# mortgage math problem

There's a $30$-year home loan for \100000$at$7$%. After$15$years the loan is paid off in order to refinance at a lower rate. The loan has a prepayment penalty of six months interest of$80$% of the remaining balance of the loan. a) How much is the remaining balance of the loan? b) If the loan can be refinanced with a$15$year loan at$6$% with no other costs, then should it be done? c) If the loan can be refinanced over$15$years with no other costs, then what interest rate would make it worthwhile? I believe I got (a) \begin{array}{|c|c|c|c|c|} \hline \text{month} &\text{payment}&\text{interest}&\text{principal}&\text{remaining}\ \\ \hline 180 &665.30&433.14&232.16&74019.66\\\hline \end{array} Not sure about b or c though. I attempted b by taking$74019.66$and making that my new loan. Find a new payment across$15$years at$6$%, which is$624.62$. I figured$665.30-624.62 = 40.68$in savings per month. One thing I don't know is does prepayment go into the new loan or do you pay out of pocket. If you pay out of pocket then you would save$\$7322.4 - \$2072.55$(prepayment penalty). ## 2 Answers I think they want you to increase the balance by the prepayment penalty (so there is no out of pocket cost), then see if the payment is higher or lower. So for (b) you figure the new payment on$74019.66+2072.55$and see if it is lower. For (c) you find the interest rate on the same balance that keeps the payment the same as today. I'm not sure I agree with this criterion for a decision in the real world-there is too much chance you will end up paying the loan off early so having the balance higher is a negative. But I think it is the way you understand the technique in the absence of other instructions. It might help to clearly lay out your assumptions. • yea, seems to make more sense like that, but wasnt sure. thanks – Matt Jan 13, 2011 at 3:47 If you do the calculations you will find that the cut off point for the new loan is$6.55$% (on a$15$year loan of$74019.66+2072.55 = 76092.21$), which gives a repayment of$664.94$(a rate of$6.56$% will give you a repayment of$665.36$, slightly more than your current repayment). However, if they are no other costs, as you suggest, you must ask yourself what is in it for the mortgage company? At$6$% interest on$76092.21$the repayment is about$642.11$giving you a monthly saving of$665.30 - 642.11 = 23.19$, which means it will effectively (ignoring complicating factors such as inflation) take$7.5$years to save back the$2072.55\$ that has been added to the amount you owe. So you should view this deal as a kind of lock-in, and make a decision with that in mind.

Anyway, be wary of making decisions based purely on mathematical calculations. There could be other factors to consider. You may find more help at Personal Finance and Money.

• I see you have posted on the site that I referenced. I'll copy my answer there as I think it's important that you realise the "lock-in" aspect of this deal. (You may spot a better deal with another company a few years down the road, then the extra added to your loan becomes important.) Jan 13, 2011 at 13:57