# Bank Loan Monthly Payment Calculation You borrowed **$500,000** at an **annual nominal interest rate of 5%**, to be repaid over **20 years** with **monthly payments**. We are to compute the monthly payment under: 1. **Monthly compounding** (standard amortized loan) 2. **Continuous compounding** (extra credit) --- ## 1. Monthly Compounding Payment **Loan Parameters:** - Principal \( P = \$500,000 \) - Annual nominal interest rate \( i = 5\% \) - Number of years \( n = 20 \) - Payments per year \( m = 12 \) - Total payments \( N = n \times m = 20 \times 12 = 240 \) - Periodic interest rate \( r = \frac{0.05}{12} = 0.0041667 \) ### Amortization Formula Monthly payment \( M \): \[ M = P \cdot \frac{r(1 + r)^N}{(1 + r)^N - 1} \] **Plug in values:** - \( r = 0.0041667 \) - \( N = 240 \) - \( P = 500,000 \) \[ M = 500{,}000 \cdot \frac{0.0041667 \times (1 + 0.0041667)^{240}}{(1 + 0.0041667)^{240} - 1} \] ### Calculation First, calculate \( (1 + r)^N \): \[ (1 + 0.0041667)^{240} \approx 2.71264 \] Now, \[ M = 500{,}000 \cdot \frac{0.0041667 \times 2.71264}{2.71264 - 1} \] \[ M = 500{,}000 \cdot \frac{0.0113027}{1.71264} \] \[ M = 500{,}000 \cdot 0.006601 \] \[ M \approx \$3,300.50 \] --- ## 2. Continuous Compounding Payment (Extra Credit) For a loan with **continuous compounding** and regular payments, the formula is: \[ M = P \cdot \frac{r_c \cdot e^{r_c N}}{e^{r_c N} - 1} \] Where: - \( r_c = \) continuous compounding rate per period (here, per month) - \( r_{annual} = 0.05 \Rightarrow r_c = \frac{0.05}{12} = 0.0041667 \) - \( N = 240 \) But the **effective annual rate** under continuous compounding is: \[ \text{EAR} = e^{0.05} - 1 \approx 0.05127 \] But since payments are monthly, use \( r_c = 0.05 \) per year and \( N = 20 \) years. The present value of a continuous-compounding loan with regular payments \( M \): \[ P = \int_{0}^{n} M e^{-i t} dt = M \cdot \frac{1 - e^{-i n}}{i} \] Solve for \( M \): \[ M = P \cdot \frac{i}{1 - e^{-i n}} \] Where: - \( P = 500,000 \) - \( i = 0.05 \) - \( n = 20 \) \[ M = 500{,}000 \cdot \frac{0.05}{1 - e^{-0.05 \times 20}} \] \[ e^{-1} \approx 0.36788 \] \[ 1 - 0.36788 = 0.63212 \] \[ M = 500{,}000 \cdot \frac{0.05}{0.63212} \] \[ M = 500{,}000 \cdot 0.079143 \] \[ M \approx \$3,957.15 \text{ (per year)} \] **Monthly payment:** \[ M_{monthly} = \frac{3,957.15}{12} \approx \$329.76 \] But this is for continuous payments (i.e., paid constantly, not in regular monthly intervals). If you want to compare to monthly discrete payments, the effective monthly rate is: \[ i_{eff, monthly} = e^{0.05/12} - 1 \approx 0.004184 \] Use the standard amortization formula with \( r = 0.004184 \): \[ M = 500{,}000 \cdot \frac{0.004184 \times (1 + 0.004184)^{240}}{(1 + 0.004184)^{240} - 1} \] \[ (1 + 0.004184)^{240} \approx 2.729 \] \[ M = 500{,}000 \cdot \frac{0.011419}{1.729} \] \[ M = 500{,}000 \cdot 0.006606 \] \[ M \approx \$3,303.20 \] --- ## **Summary Table** | Scenario | Monthly Payment (\$) | |---------------------------|---------------------| | Monthly Compounding | $3,300.50 | | Continuous Compounding* | $3,303.20 | *Assumes monthly payments calculated using the effective monthly rate from continuous compounding. --- ## **References** - [Present Value of an Ordinary Annuity](https://en.wikipedia.org/wiki/Amortization_calculator#Loan_calculation_formula) - [Continuous Compounding](https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding)

# Comprehensive Solution for Bank Loan Monthly Payment Calculation ## 1. Introduction Understanding the calculation of monthly payments for a loan is crucial in financial planning and management. This involves using formulas derived from the principles of time value of money, which considers how interest accrues on borrowed funds over time. Two scenarios will be explored in this case: monthly compounding and continuous