Question

Consider the following scenario. Suppose that you are in the market to buy a new $20,000 car. You intend to take out a loan to pay for the car. The market interest rate is at 6% annually, i.e. this is the interest rate you would get from the bank. 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: L = P 1 + i + P (1 + i) 2 + P (1 + i) 3 + ... + P (1 + i) n where n = 60. Analytically, solve for P as a function of L, i and n. [Hint: you may want to substitute λ ≡ 1 (1+i) and use the formula for the sum of a finite geometric series; notes on geometric series may be found on the lectures page of the course homepage, or within Canvas.] 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 working. You do not need to calculate and numbers or compute anything. 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) assuming that the 4 payments are $5000 each (without any interest) and the bank rate of interest is 6% p.a., the present value of the 4 payments would be:

5000 + 5000 / 1.06 + 5000 / 1.06^2 + 5000/ 1.06^3

= $18,365.06

b) L = f (P, i, n)

L = P/(1+i) + P/ (1+i)^2 … + P/(1+i)^60

= [P(1-1/(1+i)^60) ] / [1-1/(1+i)]

if we replace 1+i with Y then the function will be

L = [P(1-1/Y^60)]/(1-1/Y)

Using this we can get the function for P

P = g (L, i, n)

= L(1-1/Y) / [(1-1/Y^60)]

= L(1-1/(1+i))/[(1-1/(1+i)^60)]

c. First convert 5.1% annual rate into monthly using compounding formula:

5.1% = (1+i)^12-1

1.051 = (1+i)^12

1+i = 1.051^(1/12) = 1.004154

i = .004154 =0.4154% per month

Since L = 20000 (cost of the car), n = 60 (duration of the loan), P would be:

=20000 * (1-1/1.004154) / (1-(1/1.004154^60))

= $375.72

d. nominal value of this stream would be $375.72 * 60 = $22543.34

Present discounted value of the stream of $375.72 at 6% annual interest would be:

First convert 6% annual interest into month, using compounding formula, and we get 0.4868% per month

If we discount the stream of $375.72 of 60 instalments using 0.4868% rate of discount, we get $19508.93

Only 4 parts answered according to the guidelines.