Refinancing With a Different Loan Term

Why a New Term Changes the Comparison

When a borrower refinances, the replacement loan can have both a different interest rate and a different repayment term. A longer term generally spreads repayment over more installments, which can reduce the installment amount. But it also means more periods in which interest can accrue. The installment alone therefore does not describe the full schedule.

The calculation begins with the amount financed on the refinance date, the rate per payment period, and the number of replacement-loan payments. From these, calculate the level installment and multiply the unrounded installment by the number of payments to find the scheduled total. This is a comparison of nominal scheduled outlays, not a present-value comparison. It excludes fees, penalties, taxes, and any other amounts unless they are explicitly included in the replacement-loan principal.

Definition: Suppose a replacement loan has principal $B$ at the refinance date, an effective interest rate $i$ per payment period, and $n$ equal payments made at the ends of successive periods. If $i\ne 0$, its annuity factor is $a_{\overline{n}|}=(1-v^n)/i$, where $v=1/(1+i)$. The level payment is $R=B/a_{\overline{n}|}$, and the scheduled total is $nR$. If $i=0$, define $a_{\overline{n}|}=n$; then $R=B/n$ and the scheduled total is $B$.

The rate and payment period must match. For example, annual payments require an effective annual rate; monthly payments require an effective monthly rate. This tutorial takes the rate per payment period as given. It assumes the replacement loan is fully amortized by $n$ level end-of-period payments, with no balloon payment and no additional costs. If the actual contract has fees financed into the loan, use the resulting larger principal $B$; if costs are paid upfront, they are separate from the scheduled installments.

Why the Payment Formula Holds

At the refinance date, each future payment is discounted at the replacement loan’s rate per period. The present value of the $n$ level payments must equal the principal advanced. This is an equation of value: the loan balance is the value, at the start, of the promised repayment stream.

B=Rv+Rv^2+\cdots+Rv^n =R\frac{1-v^n}{i} =R a_{\overline{n}|}, \qquad i\ne 0.

The geometric-series formula applies because $v=1/(1+i)$, so $1-v=i/(1+i)=iv$. Thus $(v+v^2+\cdots+v^n)(1-v)=v-v^{n+1}$, and dividing by $1-v$ gives $v(1-v^n)/(1-v)=(1-v^n)/i$. Solving the value equation for $R$ gives the stated payment. When $i=0$, every discount factor is 1, so the present value is $nR$; equating it to $B$ gives $R=B/n$. The nonzero-rate expression should not be used at $i=0$, because it divides by zero.

A Reliable Comparison Process

Fix the refinance-date principal.
Use the amount financed by the replacement loan. Treat it as given here; payoff amounts and refinancing costs are not being recalculated.

Match the rate to the payment period.
Use the effective rate for one installment period and the number of installments in the new term.

Calculate the annuity factor and installment.
Use $a_{\overline{n}|}=(1-v^n)/i$ for a nonzero rate, or $a_{\overline{n}|}=n$ at zero rate. Then calculate $R=B/a_{\overline{n}|}$.

Calculate and label the scheduled total.
Multiply the unrounded installment by $n$, then round the displayed money amounts. Compare installments and totals separately.

Worked Example: A Four-Payment Replacement Loan

A borrower refinances a balance of $12,000 into a replacement loan at 5% effective annually, with four annual payments at each year-end. Find the level payment and the scheduled total. Ignore fees. Method: value the replacement payments as an annuity-immediate. This avoids using the old loan’s payment schedule for the new term.

The payment period is one year, so $i=0.05$, $n=4$, and $v=1/1.05=0.9523809524$. The discount factors, carried to eight decimal places, are:

v=0.95238095,\quad v^2=0.90702948,\quad v^3=0.86383760,\quad v^4=0.82270247.

Their sum is $a_{\overline{4}|}=3.54595050$. Equivalently, $(1-v^4)/0.05=3.54595050$. Therefore,

R=\frac{12{,}000}{3.54595050}\approx 3{,}384.1420.

The installment is $3,384.14 to the nearest cent. Using the unrounded payment, the scheduled total is $4(3{,}384.1420\ldots)=\$13{,}536.57$, rounded to the nearest cent. As a check, the present value of four unrounded payments is $3{,}384.1420\ldots(3.54595050\ldots)=\$12{,}000$, as required. If the contract instead sets every installment to exactly $3,384.14, those four rounded payments sum to $13,536.56; a lender may adjust the final installment by a cent to reconcile rounding.

Term and Rate Can Pull in Different Directions

A longer term usually lowers the payment for a fixed principal and rate because the principal is repaid over more periods. But comparing two refinancing choices also requires attention to the rate: the annuity factor depends on both $i$ and $n$. In general, one cannot infer which choice has the lower installment or lower scheduled total just from knowing which term is longer. Calculate both choices using their own rates and terms.

A scheduled total is the sum of the stated payments without discounting them back to the refinance date. It is therefore a useful measure of the nominal amount scheduled to be paid, but it does not account for the timing or present value of those payments. A lower scheduled total does not by itself establish that a choice is preferable once fees, timing, or other financial considerations matter.

Worked Example: Comparing Three- and Six-Year Terms

A $16,000 balance can be refinanced at 4% effective annually, with either three or six annual end-of-year payments. Find the level payment and scheduled total for each term, and state which has the smaller payment and which has the smaller scheduled total. Ignore fees. Method: calculate each term’s annuity factor separately. This avoids applying one term’s factor to the other schedule.

For three payments, the discount factors are $1/1.04=0.96153846$, $1/1.04^2=0.92455621$, and $1/1.04^3=0.88899636$. Thus $a_{\overline{3}|}=2.77509103$, and

R_3=\frac{16{,}000}{2.77509103}\approx 5{,}765.5765,\qquad 3R_3\approx \$17{,}296.73.

The displayed installment is $5,765.58, and the scheduled total calculated from the unrounded installment is $17,296.73, with money rounded to the nearest cent.

For six payments, the first three discount factors are as above, and the remaining factors are $1/1.04^4=0.85480419$, $1/1.04^5=0.82192711$, and $1/1.04^6=0.79031453$. Their sum with the first three is $a_{\overline{6}|}=5.24213686$. Therefore,

R_6=\frac{16{,}000}{5.24213686}\approx 3{,}052.1906,\qquad 6R_6\approx \$18{,}313.14.

The six-year option has the smaller installment: $3,052.19 rather than $5,765.58. The three-year option has the smaller scheduled total: $17,296.73 rather than $18,313.14. The comparison uses unrounded installments for the totals; in the three-payment case, multiplying the displayed rounded installment by three gives $17,296.74, one cent more. If a contract specifies cent-rounded installments, its final payment may be adjusted. The check is that each unrounded payment multiplied by its own annuity factor returns $16,000.

Zero Rate and a Longer New Term

The zero-rate case is a useful check on the method. With no interest charged, every dollar of principal is repaid through the installments, so the scheduled total equals the principal regardless of the number of payments. A longer term still changes the installment: dividing the same principal among more payments makes each payment smaller.

Worked Example: A Zero-Rate Offer Versus a Longer Loan

A $12,000 refinance balance can be paid in four annual installments at 0%, or in five annual installments at 4% effective annually. Find each installment and scheduled total, then compare the two measures. Ignore fees. Method: treat the zero-rate case separately, then use the annuity factor for the positive-rate offer. This avoids substituting zero into a formula that divides by the rate.

For the four-payment, zero-rate offer, $a_{\overline{4}|}=4$, so $R_0=12{,}000/4=\$3{,}000$. Its scheduled total is $4(3{,}000)=\$12{,}000$.

For the five-payment offer, $i=0.04$. The discount factors to eight decimal places are $0.96153846$, $0.92455621$, $0.88899636$, $0.85480419$, and $0.82192711$. They sum to $a_{\overline{5}|}=4.45182233$. Thus

R_4=\frac{12{,}000}{4.45182233}\approx \$2{,}695.5253,\qquad 5R_4\approx \$13{,}477.63.

To the nearest cent, the five-year installment is $2,695.53 and the scheduled total using the unrounded payment is $13,477.63. The five-year offer has the smaller installment, but the zero-rate four-year offer has the smaller scheduled total. The present-value check for the positive-rate offer is $2{,}695.5253\ldots(4.45182233\ldots)=\$12{,}000$. As in the earlier example, cent-rounding each payment before summing can create a small difference from the exact level-payment total.

Common Mistakes and Exam Traps

Comparing installments but not totals. The longer-term option can have the smaller installment while producing a larger scheduled total. Calculate both $R$ and $nR$ for every option.
Using the wrong number of periods. A four-year annual-payment loan has $n=4$; a four-year monthly-payment loan has $n=48$. The rate must also be expressed per payment period.
Using a zero-rate formula that divides by zero. At $i=0$, use $a_{\overline{n}|}=n$ and $R=B/n$, not $(1-v^n)/i$.
Multiplying a rounded installment and treating the result as exact. Carry the payment at full precision for $nR$, then round the money amount. If a contract rounds each installment, account for any final-payment adjustment.
Calling the scheduled total a present value. The total $nR$ adds nominal payments without discounting. The equation $B=Ra_{\overline{n}|}$ is the present-value relationship; the two quantities answer different questions.
Ignoring costs or changing the principal without explanation. If fees are financed, include them in $B$. If they are paid upfront, report them separately rather than silently adding them to scheduled installments.

Key takeaway: For each replacement term, match the effective rate to the payment period, calculate its own annuity factor, and find both the level installment and the unrounded scheduled total. A smaller installment does not imply a smaller total outlay. These calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume level end-of-period payments, no fees, and a fully amortized replacement loan unless stated otherwise. Round displayed money amounts to the nearest cent.

A $7,200 balance is refinanced at 0% with three annual payments. Find the payment and scheduled total. (Answer: $2,400.00 per payment; $7,200.00 total.)
A $10,000 balance is refinanced at 10% effective annually for two annual payments. Find the payment and scheduled total, carrying the payment unrounded for the total. (Answer: $5,761.90 per payment; $11,523.81 total.)
A $12,000 balance is refinanced at 5% effective annually for four annual payments. Find the payment and scheduled total, carrying the payment unrounded for the total. (Answer: $3,384.14 per payment; $13,536.57 total.)
A $20,000 balance can be refinanced at 5% effective annually for three annual payments or 4% effective annually for five annual payments. Find each scheduled total using unrounded installments. Which has the smaller installment, and which has the smaller scheduled total? (Answer: Three payments: about $7,344.17 each and $22,032.51 total; five payments: about $4,492.54 each and $22,462.71 total. The five-payment option has the smaller installment; the three-payment option has the smaller scheduled total, by $430.20.)

What You'll Learn