Mortgage Term Comparison Calculator

Purpose and model boundary

Use this calculator to compare a shorter fixed-rate mortgage with a longer one on the same amount borrowed. It reports the monthly principal-and-interest payment on each loan, the total interest each one costs if it runs to the end of its term, and the difference between those two interest totals.

It then does the comparison that stopping at interest saved cannot do. The shorter loan costs more every month, and that extra money is real: the borrower who takes the longer loan still has it. This calculator follows that payment difference forward to the day the shorter loan is paid off and sets two figures against each other on that same day. On one side is the balance still owed on the longer loan. On the other is what the payment difference would have grown into at a return you assume. The difference between them is the net advantage of the shorter loan, and it is the headline result.

The calculator also answers a separate question that has no counterpart in a plain payment comparison. Take the longer loan, and every month pay the shorter loan's payment on it instead of the scheduled one. This calculator reports when the loan would be gone. That date is later than the end of the shorter term, because the longer loan carries the higher rate, and the gap between the two dates is what that rate difference costs.

What this calculator does not model, plainly and specifically:

It models principal and interest only. Property taxes and homeowners insurance sit on top of both payments and are not included on either side. Mortgage insurance is not modelled at all, which matters here because the larger payment on a shorter loan reaches the point where mortgage insurance falls away sooner than the smaller payment does. Points, origination fees and closing costs are not modelled; the amount borrowed is entered directly and is the same for both loans, so a quote that buys its rate down with points is not represented.

No tax of any kind is modelled. Mortgage interest may be deductible and investment gains are usually taxed, the two are taxed differently, and leaving both out can move the comparison in either direction.

No rate changes. Both loans are fixed for their whole term, and the assumed return on the payment difference is a single constant annual figure.

The interest saved by the shorter loan is fixed by the loan terms, while the side account's return is uncertain. The headline comparison still uses the single Investment Return you enter, but the sensitivity table no longer treats that forecast as certain: Return Volatility spreads it across plausible annual returns and reports the chance and percentile range in which the shorter loan stays ahead. The analysis does not predict how long you keep the loan; tenure remains an unweighted scenario axis.

Finally, the calculator assumes the payment difference is actually invested, every month, for the whole of the shorter term. That is an assumption about behaviour, not arithmetic, and it is the one that fails most often in practice.

Inputs and units

All eight inputs are required. Investment Return must state the assumption explicitly; enter 0% when the side account earns nothing. Return Volatility and Tenure Step define the sensitivity grid around the entered return and shorter term.

The shipped state is 300,000 dollars borrowed, fifteen years at 5.500 percent against thirty years at 6.200 percent, with no assumed return.

Both terms are limited to forty years by the published input rules. A separate stored limit, the longest term the balance chart can draw, is also forty years, and the status logic below checks the longer term against that limit by name rather than against a number written into the rule.

Return Volatility and Tenure Step are required controls for the 11 × 11 sensitivity table. They do not replace Investment Return or either loan term in the headline calculation. They define the return spread across the columns and the scenario horizons down the rows.

Governing relationships

The relationships below are the method recorded in the delivered audit material. Symbols are used consistently throughout.

Let P be the amount borrowed. For a given loan, let r be its monthly rate, one twelfth of the annual rate entered, and let n be its term in whole months, the term in years multiplied by twelve and rounded to the nearest whole month. Let PMT be that loan's level monthly payment.

Level payment. A fixed-rate loan that amortizes to zero over its term pays

PMT = P × r / (1 − (1 + r)^−n)

When the monthly rate is effectively zero, the formula above divides by zero, so the model takes a separate branch and spreads the principal evenly: PMT = P / n.

Total interest. Every dollar paid over the full term, less the principal repaid:

total interest = PMT × n − P

Payment difference. With PMT_s the shorter loan's payment and PMT_l the longer loan's payment,

D = PMT_s − PMT_l

This is the money the rest of the comparison follows. Interest saved is the plain difference of the two interest totals, long less short.

Remaining balance, in closed form. After k monthly payments, a loan with monthly rate r and payment PMT still owes

B(k) = P × (1 + r)^k − PMT × ((1 + r)^k − 1) / r

floored at zero. This is evaluated rather than walked through a schedule, which is why the answer does not drift over long terms. The zero-rate branch is B(k) = P − PMT × k, also floored at zero. The model uses this twice: once on the longer loan at k equal to the shorter loan's whole term in months, which gives the balance still owed on the payoff day, and once a year for each loan to draw the balance chart.

