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An individual is 48 years old. At the end of each month, he deposits [tex]$\$[/tex] 290[tex]$ in a retirement account that pays $[/tex]5.06 \%[tex]$ interest compounded monthly.

(a) After 12 years, 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?

For the first 12 years, the individual's deposits form an ordinary annuity because the deposits are made at the end of each period. Therefore, the formula
\[
FV = PMT \left[\frac{(1+i)^n - 1}{i}\right]
\]
should be used. After 12 years, the account does not continue to behave as an annuity and a different formula should be used.

After 12 years, the value of the account will be \$[/tex] \_\_\_\_\_\_.
(Do not round until the final answer. Then round to the nearest cent as needed.)



Answer :

GinnyAnswer
To address the question step-by-step, we'll consider two parts:

(a) the value of the account after 12 years of monthly deposits, and
(b) the value of the account when the individual reaches retirement age at 65, assuming no further deposits are made after the initial 12-year period.

### Part (a): Value of the Account After 12 Years

1. Given Data:
- Monthly deposit ([tex]\(PMT\)[/tex]): \[tex]$290 - Annual interest rate: 5.06% - Compounding frequency: Monthly - Number of years of deposit: 12 years 2. Convert annual interest rate to monthly interest rate: - Annual interest rate: \(0.0506\) - Monthly interest rate (\(i\)): \( \frac{0.0506}{12} = 0.0042166667 \) 3. Calculate the total number of deposits (n): - Years of deposits: 12 - Compounding frequency (months per year): 12 - Total number of deposits (n): \(12 \times 12 = 144\) months 4. Use the Future Value of an Ordinary Annuity formula: \[ FV = PMT \left[\frac{(1 + i)^n - 1}{i}\right] \] Plugging in the numbers: \[ FV = 290 \left[\frac{(1 + 0.0042166667)^{144} - 1}{0.0042166667}\right] \] 5. Result: - After calculating, the future value of the account after 12 years is approximately: $[/tex][tex]$\boxed{57285.48}$[/tex][tex]$ ### Part (b): Value of the Account When the Individual Reaches Age 65 1. Given Additional Data: - Current age: 48 years - Age at retirement: 65 years - No further deposits after 12 years of initial deposits 2. Calculate the number of years from the end of the deposit period to retirement: - Age after deposit period: \(48 + 12 = 60\) years - Years from end of deposits to retirement: \(65 - 60 = 5\) years 3. Convert years to months for the interest compounding calculation: - Compounding frequency (months per year): 12 - Total months from end of deposits to retirement: \(5 \times 12 = 60\) months 4. Calculate the value of the account at retirement: - Initial amount after deposits (\(FV\)) = \$[/tex]57285.48
- Monthly interest rate ([tex]\(i\)[/tex]): 0.0042166667
- Total months of compounding: 60
- Future Value after compounding:
[tex]\[ FV_{\text{retirement}} = FV \times (1 + i)^n \][/tex]
Plugging in the numbers:
[tex]\[ FV_{\text{retirement}} = 57285.48 \times (1 + 0.0042166667)^{60} \][/tex]

5. Result:
- After calculating, the future value of the account at retirement age 65 is approximately: [tex]$\boxed{73737.77}$[/tex]

Thus, the value of the account after 12 years is \[tex]$57285.48, and the value of the account when the individual reaches age 65 is \$[/tex]73737.77.