# Topic 3: Effects of Unanticipated Inflation: Realized Interest Rates

Suppose that you borrow \$1000 to be paid back in a lump sum at 5 percent annual interest in 5 years. When the loan comes due you will have to pay back
\$1000 (1 + r)n = \$1000 (1.05)5 = \$1276.28.

Suppose, however, that during this five year interval the price level doubles. The amount of goods you will have to give up to pay back this loan will be only half as much as the required dollar payment would indicate because a dollar will be worth only half as much in real terms.

In real terms, therefore, you will have to pay back only \$638.14, valued in goods at the time the money was borrowed. From your point of view, this is great. You will have borrowed \$1000 worth of real goods for five years and paid back less than \$640 of real goods. The interest rate you will have actually paid (as opposed to the 5 percent you contracted for) can be found by substituting the real amount borrowed and the real amount repaid into the formula

A0 = A1 ⁄ (1 + r)n

to yield   \$1000 = \$638.14 ⁄ (1 + r)5   from which   r = [(638.14/1000)1/5] - 1 = -.085   or minus 8.5 percent.

Although you contracted to pay the individual you borrowed from an interest rate of 5 percent, that person actually ended up paying you interest at the rate of 8.5 percent per year to borrow from her. The unexpected inflation will have redistributed real wealth from your creditor to you. You are contracting to pay \$1276.28 in five years and will actually pay \$638.14 in real terms. Or to put it differently, the \$1276.28 which you pay back will buy only half as many goods as was expected when the loan was made. The present value of that difference, discounted at the market interest rate of 5 percent, is   \$638.14 ⁄ (1 + r)5 = \$500.   This figure should not be surprising because a doubling of the price level is wiping out half of the value of the loan measured in current dollars.

Of course, were you to lend \$1000 for five years to somebody under circumstances where the price level unexpectedly doubles during the term of the loan, the person you lend to will gain \$500, in current dollars, at your expense. Unexpected inflation always redistributes wealth from people who have contracted to receive fixed nominal amounts in the future to the people who have contracted to pay those fixed nominal amounts.

Unanticipated deflation has the opposite effect. The person who has borrowed a fixed nominal amount has to pay back with dollars that are worth more in terms of real goods than he/she had contracted for, and the person who is the creditor is paid an amount that is greater in real terms than anticipated so that wealth is redistributed from debtors to creditors.

When there an unanticipated movement of the price level, the real interest rate actually realized on loans will be different from the interest rate at which the loan contract was made. This realized real interest rate can be calculated quite easily in the case of one-year loans. Suppose that you borrow \$100 for one year at an agreed upon interest rate of 6 percent and that, contrary to what both you and the lender expect when you make the loan, the inflation rate turns out to be 3 percent rather than zero percent. You pay the lender \$106 at the end of the year, but that \$106 is worth only about \$103 because \$100 will buy \$3 less goods at the end of the year. The interest rate actually realized is thus only about \$3/\$100 or 3 percent. The realized real interest rate is thus approximately equal to the contracted interest rate minus the actual rate of inflation.

Unanticipated inflation has very important wealth redistribution effects in an economy. People who take out mortgages in order to buy houses at fixed interest rates end up paying back less in real terms than they had contracted for---wealth is redistributed from banks and other financial institutions (or, more correctly, the people that own them) to homeowners with mortgages. Individuals who retire on pensions that are fixed in nominal amount will find the values of those pensions in terms of the goods they buy eroded as years pass---in this case the redistribution is from pensioners to the owners of insurance companies and other financial institutions that have contracted to pay them fixed dollar amounts.

Unanticipated inflation has additional distribution effects that work through the tax system. Many countries have progressive income tax systems under which high income people pay higher percentage rates of tax on additions to their income than low income people. Since income tax rates are based on nominal rather than real income, the inflation of nominal incomes will put people in higher tax brackets, increasing the amount of taxes paid to the government in greater proportion than the increase in the price level. Real tax payments and the availability of resources to the government will therefore increase. Unless the tax system is modified to take it into account, fully anticipated inflation has these same effects.

Also, business firms are normally allowed to deduct allowances for the depreciation of their capital from their revenues in order to calculate the profits on which they must pay taxes to the government. Depreciation allowances are usually calculated as a percentage of historic cost. When inflation occurs and all nominal prices and wages rise together, these depreciation allowances based on the prices prevailing when the capital was purchased do not rise. The real value of firm's cost deductions therefore declines, leading to a rise in real taxes paid. Since the real costs of replacing depreciated capital are not lowered by inflation and real taxes increase, firm's real profits fall. Different industries will be affected differently depending upon the particular rules the tax law requires them to follow in calculating their depreciation allowances.

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