Side account. The payment difference is treated as an ordinary annuity: paid at the end of each month, for the whole of the shorter term, earning i, one twelfth of the assumed annual return. Its value on the payoff day is

S = D × ((1 + i)^n_s − 1) / i

where n_s is the shorter term in months. When the required Investment Return is 0%, this reduces to S = D × n_s, the difference simply banked. The side account is zero when D is not positive, because there is no freed-up money to follow.

Net advantage. On the shorter loan's payoff day,

net advantage = B_long(n_s) − S

A positive figure means the shorter loan leaves you better off on that day. A negative figure means the longer loan plus investing the difference does.

Months to clear the longer loan at the shorter payment. Solving the level-payment relation for the number of months when the payment is chosen rather than derived,

m = −ln(1 − P × r_l / PMT_s) / ln(1 + r_l)

where r_l is the longer loan's monthly rate. The logarithm is undefined when the chosen payment does not cover the first month's interest, that is when PMT_s is not greater than P × r_l; the model returns zero there rather than an error. The zero-rate branch is m = P / PMT_s. The same figure in years is m divided by twelve.

The sensitivity table reruns the same net-advantage relationship 121 times. Its rows move the comparison horizon five Tenure Step intervals below and above the shorter term. Its columns move Investment Return from 2.5 Return Volatility standard deviations below the entered return to 2.5 above it. Each cell is Net Advantage Short for that exact horizon and return.

Only the return axis is probability-weighted. Probability Short Wins is the weighted share of that return spread at the shorter term you entered where Net Advantage Short is above $0. Advantage P10, Advantage P50, and Advantage P90 report the unfavorable tenth, midpoint, and favorable tenth for the shorter loan. The tenure rows are scenarios and carry no probability.

Calculation sequence

1. Convert each term to whole months by multiplying the years by twelve and rounding, and convert each annual rate to a monthly rate by dividing by twelve.
2. Calculate the level monthly payment for each loan, taking the zero-rate branch when that loan's monthly rate is effectively zero.
3. Calculate each loan's total interest as its payment times its months, less the amount borrowed.
4. Subtract the longer loan's payment from the shorter loan's payment to get the payment difference, and subtract the shorter loan's interest total from the longer loan's to get the interest saved.
5. Read the required assumed return; 0% is the explicit no-growth case.
6. Calculate the balance still owed on the longer loan at the shorter loan's payoff month, in closed form and floored at zero. This is zero when the shorter term is not actually shorter.
7. Accumulate the payment difference over the shorter term at one twelfth of the assumed return to get the side account on that same day. This is zero when the payment difference is not positive.
8. Subtract the side account from the balance still owed to get the net advantage of the shorter loan.
9. Solve for the number of months the longer loan would take to clear if paid at the shorter loan's payment, and divide by twelve for the same figure in years.
10. Build the year-by-year balance for both loans and pair the two interest totals for the first two charts; expose the sensitivity table's probability-by-tenure and centre-row return slices for the third and fourth charts.
11. Build the 11 × 11 sensitivity table by rerunning Net Advantage Short for every tenure and return pair.
12. Weight the return columns to return Probability Short Wins and the 10th, 50th, and 90th percentile net advantages at the entered shorter term. Do not probability-weight tenure.
13. Evaluate the status conditions below in order and return the first one that is true.

Outputs and interpretation

Three outputs are the headline. Net Advantage Short is the one the calculator exists for: the debt still owed on the longer loan less the money accumulated in the side account, both measured on the day the shorter loan is paid off. Positive favours the shorter loan, negative favours the longer loan plus investing the difference. Interest Saved is the figure most comparisons stop at, and on its own it overstates the case for the shorter loan, because it assumes the extra payment costs you nothing. Payment Difference is how much more the shorter loan costs each month, and it is the money the whole comparison follows.

Read the three together. If interest saved is large and the net advantage is small or negative, the comparison is being carried by the assumed return rather than by the rate gap.

