Fisher Equation - Derivation

Derivation

Although time subscripts are sometimes omitted, the intuition behind the Fisher equation is the relationship between nominal and real interest rates, through inflation, and the percentage change in the price level between two time periods. So assume someone buys a $1 bond in period t while the interest rate is . If redeemed in period, t+1, the buyer will receive dollars. But if the price level has changed between period t and t+1, then the real value of the proceeds from the bond is therefore

From here the nominal interest rate can be solved for.

(1)

In expanded form, (1) becomes:

Assuming that both real interest rates and the inflation rate are fairly small, (perhaps on the order of several percent, although this depends on the application) is much larger than and so can be dropped, giving the final approximation:

.

More formally, this linear approximation is given by using two 1st order Taylor expansions, namely:

begin{align}
1/(1+x) &approx 1-x,\
(1+x)(1+y) &approx 1+x+y.
end{align}

Combining these yields the approximation:

and hence

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