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So let us agree that the fisher equation is $1 + i = (1 + r)(1 + \pi).$ Is it a definiton or good estimate. Intuition tells me this make so much sense and almost qualify as an equation that must hold. However, sources online tell me that it is a good approximation. If we accept that, can anyone suggest examples when the equation fails and why? Thank you!

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The fisher equation has its basis in the fact that the real return on an asset is the nominal return divided by the inflation rate. If you hold a bond today, it gives you back $1+r_{t+1}$ tomorrow. This is basically $\frac{1+\iota_{t}}{1+\pi_{t+1}}$ such that the promised nominal rate is deflated by the inflation rate. Rearranging terms, you get your definition. This is exact and always holds conceptually. The approximation is that $i\approx r+\pi$ which is obtained via a Taylor series expansion.

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    $\begingroup$ So in reality, it is real interest rate that gets defined? I mean, nominal interest is a given, inflation is also measured by the data in the real world, so real interest rate is the one that gets defined here, right? $\endgroup$ – Kun Oct 6 '15 at 14:52
  • $\begingroup$ Yes , you are right. $\endgroup$ – ChinG Oct 6 '15 at 14:59
  • $\begingroup$ Note it can also be defined in terms of expected inflation rather than actual inflation. This requires some model which would allow computation of that expectation, however. $\endgroup$ – transversality condition Oct 6 '15 at 20:22
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The Fisher equation is not true by definition. It is not even approximately true by definition. This view is just a misconception.

The Fisher equation is a hypothesis that is, in principle, testable. All the components in the equation are defined independently, and we can just test whether the two sides of the equation are equal.

Don't believe me? Here are the definitions of the three parts:

Real interest rate: The real interest rate $R$ is the rate at which one can transform units of consumption today into units of consumption next year. For example, suppose cookies are the only good in the economy. I offer for you to deposit a cookie with me today, and I promise you 1.03 cookies next year. That's an $R=3\%$ real interest rate.

Expected inflation: Expected inflation $E\left(\Pi\right)$ is the expected rate of increase in the consumer price index. In the cookie example, it's the expected increase in the price of a cookie.

Nominal interest rate: The nominal interest rate $I$ is the rate at which one can transform dollars today into dollars tomorrow. This would be like a bank account in the cookie economy.

The Fisher hypothesis says:

$$ 1+R = \frac{1+I}{1+E\left(\Pi\right)} $$

This could easily be false. Suppose you can deposit cookies at a 3% real interest rate, and expected inflation is 2%. That doesn't necessarily mean your bank account has to pay 5%. The bank might choose to pay less. This would be a stupid business decision, but wouldn't violate any definitions.

Now you might say, we can assume that there are many banks that compete, so dumb banks wouldn't really exist in real life. Even in that case, one could argue that the bank might choose to pay more than 5%, since you have to take the risk of inflation with a dollar deposit, while if you just bought a cookie today and deposited it, you would get paid back in cookies. So the bank needs to pay a risk premium to the customer on top of inflation compensation.

My point? The equation might be true, but it's not true by definition.

Now, even supposing the above equation is true, it is still an approximation to say that

$$ R = I - E\left(\Pi\right) $$

but this is not a bad approximation.

Why does this misconception persist? There is a tendency to define the real interest rate by using a strange version of the Fisher equation:

$$ 1+R = \frac{1+I}{1+\Pi} \quad\leftarrow\text{wrong}$$

where $\Pi$ is the actual inflation that took place between this year and next year. This is not a formula for the real interest rate, however. It is a formula for the real return you earned by putting money in your bank account and earning the nominal rate $I$. The difference? Returns are random — in this case, based on how much inflation occurred. Interest rates are fixed numbers that are observable in advance, and that goes for the real interest rate too.

By the way, the idea of a "cookie deposit" is not just a thought experiment. The U.S. Treasury sells inflation-protected bonds (TIPS) that are guaranteed to give you a certain real return on your investment. The rates on these bonds are true real interest rates.

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