Four charts accompany the results. The first plots what is still owed on each loan, year by year from the first payment out to forty years, so the shorter loan's line reaches zero while the longer loan's is still well above it. The second plots the total interest over the whole term for each loan side by side. The third shows the probability that the shorter loan is ahead across the table's tenure scenarios. The fourth holds the entered tenure and shows how the net advantage moves across the investment-return range.

The conditional table appears after all four charts. Read down to compare holding horizons and across to see how investment-return uncertainty changes the decision. Its body, probability, and percentile figures are returned by the calculation service. The page verifies the table centre against the scalar Net Advantage Short and displays the returned values; it does not calculate the probability or percentiles in the browser.

Validation and status logic

This calculator evaluates its status conditions in the order shown below, and the first condition that is true is the one returned. Later conditions are not tested once an earlier one has matched, so a set of inputs that would trip several rules reports only the first. The final row is the fallthrough: it is returned when nothing above it applies.

One status text carries a value in angle brackets. Where the fifth row shows <max term years>, this calculator substitutes the stored limit for the longest term the balance chart can draw, so the reader sees a number rather than a placeholder. That limit is forty years, and the delivered behavioural checks record the substituted message as:

• NOT VALID: the longer term is beyond the 40 years this model charts

The delivered replay cases record three of the other messages verbatim, unchanged from the table above:

• NOT VALID: there is nothing to borrow
• NOT VALID: the shorter term has to be shorter than the longer one
• CHECK: the shorter loan is priced no better than the longer one, which is unusual; confirm both quotes are from the same day

Two points about precedence are worth naming. The rate check sits below every term and amount rule, so a comparison with equal terms reports the term problem and says nothing about the rates. And the last two CHECK rows can both be true at once; when the assumed return is above the longer loan's rate and that same return has flipped the net advantage negative, the return message is returned and the negative-advantage message is not, because the return is the thing worth naming.

Assumptions and limitations

• Both loans are fixed-rate, level-payment loans for the same amount, taken on the same day, and each runs to the end of its own term. Nothing is refinanced, recast or prepaid apart from the accelerated-payoff figure, which is reported separately and does not feed the net advantage.
• Terms are rounded to whole months before anything is calculated, so a fractional year entered as a term is not carried through as a fraction.
• Only principal and interest are modelled. Property taxes, homeowners insurance, association dues, points, origination fees and closing costs are excluded from both loans.
• Mortgage insurance is not modelled. A shorter term reaches the balance at which mortgage insurance can be cancelled sooner than a longer one, and that saving is missing from the shorter loan's side of the comparison.
• No tax is modelled, on the mortgage interest or on the side account's growth.
• The headline uses one constant annual Investment Return compounded monthly. The sensitivity table adds the declared Return Volatility and reports a probability and percentile range over that return spread, but it still does not model serially varying returns, sequence risk, fees, or a separate crash process.
• The side account assumes the payment difference is contributed at the end of every month for the whole shorter term, without interruption.
• The interest saved by the shorter loan is fixed by the loan terms, while the side account's return is assumed. The net advantage subtracts the second from the first as though they were equally certain, and they are not.
• The tenure axis is an unweighted set of scenarios. Probability Short Wins varies investment return at the shorter term entered; it does not estimate when you will sell or refinance.
• Figures are nominal. Nothing is adjusted for inflation, so a dollar in year fifteen is treated the same as a dollar today.
• The comparison measures one day, the shorter loan's payoff day. It says nothing about the position at any other date, and nothing about what happens to either borrower afterwards.
• Whether either payment is affordable, or whether a lender would approve the shorter loan's larger payment, is outside the model.

Restrictions and non-computing states

The amount borrowed must be above zero. At zero or below there is no loan to compare and every figure is returned as zero.

Both terms must be at least one year and no more than forty, and the shorter term must be strictly shorter than the longer one. Equal terms are rejected rather than treated as a tie, because two identical terms are not a term comparison. A longer term beyond the forty-year charting limit is rejected rather than charted off the end of its axis.

Neither rate may be negative. Investment Return is required and may not be negative; enter 0% to model the payment difference as uninvested cash.

Two states compute and return numbers while still flagging themselves. A shorter loan priced no better than the longer one is unusual rather than impossible, so it returns figures with a CHECK. An assumed return above the longer loan's rate likewise computes, and is flagged because at that point the answer is being decided by the assumption rather than by the two quotes.

