Question

help with finance

As in Problem 1, once again, you are borrowing $6500 to purchase a car. However, now the first payment is due immediately. There will be a total of 36 monthly payments (The first payment occurs immediately. The remaining 35 occur at the end of each of the following months). The advertised interest rate is 9% APR (therefore use a monthly discount rate of r-0.75% in your computations). The payments will all be equal in size HINT: This is an annuity due problem. Make sure you understand the discussion in the book, especially Example 5B - paying attention to Footnote 3 (which applies to Parts B, C, and D). Also work the Annuity Due problems at the end of the chapter before you work this problem Part A) What is the size of the first payment? (which vou pay immediately) Part B)Now assume that the size of the car payments increases by 2% every month. size of the first payment? What is the Part C) Now assume that the size of the car payments decreases by 2% every month. What is the size of the 4th (fourth) payment? Hint: The answer is NOT $259.45 Part D) Now assume that the size of the car payments decrease by 0.75% every month. What is the size of the payment that occurs 15 months from today? Part E) Now assume that the size of the car payments increases by 0.75% every month. What is the size of the payment that occurs 15 months from today?

You don't need footnote 3 to solve it what

Answer 1

(a) Amount Borrowed = $ 6500, Borrowing Tenure = 36 months, APR = 9 % or 0.75 % per month

Let the equal monthly installments be $ K

Therefore, 6500 = K x (1/0.0075) x [1-{1/(1.0075)^(36)}] x (1.0075)

K = $ 205.16

(b) If the monthly payments increase by 2 %, it is a case of growing annuity due. The same can be solved as shown below:

Let the first payment be $ M

Therefore, 6500 = [M / (Discount Rate - Growth Rate)] x [1 - {(1+growth rate) / (1+Discount rate)}^(n)] where n is the tenure, discount rate is the interest rate per period and growth rate is the rate at which the periodic payments grow

6500 = [M / (0.0075 - 0.02)] x [1-{(1.02) / (1.0075)}^(36)] x (1.0075)

M = $ 144.32

(c) If the monthly payments decrease by 2 % the calculations remain the same with only the growth rate being - 2 %.

Let the first payment be $ N

Therefore, 6500 = [N / (0.0075 - (-0.02))] x [1-{(0.98) / (1.0075)}^(36)] x (1.0075)

N = $ 281.28

Fourth Payment = 281.28 x (1-0.02)^(3) = $ 264.74 (the power of the multiplicative factor is 3 and not 4, because all payments are received at the beginning of the period, Hence, the fourth payment comes in at the end of month 3 and not month 4)

(d) The calculations for this part will be similar to the previous part's, with the growth rate being -0.75 %

Let the first payment be $ L

Therefore, 6500 = [L / (0.0075 - (-0.0075))] x [1-{(0.9925) / (1.0075)}^(36)] x (1.0075)

L = $ 231.93

Payment Size 15 months from now (size of the 16th payment as it is an annuity due) = 231.93 x (1-0.0075)^(15) = $ 207.16

(e) If the growth rate is 0.75 % and discount rate remains fixed at 0.75%, the total present value of the payments can be depicted as 6500 = K + [K x (1.0075) / (1.0075)] + ..............+ [K x (1.0075)^(35) / (1.0075)^(35)] where $ K is the first payment

6500 = 36 K

K = $ 180.56

Payment Size 15 months from now (16th payment size as this is an annuity due) = 180.56 x (1.0075)^(15) = $ 201.97

NOTE: Footnote 3 is not required to solve this question if one is well versed with the concepts and formulas of time value of money, perpetuity, annuity and the like.