Annuities, perpetuities and loans: pricing level payments
When payments repeat in equal amounts, one formula replaces a long sum. This lesson uses it to price a loan, to show why the early payments on a long loan are mostly interest, to work out regular saving, and to value payments that never end. Apple's lease liability is read as a loan that is repaid quickly.
- Level
- Beginner
- Reading time
- About 27 minutes
- Know this first
- Present value and discounting
By the end you will be able to:
- compute the present value and future value of a level stream of payments
- work out a loan payment and split it into interest and principal
- explain why paying extra early saves so much interest
- value a perpetuity and a growing perpetuity, and say where they break down
Payments that repeat
The first two lessons dealt with single sums. Most real money moves in repeated payments: rent every month, a salary, a mortgage, a pension, interest on a bond. When the payments are equal and come at regular intervals, the stream is called an annuity. Because every payment has the same size, there is no need to discount them one at a time. A short formula adds the whole stream in a single step.
This matters most in one place: borrowing. A loan is the present value of a stream of repayments. The amount lent is the present value, the repayments are the annuity, and the interest rate is the discount rate. Once you see that, the monthly payment on a mortgage, the total interest on a car loan and the effect of paying a little extra all follow from the same relationship.
The formulas
Let PMT be the payment, r the rate per period and n the number of periods. Adding up PMT discounted by 1, 2, up to n periods gives the present value of an ordinary annuity. Compounding each payment forward to the last date gives its future value.
The loan payment is the present value formula solved for PMT. For a monthly loan, r is the yearly rate divided by 12 and n is the number of months, not years. This is the most common error with these formulas, and the calculator below does the conversion for you.
Price a loan and see where the money goes
Set the amount borrowed, the yearly rate and the length of the loan. The payment is fixed for the whole term, paid monthly. You can add an extra amount each month to see how much sooner the loan ends. The left page shows what the loan costs. The right page shows how the first payment divides between interest and principal, how much is still owed after the year you choose, and what the extra payments achieve. This is an illustration with made-up terms, not a quote.
A $250,000 loan at 6.0% over 30 years costs $1,499 a month, and $289,595 in interest over its life. In the first month $1,250 of the payment is interest and only $249 reduces the debt. After 10 years you still owe $209,214. Add an extra monthly payment to see how much interest it saves.
Four experiments. First, shorten the loan to 15 years. The payment rises from $1,499 to $2,110, an increase of 41%, yet total interest falls from $289,595 to $129,736, by 55%. Halving the term does not double the payment, because less interest is piling up. Second, return to 30 years and raise the rate from 6% to 7%. The payment rises to $1,663, and the interest over the life of the loan rises by $59,177. Third, add $200 extra each month. The loan is repaid in 22.2 years rather than 30, and $86,233 of interest is saved. Fourth, look at the balance after 10 years: $209,214 is still owed, so a third of the way through the term only 16.3% of the loan has been repaid. The early payments are mostly interest.
Why early payments are mostly interest
Each month the lender charges interest on the balance still owed. The payment is fixed, so whatever remains after interest goes to reducing the balance. At the start the balance is at its highest, so interest takes most of the payment. As the balance falls, interest shrinks and more of the same payment goes to principal. A table that follows this through the life of the loan is called an amortization schedule. The one below is for the default loan: $250,000 at 6% for 30 years.
| Year of the loan | Interest paid in the year | Principal repaid in the year | Balance at year end | Share of the year's payments that is principal |
|---|---|---|---|---|
| Year 1 | $14,916 | $3,070 | $246,930 | 17% |
| Year 5 | $14,086 | $3,900 | $232,636 | 22% |
| Year 10 | $12,725 | $5,261 | $209,214 | 29% |
| Year 15 | $10,890 | $7,096 | $177,622 | 39% |
| Year 20 | $8,414 | $9,572 | $135,009 | 53% |
| Year 25 | $5,075 | $12,911 | $77,530 | 72% |
| Year 30 | $571 | $17,415 | $0 | 97% |
In the first year $14,916 of the $17,987 paid is interest, and only $3,070 reduces the debt. In the thirtieth year interest is $571. The crossover, the first year in which more than half of the payment goes to principal, is year 19. That is why paying extra early is powerful: every extra dollar of principal removed now stops interest being charged on it for the rest of the loan.
