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# Nominal and Real Rates of Interest If you invest \$1,000 in a bank deposit offering an interest rate of 10 percent, the bank promises to pay you \$1,100 at the end of the year. But it makes no promises about what the \$1,100 will buy. That will depend on the rate of inflation over the year. If the prices of goods and services increase by more than 10 percent, you have lost ground in terms of the goods that you can buy.

Several indexes are used to track the general level of prices. The best known is the Consumer Price Index, or CPI, which measures the number of dollars that it takes to pay for a typical family’s purchases. The change in the CPI from one year to the next measures the rate of inflation.

Economists sometimes talk about current, or nominal, dollars versus constant, or real, dollars. For example, the nominal cash flow from your one-year bank deposit is \$1,100. But suppose prices of goods rise over the year by 6 percent; then each dollar will buy you 6 percent less goods next year than it does today. So at the end of the year \$1,100 will buy the same quantity of goods as 1,100/1.06 = \$1,037.74 today. The nominal payoff on the deposit is \$1,100, but the real payoff is only \$1,037.74. The general formula for converting nominal cash flows at a future period t to real cash flows is

For example, if you were to invest that \$1,000 for 20 years at 10 percent, your future nominal payoff would be 1,000×1.120 =\$6,727.50, but with an inflation rate of 6 percent a year, the real value of that payoff would be 6,727.50/1.0620 = \$2,097.67. In other words, you will have roughly six times as many dollars as you have today, but you will be able to buy only twice as many goods.

When the bank quotes you a 10 percent interest rate, it is quoting a nominal interest rate. The rate tells you how rapidly your money will grow. However, with an inflation rate of 6 percent you are only 3.774 percent better off at the end of the year than at the start.

Thus, we could say, “The bank account offers a 10 percent nominal rate of return,” or “It offers a 3.774 percent expected real rate of return.” Note that the nominal rate is certain but the real rate is only expected. The actual real rate cannot be calculated until the end of the year arrives and the inflation rate is known. The 10 percent nominal rate of return, with 6 percent inflation, translates into a 3.774 percent real rate of return. The formula for calculating the real rate of return is

1 + rnominal = (1 + rreal)(1+ inflation rate)

= 1 + rreal + inflation rate + (rreal)(inflation rate)

In our example,

1.10 = 1.03774 × 1.06

Taken from Principles of Corporate Finance by Richard A. Brealey and Stewart C. Myers (7th edition)