Fisher equation

The Fisher equation (also called the Fisher effect and the Fisher hypothesis) is an equation that describes the relationship between the inflation rate , nominal and real interest rates :

i=r+\pi

{\ displaystyle i = r + \ pi}

{\ displaystyle i = r + \ pi}

,

Where $i$ ${\ displaystyle i}$ $i$ - nominal interest rate;

r

{\ displaystyle r}

r

- real interest rate;

\pi

{\ displaystyle \ pi}

\ pi

- inflation rate.

Named after Irving Fisher .

Economic meaning

The equation in an approximate form (see # Conclusion ) describes a phenomenon called the Fisher effect. The effect is that the nominal interest rate can change for two reasons:

due to changes in the real interest rate;
due to changes in the rate of inflation.

The price level in the economy changes over time. The investor also places money at interest for a certain period. Therefore, he is interested in obtaining not only a certain income, but also to compensate for the fall in the purchasing power of money in the future.

For example, if an investor deposited a sum of money in a bank account that brings 10% per annum, then the nominal rate will be 10%. With an inflation rate of 6%, the real rate will be only 4%.

The equation can be used as the actual rate of inflation $\pi$ ${\ displaystyle \ pi}$ , and its expected value $\pi ^{e}$ ${\ displaystyle \ pi ^ {e}}$ . In the first case, the formula allows you to calculate the real rate based on the received nominal yield and actual price increases. In the second case, the investor can determine for himself the expected nominal yield based on the predicted values.

Conclusion

The equation in the above form is approximate. It is performed more precisely, the smaller the modulo value $r$ ${\ displaystyle r}$ and $\pi$ ${\ displaystyle \ pi}$ . Therefore, from a mathematical point of view, it is correct to write approximate equality:

i\approx r+\pi

{\ displaystyle i \ approx r + \ pi}

,

The exact equation is as follows:

1+i=(1+r)\times (1+\pi )

{\ displaystyle 1 + i = (1 + r) \ times (1+ \ pi)}

If you open the brackets, you get the following record:

1+i=1+r+\pi +r\pi

{\ displaystyle 1 + i = 1 + r + \ pi + r \ pi}

or

i=r+\pi +r\pi

{\ displaystyle i = r + \ pi + r \ pi}

In terms of mathematical analysis, if $r$ ${\ displaystyle r}$ and $\pi$ ${\ displaystyle \ pi}$ tend to zero, then the product $r\pi$ ${\ displaystyle r \ pi}$ is infinitesimal of a higher order. Therefore, for small (modulo) values $r$ ${\ displaystyle r}$ and $\pi$ ${\ displaystyle \ pi}$ work $r\pi$ ${\ displaystyle r \ pi}$ can be neglected. As a result, the approximate record mentioned above is obtained.

Let, for example, $r=\pi =1\%$ ${\ displaystyle r = \ pi = 1 \%}$ . Then the sum of these values is 2%, and the product is 0.01%. If you take $r=\pi =10\%$ ${\ displaystyle r = \ pi = 10 \%}$ , then the amount will be equal to 20%, and the product is 1%. Thus, with increasing values, the error in the calculations becomes more and more.

The exact record can also be converted to the following form proposed by Fisher:

r={\frac {1+i}{1+\pi }}-1={\frac {i-\pi }{1+\pi }}

{\ displaystyle r = {\ frac {1 + i} {1+ \ pi}} - 1 = {\ frac {i- \ pi} {1+ \ pi}}}

In trivial cases for $\pi =0$ ${\ displaystyle \ pi = 0}$ or $\pi =i$ ${\ displaystyle \ pi = i}$ both formulas (exact and approximate) give the same value of the real interest rate.

Literature

Vechkanov G. C., Vechkanova G. R. Macroeconomics. - SPb. : Peter, 2008. - S. 55. - (Series “Short Course”). - 3,000 copies. - ISBN 978-5-91180-108-3 .
Chetyrkin E. M. Financial mathematics. - M .: Case, 2005 .-- S. 400.

Fisher equation

Economic meaning

Conclusion

Literature

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