Some individual figures are deliberately zero rather than blank in edge states. The side account is zero when the payment difference is not positive, which happens when the shorter loan's payment is the smaller of the two. The balance still owed on the longer loan is zero when the shorter term is not shorter, because there is no payoff day to measure at. The months-to-clear figure is zero when the shorter loan's payment would not cover the first month's interest on the longer loan, a state in which no finite payoff date exists.

Errors and warnings

Three different things can go wrong, and they mean three different things.

A rejected entry happens before any calculation. The published input rules bound each field: a term outside one to forty years, a negative amount borrowed, or a negative rate is refused at the point of entry and no calculation is attempted. Nothing is returned because nothing was run. Correct the field and the calculation proceeds.

A NOT VALID or CHECK status is this calculator's own finding about a set of inputs it did accept. NOT VALID means the inputs describe a state the model cannot evaluate as a term comparison, most often equal terms, a longer term beyond the charting limit, or nothing borrowed; the accompanying figures are placeholders and should not be read. CHECK is different in kind: the figures are real and can be read, but the model is naming something about them. It names a shorter loan with no rate advantage, an assumed return above the mortgage rate, or a negative net advantage. A CHECK is not a rejection and not an error, and clearing it is not the goal; understanding why it appeared is.

A connection or calculation-service failure is neither of the above. If the calculation service cannot be reached, or a request to it does not complete, the page reports that the result is unavailable. That is a statement about availability, never a finding about your inputs, and it is never reported as zero, as OK, or as a net advantage of nothing. A blank or unavailable result should be retried, not interpreted.

The trade-off this calculator measures is the one the Consumer Financial Protection Bureau sets out in Understand the different kinds of loans available, where loan term is one of the choices a borrower makes. The CFPB describes a shorter term as carrying higher monthly payments, a typically lower interest rate, and a lower total cost, and a longer term as the reverse. Both rates are entered here rather than assumed, because the size of the gap between them is most of what decides the answer.

The monthly payment on each loan uses the standard level-payment relation for a fixed-rate loan that amortizes to zero over its term. The CFPB's How do mortgage lenders calculate monthly payments? describes the same convention: for most mortgages the principal-and-interest payment comes from a standard formula and the terms of the loan, and a fixed-rate loan making every scheduled payment is paid off exactly at the end of its term.

The accelerated-payoff figure, which takes the longer loan and pays the shorter loan's payment on it, has a floor that follows from how amortization works. A payment that does not cover the month's interest never reduces the balance. The CFPB's What is negative amortization? describes amortization as paying off a loan with regular payments so that the amount owed goes down with each payment, and describes what happens when a payment is too small to do that. This calculator reports no payoff date in that state rather than returning a misleading one.

This calculator compares principal and interest only. The CFPB's What costs come with taking out a mortgage? explains why that boundary is drawn where it is: property taxes and homeowners insurance are costs of homeownership rather than of borrowing, and association dues are usually paid separately. Those amounts are the same whichever term is chosen, so leaving them out of both sides does not move the comparison.

Mortgage insurance is not the same under both terms, and it is not modelled here. Under the CFPB's When can I remove private mortgage insurance (PMI) from my loan?, a borrower may request cancellation once the principal balance reaches 80 percent of the home's original value, and the servicer must terminate it automatically at 78 percent or at the midpoint of the amortization schedule. A shorter term reaches those points sooner, so the saving is real and is missing from the shorter loan's side of this comparison.

Tax is not modelled either, in either direction. Mortgage interest may be deductible under the rules in IRS Publication 936, Home Mortgage Interest Deduction, which would reduce the after-tax cost of the larger interest bill on the longer loan, while gains in the side account are generally taxable, which would reduce what the payment difference actually accumulates. Leaving both out can move the comparison in either direction.

The remaining relationships, the closed-form remaining balance after a given number of payments, the future value of a level monthly contribution, and the solution for the number of months to clear a balance at a chosen payment, are standard financial mathematics and are documented with worked figures in the audit material delivered with this calculator rather than taken from an outside source.

These sources give consumer and tax context. They do not supply this calculator's assumptions, and none of them certifies its result. This calculator is informational and is not financial, tax, investment or mortgage advice. The comparison turns on a return you assume and a discipline you predict; the interest saved is certain and the side account is not, and the model does not discount it for that. Take decisions about buying or financing a home with a qualified professional who knows your circumstances.