intermediate Micro

Overview of Loan Terms:
- The bank provides a loan of $50,000 to the borrower.
- The loan's monthly payment is referred to as y.
- The loan term is set for 3 years, equating to 36 months.
- The initial interest rate discussed is 10%, but a higher rate of 12% is used for calculations.
Understanding Interest Rates:
- Annual interest rate = 12%.
- Monthly interest rate = ( \frac{12}{12} ) = 1%.

Formula for Monthly Payments:
- The formula used to calculate the monthly payments is:
  $y = \frac{P \cdot r \cdot (1 + r)^{n}}{(1 + r)^{n} - 1}$
  where:
- y = monthly payment
- P = principal amount (loan amount)
- r = monthly interest rate
- n = total number of payments (months)
Applying the Formula:
- For a loan amount P = $50,000, and monthly interest rate r = 1%, over n = 36 months:
- Step 1: Calculate the monthly payment:
  - Monthly payment calculation:
  - Monthly payment interest portion = ( 50000 \times 0.01 = 500 )
- Step 2: Calculate denominator component:
  - Calculate (1 + r)^{36}:
  - ( (1 + 0.01)^{36} \approx 1.431 )
  - Calculate ( 1 - \frac{1}{(1 + r)^{36}} = 1 - \frac{1}{1.431} \approx 0.301 )
- Step 3: Calculate monthly payment:
  - ( y = \frac{500}{0.301} \approx 1661.51 )
Final Calculation:
- The resulting monthly payment the borrower must make is approximately $1,661.51.

Real-life Scenarios:
- Example of transitioning from renting to mortgage payments:
- If renting costs $3,000 a month, the potential home buyer would determine how much they can borrow to match their rental payment.
Loan Calculations for Different Situations:
- When purchasing another item (e.g., car or house), similar calculations apply to determine the maximum loan amount while keeping monthly payments equal.
Bond Pricing Concept:
- Financial implications of bonds:
- When purchasing a bond or stock, the present value is calculated based on future expected dividends or coupon payments over a set period, factoring in the rate of return.
- Formula for Present Value:
  - ( PV = \frac{C}{i} ) where C is the cash flow and i is the interest rate.
Impact of Interest Rates on Loan Payments:
- An increase in interest rates results in a decrease in present value, which decreases the worth of bonds and increases monthly payments for borrowers.

Definition of Future Value:
- Future value considers how much an investment will grow over time, calculated using the base amount plus interest over a period.
- Example formula:
  $FV = PV \times (1 + i)^{t}$
  where:
- FV = future value
- PV = present value
- i = interest rate
- t = time period in years
Example of Exponential Growth in Population:
- Utilizes similar formulas:
- Future population can be modeled using the growth rate.
  - If population growth is projected at 10%, the population next year would increase based on current value and 10%. This exponential growth continues year over year.

Using Excel for Loan Payment Calculations:
- Formula used: PMT function in Excel calculates monthly payments.
- Inputs for PMT Function:
- Monthly rate, number of periods, principal amount.
- Example Calculation:
- Monthly payment can be verified using Excel, calculated for different scenarios (e.g., mortgages, car payments).
Example Values for Mortgage Payments:
- Mortgage for borrowing $2,000,000 at 10% over 30 years yields a monthly payment of approximately $17,005.51.
- Car loan of $50,000 with an interest rate of 1% over 36 months equals a specific monthly payment, confirming all calculated figures align correctly with financial principles.
Comparison with Renting:
- Decision making based on potential mortgage payments versus renting and evaluating long-term financial implications.