Final answer:
The purchase price of a car with a monthly payment of $350 over 5 years at a 4.5% interest rate compounded daily can be calculated using the present value formula for an annuity. An effective monthly interest rate is first determined from the annual rate, to account for daily compounding, and then used in the present value formula. This shows the total cost considering interest over the loan term.
Step-by-step explanation:
To calculate the purchase price of a car with a monthly payment of $350 over 5 years at a nominal interest rate of 4.5% compounded daily, we need to use the present value formula for an annuity:
PV = Pmt * [(1 - (1 + i)^(-n)) / i]
where:
- PV is the present value, or the amount that a series of future payments is worth now
- Pmt is the periodic payment amount
- i is the interest rate per period
- n is the total number of payments or periods
For daily compounding, the effective interest rate per month needs to be calculated because payments are monthly.
The formula to convert a nominal annual interest rate to an effective periodic interest rate is:
i_effective = (1 + r/n)^(n*p) - 1
where:
- r is the nominal annual interest rate
- n is the number of times the interest is compounded per year
- p is the number of periods per year
In this situation:
- r = 0.045 (4.5% expressed as a decimal)
- n = 365 (daily compounding)
- p = 12 (monthly payments)
Once the effective monthly interest rate is calculated, it can be plugged into the present value formula along with the number of payments (n = 60 for 5 years) to find the purchase price of the car (PV). This approach is critical for understanding the implications for car financing, as it determines how much one is actually paying for a vehicle when considering the interest over the term of the loan. The longer the loan term or the higher the interest rate, the more a consumer will pay in the long run.