Question

You need a 30-year, fixed-rate mortgage to buy a new home for $290,000. Your mortgage bank will lend you the money at a 5.85% APR (semi-annual) for this 360-month loan. However, you can afford monthly payments of only $1,300, so you offer to pay off any remaining loan balance at the end of the loan in the form of a single balloon payment.

Can someone please explain how to solve this without using excel. Confused with the semi-annual part. Thanks!

Answer 1

APR is compounded semi-annually, but payments are on monthly basis, hence we need to calculate Effective Annual Rate (EAR) then monthly rate.

Effective Annual Rate of interest

i = nominal interest rate

n = number of componding periods

Componding is semi annual so compounding period = 2, APR = i = 5.85%

= ( 1.02925 )² - 1

= 1.05936 - 1

= 0.05936 Or 5.936%

Payments are on monthly basis hence need a monthly rate which leads to Effective annual rate 5.936% as calculate above.

Effective monthly rate =

= ( 1 + 0.05936 )^1/12 - 1

= ( 1.05936 )^1/12 - 1

= 1.0048170 - 1

= 0.0048170 Or 0.4817%

We need to calculate present value of annuity for 360 months, monthly payment of $1,300 at the monthly rate 0.4817%.

  

= $222,030.38

Ballon Payment = Loan - Present Value of Equity

= $290,000 - $222,030.38

= $67,969.62

Note:

1. Rounding off of calculation can change answer at some points.
2. As in question is not mention given APR is semiannually specifically, then we can simply start our calculation by showing monthly rate 5.85%/12 = 0.4875% and further calculation on this basis.