Fisher equation

In economics, the Fisher equation describes the relationship, named after Irving Fisher , between nominal interest rate , real interest rate and expected inflation rate . The Fisher equation states that the nominal interest rate is roughly the sum of the real interest rate and the expected inflation rate:

{\ displaystyle i_ {t} \ approx r_ {t} + \ pi _ {t + 1} ^ {e},}

where denotes the nominal interest rate in the current period , the real interest rate and the expected inflation rate for the coming period . ${\ displaystyle i_ {t}}$ ${\ displaystyle t}$ ${\ displaystyle r_ {t}}$ ${\ displaystyle \ pi _ {t + 1} ^ {e}}$ ${\ displaystyle t + 1}$

background

In order to understand the economic background of the Fisher equation, it is helpful to consider the following thought experiment.

An example with perfect foresight

An economic operator has 100 euros at his disposal, which he would like to invest for one year. The world is free of surprises, i. H. the future development of economic variables is known to all actors (perfect foresight). Irving has various options for investing the 100 euros. One way is to lend the money at an interest rate of . For example, if the interest rate is 4% ( ), he will get his 100 euros back in one year and additional euros in interest, so that he has a total of euros. ${\ displaystyle i_ {t}}$ ${\ displaystyle i_ {t} = 0.04}$ ${\ displaystyle 100 \ times 0.04 = 4}$ ${\ displaystyle 100 \ times (1 + i_ {t}) = 104}$

Another option for Irving is to invest the 100 euros in a profitable project, such as growing wheat. We assume that a unit of wheat costs 1 euro today and that sowing and tending the field results in a yield increase of 3%, so that 103 units of wheat can be harvested in one year.

Which of the two alternatives is better? That depends on how the price of a unit of wheat will develop. Due to the perfect foresight, it is now known that a unit of wheat in one year will no longer cost 1 euro, but 1.02 euro. We are therefore assuming a rate of change in prices (inflation rate) of 2% ( ). From this it follows that in one year Irving will calculate the 103 units of wheat for 103 units of wheat times 1.02 euros per unit of wheat, i.e. H. can sell for about 105 euros (exactly 105.06 euros). So it is beneficial to invest the money in growing wheat and not lend it out. ${\ displaystyle \ pi _ {t + 1} = 0.02}$

Rational economic operators recognize this connection and under the given circumstances do not lend money for 4% interest, but prefer to invest it in wheat cultivation. Actors in need of money will now offer a higher interest rate to find someone to lend them money. An equilibrium is only achieved when both alternatives lead to the same yield after one year. As long as one of the two alternatives promises a higher yield than the other, no one will be willing to choose the other alternative. This leads to adjustment processes, such as the increase in interest rates for investments just described. Other adaptation processes are also conceivable. As long as the yield from wheat cultivation is higher than that from an investment, more and more players will invest in wheat cultivation. This increases the wheat supply in the coming period, so that the wheat price will no longer rise by 2% in the coming period, but by a smaller percentage due to the larger supply. If the inflation rate is only 1%, the equilibrium described by the Fisher equation results again: both alternatives offer an interest rate of 4%. In wheat cultivation, this 4% is composed of a 3% increase in yield (real interest rate) plus a 1% increase in price (inflation rate).

But the future is uncertain

Of course, nobody today knows exactly how high the price of wheat will be in a year. Therefore, in the current period t, an expectation must be formed about how high the wheat price will be in a year and what that means for the inflation rate. This expected inflation rate can then be used to compare the two alternatives described above, and the Fisher equation presented above results as a characterization of the economic equilibrium between nominal interest rate, real interest rate and expected inflation rate.

The ex post real interest rate

In contrast to the nominal interest rate, the real interest rate and the inflation expectations of economic operators are not observable quantities. If one nevertheless wants to determine the level of the real interest rate in a certain period t , one can approximately consider the so-called ex-post real interest rate. This results from the Fisher equation if one replaces the expected inflation rate with the actual inflation rate, which, however, is only available ex post, i.e. H. later from period t + 1 , knows: ${\ displaystyle r_ {t}}$

{\ displaystyle r_ {t} \ approx i_ {t} - \ pi _ {t + 1}}

It is assumed that there are no systematic expectation errors about the inflation rate. Alternatively, survey values can be used for the expected inflation rate or interest rate differences between bonds with an inflation hedge and bonds without an inflation hedge can be compared.

