Comparing Refinancing Offers With Fees and Rate Changes

Why Refinancing Offers Need a Common Basis

A refinancing offer can advertise a lower rate or a smaller monthly payment while also charging a different fee or extending the repayment term. Comparing only the stated rate or installment can therefore give the wrong answer. A sound comparison begins with the actual amount needed to complete the refinancing, then places each offer’s fees and future payments on a common valuation date.

The contractual rate on an offer determines its scheduled payments. It does not, by itself, determine how to value those payments when comparing two offers. For that, the borrower needs one common valuation assumption, such as a stated opportunity rate. Applying the same assumption to every offer makes the comparison consistent. If no valuation rate is specified, there is no unique present-value comparison; a nominal total or another decision criterion would need to be stated instead.

Definition: A common-date refinancing comparison values every relevant cash flow from each alternative at the same date, using the same valuation interest rate. Relevant cash flows can include an old-loan payoff, fees, new-loan payments, and—if the comparison ends before a loan is paid off—its outstanding balance at that endpoint.

Use the actual payoff amount due on the transaction date, not the original amount borrowed, when determining how much the new loan must cover. State clearly whether each fee is paid immediately or added to the new loan balance. A fee paid immediately is a separate transaction-date cash flow; a fee financed into the loan is already reflected in its principal and later payments.

Set Up the Cash Flows Before Comparing

Assume level payments are made at the end of each month. If the new loan principal is $L$, its monthly contractual rate is $i$, and it is repaid in $n$ monthly payments, the payment $R$ is

R=\frac{L}{a_{\overline{n}|i}} \qquad\text{where}\qquad a_{\overline{n}|i}=\frac{1-(1+i)^{-n}}{i}.

Here, $a_{\overline{n}|i}$ is the present value at the contractual rate of one dollar paid at the end of each of $n$ periods. If a quoted rate is nominal convertible monthly, divide it by 12 to obtain the monthly rate. If a rate is effective annual, convert it to a monthly effective rate before using a monthly annuity factor. Do not use an annual rate directly in a formula whose periods are months.

For a comparison made at the refinancing date, let $j$ be the common monthly valuation rate. If an offer includes an upfront fee $F$, has level payment $R$, and fully repays the loan in $n$ months, its value of costs at that date is

C_0=F+R\,a_{\overline{n}|j}.

This formula treats the fee as paid at time 0. If the fee is financed, do not add it again as $F$. Instead, include it in $L$, calculate the payment using that larger principal, and value those payments. When the old-loan payoff is financed, it is likewise part of the new principal. If both offers cover the same payoff, that common amount is not a separate cash payment by the borrower on top of the new loan; it is the amount the new financing must cover.

Fix the transaction date and comparison endpoint.
Record when fees and payments occur. If an offer still has a balance at the chosen endpoint, include that balance there.

Calculate each offer’s contractual payment.
Use that offer’s own rate, term, and financed principal. Keep the payment-rate calculation separate from the common valuation.

Value each cash flow consistently.
Discount all alternatives using the same stated valuation rate and bring the results to the same date.

Check the fee and balance treatment.
A financed fee is in the principal; an upfront fee is a separate cash flow. An unpaid endpoint balance is a cash flow too.

Why the Common-Date Formula Works

The annuity factor follows directly from discounting each payment separately. At a monthly valuation rate $j$, the payments $R$ occur at times 1 through $n$, so their value at time 0 is the sum of a geometric series:

R\left(v+v^2+\cdots+v^n\right) =R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{j} =R\,a_{\overline{n}|j}, \qquad v=\frac{1}{1+j}.

The equality uses $1-v=jv$. Adding an upfront fee gives $F+R\,a_{\overline{n}|j}$. If the comparison is made at an interim date $m$, the same reasoning applies: value cash flows through month $m$ at the comparison date and add the outstanding balance at month $m$. Equivalently, discount those amounts back to the transaction date using $j$. Omitting the remaining balance would value only part of the obligation.

Worked Example: A Lower Rate With an Upfront Fee

At the transaction date, a borrower must refinance a payoff of $10,000. Both offers repay the new loan with 12 end-of-month payments. Offer A charges a nominal rate of 12% convertible monthly and a $200 fee paid at the transaction date. Offer B charges a nominal rate of 6% convertible monthly and adds a $400 fee to the new loan principal. Compare the offers using a common valuation rate of 6% nominal convertible monthly. Assume no other charges and no early payoff.

