The Bureau of Labor Statistics has a website (www.bls.gov) that contains a Consumer Price Index inflation calculator. This calculator uses the average CPI to adjust the purchasing power of money over different periods of time. The CPI index value has been calculated every year since 1913. The calculator indicates that
23,930,909 in 2016. What was the average inflation rate over this 103-year time period? The average inflation rate was [insert answer] per year.
The Correct Answer and Explanation is:
To calculate the average inflation rate over a 103-year period using the future value, present value, and number of years, the compound interest formula is used:FV=PV×(1+r)tFV = PV \times (1 + r)^tFV=PV×(1+r)t
Where:
- FVFVFV = Future Value = $23,930,909
- PVPVPV = Present Value = $1 (assumed to represent 1913 value)
- ttt = 103 years (from 1913 to 2016)
- rrr = average annual inflation rate (unknown)
Solving for rrr:23,930,909=1×(1+r)10323,930,909 = 1 \times (1 + r)^{103}23,930,909=1×(1+r)103
Take the natural logarithm on both sides:ln(23,930,909)=103ln(1+r)\ln(23,930,909) = 103 \ln(1 + r)ln(23,930,909)=103ln(1+r)ln(23,930,909)≈16.987\ln(23,930,909) \approx 16.987ln(23,930,909)≈16.98716.987103=ln(1+r)\frac{16.987}{103} = \ln(1 + r)10316.987=ln(1+r)ln(1+r)≈0.1649\ln(1 + r) \approx 0.1649ln(1+r)≈0.1649
Exponentiating both sides:1+r=e0.1649≈1.17931 + r = e^{0.1649} \approx 1.17931+r=e0.1649≈1.1793r≈0.1793 or 17.93%r \approx 0.1793 \text{ or } 17.93\%r≈0.1793 or 17.93%
The average inflation rate was approximately 17.93% per year.
Explanation
Inflation refers to the general increase in prices over time, reducing the purchasing power of money. When analyzing how much an amount of money in the past would be worth in the future (or vice versa), financial formulas that involve compound interest are typically used. In this case, the Consumer Price Index (CPI) calculator from the Bureau of Labor Statistics indicates that one dollar in 1913 had the same purchasing power as $23,930,909 in 2016. This massive increase reflects inflation over a 103-year period.
To determine the average annual rate of inflation, the compound interest formula is used. This formula assumes consistent percentage growth year over year. The known values include the final amount ($23,930,909), the starting value ($1), and the time period (103 years). The goal is to solve for the rate that, when compounded annually, transforms one dollar into nearly 24 million dollars.
By isolating the rate in the formula and applying logarithms, the exact average rate is found to be about 17.93% per year. This value may seem surprisingly high because it represents an average across a long time span where compounding causes exponential growth. In reality, actual yearly inflation fluctuates and is usually far lower annually. However, the enormous final value amplifies the apparent average when viewed over a full century.
This result highlights the significant impact of inflation on money’s value over time and underlines the importance of long-term financial planning, especially in the context of saving and investing.
