# How Much Will I Save by Refinancing?

Original Loan
Principal \$
Annual Rate %
Length years
Refinancing
Refinance when? after years
For how long? years
New Annual Rate %
Total Refi Fees \$
Fees are... paid up front
rolled into loan

Home owners refinance loans for several reasons. Some buyers decide to extend the length of their loan so that they can make smaller monthly payments. Some decide to increase their monthly payments so that they can pay off their mortgage more quickly. Others become eligible for lower annual interest rates, which means refinancing can save them money.

Whatever the reason for refinancing a mortgage, the amount of total interest paid will be different from the interest paid in the original loan terms. When home owners are offered significantly lower interest rates, they will usually pay less in total interest charges by refinancing. If you know the original conditions of your loan and the refinance conditions, you can calculate the savings (or loss!).

Use the refinancing calculator on the left or apply the mortgage formulas below.

### Key Equations: Original Loan

If the original amount borrowed is P, the annual interest rate i (expressed as a decimal), and the number of years in the lending period N, then the monthly payments M are given by the formula

M = (Pi/12)(1 + i/12)12N/[(1 + i/12)12N - 1].

The total interest paid over the course of the loan is the sum of the payments minus the amount borrowed, or 12NM - P.

Due to amortization, the amount of principal versus interest in each monthly payment is not the same from month to month. At the beginning of a loan, a greater portion of your monthly payment goes toward interest than at the end of the borrowing term. To calculate the principal balance paid at the end of K years, you must use the formula

PPaid@K = (12M/i - P)[(1 + i/12)12K - 1].

The principal balance owed at the end of K years is P - PK, or

POwed@K = P - (12M/i - P)[(1 + i/12)12K - 1].

The interest paid thus far at K years is 12KM - PPaid@K, or

IPaid@K = 12KM - (12M/i - P)[(1 + i/12)12K - 1].

### Key Equations: Refinanced Loan

When you refinance a loan, you negotiate new loan terms based on the unpaid portion of the principal balance. For example, if you refinance your mortgage after K years, the new balance is POwed@K, carried over from the old loan.

If you refinance at a new annual rate of r (expressed as a decimal) for L years, the new monthly payments F are given by the equation

F = (POwed@K)(r/12)(1 + r/12)12L/[(1 + r/12)12L - 1].

The interest you will pay on this new loan is 12LF - POwed@K.

The total interest paid on a refinanced loan is equal to

IPaid@K + (12LF - POwed@K).

Unfortunately, refinancing entails dozens of fees. These may include
• pre-payment penalty on the original loan
• appraisal fee
• loan origination fee
• application fee
• credit report
• land survey
• legal fees
plus several other small fees depending on your lender. Closing costs and fees can either be paid up front or financed, that is, the lender rolls the fees into the balance of new loan. If you pay refinancing fees of X dollars up front, then the total interest and fees for a refinanced loan is

IPaid@K + (12LF - POwed@K) + X.

To compute your total savings, take the total interest that you would have paid under the original loan terms and subtract the total interest and fees that you pay with a refinanced loan. If the answer is a positive number, it means that refinancing has indeed saved you money. If the answer is negative, it means that refinancing actually costs you more in the long run.

### Other Important Considerations

Keep in mind that when you refinance to a lower rate, you forgo any some tax-deductible interest, which means that you will pay more in taxes, or receive a smaller refund.

Refinancing only makes good financial sense if you plan to stay in your home for a long time. If you sell your house only a few years after refinancing and buy a new property, the savings will shrink, and in the same cases it may even cost you more.