Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=30,000+500 t, r=7 \%, t_{1}=6 \text { years } $$

Step-by-Step Solution

Answer

The present value of the income stream over six years, given an annual inflation rate of 7%, should be the sum of the present values of the increasing annual incomes.

Step 1: Understand the income stream

The income stream \(c = 30,000 + 500 t\) means that the annual income increases by $500 every year. The total income over \(t_{1}\) years will therefore be the sum of these increasing annual incomes.

Step 2: Convert the inflation rate into a decimal

The annual inflation rate \(r = 7\%\) is equivalent to 0.07 when expressed as a decimal.

Step 3: Calculate the present value of each year’s income

The present value of the income for each year is calculated by dividing that year’s income by (1 + r) raised to the power of that year number. This reflects the decrease in purchasing power of the income due to inflation.

Step 4: Sum the present values

To find the present value of the entire income stream over \(t_{1}\) years, sum the present values of the income for each year. This will give the total present value of the income stream, considering the annual inflation rate.

Key Concepts

Income Stream Analysis

In finance, analyzing an income stream is crucial when deciding on investments or planning for future financial needs. The given exercise involves an income stream of the form \(c = 30,000 + 500t\). This means an annual income that starts at \(30,000 and increases by \)500 each year. It is important to understand the following concepts when analyzing such an income stream:

• Base Income: The income starts from a base amount, here \(30,000. This is the income for year 0, or the starting point.
• Incremental Increase: Each year the income increases by a fixed amount, in this case \)500, reflecting raises or increased revenue.
• Total Income Calculation: Over a specific period, such as the 6 years given, calculate each year's income by substituting \(t\) with the corresponding year number and adding these for the total.

Understanding these aspects helps in predicting future income and making informed financial decisions.

Inflation Impact on Value

Inflation reduces the purchasing power of money over time, meaning what you can buy with a dollar today will not be the same in the future. It is pivotal to factor how inflation impacts the value of future income:

• Inflaction Rate Conversion: The inflation rate must be converted to a decimal for calculations. Here, 7% becomes 0.07.
• Real Value Decline: Over 6 years, inflation reduces the real value of each dollar earned, necessitating present value calculations to determine how much future earnings are worth today.
• Yearly Adjustments: Calculate the reduced value for each year's income by dividing the income by \((1 + r)^t\), where \(r\) is the inflation rate, and \(t\) is the year number.

This shows the real value and purchasing power once the inflation impact is considered.

Time Value of Money

The concept of the time value of money (TVM) is that a dollar today is worth more than a dollar in the future. This key principle in finance is crucial for understanding present value calculations.

• Present Value Principle: TVM allows us to determine the present value of a future income stream by considering inflation and other factors.
• Discounting Future Income: Each future income amount is discounted back to the present value using the formula given by \(PV = \frac{FV}{(1 + r)^t}\), where \(PV\) is present value, \(FV\) is future value, \(r\) is the discount rate, and \(t\) is time in years.
• Total Present Value Calculation: To find the present value over multiple years, sum up the present values of each year. This ensures you understand what all future earnings are worth in today's terms.

Applying this concept helps investors and individuals plan effectively for future financial situations.