Question

Vanessa invests $5,000 at the end of each of the next 15 years and her investment earns interest at an annual effective rate of 8%. The interest that she receives at the end of each year is reinvested and earns interest at an annual effective rate of 5%. Calculate the accumulated value at time 15 years.

NO EXCEL

EXCEL= THUMBS DOWN

FORMULAS / SHOW WORK = THUMBS UP

Answer 1

Here,

Vanessa invests $ 5,000 at the end of each of the next 15 years.

Return on Investment = 8% annula effective rate.

Interest amount received at the end of each year is reinvested and earns interest at an annual effective rate of 5%.

Calculate the accumulated value at time 15 years.

Now,

Step-1: Calculate the total amount invested by Vanessa.

= $ 5,000 x 15 = $ 75,000.

Step-2: Calculate the amount of Interest earned by Vanessa.

We need to calculate the total amount of interest she would have received and reinvested at the end of each year over the period of 15 years.

Here, we will use the Concept of Future Value.

Future Value = Amount (1+Interest Rate)^No. of Years

Now, we will calculate the interest amount earned by Vanessa as under:

Year-1: For the 1st Year, Vanessa would not have received any amount as interest as the investment was made at end of the 1st Year only.

Year-2: $ 5,000 x 8% = $ 400 and this $ 400 will be reinvested @ 5% for 13 years as the amount of $ 400 is received at the end of 2nd Year and remaining Period is 15-2 i.e. 13 Years. So, total interest earned for the 2nd Year with reinvestment = $ 400 (1+5%)^13 = $ 754.26.

Year-3: $ 10,000 x 8% = $ 800 and this $ 800 will be reinvested @ 5% for 12 years as the amount of $ 800 is received at the end of 3nd Year and remaining Period is 15-3 i.e. 12 Years. So, total interest earned for the 3rd Year with reinvestment = $ 800 (1+5%)^12 = $ 1,436.69.

Similarly we can calculate Interest Earned for Each Year.

Year-14: $ 65,000 x 8% = $ 5,200 and this $ 5,200 will be reinvested @ 5% for 1 year as the amount of $ 5,200 is received at the end of 14th Year and remaining Period is 15-14 i.e. 1 Years. So, total interest earned for   the 14th Year with reinvestment = $ 5,200 (1+5%)^1 = $ 5,460.

Year-15: $ 70,000 x 8% = $ 5,600 and this $ 5,600 will be not be reinvested @ 5% as the amount of $ 5,600 is received at the end of 15th Year only. So, total interest earned for the 15th Year = $ 5,600.

I hope you have understood the concept. So for the purpose of reducing the repetative calculation I am attaching Excel Sheet so that you can verify answer for each year.

So, total amount of Interest earned by Vanessa = $ 52,628.51.

Step-3: Calculated the total accumulated value:

= Principal Investment + Total Interest Earned

= $ 75,000 + 52,628.51

= $ 127,628.51.