Topic 4: Effects of Anticipated Inflation: The Fisher Equation


We now consider a situation where everyone knows what the inflation rate will be between this year and next. Suppose, for example that you are lending $100 for one year and you expect that the inflation rate over the next year will be 10 percent. You have to charge 10 percent interest just to cover the loss in real value of the principal during the year---the $100 you will receive on repayment at the end of the year will buy only $90 worth of goods. You also want to receive real interest on the loan at, let us say, 5 percent so you will have to charge an actual interest rate of 15 percent---5 percent real interest and 10 percent to cover expected inflation.

The person borrowing the $100 from you will be willing to pay interest at 15 percent per year because 10 of the 15 percentage points will be compensated for by the expected reduction in the amount of real goods that will have to be paid back to discharge the loan. This assumes, of course, that the borrower also expects the inflation rate to be 10 percent per year and is willing to borrow from you at a real interest rate of 5 percent per year.

In this case we can say that the contracted real rate of interest (sometimes called the "ex ante" real rate) is 5 percent per annum. The realized (or "ex post") real interest rate will depend on the rate of inflation that actually occurs, which will normally differ from the inflation rate you and the borrower are expecting.

If the inflation rate turns out to be higher than expected, the realized real interest rate will be below the contracted real interest rate and there will be a redistribution of wealth from you to the borrower. If the inflation rate turns out to be lower than expected, the ex post real interest rate will be above the ex ante real rate and you will gain at the borrower's expense. If the actual and expected inflation rates turn out to be the same, there will be no wealth redistribution effect. Only the portion of inflation or deflation that is unanticipated leads to transfers of wealth between debtors and creditors---the rest is accounted for in the rate of interest specified in the loan contract.

We can now establish the approximate relationship between nominal interest rates and the expected rate of inflation. The lender will require, and the borrower will be willing to pay, an interest rate equal to the real rate of interest that can be obtained by investing in cars, clothes, houses, etc., plus (minus) the expected rate of decline (increase) in the real value of the fixed amount that the borrower must repay due to inflation (deflation). The nominal interest rate must thus equal the real rate plus the expected rate of inflation

1.       i = r + τe

where τe is the annual rate of inflation expected during the term of the loan, and r is the contracted real interest rate. Of course, the nominal interest rate   i   is also a contracted rate. Equation 1 is called the Fisher Equation, after economist Irving Fisher (1867-1947).

A similar equation can be written to express the relationship between the nominal interest rate, the realized real interest rate and the actual rate of inflation that occurs over the term of the loan.

2.       i = rr + τ

where τ is the actual rate of inflation that occurs during the term of the loan, and rr is the realized real interest rate.

We can subtract Equation 2 from Equation 1 to obtain

                        r + τe - rr - τ = 0

from whence

2.       rr - r = τe - τ

A higher rate of inflation than expected lowers the realized real real interest rate below the contracted real interest rate. The lender loses and the borrower gains. A lower rate of inflation than expected raises the realized real interest rate above the contracted real interest rate. The borrower loses and the lender gains.

Time for a test. Before accessing the answer provided you should first come up with an answer of your own.

Question 1
Question 2
Question 3

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