# What is the formula to calculate loans at variable rates and what is the function of PV (present value) in calculations?

Yesterday, I tried during the whole day to solve that problem:

You have been living in the house you bought 6 years ago for \$250,000. At that time, you took out a loan for 80% of the house at a fixed rate 25-year loan at an annual stated rate of 9.5%. You have just paid off the 72th monthly payment. Interest rates have meanwhile dropped steadily to 6.0% per year, and you think it is finally time to refinance the remaining balance. But there is a catch. The fee to refinance your loan is \$4,000. Should you refinance the remaining balance? How much would you save/lose if you decided to refinance?

Yes, gain \$53,229.73
No, lose \$49,229.73
No, lose \$53,229.73

I found that 'I' could save... \$85,334.06.

I used calculators which do that work... Some give the same answer as us. Other ones give different answers.

I must admit that I do not have a clue! I searched a formula and did not find any that looked both complete and easy to understand.

Also, I found on forums that Present Value (PV) was an element to take into account. But Present Value of what ? And when ?

Finally, how does the Principal interfere ?

2. explain how you reached that conclusion

Best regards,

• Can you add a country tag? Just curious, is this a situation you are in (and subject to a different set of rates I'm used to), or is this homework? Commented Apr 16, 2016 at 17:01

The first step is pick a date for calculating the cost/benefit. The logical one in this case is the date of the 72nd payment, the moment when you're trying to reach this decision.

The second step is to define two scenarios that describe your two options. Along the way, you'll need to use various values from the current and hypothetical payment schedules. It's Important that the two scenarios are as similar as possible, to highlight any differences as real costs or benefits.

First, find the monthly payment you're making, and the balance owed after the 72 payment.

The first scenario is easy, the status quo: keep making the current monthly payment for 19 more years.

The second scenario is more complex, because we need to match the cash flow in the first scenario. The first step is to take the balance owing after the 72nd payment, and add \$4,000 to that. Then, find the monthly payment needed to pay off a 19 year 6% mortgage for this new amount.

You could stop here. You could say that the re-finance saves you \$x per month compared to the status quo, or that you need to pay \$y more per month.

However, this has you making the major mistake of cost analysis: adding or comparing amounts of money at different times. So lets continue with scenario #2, to quantify the actual cost or benefit.

Suppose you find that the new schedule has you paying less per month, by \$X. So take this extra \$x and find out: how much can I borrow, today, the 72nd payment if I will pay it off with \$x per month at 6% for 19 years.

So now the two scenarios give you exactly the same cash flow. But in scenario #2, you're sitting there with the loan cash in you pocket. That's the benefit of the refinancing scheme.

If you find that you need to pay more per month, the calculation is the same, except you're out of pocket lending money at 6% for 19 years, to get the regular payment needed to bulk up your payments.

Another, perhaps simpler way to do the same calculation:

Find out how big a loan your current payments, for 19 years at 6%, will get you, and use the loan to pay off the original mortgage. If you have money left over, great. If you need to put in extra cash to pay off the original loan, not so great...