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

In summary, considering a constant inflation rate of \(6\%\) over the 16 years, Rosalie's one-time payment of \(\$20,000\) will have a future value of \(\$59,926\). However, its buying power in today's dollars will be \(\$20,000\), which will be significantly reduced due to inflation. To maintain the same buying power, Rosalie will need more than \(\$59,926\) in her retirement savings.

Step-by-step solution

Understand the Model Formula for Inflation

The formula to find the future value (FV) considering a constant inflation rate is given by: FV = PV * (1 + r)^n Where: - FV is the future value (value after inflation) - PV is the present value (value in today's dollars) - r is the inflation rate per period (in this case, per year) - n is the number of periods (years until retirement, in this case) We have PV as \(\$ 20,000\), r as \(6\%\) (or \(0.06\) when expressed as a decimal), and n as 16 years. We will use the same formula to convert the future value back to the present value after finding it.

Calculate the Future Value Considering Inflation

First, we need to calculate the future value (FV) of Rosalie's retirement payment, considering the inflation rate. Using the formula (FV = PV * (1 + r)^n): FV = \(\$ 20,000\) * (1 + 0.06)^16 Calculating (1 + 0.06)^16: (1 + 0.06)^16 = 2.9963 (rounded to 4 decimal places) Now, multiply this by the present value: FV = \(\$ 20,000\) * 2.9963 = \(\$ 59,926\) (rounded to the nearest dollar) So, considering a constant inflation rate of \(6\%\), the \(\$20,000\) payment will have a future value of \(\$ 59,926\) when Rosalie retires.

Convert the Future Value Back to the Present Value

Now that we have found the future value, we need to convert it back to the present value to determine its buying power in today's dollars. To do this, we need to rearrange the formula we used before (FV = PV * (1 + r)^n): PV = FV / (1 + r)^n We have already calculated FV as \(\$ 59,926\) and have r as \(0.06\) and n as 16 years: PV = \(\$ 59,926\) / (1 + 0.06)^16 We calculated (1 + 0.06)^16 in Step 2: PV = \(\$ 59,926\) / 2.9963 = \(\$ 20,000\) Amazingly, the present value of that payment remains the same at \(\$20,000\). However, it is important to note that the buying power of \(\$20,000\) in 16 years is drastically reduced due to inflation. The same amount will have a lesser value in terms of real purchasing power in the future. To maintain the same buying power, Rosalie will need more than \(\$59,926\) in her retirement savings.

Key concepts

Present Value

In financial terms, the present value (PV) refers to the worth of an expected sum of money at the current time, based on a specific discount rate or interest rate. The concept of present value allows individuals and businesses to understand how much a future sum of money is valued today.
It’s an essential tool used in finance to assess investments, calculate loan payments, or as in the case of Rosalie, to understand how much her future payment is worth right now.
The formula used for calculating present value comes in handy to adapt future cash flows, facilitating the comparison of the worth of money across different time periods:

• PV = \( \text{FV} / (1 + r)^n \)
• FV is the future value or the sum that one plans to receive in the future,
• \( r \) is the interest or discount rate per period,
• \( n \) is the number of periods until the value is realized.

This formula allows Rosalie or anyone trying to determine the present value to make informed financial and investment decisions.

Future Value Calculation

Future value (FV) takes today's monetary value and projects how much that money will grow to be in the future, given a certain interest or inflation rate over time. In essence, it involves calculating the increased worth of a sum over time due to the effect of compounding interest or inflation growth.
The basic formula used for calculating the future value when inflation is taken into account is:

• FV = \( \text{PV} \times (1 + r)^n \)
• Where \( \text{PV} \) is the present value,
• \( r \) represents the rate of interest or inflation,
• \( n \) is the total number of periods the money will be subject to this rate.

In the scenario presented in the original exercise with Rosalie, understanding future value is crucial to determining how much her retirement money will increase in its nominal value by the time it is received given a steady inflation rate of 6%. The calculated future value suggests that Rosalie’s payment will grow but, importantly, the real purchasing power might not increase due to inflation.

Buying Power

Buying power indicates the real goods and services that can be purchased with a specific sum of money. As time progresses, inflation can erode one’s buying power, making it inexpensive in the future than it is today.
Inflation adjustments, such as Rosalie's retirement payment, show that while the nominal value might increase in the future, the actual value – what you can buy with it – might not increase. This aspect is crucial for anyone planning for the future, like retirees or investors, to understand.
The calculation of buying power in today's terms provides a tangible representation of how inflation will affect their savings or income over time:

• Buying power in the future: Calculated using present value to deduce how much current buying capability future money will have.
• Inflation diminishes the value: Even though the money amount may rise, the volume of goods or services that can be bought may decline.
• Long-term financial planning: A crucial component for individuals planning for future savings or retirement based on a certain level of comfort in the present.

In Rosalie's situation, her substantial future sum of $20,000 will not go as far in buying power because prices will have risen over the 16 years, implying the same amount will cover fewer items than it does today. Understanding buying power helps individuals set realistic savings goals and manage future financial outcomes.