Assuming a tax rate of 40 percent, compute the before-tax real interest rate and the after-tax real interest rate for each of the following cases. a. The nominal interest rate is 10 percent, and the inflation rate is 5 percent. b. The nominal interest rate is 6 percent, and the inflation rate is 2 percent. c. The nominal interest rate is 4 percent, and the inflation rate is 1 percent.

Understanding the Fisher Equation is vital when analyzing investments and savings. It's named after economist Irving Fisher, who introduced the concept. The equation acts as a bridge between the nominal interest rate and the real interest rate by factoring in inflation. Simply put, the Fisher Equation is expressed as:
\[ \text{Before-Tax Real Interest Rate} = \text{Nominal Interest Rate} - \text{Inflation Rate} \]
For example, if you have a nominal interest rate of 10 percent and the inflation rate is 5 percent, the equation shows that the before-tax real interest rate is \( 10\text{\texttt{\textasciipercent}} - 5\text{\texttt{\textasciipercent}} = 5\text{\texttt{\textasciipercent}} \). This is crucial in understanding how much the value of money changes in terms of purchasing power after considering the effects of inflation.

The nominal interest rate is the percentage increase in money you see in your bank account at the end of a given period, such as a year, without adjusting for inflation. This is the 'face value' rate offered on an investment or loan, which does not consider any external economic factors.
For instance, if you deposit \(1000 in a savings account at a 6 percent nominal interest rate, after one year, you'd theoretically have \)1060. However, this does not tell you your actual purchasing power unless you adjust it for the inflation rate, revealing the real interest rate.

The inflation rate measures how fast prices for goods and services rise over time, reducing the purchasing power of money. It is usually expressed as a percentage and can significantly impact financial decisions. If the inflation rate is high, your real interest earnings on savings could be lower than they appear, or even negative.
In our exercise, inflation rates of 5 percent, 2 percent, and 1 percent are used in the different scenarios to illustrate how inflation reduces the effective earnings from interest on deposits or loans.

The after-tax real interest rate provides a more precise picture of an investor's earnings by considering both taxes and inflation. It's what you effectively earn on an investment after paying income tax on the interest earned and adjusting for the inflation rate. The formula to calculate it takes into account the tax rate and the nominal interest rate.
\[ \text{After-Tax Real Interest Rate} = \text{Before-Tax Real Interest Rate} - (\text{Tax Rate} \times \text{Nominal Interest Rate}) \]
Applying this to the examples given, one would consider a 40 percent tax rate to find the after-tax rate of your return, which is substantially different from the before-tax rate due to the impact of taxation.

The tax rate is the percentage at which income or profits are taxed by the government. It plays a significant role in financial planning as it determines the portion of your investment returns or income that you must pay in taxes, thus reducing your actual gains.
For example, if you are subject to a 40 percent tax rate, then 40 percent of the interest you earn on an investment will be claimed by the government. When calculating the after-tax real interest rate, one subtracts the product of this tax rate and the nominal interest rate from the before-tax real interest rate, providing insight into the actual earnings after considering taxation.