Let's carefully evaluate the problem step-by-step to determine if the person received an after-tax real interest rate of 5%.
1. Nominal Interest Rate:
- The nominal interest rate received: [tex]\( 4\% \text{ or } 0.04 \)[/tex]
2. Tax Rate:
- The tax rate is: [tex]\( 25\% \text{ or } 0.25 \)[/tex]
3. Inflation Rate:
- The inflation rate is: [tex]\( -2\% \text{ or } -0.02 \)[/tex] (deflation)
4. After-Tax Nominal Interest Rate:
- The after-tax nominal interest rate can be calculated by reducing the nominal interest rate due to taxes:
[tex]\[
\text{After-Tax Nominal Interest Rate} = \text{Nominal Interest Rate} \times (1 - \text{Tax Rate})
\][/tex]
- Plugging in the values:
[tex]\[
\text{After-Tax Nominal Interest Rate} = 0.04 \times (1 - 0.25) = 0.04 \times 0.75 = 0.03 \text{ or } 3\%
\][/tex]
5. Real Interest Rate:
- The formula to find the real interest rate is:
[tex]\[
\text{Real Interest Rate} = \frac{(1 + \text{After-Tax Nominal Interest Rate})}{(1 + \text{Inflation Rate})} - 1
\][/tex]
- Substituting in the values we have:
[tex]\[
\text{Real Interest Rate} = \frac{(1 + 0.03)}{(1 - 0.02)} - 1 = \frac{1.03}{0.98} - 1 \approx 1.0510204081632653 - 1 = 0.0510204081632653 \text{ or } 5.102\%
\][/tex]
6. Comparison with Given Real Interest Rate:
- The calculated real interest rate of approximately [tex]\( 5.102\% \)[/tex] is slightly above [tex]\( 5\% \)[/tex].
Therefore, saying that the person received an after-tax real interest rate of [tex]\( 5\% \)[/tex] is false. The correct real interest rate received is closer to [tex]\( 5.102\% \)[/tex].
