if Greece owes $1 billion today, in a mere 28 months it would owe $2 billion (presuming the state borrows at this rate to make its interest payments). In 56 months, it would owe $4 billion. In 7 years, it would owe $8 billion. If Greece owes 100% of GDP today, then in 8 years it would owe 800% in 7 years. At the same time, with economic catastrophe hanging over its head Greece shall experience negative growth. Clearly this divergence is unsustainable, given that most developed states rarely enjoy growth rates greater than 5% (I.e. doubling every 14 years) and often less than 3% (I.e. doubling every 24 years).

To put this in simplistic but tangible terms, the cost of apples shall double in Greece every two years, yet assuming an optimistic 3% growth wages will double every 24 years (the growth in Greece is actually -5.5% at the moment, meaning wages in aggregate shall be decreasing).