Question

At the end of each​ quarter, a 52​-year-old individual puts ​$1300 in a retirement account that pays 9.1​% interest compounded quarterly.
​(a) When the individual reaches age​ 60, what is the value of the​ account?
​(b) If no further deposits or withdrawals are made to the​ account, what is the value of the account when the individual reaches age​ 65?
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Part 1
​(a) When the individual reaches age​ 60, the value of the account will be ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)
​(b) When the individual reaches age​ 65, the value of the account will be ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

Answer 1

Given: deposit R = $1300 each quarter, nominal rate 9.1% compounded quarterly so i = 0.091/4 = 0.02275 per quarter.

(a) From age 52 to 60 is 8 years = 32 quarters. Future value of an ordinary quarterly annuity:
FV60 = R * [((1+i)^n − 1) / i], with n = 32.

Compute (1 + i)^32 ≈ 1.02275^32 ≈ 2.054108832
So FV60 ≈ 1300 * (2.054108832 − 1) / 0.02275
FV60 ≈ 1300 * (1.054108832 / 0.02275) ≈ 1300 * 46.3352999 ≈ $60,235.89

(b) From age 60 to 65 is 5 years = 20 quarters. The balance grows with no deposits:
FV65 = FV60 * (1+i)^{20} = 60,235.89 * 1.02275^{20}

Compute 1.02275^{20} ≈ 1.56815745
FV65 ≈ 60,235.89 * 1.56815745 ≈ $94,459.36

Answers:
(a) $60,235.89
(b) $94,459.36