Question

Two brothers each open IRAs in 2009 and plan to invest $3,000 per year for the next 30 years. John makes his first deposit on January 1, 2009, and will make all future deposits on the first day of the year. Bill makes his first deposit on December 31, 2009, and will continue to make his annual deposits on the last day of each year. At the end of 30 years, the difference in the value of the IRAs (rounded to the nearest dollar), assuming an interest rate of 7% per year, will be
A) $19,837.
B) $12,456.
C) $6,300.
D) $210.

Answer 1

Answer:

Future value of John's investment

FV = A(1+r)n+1 - (1+r)

                   r

Fv = $3,000((1 + 0.07)30+1 - (1 +0.07))

                           0.07

FV = $3,000((1.07)31 - (1.07)

                            0.07

FV = $3,000 x 101.0730414

FV = $303,219

Future value of Bill's investment

FV = A((1 + r)n - 1)

                r

FV = $3,000 ((1 + 0.07)30 - 1)

                          0.07

FV = $3,000((1.07)30 - 1)

                        0.07

FV = $3,000 x 94.46078632

FV = $283,382

The difference in the value of IRAs

= $303,219 - $283,382

= $19,837

The correct answer is A

Explanation:

In the first case, we need to apply future value of annuity due formula since deposits are made at the beginning of each year.

In the second case, we need to apply future value of an ordinary annuity formula since deposits are made at the end of each year.