Exact formulation of the original Fisher equation (nominal arbitrage argument)

The exact version of the original Fisher equation can be derived from a nominal arbitrage argument in which the nominal returns of a financial investment and the expected nominal returns of a real investment are equated: ${\ displaystyle (1 + i_ {t})}$

{\ displaystyle (1 + i_ {t}) = (1 + r_ {t}) {\ frac {P_ {t + 1} ^ {e}} {P_ {t}}}}

${\ displaystyle P_ {t}}$ denotes the price of the real good (wheat in the example above) in the current period and the corresponding price in the following period . The superscript indicates that it is an expectation. Along with the definition of the net inflation rate, ${\ displaystyle t}$ ${\ displaystyle P_ {t + 1}}$ ${\ displaystyle t + 1}$ ${\ displaystyle e}$

{\ displaystyle \ pi _ {t + 1} = {\ frac {P_ {t + 1}} {P_ {t}}} - 1}

follows the exact Fisher equation

{\ displaystyle (1 + i_ {t}) = (1 + r_ {t}) (1+ \ pi _ {t + 1} ^ {e})}

The approximate version is obtained by multiplying the right side

{\ displaystyle (1 + i_ {t}) = 1 + r_ {t} + \ pi _ {t + 1} ^ {e} + r_ {t} \ pi _ {t + 1} ^ {e}}

and neglecting multiplication : ${\ displaystyle r_ {t} \ pi _ {t + 1} ^ {e}}$

{\ displaystyle i_ {t} \ approx r_ {t} + \ pi _ {t + 1} ^ {e}}

Both the real interest rate and the inflation rate are measured here as a decimal fraction, i.e. H. corresponds to an expected inflation rate of 2% percent , so that the product is negligibly small for realistic orders of magnitude of the real interest rate. ${\ displaystyle \ pi _ {t + 1} ^ {e} = 0.02}$ ${\ displaystyle r_ {t} \ pi _ {t + 1} ^ {e}}$

The approximate version is mainly used for illustrative and theoretical presentations. The exact version should always be used for practical calculations.

Alternative formulation of the Fisher equation (real arbitrage argument)

If the economic agents are not interested in nominal but in real returns, the arbitrage argument to be used is a real one. In this case, the expected real return on a nominal investment must be equated with net interest and the real return on a real investment: ${\ displaystyle i_ {t}}$

{\ displaystyle E_ {t} \ left [(1 + i_ {t}) {\ frac {P_ {t}} {P_ {t + 1}}} \ right] = (1 + r_ {t})}

${\ displaystyle E_ {t}}$ denotes the conditional expected value. Along with the definition of the net inflation rate, the exact Fisher equation follows in real form as

{\ displaystyle E_ {t} \ left [{\ frac {1 + i_ {t}} {1+ \ pi _ {t + 1}}} \ right] = (1 + r_ {t})}

This equation results, for example, in utility-based general equilibrium models under risk neutrality or when the covariance of the stochastic discount factor with inflation is zero. The difference to the original Fisher equation is that the effect of inflation on the expected real return is relevant and not the expected inflation per se. This distinction is based on Jensen's inequality , which affects the expected value.

This distinction is only irrelevant if the expected value of inflation occurs with certainty, i.e. H.

${\ displaystyle E_ {t} \ left [{\ frac {1} {1+ \ pi _ {t + 1}}} \ right] = {\ frac {1} {1 + E_ {t} \ pi _ { t + 1}}}}$

or "certainty equivalence" holds. The latter is the case, for example, when the Fisher equation is linearized as in the derivation of the approximate version. In this case, Jensen's inequality does not apply, so that the approximate version of the original, nominal Fisher equation and the approximate version of the real Fisher equation are identical.

literature

Irving Fisher dealt mainly with the interest theory in the following work:

Irving Fisher: The theory of interest . Macmillan, New York 1930.

The example described above is based on the presentation of the theory of the monetary interest rate in the following textbook by Rudolf Richter:

Rudolf Richter: Monetary Theory. Lecture on the basis of general equilibrium theory and institutional economics . 2nd edition, Springer, Berlin 1990, ISBN 978-3-540-51750-4 .

The Fisher equation is the subject of major textbooks on macroeconomics , monetary theory, and monetary policy .

Individual evidence

↑ Simon Benninga, Aris Protopapadakis: Real and Nominal Interest Rates under Uncertainty: The Fisher Theorem and the Term Structure . In: Journal of Political Economy . tape 91 , no. 5 , October 1, 1983, ISSN 0022-3808 , p. 856–867 , doi : 10.1086 / 261185 ( uchicago.edu [accessed April 17, 2018]).
^ Rogoff, Kenneth S. and Obstfeld, Maurice: Foundations of international macroeconomics . MIT Press, Cambridge, Mass. 1996, ISBN 0-262-15047-6 , pp. Chapter 8.7.2 .

[1] Simon Benninga, Aris Protopapadakis: Real and Nominal Interest Rates under Uncertainty: The Fisher Theorem and the Term Structure . In: Journal of Political Economy . tape 91 , no. 5 , October 1, 1983, ISSN 0022-3808 , p. 856–867 , doi : 10.1086 / 261185 ( uchicago.edu [accessed April 17, 2018]).

[2] Rogoff, Kenneth S. and Obstfeld, Maurice: Foundations of international macroeconomics . MIT Press, Cambridge, Mass. 1996, ISBN 0-262-15047-6 , pp. Chapter 8.7.2 .