1. You want to buy a new car for $30,000 and want to finance $20,000 of the purc

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Question

1. You want to buy a new car for $30,000 and want to finance $20,000 of the purchase price (you will put $10,000 down out of your own pocket). You want to get an idea of the difference between the monthly payments on a 4-year (i.e., 48-month) versus a 5-year (60-month) loan. The stated rate on the 4-year loan is 3 percent (.03) while the stated rate on the 5-year loan is 3.5 percent (.035); both are compounded monthly. What are the monthly payments of each of these loans? Both loans require the first payment to be made exactly one month from now. Some loans allow you to make payments every half month rather than every month. Consider the 5-year loan above. If instead of making 60 monthly payments (with the first payment exactly a month from now) you make 120 half-monthly payments (with the first payment exactly one half month from now), what is the difference in the total monthly payment between these two cases?

Some loans allow you to make payments every half month rather than every month. Consider the 5-year loan above. If instead of making 60 monthly payments (with the first payment exactly a month from now) you make 120 half-monthly payments (with the first payment exactly one half month from now), what is the difference in the total monthly payment between these two cases?

2. Calculate the present value of an annuity that makes annual payments of $1,000,000 every year for 9 years, with the next payment being made exactly one year from now.

3. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 7 percent (g = .07).

4. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 9 percent (g = .09).

5. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 9.5 percent (g = .095).

6. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 9.9 percent (g = .099).

7. Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 10.5 percent (g = .105). Hint: Consider the trend in the present values as the growth rate increases by comparing your answers to a through d. If you use the standard growing perpetuity formula when g > r, you get a silly answer. Using your intuition from your answers to a through d, what is the real answer?

8. The effective annual discount rate is 5% (r = .05) at all maturities. What is the present value of a stream of payments that starts at $100 at t = 1 (time in years) and grows at 3 percent for 5 years (to t = 6), then growing at 10 percent for 10 years (to t = 16), then growing thereafter at 1 percent?

Explanation / Answer

Solution :

Monthly payments on a 4 year loan :

A = P X r ( 1+r)n / ( 1+r)n -1

A = 20,000 X .03/12 ( 1+0.03/12)12 x4 / ( 1+0.03/12)12x4 -1

A = $ 442.71

The monthly payment on a 4 year loan with a interest of 3% compounded monthly = $ 442.71

Monthly payments on a 5 year loan:

A = 20,000 X 0.035/12X (1+0.035/12)12x5 / (1+0.035/12)12x5 -1

A = $ 363.77

The monthly payment on a 5 year loan with a interest of 3.5 % compounded monthly = $ 363.77

The difference between these 2 payments are as follows:

For 4 year loan total payment =$ 442.71 X 48 = $ 21,250.08

For 5 year loan total payment =$ 363.77 X 60 = $ 21,826.20

The difference = $ 576.12 i.e 4 year loan is better than 5 year term loan.

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If 120 half monthly payments are made on a 5 year term loan

Monthly payment = 20,000 X 0.0175 / 24 ( 1+0.0175/24)24x5 / (1+0.0175/24)24x5 -1

Monthly payment = $ 159.50

Total payment = $ 19,140

Difference in total payment = $ 21,826.20 - $ 19,140

Difference in total payment = $ 2,686.20

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Calculate the present value of an annuity that makes annual payments of $1,000,000 every year for 9 years, with the next payment being made exactly one year from now.

P(PVA) = A [ (1+r)n -1] / r ( 1+r)n

Interest rate not given in the problem , hence please assuming an interest rate of 5%

PVA = $1,000,000 [ ( 1+0.05)9 -1] / 0.05 ( 1+0.05)9

PVA = $ 7,107,821.60

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Calculate the present value of a growing perpetuity that makes one payment per year with the first payment, made in exactly one year from now, being $1000. Let the payments grow at an annual rate of 7 percent (g = .07).

The present value of a growing perpetuity formula is given by

PV = c/ r - g

C = annual payment

r = interest rate

g = growth rate

To apply the growing perpetuity formula the interest rate should be greater than the growth rate.

If g were greater than r, however, the cash flow 1 period later would be worth more even in today

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