Question

An investor has $60,000 to invest in a $280,000 property. He can obtain either a $220,000 loan at 9.5 percent for 20 years or a $180,000 loan at 9 percent for 20 years and a second mortgage for $40,000 at 13 percent for 20 years. All loans require monthly payments and are fully amortizing

a. Which of alternative should the borrower choose, assuming he will own the property for the full loan term?

b. Would your answer change if the borrower plans to own the property only five years?

c. Would your answers to (a) and (b) change if the second mortage had a 10 year term?

Answer 1

Solution:-

1st Alternative : $220,000 loan at 9.5% for 20 years

monthly interest rate = 9.5%/12 = 0.007916

No. of payments = 20*12 = 240

Payment (A) per month is given by

[A/0.00791667] * (1-1/1.00791667²⁴⁰) = $220,000

[A/0.00791667] * (1-1/6.635008) = $220,000

A/0.00791667 * 0.84928 = $220,000

A * 107.27 = $220,000

=> A = $220,000 / 107.27

= $2,050.69

2nd Alternative :

i) $180,000 loan at 9% for 20 years

Monthly interest rate = 9%/12 = 0.0075,No. of payments = 240

Payment (A) per month is given by

[A/0.0075] * (1-1/1.0075²⁴⁰) = $180,000

[A/0.0075] * (1-1/6.0091) = $180,000

A/0.0075 * 0.83358 = $180,000

A * 111.144 = $180,000

=> A = $180,000/111.144

= $1,619.52

ii)$40,000 loan at 13% for 20 years

Monthly interest rate = 13%/12 = 0.010833,No. of payments = 240

Payment (A) per month is given by

[A/0.010833] * (1-1/1.010833²⁴⁰) = $40,000

[A/0.010833] * (1-1/13.2757) = $40,000

A/0.010833 * 0.9246 = $40,000

A * 85.3571 = $40,000

=> A = $40,000/85.3571

= $468.62

Total monthly payment in 2nd Alternative = $1,619.52+$468.62 = $2,088.14.

a) As the total monthly payment in the 1st alternative is less than in the 2nd alternative,the 1st alternative has a lesser effective rate of interest.Hence,one should choose 1st alternative and borrow $220,000 at 9.5% if one wants to own the property for full term.

b)If property is to be owned only for 5 years,even then the 1st alternative is better,as the balance mortgage amount will have to be paid after 5 years,which in the case of 2nd alternative will be higher,

Total amount paid in first alternative for 5 years = $2,050.69*60= $123,041.4

Total amount paid in 2nd alternative for 5 years = $2,088.14*60= $125,288.4

Balance on mortgage left after 5 years(1st alternative) = 2050.69/0.00791667*91-1/1.00791667¹⁸⁰)

= $196,383.85

Balance on mortgage left after 5 years(2nd alternative)

= 1,619.51/0.0075*(1-1/1.0075¹⁸⁰) + 468.663/0.10833*(1-1/1.010833¹⁸⁰

= $196,711.47

So, in case of 1st alternative,not only the payments for 5 years be less,the loan outstanding after 5 years will also be lesser.Hence 1st alternative is better and THE ANSWER WILL NOT CHANGE.

c) The answer will NOT CHANGE even if the loan terms is changed to 10 years as in this case,the monthly payment will be much more but the loan balance after 5 years will be lesser.However,the effective loan will still be more than that of the 1st alternative.

Hence even in this case the 1st alternative is better.