What is the accumulated future value of those annual amounts over 21 years if the annual earnings were invested regularly and provided a 4 percent return each year?

Question

Throughout this book, we will present a continuing narrative about Victor and Maria Hernandez. Following is a brief description of the lives of this couple.

Victor and Maria, both in their late 30s, have two children: Jacob, age 13, and Nicholas, age 15. Victor has had a long sales career with a retail appliance store in Fargo, North Dakota earning $53,000 annually. Maria works as a medical records assistant earning $29,000.

1. Victor and Maria regularly buy and sell a number of items on eBay, Craig's List, and through the free community newspaper, from which they earn about $5,000 each year. What is the accumulated future value of those annual amounts over 21 years if the annual earnings were invested regularly and provided a 4 percent return each year? (Hint: Use Appendix A.3.) Round your answer to nearest dollar. Round Future value of Series of Equal Amounts in intermediate calculations to four decimal places
2. What would Victor and Maria's annual income be after 21 years if they both received an average 5 percent raise over their current $82,000 salary ($53,000 + $29,000) every year? (Hint: Use Appendix A.1.) Round your answer to nearest dollar. Round Future value of a Single Amount in intermediate calculations to four decimal places.

Answer 1

1. It is given that they earn $5000/year with 4% interest. After one year, they would have made $5000*1.04, and the year after, $5000 + ($5000*1.04)*1.04. One could think of the first term as income for that year, and the second term as accumulated investment earnings. Thus over T years we can define the sequence,

a(-1) = 0

a(t) = W_0 + a(t-1)*(1+r)

where W_0 is initial wealth, r is interest rate. So the future value is then simply ∑a(t) for t =0,...,T.

1. 
   
2. It is given they earn $82,000/year, and receive a 5% raise every year. This means after one year, they will earn $82,000*1.05, and the year after, ($82,000*1.05)*1.05. Over T years,

FV = W_0*(1+r)**T

where this value represents their income T years from now.