The Mathematics of Refinancing
Dr. Youngna Choi and Crystal Dahlhaus (2009)
Presented by Amara Yeb
What is Refinancing?

Finance something again

Continue the same loan at a lower interest rate

Paying off an existing loan and replacing it with a new
one at a lower interest rate
Modeling Objective

How to make an intelligence decision on
refinancing

Model the effects of refinancing loans at lower
interest rates
Questions to be Answered
• Should the borrower always refinance if there is a
lower interest rate?
• Can various fees make refinancing at lower
interest rate worse than the original loan?
Basic Financial Background

Compound interest
𝑆𝑛 = 𝑆0 (1 + 𝑖)𝑛

Present Value
𝑆0 =

Net Present Value
𝐶
𝐶
𝐶
𝑆𝑛
(1+𝑖)𝑛
𝐶
1
2
𝑛−1
𝑛
𝐶0 + (1+𝑖)
+ (1+𝑖)
2 +…+(1+𝑖)𝑛−1 +(1+𝑖)𝑛

Effective Rate of Interest is when the NPV is zero

Payment 𝐴 = 𝑃 1−(1+𝑖)−𝑁
𝑖
Model Assumptions

Specifically looking at automobile (Short term loan) and
Real Estate (Long Term Loan)

Compound interest is used and assume that the interest
is compounded at the end of each term

Number of payments made by the time of refinancing:
N-n

Interest rate per month for refinancing: 𝑖𝑛𝑒𝑤 which is
lower than 𝑖
Model Assumptions (Cont.)

All the equity built to date is used for refinancing

Refinancing fee: F

Discount points for refinancing: x

𝐷𝑛 is remaining debt at time 𝑛

𝑄𝑛 is new loan principal

𝐵𝑛 , 𝐶𝑛 are new monthly payment
Modeling Strategy

Picking up the old loan: Short-term Refinancing

Starting a new loan: long-term Refinancing

Amortization with combined Fees are used
Picking up the Old Loan
Starting a New Loan
Result

A lower Interest rate alone does not guarantee a
profitable refinancing (because there are other factors
that should be taken into consideration such as the
remaining number of payments, the refinancing fee
etc.)

Refinancing is not always a good idea, because we might
end up paying more interest throughout the refinanced
loan.

This is not bad if there is a tax exemption on the
interest.
Conclusion

When refinancing, all the variables of the NPV function
should be considered.

Other investment opportunity should be considered (ex.
The borrower may take advantage of the lower monthly
payment and invest the savings for higher return.

If there is no tax benefit and/or other investment
options, the borrower should keep the old loan.
Reference

Choi, Youngna, and Crystal Dahlhaus, “The Mathematics
of Refinancing,” UMAP Journal 30(4) (2009): 429-456.