Let's break down the solution step by step: --- ### **A. Annual Investment Needed** **Given:** - Current age = **25 years** - Retirement age = **65 years** - Years to invest = **65 - 25 = 40 years** - Future Value (FV) = **$1,000,000** - Interest rate (r) = **7% per year** - Initial investment = **$0** - Payments are made **at the end of each year** (ordinary annuity) #### **Step 1: Use the Future Value of an Ordinary Annuity Formula** \[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) = Future Value ($1,000,000) - \( PMT \) = Payment per period (what we want to find) - \( r \) = annual interest rate (7% = 0.07) - \( n \) = number of periods (40 years) #### **Step 2: Plug in the Values** \[ 1,000,000 = PMT \times \frac{(1 + 0.07)^{40} - 1}{0.07} \] Calculate \((1 + 0.07)^{40}\): \[ (1.07)^{40} \approx 14.974 \] \[ \frac{(1.07)^{40} - 1}{0.07} = \frac{14.974 - 1}{0.07} = \frac{13.974}{0.07} \approx 199.63 \] #### **Step 3: Solve for PMT** \[ 1,000,000 = PMT \times 199.63 \] \[ PMT = \frac{1,000,000}{199.63} \approx \$5,010.38 \] --- ### **Final Answer for A:** **You need to invest approximately \$5,010.38 at the end of each year to reach \$1,000,000 by age 65.** --- ### **B. Monthly Investment Needed** Now, let's find the monthly investment amount. #### **Step 1: Convert Parameters to Monthly** - Number of months: \( 40 \times 12 = 480 \) - Monthly interest rate: \( \frac{0.07}{12} = 0.005833 \) #### **Step 2: Use the Future Value of an Ordinary Annuity Formula (monthly)** \[ FV = PMT \times \frac{(1 + r_{monthly})^{n_{months}} - 1}{r_{monthly}} \] Plug in the values: \[ 1,000,000 = PMT \times \frac{(1 + 0.005833)^{480} - 1}{0.005833} \] Calculate \((1.005833)^{480}\): \[ (1.005833)^{480} \approx 16.445 \] \[ \frac{16.445 - 1}{0.005833} = \frac{15.445}{0.005833} \approx 2,647.81 \] #### **Step 3: Solve for PMT** \[ PMT = \frac{1,000,000}{2,647.81} \approx \$377.60 \] --- ### **Final Answer for B:** **You need to invest approximately \$377.60 at the end of each month to reach \$1,000,000 by age 65.** --- #### **Summary Table** | Investment Frequency | Amount to Invest | |---------------------|------------------| | Yearly | \$5,010.38 | | Monthly | \$377.60 | Let me know if you need further explanation or calculations!

Let's carefully recalculate Part B (monthly investment needed) with detailed steps: --- ### **Part B: How much money would you need to invest each month?** **Goal:** To accumulate **\$1,000,000** in **40 years** (from age 25 to 65) with **monthly investments** at an interest rate of **7% annually**, compounded monthly. --- ### **Given Data:** - Future Value, \( FV = \$1,000,000 \) - Annual interest rate, \( r_{annual} = 7\% = 0.07 \) - Total years, \( n_{years} = 40 \) - Number of months, \( n_{months} = 40 \times 12 = 480 \) - Monthly interest rate, \( r_{monthly} = \frac{0.07}{12} \approx 0.0058333 \) --- ### **Step 1: Use the Future Value of an Ordinary Annuity Formula (monthly payments):** \[ FV = PMT \times \frac{(1 + r_{monthly})^{n_{months}} - 1}{r_{monthly}} \] Where: - \( PMT \) = monthly payment (what we want to find) --- ### **Step 2: Calculate \( (1 + r_{monthly})^{n_{months}} \):** \[ (1 + 0.0058333)^{480} \] Calculate: \[ \ln(1 + 0.0058333) \approx 0.005820 \] \[ 0.005820 \times 480 \approx 2.7936 \] Exponentiate: \[ e^{2.7936} \approx 16.36 \] Alternatively, directly: \[ (1.0058333)^{480} \approx 16.36 \] --- ### **Step 3: Calculate the annuity factor:** \[ \frac{(1 + r_{monthly})^{n_{months}} - 1}{r_{monthly}} = \frac{16.36 - 1}{0.0058333} = \frac{15.36}{0.0058333} \approx 2,634.48 \] --- ### **Step 4: Solve for \( PMT \):** \[ PMT = \frac{FV}{\text{Annuity factor}} = \frac{1,000,000}{2,634.48} \approx \$379.76 \] --- ### **Final Answer:** **You need to invest approximately \$379.76 each month at the end of the month to reach \$1,000,000 in 40 years at 7% annual interest compounded monthly.** --- ### **Summary:** | Investment Type | Monthly Investment | |------------------|---------------------| | Monthly | **\$379.76** | --- **Note:** Slight variations in the exponential calculations can occur depending on rounding. Using precise calculator functions or financial calculator software can improve accuracy. Let me know if you'd like further clarification!