Question

Rosalie the Retiree knows that when she retires in 16 years, her company will give her a one-time payment of \(\$ 20,000 .\) However, if the inflation rate is \(6 \%\) per year, how much buying power will that \(\$ 20,000\) have when measured in today's dollars? Hint: Start by calculating the rise in the price level over the 16 years.

Short answer

The buying power of \(\$ 20,000\) in today's dollars considering an inflation rate of 6% per year for 16 years is approximately \(\$ 8,414.16\).

Step-by-step solution

Understand the given information

We know that Rosalie will receive \(\$ 20,000\) in 16 years, and the annual inflation rate is 6%. We need to find the buying power of \(\$ 20,000\) in today's dollars.

Define the formula

In order to find the value of the amount in today's dollars, we should use the future value formula, which is: \[P = F \times (1 + r)^{-n}\] Where: - \(P\) = Present value (the value of the amount in today's dollars) - \(F\) = Future value (the amount Rosalie will receive in 16 years, which is \(\$ 20,000\)) - \(r\) = Annual inflation rate (6% or 0.06 as a decimal) - \(n\) = Number of years (16)

Substitute the given values in the formula

Now, we will substitute the given values into the formula: \[P = \$ 20,000 \times (1 + 0.06)^{-16}\]

Calculate the present value

To find the present value (P), we will perform the calculations inside the parentheses first, and then raise it to the power of -16, and finally, multiply it by $20,000: \[P = \$ 20,000 \times (1.06)^{-16}\] \[P = \$ 20,000 \times 0.420708\] \[P \approx \$ 8,414.16\]

Interpret the result

The buying power of \(\$ 20,000\) in today's dollars considering an inflation rate of 6% per year for 16 years is approximately \(\$ 8,414.16\). This means that the one-time payment of \(\$ 20,000\) that Rosalie will receive in 16 years will be equivalent to only \(\$ 8,414.16\) in terms of today's purchasing power.

Key concepts

Future Value Analysis

When we talk about future value analysis, we are trying to estimate what a certain amount of money will be worth in the future.
Future value helps us understand how much the current investment or savings will grow over time, given a specific interest or inflation rate.
For example, if you deposit money in a savings account today, understanding the future value tells you how much money you will have at a future date.

• The formula for future value is: \[ F = P \times (1 + r)^n \]where:
  • \( F \) = Future Value
  • \( P \) = Present Value or initial amount
  • \( r \) = Interest or inflation rate
  • \( n \) = Number of periods (years)

In Rosalie's case, even though she's expecting to receive $20,000 in the future, which looks like a substantial amount, future value analysis deems it necessary to adjust for inflation to determine its actual equivalent worth at that time.

Present Value Concept

The present value concept is the reverse of future value. It tells you how much a future sum of money is worth in today's terms.
This concept is used to account for the fact that money today is more valuable than the same amount in the future, due to potential earning opportunities and inflation.

• The present value formula is:\[ P = F \times (1 + r)^{-n} \]where:
  • \( P \) = Present Value
  • \( F \) = Future Value
  • \( r \) = Discount or inflation rate
  • \( n \) = Number of periods (years)

Calculating the present value helps Rosalie understand that her promised \(20,000 will only have the buying power of \)8,414.16 today due to the effect of inflation over time. It demonstrates that money decreases in purchasing power over the years when inflation persists.

Inflation Rate Impact

Inflation rate impacts the buying power of money by decreasing its value over time. This means that goods and services become more expensive as years go by.
The inflation rate is a crucial factor to consider when planning for future financial needs.
Inflation eats away at the value of money, making it necessary to adjust financial calculations to understand what a fixed amount of money will be able to purchase in the future.

• An inflation rate of 6% per year implies that something purchased today for $1 would cost approximately $1.06 next year.
• Over long periods, even moderate inflation can significantly increase the prices.
• In Rosalie's scenario, a 6% annual inflation rate over 16 years greatly reduces the future value of $20,000 in terms of its present buying power.

Understanding inflation's impact allows individuals to plan better by determining how much additional income or savings they would need to maintain their desired lifestyle in the future.