Question 15 of \( 15,5 \operatorname{tep} 1 \) of 1 11/15 Correct Suppose you wish to retire at the age of 60 with \( \$ 80,000 \) in savings. Determine your monthly payment into an IPA if the APR is \( 5.5 \% \) compounded monthly and you begin making payments at 30 years old. Round your answer to the nearest cent, if necessary. Formulas Present Value with Compound \( \quad P V=\frac{A}{\left(1+\frac{r}{n}\right)^{2 m}} \) Annuity Payment Amount Future Value of an Annuity \( \mathrm{PMT}=\mathrm{FV} \cdot \frac{\left(\frac{r}{n}\right)}{\left[\left(1+\frac{r}{n}\right)^{e t}-1\right]} \) \[ \mathrm{FV}=\mathrm{PMT} \cdot \frac{\left[\left(1+\frac{r}{n}\right)^{n}-1\right]}{\left(\frac{r}{n}\right)} \] Keypad

To determine the **monthly payment** needed to accumulate **\$80,000** by the age of **60**, starting at **30 years old**, with an **APR of 5.5%** compounded monthly, we'll use the **Future Value of an Annuity** formula. ### Given: - **Future Value (FV):** \$80,000 - **Annual Percentage Rate (APR):** 5.5% - **Compounded Monthly:** \( n = 12 \) times per year - **Investment Period:** 30 years (\( t = 30 \)) - **Total Number of Payments (N):** \( 30 \times 12 = 360 \) months ### Formula: \[ \text{FV} = \text{PMT} \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \] Where: - \( \text{PMT} \) = Monthly payment - \( r \) = Annual interest rate (decimal) - \( n \) = Number of compounding periods per year - \( t \) = Number of years ### Steps to Calculate PMT: 1. **Convert APR to Monthly Rate:** \[ \frac{r}{n} = \frac{5.5\%}{12} = 0.0045833333 \] 2. **Calculate \(\left(1 + \frac{r}{n}\right)^{nt}\):** \[ \left(1 + 0.0045833333\right)^{360} \approx 5.4608 \] 3. **Rearrange the Future Value Formula to Solve for PMT:** \[ \text{PMT} = \frac{\text{FV} \times \frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nt} - 1} \] 4. **Plug in the Values:** \[ \text{PMT} = \frac{80,000 \times 0.0045833333}{5.4608 - 1} \approx \frac{366.6666664}{4.4608} \approx 82.14 \] ### **Final Answer:** You need to make a **monthly payment of approximately \$82.14** into the investment account to reach your retirement savings goal of \$80,000 by age 60.