Masako buys a house in 1999. She obtains a mortgage that carries an annual interest rate of 12 per cent, and makes payments of $880 per month. The CPI in 1999 is 100, in 2000 it is 110, and in 2001 it is 120. What is the real interest rate in 2001?

Question

BlackTom AI · Accepted Answer

The Fisher equation links nominal and real interest rates by adjusting for inflation: (1 + real) = (1 + nominal) / (1 + inflation). CPI gives the price level used to compute the inflation rate for the period of interest.

1) Relevant knowledge points
- Nominal interest rate = stated rate (12% per year here).
- Real interest rate = nominal rate adjusted for inflation (purchasing-power change).
- Year‑to‑year inflation from CPIt–1 to CPIt = (CPIt − CPIt–1) / CPIt–1. For “in 2001” we use the inflation from 2000 to 2001.

2) Step-by-step reasoning and calculations
- CPI(2000) = 110, CPI(2001) = 120.
- Inflation rate for 2001 (year-over-year) = (120 − 110) / 110 = 10 / 110 = 1/11 ≈ 0.090909 = 9.0909%.
- Nominal annual rate = 12% = 0.12.

Apply the Fisher equation exactly:
- (1 + real) = (1 + 0.12) / (1 + 0.090909) = 1.12 / (12/11) = 1.12 × (11/12).
- Compute numerator: 1.12 × 11 = 12.32; divide by 12 gives 12.32 / 12 = 1.026666...
- Therefore real = 1.026666... − 1 = 0.026666... = 2.6666...%.

Using the common linear approximation real ≈ nominal − inflation gives 0.12 − 0.090909 ≈ 0.02909 ≈ 2.91%, which is close to the exact 2.667%.

3) Why the correct answer is correct
Rounded to the nearest whole percent (as the answer choices are whole percents), the exact real rate 2.6667% rounds to 3%. The approximation nominal − inflation using 9% inflation also yields 3%, matching the provided choice.

4) Why the other options are incorrect
- 4% would require a much lower inflation (about 8%) or a different nominal rate; it does not match the CPI change.
- 21% is implausible (it is larger than the nominal rate and would imply negative inflation or conceptual error).
- 9% confuses inflation with the real interest rate (9% is the inflation rate, not the real return).

Conclusion: The real interest rate in 2001 is approximately 3 per cent, which matches the provided answer.

Masako buys a house in 1999. She obtains a mortgage that carries an annual interest rate of 12 per cent, and makes payments of $880 per month. The CPI in 1999 is 100, in 2000 it is 110, and in 2001 it is 120. What is the real interest rate in 2001?单项选择题

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