Question

1.

You would like to buy a house that costs$ 350 comma 000$350,000.You have$ 50 comma 000$50,000

in cash that you can put down on the​ house, but you need to borrow the rest of the purchase price. The bank is offering you a​ 30-year mortgage that requires annual payments and has an interest rate of

7 %7%

per year. You can afford to pay only

$ 23 comma 210$23,210

per year. The bank agrees to allow you to pay this amount each​ year, yet still borrow

$ 300 comma 000$300,000.

At the end of the mortgage​ (in 30​ years), you must make a balloon​ payment; that​ is, you must repay the remaining balance on the mortgage. How much will be this balloon​ payment?

​Hint: The balloon payment will be in addition to the 30th payment.

2.

You are thinking of building a new machine that will save you

$ 5 comma 000$5,000

in the first year. The machine will then begin to wear out so that the savings decline at a rate of

3 %3%

per year forever. What is the present value of the savings if the interest rate is

8 %8%

per​ year?

The present value of the savings is ​$nothing. ​(Round to the nearest​ dollar.)

Answer 1

1.

Annual Payment = P x r/ [1 – (1+r)-n]

P = principal of loan = $ 300,000

r = Rate per period = 7 %

n = 30 periods

Annual Payment = $ 300,000 x 0.07/ [1 – (1+0.07)-30]

                             = $ 210,00/ [1 – (1.07)-30]

                            = $ 21,000/ (1 – 0.13136711715459)

                            = $ 21,000/ 0.86863288284541

                            = $ 24,175.921053333 or $ 241,759.92

Annual deferred payment = $ 241,759.92 - $ 23,210 = $ 965.92

Future value of these 30 deferred payments can be computed using formula for FV of annuity as:

FV = P x [(1+r) n -1/r]

FV = Future value of balance payments

P = periodic cash balance = $ 965.92

r = Rate of interest = 0.7

n = Number of periods = 30

P = $ 965.92 x [(1+0.07)30 -1/0.07]

= $ 965.92 x [(1.07)30 -1/0.07]

= $ 965.92 x [(7.61225504266203-1)/0.07]

= $ 965.92 x (6.61225504266203/0.07)

= $ 965.92 x 94.4607863237433

= $ 91,241.5627258301 or $ 91,242

Total balloon payment = 30th payment + Future value of deferred payments

                                   = $ 23,210 + $ 91,242 = $ 114,452

The balloon payment at the end of 30 years will be of $ 114,452

2.

FV of decreasing perpetuity annuity = P/ (r + g)

P = Periodic Cash Flow = $ 5,000

r = Rate of interest = 0.08

g = Decrease in growth rate = 0.03

FV = $ 5,000/ (0.08 + 0.03)

    = $ 5,000/0.11

    = $ 45,454.54545 or $ 45,455

Present value of savings is $ 45,455