compounding. ### Explanation The introduction establishes the foundational concepts necessary for loan payment calculations, emphasizing the significance of the time value of money and different compounding methods. This understanding is essential for accurately calculating the expected monthly payment for the loan in question. --- ## 2. Presentation of Relevant Formulas and Representation of Given Data ### Given Data - **Principal (P)**: $500,000 - **Annual nominal interest rate (i)**: 5% or 0.05 - **Loan term (n)**: 20 years - **Payments per year (m)**: 12 (monthly payments) ### Formulas 1. **Monthly Payment (M) for Monthly Compounding:** \[ M = P \cdot \frac{r(1 + r)^N}{(1 + r)^N - 1} \] Where: - \( r = \frac{i}{m} \) (monthly interest rate) - \( N = n \times m \) (total payments) 2. **Monthly Payment (M) for Continuous Compounding:** \[ M = P \cdot \frac{i}{1 - e^{-i n}} \] Where: - \( n \) is in years. ### Explanation The provided formulas are essential for solving the loan payment problem. The first formula calculates monthly payments based on amortization, allowing borrowers to understand their repayment obligations. The second formula for continuous compounding demonstrates a different approach to interest calculations, which is relevant for the extra credit portion of the question. --- ## 3. A Detailed Step-by-Step Solution ### Monthly Compounding Payment Calculation 1. **Calculate the monthly interest rate (r)**: \[ r = \frac{0.05}{12} = 0.0041667 \] 2. **Calculate the total number of payments (N)**: \[ N = 20 \times 12 = 240 \] 3. **Apply the amortization formula**: \[ M = 500{,}000 \cdot \frac{0.0041667(1 + 0.0041667)^{240}}{(1 + 0.0041667)^{240} - 1} \] 4. **Calculate \((1 + r)^N\)**: \[ (1 + 0.0041667)^{240} \approx 2.71264 \] 5. **Substitute back into the payment formula**: \[ M = 500{,}000 \cdot \frac{0.0041667 \times 2.71264}{2.71264 - 1} \] \[ M = 500{,}000 \cdot \frac{0.0113027}{1.71264} \] \[ M \approx 500{,}000 \cdot 0.006601 \approx 3,300.50 \] ### Explanation This step-by-step breakdown illustrates how to compute the monthly payment for a loan with monthly compounding. Each calculation builds upon the previous one, demonstrating the relationships among principal, interest rate, and loan term. --- ### Continuous Compounding Payment Calculation (Extra Credit) 1. **Apply the continuous compounding formula**: \[ M = 500{,}000 \cdot \frac{0.05}{1 - e^{-0.05 \times 20}} \] 2. **Calculate \(e^{-1}\)**: \[ e^{-1} \approx 0.36788 \] \[ 1 - 0.36788 = 0.63212 \] 3. **Substitute and calculate**: \[ M = 500{,}000 \cdot \frac{0.05}{0.63212} \approx 500{,}000 \cdot 0.079143 \approx 39,571.50 \] 4. **Convert annual payment to monthly**: \[ M_{monthly} = \frac{39,571.50}{12} \approx 3,297.63 \] ### Explanation This calculation demonstrates how to derive the monthly payment for a loan with continuous compounding. By understanding the continuous compounding formula, one can see how it differs from standard amortization and how it impacts the monthly payment. --- ## Conclusion The monthly payment for a loan of $500,000 at an interest rate of 5% over 20 years is approximately **$3,300.50** when using monthly compounding. For continuous compounding, the effective monthly payment is about **$3,297.63**. This analysis highlights the impact of different compounding methods on loan repayment amounts, underscoring the importance of understanding the nuances of financial calculations. --- ## Summary Table | Scenario | Monthly Payment (\$) | |---------------------------|---------------------| | Monthly Compounding | $3,300.50 | | Continuous Compounding | $3,297.63 | --- ## References - [Amortization Calculator](https://en.wikipedia.org/wiki/Amortization_calculator#Loan_calculation_formula) - [Continuous Compounding](https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding)