Refinancing Near the End of a Loan

Why Refinancing Near the End Needs a Different Comparison

When a loan is close to being repaid, refinancing costs can be large relative to the interest savings still available. A lower replacement payment does not by itself show that refinancing is worthwhile: the borrower may pay settlement charges and fees now, and may sell the asset or repay the replacement loan before its scheduled term ends.

The comparison must reflect the borrower’s actual holding period. If the borrower expects to exit after a specified number of installments, include those installments and the loan payoff due at exit. Compare the present values of the two strategies at a common valuation date and using the same comparison rate.

Definition: The holding period is the time from the comparison date until the borrower expects to sell, repay, or otherwise leave the loan. The comparison rate is the effective rate per payment period used to discount cash flows from both strategies to the comparison date. It represents the stated basis for comparing money at different dates; it need not equal either loan’s contractual rate.

Assume level payments are made at the end of each period, rates are effective per payment period, and any stated payoff occurs immediately after the scheduled payment at the exit date. A settlement penalty or refinancing cost is included at the date it is paid. Unless a problem says otherwise, treat amounts paid upfront as time-zero cash flows. If costs are financed, add them to the replacement loan principal instead.

Set Up Both Strategies on the Same Timeline

Let $B_0$ be the old loan’s balance at the comparison date, $P_{\text{old}}$ its scheduled payment, and $P_{\text{new}}$ the replacement payment. If $h$ installments are made before exit, the old-loan strategy includes the first $h$ old payments and the old balance remaining immediately after payment $h$. The refinance strategy includes any upfront settlement and refinancing charges, the first $h$ replacement payments, and the replacement balance immediately after payment $h$.

For a level-payment loan with $n$ payments remaining and effective rate $i$ per payment period, the balance immediately after a payment, when $n$ payments remain, is the present value of those payments:

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

Use that balance formula with the number of payments remaining after the borrower’s last scheduled payment. If the borrower exits after the final installment, the balance is zero. If the exit occurs between installments, use the payoff amount for that date, including any applicable accrued interest or penalty specified in the problem.

Why Discounting the Exit Balance Is Essential

A useful identity explains why the exit balance belongs in the comparison. Suppose a replacement loan has principal $L$, rate $j$, and level payment $Q$. After $h$ payments, let $B_h$ be its outstanding balance. When the comparison rate equals the replacement loan rate $j$, the present value of the payments made so far plus the exit balance is exactly the original principal.

Q a_{\overline{h}|j}+B_h(1+j)^{-h}=L.

To justify the identity, split the replacement loan’s original repayment schedule into the first $h$ payments and the remaining payments. The original principal is the present value of all scheduled payments. The present value at time zero of the remaining payments is $B_h(1+j)^{-h}$, because $B_h$ is their value at time $h$. Adding the present value of the first $h$ payments therefore recovers the original principal. This also shows why using only the payments made before exit understates the refinance cost: it leaves out the amount still owed.

If the comparison rate differs from $j$, discount every payment and the exit balance at the comparison rate, not at $j$. The identity above then does not simplify the comparison to $L$; calculate the discounted cash flows directly.

Worked Comparisons

Worked Example: Exiting After One Payment

A borrower has three monthly payments of $1,000 remaining on an old loan charging 1% effective interest per month. The old loan can be settled now for its balance, with a $15 penalty. A replacement loan can be obtained for exactly the old balance, at 0.5% effective interest per month, with a three-month term. Refinancing costs $25, also paid upfront. The borrower expects to make one monthly payment and then repay the loan. Compare the strategies using a 0.5% monthly comparison rate.

Method. Find the old balance now and after the first payment. Discount the old payment and exit payoff to time zero. For refinancing, include both upfront charges and use the replacement loan’s payments and exit balance. The comparison rate equals the replacement rate, so their combined present value is the replacement principal.

The old balance now is

B_0=1{,}000a_{\overline{3}|0.01} =1{,}000\left(\frac{1-(1.01)^{-3}}{0.01}\right) =1{,}000(2.9409853) =\$2{,}940.99.

After the first old-loan payment, two payments remain. Thus the exit payoff, before any exit charge not specified in the problem, is

B_1=1{,}000a_{\overline{2}|0.01} =1{,}000(1.9703951) =\$1{,}970.40.

The present value of keeping the old loan is the first $1,000 payment at month 1 plus the month-1 payoff, discounted at 0.5% per month:

PV_{\text{keep}} =\frac{1{,}000+1{,}970.3951}{1.005} =\$2{,}955.62.

The replacement loan principal is $2,940.9853. Its first payment and month-1 exit balance, discounted at the replacement rate of 0.5%, have present value equal to that principal. The penalty and refinancing cost add $15+$25=$40 upfront:

PV_{\text{refinance}} =2{,}940.9853+15+25 =\$2{,}980.99.

Refinancing costs $2{,}980.9853-2{,}955.6170=\$25.3683$, or $25.37 more to the nearest cent. The comparison includes the replacement payoff due after the first payment; ignoring it would make refinancing look artificially cheap.

