Final answer:
To calculate the future value at year 20 of company income, first determine the future value of $8000 annually for the first 5 years using the future value formula. Then, calculate the decreasing annual income from year 6 to 20, taking into account the 15% annual decrease and 10% interest rate, and sum all future values for a final total.
Step-by-step explanation:
To calculate the future value in year 20 of a company's income that is $8000 for the first 5 years and then decreases by 15% per year starting from year 6, we will use the future value formula for multiple cash flows. We'll calculate the future value of the cash flows from years 1 through 5 separately from the cash flows from years 6 through 20.
The future value of an income of $8000 per year for the first 5 years is simply calculated by the annual income multiplied by the sum of the individual future value factors for each year:
Future Value (FV) = Payment × [(1 + Interest rate)^numbers of years t]
For the first 5 years, the future value of the income is:
FV = $8000 × [(1 + 0.10)^5 + (1 + 0.10)^4 + (1 + 0.10)^3 + (1 + 0.10)^2 + (1 + 0.10)^1]
Beginning from year 6, the income decreases by 15% per year. To handle the decreasing income, we must calculate each year's income separately, applying the 15% decrease from the previous year's income and then calculating its future value for year 20.
For years 6 to 20, the income for any year 'n' is $8000 × (1 - 0.15)^(n-5). To find the future value of each income at year 20, we calculate:
FV = Income at year n × (1 + 0.10)^(20-n)
After computing the future values for each year, we sum all these calculated future values from years 6 to 20 to get the total future value at year 20 for these years. Lastly, we add the total future value from the first 5 years to the sum from years 6 to 20 for the final answer.