Method. First determine each offer’s contractual monthly rate and payment. Then value the payment stream at the common monthly valuation rate. This separates the rate used to set the installment from the rate used to compare costs.

Offer A payment. The monthly contractual rate is $0.12/12=0.01$, and its principal is $10,000 because the fee is paid separately. The annuity factor is

a_{\overline{12}|0.01} =\frac{1-(1.01)^{-12}}{0.01} \approx 11.2550775.

Thus $R_A=10{,}000/11.2550775\approx \$888.49$. The common monthly valuation rate is $j=0.06/12=0.005$, for which

a_{\overline{12}|0.005} =\frac{1-(1.005)^{-12}}{0.005} \approx 11.6189322.

The present value of Offer A’s payments is approximately $888.488\times11.6189322=\$10{,}323.28$, using the unrounded payment in the calculation. Add the upfront fee once: $C_{0,A}=10{,}323.28+200=\$10{,}523.28$.

Offer B payment. Its contractual monthly rate is $0.06/12=0.005$. Because the $400 fee is financed, the principal is $10{,}000+400=\$10{,}400$. Its payment is $R_B=10{,}400/11.6189322\approx \$895.08$. Since its contractual rate equals the common valuation rate, the present value of those payments is the financed principal:

C_{0,B}=895.076\ldots\times11.6189322=\$10{,}400.00.

Offer B has the lower common-date cost by $\$10{,}523.28-\$10{,}400.00=\$123.28$, even though its monthly payment is slightly higher. The comparison includes the financed fee through the larger payment and does not add that fee a second time.

Check. Both new loans cover the same $10,000 payoff. Offer A’s additional $200 is paid immediately; Offer B’s additional $400 is included in its $10,400 principal. The cash-flow treatment matches the stated fee arrangements.

Fees: Paid Upfront or Added to the Loan

A fee’s timing matters. A dollar paid at the transaction date and a dollar repaid gradually through a loan are not identical cash-flow patterns. Their values can coincide under particular assumptions, but that is a result of the comparison—not permission to count the fee twice or ignore its timing.

Worked Example: The Same Fee Paid Now or Financed

A borrower refinances a $6,000 payoff into a 12-month loan at a nominal rate of 6% convertible monthly. One offer charges a $60 fee paid upfront. A second offer charges the same $60 fee, but adds it to the principal. Compare the offers at the transaction date using a common valuation rate of 6% nominal convertible monthly. Assume level end-of-month payments and no other charges.

Upfront-fee offer. Its monthly rate is $0.06/12=0.005$, and its principal is $6,000. The monthly payment is $6{,}000/11.6189322\approx \$516.40$. At the common valuation rate, the payments have a value of exactly $6,000 before rounding, because they repay a $6,000 loan at that same rate. Including the fee gives a total value of costs of $6{,}000+60=\$6{,}060$.

Financed-fee offer. Its principal is $6{,}000+60=\$6{,}060$. Its payment is $6{,}060/11.6189322\approx \$521.56$. At the same valuation rate, the value of its 12 payments is $6,060. The fee is already included in that value, so no separate $60 is added.

Under these assumptions, the two offers have the same transaction-date value of costs: $6,060. The financed-fee offer has a higher monthly payment, while the upfront-fee offer requires cash at the transaction date. The equality of present values depends on the valuation rate matching the loan rate; at a different valuation rate, the values need not be equal.

Check. The first offer’s ledger contains $60 at time 0 plus payments on $6,000. The second contains payments on $6,060 and no separate fee cash flow. Adding $60 to the second offer’s present value would count the fee twice.

Different Terms Require an Endpoint Balance

If offers have different terms, comparing them at an intermediate date may help answer a specific question, such as which offer has the lower cost through the first year. But a loan that continues beyond that date still represents an obligation. Include its outstanding balance at the comparison endpoint, valued consistently with the other cash flows.

Worked Example: A Shorter Term Versus a Lower Payment

Two offers each refinance a $6,000 payoff at a nominal rate of 6% convertible monthly. Offer A has 12 monthly payments and a $100 upfront fee. Offer B has 24 monthly payments and a $50 upfront fee. Compare them at the transaction date using a common valuation rate of 6% nominal convertible monthly. To compare Offer B at month 12, include its outstanding balance then. Assume level end-of-month payments and no other charges.

Offer A. The monthly rate is $0.005$, and $a_{\overline{12}|0.005}\approx11.6189322$. Its payment is $6{,}000/11.6189322\approx\$516.40$. At the common valuation rate, the 12 payments have a value of $6,000. Adding the upfront fee gives $C_{0,A}=\$6{,}100$.

