Your friend is buying a 12375 $ car with 3000 down payment to be paid immediatel

Question

Your friend is buying a 12375 $ car with 3000 down payment to be paid immediately and the rest is to be paid over the next three years. your friend was give two options

Option 1: Pay monthly payments of 325$ each for the next 3 years.

Option 2: Pay quarterly payments of 975$ each the next 3 years

Find the effective annual intrest rate foroption 2 (12 quarterly payments) is closest to :

A- 14.7%

B- 16.3%

C- 13.1%

D- 3.5%

and which option is the best for your friend financially.

Please show the full steps.

Explanation / Answer

Correct Answer:

A.14.7%

Working note:

For option 2:

Quarterly payment = $975

Time = 12 quarters (3 years)

Value of the loan = 12375-3000 = $9375

Let, quarterly interest rate is R.

Then,

9375 = 975*(1-1/(1+R)^12)/R

At R = 3%

Present value of the loan = $9705.15

At R = 4%

Present value of the loan = $9150.45

As per the method of interpolation,

R = 3% + ((PV of loan at 3% - 9375)/( PV of loan at 3%- PV of loan at 4%))*(4%-3%)

R = 3% + ((9705.15- 9375)/( 9705.15 - 9150.45))*(4%-3%)

R = 3.59%

Effective annual interest rate = 4*3.59% = 14.36% that is closest to 14.7%.

For option 1:

At this rate of 14.7%

Monthly rate = 14.7%/12 = 1.225%

Present value of payment = 325*(1-1/(1+1.225%)^36)/.01225

Present value of payment = $9415

It is better to opt for the option 2, as with the given interest rate as it only requires $9375 in present value, but option 1 takes more money for the payment that is $9415.

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