You are going to buy a $18,000 car. The dealer offers you $2000 cash back of nothing down and 0% financing for 60 months. If you take the financing, starting in one month you will pay one-sixtieth of $18,000 each month for 60 months. In some sense that is not really "0% financing" because you could have bought the care for, effectively, $16,000 and you will be making $18,000 in payments. Use the present value formula to determine the actual finance rate. [Hint: Do not expect to solve it algebraically.]
Present value Formula: A={R[1-(1+i)^-n]} / i
where A is the present value, R is the amount of each payment, i is the rate per time period, and n is the time period.
Please respond somebody!!!
16000 = 300(1 - (1+i)^-60)/i
53.3333 = (1 - (1+i)^-60)/i
After a few trial-and-error attempts with my calculator, I "sandwiched" the rate i between .004 and .0038
so
.004 53.24886
i 53.33333
.0038 53.561
we can now set up an interpolation ratio
(i - .004)/(.0038 - .004) = (53.3333-53.24886)/(53.561-53.24886)
i = .003946
check:
300(1 - (1.003946)^-60)/.003946 = 15999.86
not bad
so the annual rate is approx .003946 x 12 = .04735
or 4.735%
To determine the actual finance rate using the present value formula, we need to rearrange the formula to solve for i (the rate per time period).
The present value formula can be rearranged as:
i = [R/(A/R-1)]^(1/n) - 1
where A is the total amount to be paid, R is the amount of each payment, i is the rate per time period, and n is the time period.
In this case, the total amount to be paid (A) is $18,000 and the amount of each payment (R) is ($18,000/60) = $300. The time period (n) is 60 months.
Substituting these values into the formula, we have:
i = [300 / (18000/300 - 1)]^(1/60) - 1
Simplifying,
i = [300 / (60 - 1)]^(1/60) - 1
i = [300 / 59]^(1/60) - 1
Using a calculator, we can evaluate [300 / 59]^(1/60) which equals approximately 1.00466028.
i = 1.00466028 - 1
i ≈ 0.00466028
Therefore, the actual finance rate is approximately 0.466028% per month.