## Step 1: Define Variables - **Future Value Needed (\(FV\))**: $50,000 (in 5 years) - **Current Savings (\(PV\))**: $20,000 - **Quarterly Contribution (\(PMT\))**: $500 (at end of each quarter) - **Number of (\(n\))**:5 -Contribution Frequency**: Quarterly (\(4\) per year) - **Compounding Frequency**: Monthly (\(12\) per year) - **Unknown**: Nominal annual interest rate (\(r\)), compounded monthly --- ## Step 2: Calculate Number of Periods - **Total months**: \(5 \times 12 = 60\) - **Total quarters**: \(5 \times 4 = 20\) --- ## Step 3: Set Up Future Value Equation Since the account compounds monthly, but contributions are made quarterly, we need to: - Calculate the **future value of the initial lump sum** (\(PV\)), compounded monthly for 5 years. - Calculate the **future value of a quarterly ordinary annuity**, with each payment growing for a different number of months. Let \(i\) be the **monthly interest rate** (\(r/12\)). ### (a) Future Value of Initial Lump Sum \[ FV_{\text{lump sum}} = PV \times (1 + i)^{60} \] --- ### (b) Future Value of Quarterly Contributions Each $500 payment is deposited every 3 months (quarter), so the first payment will compound for \(60-3=57\) months, the next for \(54\), ..., the last for \(0\) months. \[ FV_{\text{annuity}} = PMT \times \sum_{k=1}^{20} (1 + i)^{60-3k} \] --- ### (c) Total Future Value \[ FV = FV_{\text{lump sum}} + FV_{\text{annuity}} = 50,000 \] --- ## Step 4: Express Everything in Terms of \(i\) \[ 20,000 \times (1+i)^{60} + 500 \times \left[(1+i)^{57} + (1+i)^{54} + \cdots + (1+i)^{0}\right] = 50,000 \] The sum is a **finite geometric series** where each term is \( (1+i)^{60-3k} \), \( k=1 \) to \( 20 \). Let’s express the series: - First term: \( (1+i)^{57} \) - Number of terms: 20 - Common ratio: \( (1+i)^{-3} \) So: \[ S = (1+i)^{57} + (1+i)^{54} + \cdots + (1+i)^{0} \] This is: \[ S = \frac{(1+i)^{57} \left[1 - \left((1+i)^{-3}\right)^{20}\right]}{1 - (1+i)^{-3}} \] \[ = \frac{(1+i)^{57} [1 - (1+i)^{-60}]}{1 - (1+i)^{-3}} \] --- ## Step 5: Combine and Set Up Final Equation \[ 20,000 (1+i)^{60} + 500 \times \frac{(1+i)^{57} [1 - (1+i)^{-60}]}{1 - (1+i)^{-3}} = 50,000 \] --- ## Step 6: Solve for \(i\) Numerically This equation cannot be solved algebraically; we must use **trial and error** or a **numerical solver**. Let’s denote \( i \) as the monthly rate, \( r = 12i \) as the nominal annual rate. Let’s try with some estimated values. --- ### (a) Guess \( i = 0.004 \) (i.e., 0.4% per month = 4.8% annual) 1. \( (1+i)^{60} = (1.004)^{60} \approx 1.270 \) 2. \( (1+i)^{57} = (1.004)^{57} \approx 1.247 \) 3. \( (1+i)^{-60} = 1/1.270 \approx 0.787 \) 4. \( (1+i)^{-3} = 1/(1.004)^3 = 1/1.012 \approx 0.988 \) 5. Denominator: \( 1 - 0.988 = 0.012 \) 6. Numerator: \( 1.247 \times (1 - 0.787) = 1.247 \times 0.213 = 0.266 \) 7. Annuity FV: \( 500 \times 0.266 / 0.012 = 500 \times 22.17 = 11,085 \) 8. Lump sum FV: \( 20,000 \times 1.270 = 25,400 \) 9. Total FV: \( 25,400 + 11,085 = 36,485 \) (**Too low**) --- ### (b) Try \( i = 0.01 \) (1% per month = 12% annual) 1. \( (1.01)^{60} \approx 1.819 \) 2. \( (1.01)^{57} \approx 1.751 \) 3. \( (1.01)^{-60} \approx 0.549 \) 4. \( (1.01)^{-3} \approx 0.971 \) 5. Denominator: \( 1 - 0.971 = 0.029 \) 6. Numerator: \( 1.751 \times (1 - 0.549) = 1.751 \times 0.451 = 0.789 \) 7. Annuity FV: \( 500 \times 0.789 / 0.029 = 500 \times 27.21 = 13,605 \) 8. Lump sum FV: \( 20,000 \times 1.819 = 36,380 \) 9. Total FV: \( 36,380 + 13,605 = 49,985 \) (**Very close**) --- ## Step 7: Calculate the Nominal Annual Rate At \( i = 0.01 \), \( r = 12 \times 0.01 = 0.12 = 12\% \) --- ## Step 8: Final Answer **Nominal annual interest rate, compounded monthly, required:** \[ \boxed{12\%} \] --- ### **Summary Table** | Step | Value | |---------------|---------------| | Needed FV | $50,000 | | Lump Sum | $20,000 | | Contribution | $500/quarter | | Time | 5 years | | Rate Needed | **12% nominal, compounded monthly** |

Certainly! Here's a detailed, step-by-step solution with all calculations: --- # Problem Recap - **Goal**: Save \$50,000 in 5 years. - **Current savings**: \$20,000. - **Quarterly contribution**: \$500. - **Interest compounded monthly**. - **Find**: The nominal annual interest rate \( r \) (compounded monthly) required to reach the goal. --- # Step 1: Define Variables - \( FV = \$50,000 \) - \( PV = \$20,000 \) - \( PMT = \$500 \) (quarterly) - Time: \( T = 5 \) years - Number of months: \( n_{months} = 5 \times 12 = 60 \) - Number of quarters: \( n_{quarters} = 5 \times 4 = 20 \) --- # Step 2: Convert to Monthly Interest Rate Let \( i \) = monthly interest rate (unknown). - Annual nominal rate: \( r \), compounded monthly, so: \[ i = \frac{r}{12} \] --- # Step 3: Calculate the Future Value of the Initial Lump Sum \[ FV_{PV} = PV \times (1 + i)^{60} \] --- # Step 4: Future Value of Quarterly Contributions Contributions are made every 3 months (\( \text{quarter} \)), so: - Each contribution of \$500 is invested for a different number of months. - The first contribution (at end of first quarter) will compound for \( 60 - 3 = 57 \) months. - The second for \( 54 \) months, ..., the last for 0 months. The sum of the future values of all contributions: \[ FV_{PMT} = PMT \times \sum_{k=1}^{20} (1 + i)^{60 - 3k} \] This is a finite geometric series with: - First term: \( (1 + i)^{57} \) (when \( k=1 \)) - Ratio: \( (1 + i)^{-3} \) Number of terms: 20 Sum: \[ S = (1 + i)^{57} \times \frac{1 - [(1 + i)^{-3}]^{20}}{1 - (1 + i)^{-3}} \] --- # Step 5: Set Up the Equation Total future value at the end: \[ FV = FV_{PV} + FV_{PMT} = 50,000 \] Expressed in terms of \( i \): \[ 20,000 \times (1 + i)^{60} + 500 \times \frac{(1 + i)^{57} \left[ 1 - (1 + i)^{-60} \right]}{1 - (1 + i)^{-3}} = 50,000 \] --- # Step 6: Numerical Solution via Trial We'll estimate \( i \): ### Trial 1: \( i = 0.004 \) (0.4% per month, 4.8% annually) - \( (1 + 0.004)^{60} \approx e^{60 \times \ln(1.004)} \approx e^{60 \times 0.00399} \approx e^{0.239} \approx 1.270 \) - \( (1 + 0.004)^{57} \approx e^{57 \times 0.00399} \approx e^{0.227} \approx 1.255 \) - \( (1 + 0.004)^{-60} \approx 1/1.270 \approx 0.787 \) - \( (1 + 0.004)^{-3} \approx 1/1.012 \approx 0.988 \) Calculate denominator: \[ 1 - 0.988 = 0.012 \] Calculate numerator: \[ 1.255 \times (1 - 0.787) = 1.255 \times 0.213 \approx 0.267 \] Calculate annuity FV: \[ 500 \times \frac{0.267}{0.012} \approx 500 \times 22.25 = 11,125 \] Calculate lump sum FV: \[ 20,000 \times 1.270 = 25,400 \] Total: \[ 25,400 + 11,125 = 36,525 \] **Result**: Too low (< \$50,000). Increase \( i \). --- ### Trial 2: \( i = 0.01 \) (1% per month, 12% annually) - \( (1.01)^{60} \approx e^{60 \times 0.00995} \approx e^{0.597} \approx 1.817 \) - \( (1.01)^{57} \approx e^{57 \times 0.00995} \approx e^{0.567} \approx 1.763 \) - \( (1.01)^{-60} \approx 1/1.817 \approx 0.550 \) - \( (1.01)^{-3} \approx 1/1.030 \approx 0.971 \) Denominator: \[ 1 - 0.971 = 0.029 \] Numerator: \[ 1.763 \times (1 - 0.550) = 1.763 \times 0.45 \approx 0.793 \] Annuity FV: \[ 500 \times \frac{0.793}{0.029} \approx 500 \times 27.34 \approx 13,670 \] Lump sum FV: \[ 20,000 \times 1.817 = 36,340 \] Total FV: \[ 36,340 + 13,670 = 50,010 \] **Result**: Very close to target. --- # **Final Calculation** - \( i \approx 0.01 \) (monthly rate) corresponds to a nominal annual rate: \[ r = 12 \times i = 12 \times 0.01 = 0.12 = 12\% \] --- # **Answer:** \[ \boxed{ \textbf{Nominal annual interest rate } r \approx \mathbf{12\%} } \] This interest rate, compounded monthly, is sufficient to grow your savings to \$50,000 in 5 years given your initial amount and quarterly contributions.