Question

Suppose you place \(\$ 10,000\) in a retirement fund that earns a nominal interest rate of 8 percent. If you expect inflation to be 5 percent or lower, then you are expecting to earn a real interest rate of at least a. 1.6 percent. b. 3 percent. c. 4 percent. d. 5 percent.

Short answer

The real interest rate we expect to earn is at least 1.6%.

Step-by-step solution

Insert Given Values to The Equation

Insert the given nominal interest rate and inflation rate into the equation: \(1 + 0.08 = (1 + r_{real})(1 + 0.05)\)

Simplify the Equation

Simplify the equation by performing the arithmetic on the left side: \(1.08 = (1 + r_{real})(1.05)\)

Solve for Real Interest Rate

To solve for \(r_{real}\), we will first divide both sides of the equation by 1.05: \(r_{real} + 1 = \frac{1.08}{1.05}\) Now subtract 1 from both sides of the equation: \(r_{real} = \frac{1.08}{1.05} - 1\) Calculate the right side of the equation: \(r_{real} = 0.0286\)

Convert Real Interest Rate to Percentage

To express this number as a percentage, multiply it by 100: \(r_{real} = 0.0286 \times 100 = 2.86\%\) Since the real interest rate is approximately 2.86%, the closest option is: a. 1.6 percent. So the real interest rate we expect to earn is at least 1.6%.

Key concepts

Nominal Interest Rate

When you deposit money into a savings account or invest in a bond, the rate of interest advertised by the bank or institution is typically the nominal interest rate. This rate does not take into account inflation or the general increase in prices over time. In essence, the nominal interest rate is the raw rate of return on your investment without adjusting for any external economic factors.

For example, if you were told that your savings account offers an 8% nominal interest rate, this means that for every \(100 you invest, you'd earn \)8 over the period specified, assuming no compounding. However, this does not reflect your true earning power if the cost of goods and services is rising due to inflation. To understand how much you're genuinely earning, you need to consider the real interest rate, which adjusts for inflation.

Inflation Rate

Inflation is the rate at which the general level of prices for goods and services is rising, and subsequently, the purchasing power of currency is falling. Central banks attempt to limit inflation, and avoid deflation, in order to keep the economy running smoothly.

For instance, if the inflation rate is 5%, it means that on average, the cost of goods and services has increased by 5% over a set period, typically a year. If you have money in a savings account or investment, inflation can erode the value of your earnings. When your nominal interest rate is close to the inflation rate, your real earnings could be near zero or even negative—in other words, you could be losing purchasing power even though your investment's balance is growing.

Real Interest Rate Calculation

To determine the true increase in your purchasing power from an investment, you need to calculate the real interest rate. The real interest rate essentially measures the actual return on your investment after adjusting for the effect of inflation. The calculation can be expressed with the formula:
\[ r_{real} = \frac{1 + r_{nominal}}{1 + r_{inflation}} - 1 \]
where \( r_{nominal} \) is the nominal interest rate and \( r_{inflation} \) is the inflation rate. By rearranging the formula from the given example, we can solve for the real interest rate to see the actual earnings on an investment. If a fund offers a nominal interest rate of 8% and the inflation rate is 5%, the calculation would be as follows:
\[ r_{real} = \frac{1.08}{1.05} - 1 \]
Upon solving, you would find that the real interest rate is approximately 2.86%. This means that after accounting for inflation, your investment will earn slightly less than 3% in real terms over the period in question. Understanding this figure is crucial for making smart investment decisions and ensuring that your money retains its purchasing power over time.