You have just purchased a Kia with a 20,000 price tag. The...

You have just purchased a Kia with a $\$ 20,000$ price tag. The dealer offers to let you pay for your car in five equal annual installments, with the first payment due in a year. a. If the dealer finances your purchase at an interest rate of $10 \%$, how much will your annual payment be? b. How much would your payment be if you had purchased a $\$ 40,000$ Camry instead of a $\$ 20,000$ Kia? c. How much would your payment be if you arranged to pay in 10 annual installments instead of $5 ?$ Is your payment cut in half? Why or why not? d. How much would your payment fall if you paid $\$ 10,000$ down at the time of purchase?

You have just purchased a Kia with a $\$ 20,000$ price tag. The dealer offers to let you pay for your car in five equal
annual installments, with the first payment due in a year.
a. If the dealer finances your purchase at an interest rate of $10 \%$, how much will your annual payment be?
b. How much would your payment be if you had purchased a $\$ 40,000$ Camry instead of a $\$ 20,000$ Kia?
c. How much would your payment be if you arranged to pay in 10 annual installments instead of $5 ?$ Is your
payment cut in half? Why or why not?
d. How much would your payment fall if you paid $\$ 10,000$ down at the time of purchase?

Key Concepts

Time Value of Money

The time value of money is the underlying principle in finance that a sum of money has different values at different points in time due to its potential earning capacity. This concept is crucial when evaluating loans and investments, as future cash flows must be discounted to their present value using an appropriate interest rate to accurately compare or determine payment amounts.

Annuity

An annuity is a series of equal payments made at regular intervals over a specified period. In financial calculations, annuities are used to structure loans, mortgages, and investments, making it essential to understand how to compute the present and future values of these repetitive cash flows, particularly when payments are scheduled at the end of each period (ordinary annuity).

Present Value of an Annuity

This concept refers to the current value of a series of future annuity payments, calculated by discounting each payment back to the present using a specific interest rate. It is key in determining the periodic payment required to amortize a loan or to equate the series of future payments to a lump-sum amount, like a car's total price after accounting for financing details.

Effect of Loan Duration and Payment Frequency

Altering the number of payment periods changes the dynamics of how interest is accrued on the principal. Extending the payment duration does not necessarily cut the payment in half; instead, it spreads the payments over more periods, and the periodic payment must be recalculated based on the changed term while still accounting for the interest accumulation over time.

Down Payment Impact

A down payment is an upfront partial payment that reduces the total amount financed by a loan. This reduction decreases the principal on which interest is calculated, leading to a lower periodic payment and total interest expense over the life of the loan. Understanding its impact is essential for budgeting and evaluating loan affordability.

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