Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Inflation comparisons In 1974 , Johnny Miller won 8 tournaments on the PGA tour and accumulated \(\$ 353,022\) in official season earnings. In 1999 , Tiger Woods accumulated \(\$ 6,616,585\) with a similar record. (a) Suppose the monthly inflation rate from 1974 to 1999 was \(0.0025(3 \% / \mathrm{yr})\). Use the compound interest formula to estimate the equivalent value of Miller's winnings in the year 1999 . Compare your answer with that from an inflation calculation on the web (e.g., bls.gov/cpi/home.htm). (b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. (c) What type of function did you use in part (a)? part (b)?

Step-by-Step Solution

Answer

(a) $353,022 in 1974 is $788,501 in 1999; Woods's $6,616,585 far exceeds this. (b) About 1.8% monthly interest needed. (c) Exponential function used in both parts.

Step 1: Understanding Compound Interest Formula

To determine the future value of Johnny Miller's earnings with inflation, we'll use the compound interest formula: \[ A = P (1 + r)^n \] where \(A\) is the future value, \(P\) is the initial principal balance (Miller's earnings in 1974), \(r\) is the monthly interest rate, and \(n\) is the total number of compounding periods. Here, \( r = 0.0025 \) and \( n = 12 \times 25 \) since we are compounding monthly over 25 years.

Step 2: Calculating Future Value for Miller's Earnings

We substitute the values into the formula:\[ A = 353,022 \times (1 + 0.0025)^{12 \times 25} \] First, calculate the compounding factor:\( 1 + 0.0025 = 1.0025 \).Find \( (1.0025)^{300} \) using a calculator and then multiply by \( 353,022 \). This gives us the inflated value of Miller's 1974 earnings in 1999.

Step 3: Comparing Equivalent Value

Perform the calculation step (using a scientific calculator): \[ A \approx 353,022 \times (1.0025)^{300} \approx 788,501 \] So, according to this calculation, Miller's \\(353,022 in 1974 would be approximately \\)788,501 in 1999 dollars. Now compare this amount to Woods's \$6,616,585.

Step 4: Calculating Required Interest Rate for Equivalence

For Miller's 1974 earnings of \(353,022 to equal Woods's 1999 earnings of \)6,616,585, use the compound interest formula rearranged for \( r \): \[ 6,616,585 = 353,022 (1 + r)^{300} \] Solve for \( r \): \( \frac{6,616,585}{353,022} = (1 + r)^{300} \), then take the 300th root to isolate \( 1 + r \), yielding \( r \), which is the annual rate needed.

Step 5: Solving for the Required Rate

Calculate the left-hand side:\[ \frac{6,616,585}{353,022} \approx 18.75. \]Then, find the 300th root:\[ (1 + r) = 18.75^{1/300} \approx 1.018. \]Thus, \( r \approx 0.018 \) or approximately 1.8% per month, which translates to a higher annual rate, considering compounding.

Step 6: Identifying Function Types

For part (a), the function used is an exponential growth function given by the compound interest formula. For part (b), the same exponential model is used to solve for the growth rate that would equate the future values.

Key Concepts

Exponential Growth

Exponential growth is a critical concept in understanding how values change over time, especially in the realm of finance. Imagine you have an amount of money growing steadily over time. Instead of growing by a fixed increment, it grows by a percentage of its current size. This is what we call exponential growth. It accumulates gradually but accelerates rapidly as time passes.

• Financial Growth: In the context of compound interest, the principal amount increases based on a fixed percentage applied repeatedly over a series of compounding periods. This application is straightforward yet powerful over the long term.
• Mathematical Representation: The mathematical model for exponential growth used in compound interest is expressed through the formula: \[ A = P (1 + r)^n \]Here, A represents the future amount, P is the principal, r is the interest rate per period, and n is the total number of periods.

Understanding exponential growth is essential for grasping the differences in calculated future values, such as how inflation impacts earnings over decades.

Inflation Rate

The inflation rate is a measure that reflects the decline in purchasing power of a nation’s currency. When prices rise, each unit of currency buys fewer goods and services, which is essentially inflation. It’s crucial to consider in financial calculations because it helps us understand true value over time.

• Affect on Value: For example, Johnny Miller's earnings needed adjustment for inflation to compare with 1999 values sensibly. The monthly inflation rate used was 0.0025, implying a 3% annual inflation.
• Impact on Purchasing Power: As inflation compounds over time, what seemed like significant past earnings may buy much less in the future due to persistent inflationary pressure.

Recognizing inflation is vital when assessing long-term financial equivalence to ensure accurate comparison between different time periods.

Future Value Calculation

Calculating the future value of a sum of money involves estimating its value at a future date, taking into account interest rates or inflation over time. It's a fundamental calculation in finance, used to assess investment value or compare monetary amounts from different periods.

• Compound Interest Formula: To calculate the future value in the context of compound interest and inflation, the formula used is \[ A = P (1 + r)^n \], where A stands for the future value obtained.
• Steps Involved: The process involves substituting known values (such as initial earnings, interest rate, and duration) into the formula to arrive at the future equivalent value. For instance, Johnny Miller's earnings from 1974 were adjusted for 1999 by applying this formula.

Being comfortable with future value calculations is essential for making informed financial decisions and determining the true worth of time-separated values.

Annual Interest Rate

The annual interest rate is a key component in the compound interest formula. It determines how much money will grow annually, considering the compounding effect. Knowing this rate can influence investment decisions and help estimate how long money will take to grow to a desired amount.

• Importance: An accurate calculation of the annual interest rate is necessary to assess investments' growth potential and compare financial products.
• Rearranging the Formula: Sometimes, you may need to find this rate for desired future values. By rearranging the formula, you can solve for the interest rate, giving insights into necessary growth rates for target achievements.
• Practical Application: In the example exercise, calculating the equivalent of Miller's earnings compared to Woods's required determining the implicit annual rate of interest using the compound interest setup.

Understanding annual interest rates is crucial for making strategic financial plans, ensuring realistic growth expectations, and gauging the impact of compounding periods on future values.