Loan calculator: payment, total interest and the true APR

A personal loan calculator turns four numbers -- amount borrowed, rate, term and any up-front fee -- into the two figures that decide whether a loan is worth taking: the monthly payment you are committing to, and the total amount you hand back.

The payment comes from the same amortizing annuity formula that governs a mortgage or a car loan. What changes on a personal loan is the fee. Many lenders deduct an origination fee from the advance, so you repay interest on money you never received. That gap is why a quoted rate and an effective annual percentage rate are different numbers.

This page sets out the formula, works one loan through by hand, shows how the fee is converted into an effective rate numerically, and quantifies what happens when you stretch the term or pay extra. Every rate below is an assumption for illustration -- lenders price loans individually.

#The formula for a fixed-rate installment loan

Every payment covers the interest accrued that month, and whatever is left reduces the balance. The payment that clears the balance exactly on the final due date is:

Formula: P = A * r / (1 - (1 + r)^-n) where P is the monthly payment, A is the amount borrowed, r is the monthly rate (annual rate / 12, as a decimal), and n is the number of monthly payments.

This is the same relationship as the mortgage version written the other way round; both give an identical payment. Total interest is then simply P * n - A, and total repaid is P * n.

The origination fee sits outside that formula. It is deducted from the advance, so:

Formula: Cash received = A - fee, while the repayment schedule is still built on the full A.

#A worked example, done by hand

Take $18,000 borrowed over 5 years at an assumed 9.5% APR, with a 3% origination fee of $540 deducted at funding.

1. Monthly rate: 0.095 / 12 = 0.0079166667. Payments: 5 x 12 = 60.
2. Discount factor: (1.0079166667)^-60 = 0.62304928.
3. Denominator: 1 - 0.62304928 = 0.37695072.
4. Numerator: 18,000 x 0.0079166667 = $142.50.
5. Payment: 142.50 / 0.37695072 = $378.03 a month.
6. Total repaid: 378.03 x 60 = $22,682.01. Total interest: 22,682.01 - 18,000 = $4,682.01, or 26.0% of the amount borrowed.

The first payment is $142.50 interest and $235.53 principal, leaving $17,764.47 owed. As on any amortizing loan, the interest share falls every month.

Worked example: With the $540 fee deducted, you receive $17,460 but still repay $378.03 a month for 60 months. The rate that makes $17,460 today equal that stream is 10.82%, not 9.5%. The fee costs about 1.32 percentage points of effective rate.

#Quoted rate versus effective APR

A quoted rate prices the loan on its face amount. An effective APR prices it on the money that actually reached your account. When nothing is deducted, they are the same; every dollar of fee pushes them apart.

The calculator finds the effective rate the way a spreadsheet's internal rate of return function does: by search, not by algebra. There is no closed-form solution for r in the payment equation, so it brackets the answer and narrows it.

1. Start with a low guess and a high guess that certainly straddle the answer.
2. Take the midpoint and compute the payment that midpoint would produce on the cash actually received.
3. If that payment is smaller than the real payment, the true rate is higher, so move the low bound up. Otherwise move the high bound down.
4. Repeat. Each pass halves the interval, so a few dozen passes fix the rate to far more precision than any lender quotes.

Multiply the resulting monthly rate by 12 and you have the effective annual rate. Below, the loan is $18,000 over 60 months at an assumed 9.5% nominal rate; only the fee changes.

Two things follow. The payment never changes, so a fee is invisible in the monthly figure -- it only shows up in what you receive. And the shorter the term, the more a fee hurts, because the same cost is spread over fewer payments. A 3% fee on this loan adds about 3.06 points of effective rate over 24 months, 2.10 over 36 months, 1.32 over 60 months and 0.98 over 84 months.

#Where each input comes from

Amount borrowed. The face amount on the note, not the cash you expect to receive. If you need a specific sum in hand and the lender deducts a fee, you must borrow more than that sum.

APR. Use the figure in a written offer. Under federal disclosure rules the APR a lender quotes is meant to include certain finance charges, so if your quote already reflects the fee, leave the fee field at zero to avoid counting it twice.

Term and term unit. Enter years or months to match how the offer is written. A "60-month" and a "5-year" loan are identical here.

Origination fee. Only enter it if it is deducted from the advance. A fee added to the balance is different: in that case, raise the amount borrowed instead.

The payment is most sensitive to the term, and the total cost is most sensitive to the rate. The fee is the smallest of the three in dollars but has an outsized effect on short loans.

#What a longer term actually costs

Extending a term lowers the payment because the same principal is spread over more months. It raises total interest because the balance stays outstanding longer. Below: $18,000 at an assumed 9.5%, no fee.

Going from three years to seven cuts the payment by roughly half but nearly triples the interest. Rate has a similar shape in the other direction, holding the 5-year term fixed:

#Paying extra, and what it removes

On a true simple-interest installment loan, any payment above the scheduled amount goes straight to principal. Next month's interest is charged on a smaller balance, so the saving compounds through the rest of the schedule. Using the same $18,000 at an assumed 9.5% over a 60-month schedule:

An extra $100 a month -- about 26% more than the required payment -- ends the loan 15 months early. The same principle drives the credit card payoff calculator, though a card's balance is revolving rather than fixed.

#Simple interest versus precomputed interest

This calculator models a simple-interest loan: interest accrues on the balance outstanding, day by day or month by month, and prepayment reduces it.

A precomputed loan works differently. All the interest for the full term is calculated at signing and added to the balance, and you owe that total whether you take the whole term or not. Early payoff triggers a rebate of unearned interest calculated by a rule set in the contract, and older rebate methods return less than a straight simple-interest recalculation would. Some vehicle and small-dollar consumer loans still use this structure.

Two questions separate them: does the payoff quote fall as you pay early, and does the contract mention a rebate of unearned finance charges? If the loan is precomputed, the prepayment table above overstates the saving.

#How to read the result

The monthly payment is a fixed obligation. Test it against your take-home pay at a level you could still meet in a poor month.

The effective APR is the number to use when comparing offers. Two loans with the same payment are not equivalent if one hands you less cash. And the total interest figure is the price of the borrowing, worth comparing directly against the cost of the thing being financed -- see the cost per use calculator for that comparison.

#What this model leaves out

• Fees other than origination. Documentation, late, returned-payment, and payment-processing charges are not included.
• Prepayment penalties. Some contracts charge for early payoff, which would erase part of the saving in the prepayment table.
• Variable rates. The model assumes one rate for the whole term.
• Payment timing and day counts. It assumes 12 equal monthly periods a year. A lender accruing daily will produce slightly different interest, depending on the exact funding and due dates.
• Insurance products and add-ons. Credit life or payment protection premiums bundled into the loan raise the real cost.
• Approval. Nothing here evaluates credit, income or eligibility.

#Common mistakes

Comparing loans by payment. A lower payment usually means a longer term, which almost always means more total interest. Compare rate, total repaid and cash received.

Double-counting the fee. If the offer's APR already includes the origination fee, entering the fee again overstates the effective rate.

Borrowing the amount you need rather than the amount that nets it. With a 3% deducted fee, receiving $18,000 requires borrowing about $18,557.

Assuming extra payments always shorten the loan. Some servicers apply extra amounts to the next installment rather than to principal unless you specify otherwise.

Ignoring what the loan replaces. Consolidating higher-rate debt at a lower rate helps only if the balance is not rebuilt behind it. The compound interest calculator shows the same arithmetic working in your favor rather than against you, and the full calculator index covers the surrounding decisions.