Question

a. [3 pts] Consider the simple loan case. Suppose that the dealership allows you to pay the car off in four installments of $5,000, with each installment due once a year. The first payment is due the day that you purchase the car; the remaining installments are then paid on the same date each consecutive year, for the remaining three years. What is the present discounted value of the payments you make for the car?

b. [5 pts] Now suppose that the dealership instead offers you a financing package. The package involves making monthly payments of $P over 60 months at a monthly interest rate of i% for a loan amount of $L. In other words: where n = 60. Analytically, solve for P as a function of L, i and n. Note: for this question, you need to derive an expression P = g (L,i,n); you will not get any credit if you do not show your work.

c. [5 pts] Suppose that the 60 month loan in part (b) is financed at a 5.1% annual interest rate. Using your answer to part (b), solve for how much you would have to pay each month, P, assuming that the 5.1% interest rate is the yield to maturity. [Hints: You need to convert the 5.1% annual rate into a monthly rate first, and then you can use your answer for part b; also don’t forget that these interest rates are compound interest rates, so the gross annual rate is the gross monthly interest rate raised to the power of 12]

d.[3 pts] Based on your answer to part c, what is the nominal amount (or total dollar value)that you would pay with a 5.1% interest rate? What would be the present discounted value of those stream of payments?

e.[4 pts] Suppose now that because of your credit score, you are unable to qualify for the 5.1% interest rate. Instead you can take out a loan from the bank at the market rate. What is the nominal amount that you would pay with the bank loan? What is the present discounted value of those stream of payments?

Answer 1

a. Since the discount rate is not given the result would be a formula as below

b. The present cost of the car should be equal to the annuity payments for the 60 months

Which can be solved by taking P common and adding the Geometric progression  to

C. Apply the formula P= L X 0.18917

D.There is some data missing as there is no principal or loan amount

E. Present value will always be equal to the Loan amount as discounting factor and interest rates are same