Compounding

Dr. Zoidberg decided to buy a 2019 Tesla Model S P100D Hatchback for $130,000. After paying a down payment and taxes, Dr. Zoidberg can finance the rest of the purchase price with a loan of 100,000 for 5 years at a promotional finance rate of 3% APR compounded quarterly. Please answer the following questions:

A. What is the effecting annual rate changed on this loan?
B. What would be the quarterly payment on this loan?
C. Dr. Zoidberg also discovers that instead of the special promotional rate he can make
an additional down payment of $20,000 that would lower his loan amount accordingly (i. e. by $20,000). At what APR would Dr. Zoidberg have the same quarterly payment with this option as with the initial promotional rate of 3%?
D Dr. Zoidberg finds that he can get 1.5% APR if he elects option (c). What will his quarterly payment be under this option?
E. Now assume that that payment frequency changes to annual, preserving the same EAR. What is his payment now?

Task A:

What is the effecting annual rate changed on this loan?

Answer is 3.03%

Task B:

What would be the quarterly payment on this loan?

Answer is $5,403.06

Task C:

Dr. Zoidberg also discovers that instead of the special promotional rate he can make  an additional down payment of $20,000 that would lower his loan amount accordingly (i.e. by $20,000). At what APR would Dr. Zoidberg have the same quarterly payment with this option as with the initial promotional rate of 3%?

Answer is 12.21%

Task D

Dr. Zoidberg finds that he can get 1.5% APR if he elects option (c). What will his quarterly payment be under this option?

The answer is $4,159.37

Task E:

Now assume that that payment frequency changes to annual, preserving the same EAR. What is his payment now?

The answer is $21,835.46

Explanation:

Solution:

Effective annual rate = (1 + (APR/n))ⁿ - 1

where

n = number of compounding periods per year = 4 (compounding quarterly)

APR = 3%

Effective annual rate = (1 + (3%/4))⁴ - 1

Effective annual rate = 3.03% (answer).

Solution:

Quarterly loan payment is calculated using PMT function in Excel :

Rate = 3% / 4   (converting annual rate into Quarterly rate)

nper = 5*4 (5 year loan with 12 Quarterly payments each year)

pv = 100000 (loan amount)

PMT Formula = PMT(3%/4,5*4,100000)

PMT is calculated to be $5,403.06 (answer)  

Note: PMT calculation has been attached.

Solution

The quarterly rate to have the same quarterly payment is calculated using RATE function in Excel :

nper = 5*4 (5 year loan with 12 Quarterly payments each year)

pmt = -5403.06 (Quarterly payment. This is entered with a negative sign because it is a payment)

pv = 80000 (loan amount)

RATE is calculated to be 3.05%. This is the quarterly rate. To get APR, we multiply by 4.

Formula for APR = RATE(5*4,C1,80000)*4

APR = 12.21% (answer)

Solution:

Quarterly loan payment is calculated using PMT function in Excel :

rate = 1.5% / 4   (converting annual rate into Quarterly rate)

nper = 5*4 (5 year loan with 12 Quarterly payments each year)

pv = 80000 (loan amount)

PMT formula: PMT(1.5%/4,5*4,80000)

PMT is calculated to be $4,159.37

Solution:

PMT = PMT(3%,5,100000)

PMT = $21,835.46

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