# Fisher equation

In financial mathematics and economics, the Fisher equation expresses the relationship between nominal interest rates and real interest rates under inflation. Named after Irving Fisher, an American economist, it can be expressed as real interest rate ≈ nominal interest rate − inflation rate.[1][2] In more formal terms, where ${\displaystyle r}$ equals the real interest rate, ${\displaystyle i}$ equals the nominal interest rate, and ${\displaystyle \pi }$ equals the inflation rate, the Fisher equation is ${\displaystyle r=i-\pi }$. [3][4] It can also be expressed as ${\displaystyle i=r+\pi }$[5][6] or ${\displaystyle (1+i)=(1+r)(1+\pi )}$.[7][8]

## Applications

### Borrowing, lending and the time value of money

When loans are made, the amount borrowed and the repayments due to the lender are normally stated in nominal terms, before inflation. However, when inflation occurs, a dollar repaid in the future is worth less than a dollar borrowed today. To calculate the true economics of the loan, it is necessary to adjust the nominal cash flows to account for future inflation.[9]

### Inflation-indexed bonds

The Fisher equation can be used in the analysis of bonds. The real return on a bond is roughly equivalent to the nominal interest rate minus the expected inflation rate. But if actual inflation exceeds expected inflation during the life of the bond, the bondholder's real return will suffer. This risk is one of the reasons inflation-indexed bonds such as U.S. Treasury Inflation-Protected Securities were created to eliminate inflation uncertainty. Holders of indexed bonds are assured that the real cash flow of the bond (principal plus interest) will not be affected by inflation.[10]

### Cost–benefit analysis

As detailed by Steve Hanke, Philip Carver, and Paul Bugg (1975),[11] cost benefit analysis can be greatly distorted if the exact Fisher equation is not applied. Prices and interest rates must both be projected in either real or nominal terms.

### Monetary policy

The Fisher equation plays a key role in the Fisher hypothesis, which asserts that the real interest rate is unaffected by monetary policy and hence unaffected by the expected inflation rate. With a fixed real interest rate, a given percent change in the expected inflation rate will, according to the equation, necessarily be met with an equal percent change in the nominal interest rate in the same direction.[citation needed]

## References

1. ^ Cooper, Russell and John, A. Andrew. Theory and Applications of Macroeconomics. Creative Commons. Retrieved 4 April 2021.{{cite book}}: CS1 maint: multiple names: authors list (link)
2. ^ Fisher, Irving (1907). The Rate of Interest. Mansfield Centre, CT: Martino Publishing (2009); MacMillan (1907). p. Cover. ISBN 9781578987450.
3. ^ Cooper and Andrew op cit.
4. ^ Fisher op cit.
5. ^ Cooper and Andrew op cit.
6. ^ Fisher op cit.
7. ^ Cooper and Andrew op cit.
8. ^ Fisher op cit.
9. ^ Cooper and Andrew op cit.
10. ^ Neely, Michelle Clark. "The Name Is Bond—Indexed Bond". Federal Reserve Bank of St. Louis. Retrieved 5 April 2021.
11. ^ Hanke, Steve H. (1981). "Project evaluation during inflation, revisited: A solution to Turvey's relative price change problem". Water Resources Research. 17 (6): 1737–1738. Bibcode:1981WRR....17.1737H. doi:10.1029/WR017i006p01737.