Question

1. You plan to take out a $700,000 mortgage for 30 years with monthly payments.
   1. If you don’t have a down payment, they will offer you an interest rate of 5.7%. If you can make a down payment of at least $20,000 they will lower the interest rate to 5.2%. Calculate the monthly payment for each loan.
   2. 1. You would like to see how long it will take you to come up with a down payment. You already have $6,000 and can put away $500 every quarter at 4% interest compounded quarterly. How long will it take you to save up to $20,000?

Answer 1

According to given information the mortgage value is $700000 .

If no down payment is there then

Period   t = 30 x 12 = 360 monthly payments

Rate of interest r = 5. 7%

Rate of interest r = 5.7 / 100 = 0.057 => 0.057/12 = 0.00475

Now we can use the below formula to find the monthly payment (PMT)

PMT = [ p x r x (1+r)^t ] / [(1+r)^t-1]

PMT = [700000 x 0.00475 x (1+0.00475)³⁶⁰ ] / [(1+0.00475)³⁶⁰-1]

PMT = [3325 x (1.00475)³⁶⁰ ] / [(1.00475)³⁶⁰-1]

PMT = [3325 x 5.50662] / [5.50662 - 1]

PMT = [18309.5115] / [4.50662]

PMT = 4062.80349 ~ 4062.8

So Monthly payment amount is $4062.8

If down payment of $20000 offers then

According to given information the mortgage value is $700000 .

Down payment = $20000

Then mortgage value = 700000 – 20000 = 680000

If no down payment is there then

Period   t = 30 x 12 = 360 monthly payments

Rate of interest r = 5.2%

Rate of interest r = 5.2 / 100 = 0.052 => 0.052/12 = 0.004333

Now we can use the below formula to find the monthly payment (PMT)

PMT = [ p x r x (1+r)^t ] / [(1+r)^t-1]

PMT = [680000 x 0.004333 x (1+0.004333)³⁶⁰ ] / [(1+0.004333)³⁶⁰-1]

PMT = [2946.44 x (1.004333)³⁶⁰ ] / [(1.004333)³⁶⁰-1]

PMT = [2946.44 x 4.74224] / [4.74224 - 1]

PMT = [13972.7256] / [3.74224]

PMT = 3733.7866 ~ 3733.8

So Monthly payment amount is $3733.8

Now we need to calculate the time period to save $20,000 with an initial amount of $6000 to make a down payment.

Each quarter we make $500

now we can use the formula of ordinary annuity with initial deposit is

FV = Pv(1 + (r/n))^nt + [ C x [ ( 1 + (r/n) )^nt-1 ] / ( r/n )]

Fv = future value =$20000

C = payment flow per period = $500

Pv = initial deposit = $6000

Rate of interest = r = 4% = 4/100 = 0.04

Compounding frequency n = quarterly =4

Number of years t

20000 = 6000(1 + (0.04/4))^4t + [ 500 x [ ( 1 + (0.04/4))^4t-1 ] / (0.04/4)]

20000 = 6000(1.01))^4t + [ 500 x [ ( 1.01))^4t-1 ] / (0.01)]

20000 = 6000 (1.01)^4t + [ 500 (1.01)^4t-500] / (0.01)]

20000 = 6000 (1.01)^4t + [ (500/0.01) (1.01)^4t-(500/0.01)]

20000 = 6000 (1.01)^4t + 50000 (1.01)^4t-50000

20000 + 50000 = 6000 (1.01)^4t + 50000 (1.01)^4t

70000 = (6000 + 50000 ) (1.01)^4t

70000 = (56000) (1.01)^4t

70000 /56000 = (1.01)^4t

1.25 = (1.01)^4t

Now apply in both sides

Ln(1.25) = ln(1.01)^4t

Ln(1.25) = 4t in(1.01)

0.22314 = 4t (0.00995)

0.22314 /0.00995 = 4t

22.4261 = 4t

Now t = 22.4261 / 4 = 5.60 years

So required time period to get $20000 is 5.60 years