Offer B payment and balance. The 24-month annuity factor is $a_{\overline{24}|0.005}=a_{\overline{12}|0.005}(1+v^{12})$, where $v=1/1.005$. Using $v^{12}\approx0.94190534$, this factor is approximately $22.56287$. Therefore, the monthly payment is $6{,}000/22.56287\approx\$265.92$.

After 12 payments, 12 payments remain. The outstanding balance at month 12 is the value then of those remaining payments at the contractual rate: $B_{12}=R_Ba_{\overline{12}|0.005}\approx265.9237\times11.6189322=\$3{,}089.75$.

At time 0, Offer B’s first 12 payments have a value of $R_Ba_{\overline{12}|0.005}\approx\$3{,}089.75$. Its month-12 balance, discounted to time 0, has a value of approximately $v^{12}B_{12}=\$2{,}910.25$. Together, the payments and balance have a value of $6,000 before rounding; including the $50 upfront fee gives $C_{0,B}=\$6{,}050$.

Offer B has the lower common-date cost by $50, even though it takes twice as long to repay and leaves a balance after the first year. Its lower monthly payment alone would not establish that it is cheaper. The endpoint balance is essential to a comparison restricted to the first 12 months.

Common Traps in Offer Comparisons

A frequent error is to compare only monthly payments. In the first worked example, Offer B’s payment is about $895.08, higher than Offer A’s $888.49, yet Offer B has the lower common-date cost. The payment amounts alone leave out the fee and the timing and value of every payment.

Another error is to discount each offer at its own contractual rate. That may describe a loan’s own amortization, but it does not provide a consistent comparison between alternatives. Use each offer’s contractual rate to calculate its payment, then apply one stated valuation rate to all offers. If the task instead asks for a nominal total, state that no discounting is being used and compare complete cash-flow totals on a consistent endpoint.

Wrong: Add a financed fee to the new principal, calculate the payments, and then add the same fee again as an upfront cost.
Correction: Include a fee either in the financed principal or as a separate cash flow according to its timing, not both.
Wrong: Compare payments over a limited period and ignore the remaining balance on a longer loan.
Correction: Include the outstanding balance at the endpoint, or compare each fully amortized schedule through its full term.
Wrong: Use an annual rate as though it were a monthly rate.
Correction: Convert the rate to the payment period before calculating payments or present values.

Key takeaway: Keep two rate roles separate: the contractual rate sets each offer’s payments, while one common valuation rate compares the cash flows. Include every fee once, use the actual payoff amount, and include an outstanding balance whenever the comparison ends before repayment is complete.

Check Your Understanding

For each problem, state the valuation basis and check the timing of any fee or remaining balance.

A borrower refinances an $8,000 payoff into a 12-month loan at 6% nominal convertible monthly. Offer A charges $100 upfront. Offer B charges $40, added to the principal. At a common valuation rate of 6% nominal convertible monthly, which offer has the lower transaction-date value of costs, and by how much? (Answer: Offer B, by $60; the values are $8,100 and $8,040.)
A $9,000 payoff is refinanced for 12 monthly payments at 6% nominal convertible monthly. A $90 fee is financed. Find the monthly payment to the nearest cent. (Answer: $782.34.)
A $5,000 payoff is refinanced for 10 monthly payments at a monthly rate of 1%. One offer charges $75 upfront; the other finances a $75 fee. At a common valuation rate of 1% per month, compare their transaction-date costs. (Answer: They are equal at $5,075.)
Offer A repays a $6,000 loan in 12 months at 6% nominal convertible monthly and charges a $40 upfront fee. Offer B uses the same rate, repays in 24 months, and charges a $20 upfront fee. Compare at month 12 using the same 6% nominal convertible monthly valuation rate; Offer B’s balance then is $3,089.75. Which has the lower value of costs at time 0, and by how much? (Answer: Offer B, by $20; the values are $6,040 and $6,020.)
A borrower refinances $10,000 into 12 payments. Offer A charges 12% nominal convertible monthly, has a $50 upfront fee, and a payment of $888.49. Offer B charges 6% nominal convertible monthly, has a $300 upfront fee, and a payment of $860.66. Use a common valuation rate of 6% nominal convertible monthly and $a_{\overline{12}|0.005}=11.6189322$. Which offer has the lower transaction-date value of costs, and by how much? (Answer: Offer B, by about $73.31; the values are about $10,373.31 for A and $10,300.00 for B.)

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