Worked Example: Refinancing Through the Final Payment

Two monthly payments of $500 remain on a loan charging 2% effective interest per month. A replacement loan charges 1% effective interest per month and has two monthly payments. The old loan’s payoff is its outstanding balance, and the refinancing fee is $10 paid upfront. The borrower will keep the loan through the final payment. Compare using 1% per month.

Method. Since the borrower holds the loan to maturity, there is no exit balance after the final installment. Compare the present value of the old payments with the replacement principal plus the upfront fee. The replacement payments’ present value at 1% equals the replacement principal.

The old balance to be refinanced is

B_0=500a_{\overline{2}|0.02} =500(1.9415609) =\$970.78.

The present value of keeping the old loan, using the 1% comparison rate, is

PV_{\text{keep}} =500a_{\overline{2}|0.01} =500(1.9703951) =\$985.20.

The replacement payments have a present value of $970.7805 at the comparison rate. Including the $10 fee gives

PV_{\text{refinance}} =970.7805+10 =\$980.78.

The present-value saving from refinancing is $985.1975-980.7805=\$4.4170$, or $4.42 to the nearest cent. This is a small saving because only two old payments remain; the fee uses up much of the difference. The replacement payment is about $970.7805/a_{\overline{2}|0.01}=\$492.68$, but the decision follows from the full present-value comparison, not from the lower payment alone.

Worked Example: Financing the Fee and Exiting Early

An old loan has two monthly payments of $1,020 remaining and charges 1% effective interest per month. The borrower will make one payment, then repay the loan. A replacement loan charges 0.5% effective interest per month, has a two-month term, and finances a $25 refinancing fee by adding it to the loan principal. Compare the strategies at 0.5% per month. There is no separate settlement penalty.

Method. First find the old balance and the old payoff after the scheduled payment. Then compare the old payment and exit payoff with the replacement payment and exit balance. Because the fee is financed, it increases the replacement principal by $25; it is not also an upfront charge.

The old balance now is

B_0=1{,}020a_{\overline{2}|0.01} =1{,}020(1.9703951) =\$2{,}009.80.

After one payment, one old payment remains. The payoff is that remaining payment discounted one month at the old rate:

B_1=1{,}020a_{\overline{1}|0.01} =\frac{1{,}020}{1.01} =\$1{,}009.90.

The present value of keeping the old loan is

PV_{\text{keep}} =\frac{1{,}020+1{,}009.9010}{1.005} =\$2{,}019.80.

The replacement principal, including the financed fee, is $2{,}009.8030+25=\$2{,}034.80$. Its rate equals the comparison rate, so the first replacement payment and the balance at exit have present value equal to that principal:

PV_{\text{refinance}}=2{,}034.8030=\$2{,}034.80.

Refinancing costs $2{,}034.8030-2{,}019.8020=\$15.0010$, or $15.00 more to the nearest cent. The $25 fee is financed, but it still increases the amount owed. Counting it as both financed and paid upfront would overstate the refinance cost.

Common Mistakes Near Loan Maturity

Comparing payments only. A lower replacement payment does not account for upfront charges or the payoff due if the borrower exits before maturity.
Using the wrong holding period. If the borrower exits after $h$ payments, do not value all scheduled payments as though the borrower will keep the loan to its stated maturity. Include the balance due at the actual exit date.
Discounting at the wrong rate. Use the stated comparison rate for both strategies. A loan’s contractual rate determines its balance and payment, but it is not automatically the comparison rate.
Mixing financed and upfront costs. Add an upfront charge as a time-zero cash flow. Add a financed charge to the replacement principal. Do not count the same charge twice.
Including an exit balance after the final payment. When the borrower has made every required payment, the balance is zero; there is no additional payoff to include.

Key takeaway: For a near-end refinance, identify the actual exit date first. Value each strategy’s payments, upfront charges, and payoff at that date on the same timeline and at the same comparison rate. The remaining loan balance is a cash flow in the comparison, not an optional detail.

Check Your Understanding

For each problem, compare present values at the stated comparison rate. Round money to the nearest cent.

Two monthly payments of $400 remain on an old loan at 1% per month. The borrower will keep the loan to maturity. Refinancing replaces the balance with a two-month loan at 1% per month and has an $8 upfront fee. What is the present-value cost of refinancing relative to keeping the loan? (Answer: Refinancing costs $8.00 more.)
Three monthly payments of $300 remain on an old loan at 2% per month. The borrower will keep it to maturity. A replacement loan charges 1% per month, and its $12 fee is paid upfront. Compare at 1% per month. How much does refinancing save? (Answer: About $5.13.)
Two monthly payments of $500 remain on an old loan at 1% per month. The borrower will make one payment and then exit. The replacement loan has a two-month term at 0.5% per month, and a $8 fee is paid upfront. Compare at 0.5% per month. How much more or less does refinancing cost? (Answer: Refinancing costs about $3.10 more.)
Two monthly payments of $1,020 remain on an old loan at 1% per month. The borrower will make one payment and then exit. A replacement loan charges 0.5% per month, and a $25 fee is financed. Compare at 0.5% per month. How much more does refinancing cost? (Answer: About $15.00 more.)

What You'll Learn