The same formulas for saving
The future value formula works for regular saving. If you put $500 a month into an account earning 7% a year, which is 7% ÷ 12 per month, for 30 years, the account grows to $609,985. You paid in $180,000, so $429,985 came from interest. The same calculation run backwards answers a retirement question: how much do you need to save each month to reach a target? To reach $500,000 in 30 years at 7%, the monthly saving is $410. Starting 10 years later, with only 20 years to go, it rises to $960. Time again matters more than the amount.
Perpetuities: payments that never end
If a payment continues forever, the formula simplifies. As n grows, (1 + r)^−n shrinks toward zero, and the annuity formula reduces to the payment divided by the rate. A perpetuity paying $100 a year at a 5% discount rate is worth $100 ÷ 0.05 = $2,000. Few payments truly last forever, but the formula is a useful approximation for very long streams, and it is the starting point for valuing shares and for estimating the value of a business beyond a forecast period.
If the payment grows at a steady rate g each year, the value is the next payment divided by (r − g), which requires that g is less than r. The table shows how sensitive that is. The next payment is $100 and the discount rate is 8%.
| Growth of the payment | Value, next payment $100, rate 8% | Gap between rate and growth |
|---|---|---|
| 0% a year | $1,250 | 8 points |
| 2% a year | $1,667 | 6 points |
| 4% a year | $2,500 | 4 points |
| 6% a year | $5,000 | 2 points |
Moving growth from 0% to 4% doubles the value, from $1,250 to $2,500, because the gap between rate and growth halves from eight points to four. As g approaches r the value races toward infinity. That is a warning, not a result: no payment can outgrow the economy forever, and anyone using the growing perpetuity must keep g sensibly below r.
Apple's leases: a loan repaid fast
The lease liability from the previous lesson is a loan in everything but name. The lender is the landlord, the amount borrowed is the $12,430 million present value, the discount rate is 3.1%, and the payments in Apple's lease note are the repayments. Apple's note gives the payments by year: $1,991 million in fiscal 2025, falling to $1,223 million in fiscal 2029, then $6,400 million over the years after that.
Run the loan view. In the first year interest is about 3.1% of $12,430 million, which is $385 million, and the payment is $1,991 million, so about $1,606 million, or 81% of the first year's payment, repays principal. Compare that with the 30-year mortgage above, where 83% of the first year's payments went to interest. A mortgage is paid slowly with level payments; Apple's leases are shorter and front-loaded, so the balance falls quickly. Apple states a weighted-average remaining lease term of 10.3 years.
A level annuity does not describe the stream well. Spread over 10.3 years at 3.1%, a level payment that repays $12,430 million is about $1,428 million a year, well below the $1,991 million paid in the first year. The disclosed schedule is front-loaded: payments fall from $1,991 million in fiscal 2025 to $1,223 million in fiscal 2029, and the $6,400 million after that, on the nine-year assumption of the previous lesson, is about $711 million a year. The lesson here is a general one. Annuity formulas assume level payments, and a real schedule that is not level has to be discounted payment by payment, as in the previous lesson.
What the annuity formulas cannot tell you
- They assume a fixed rate and fixed payments. A variable-rate loan changes its payment when the rate changes, and the formula applies only between changes.
- They leave out fees. Origination fees, insurance and taxes added to a mortgage payment make the real cost higher than the formula says.
- They assume payments are made on time. Late fees and penalties are not in the schedule.
- Growing perpetuities are fragile. A small change in g or r changes the value a great deal, so treat the answer as a range, not a figure.
- Early repayment may carry a cost. Some loans charge a penalty for paying early, which reduces the saving shown above.
Start of the period or end of the period
An annuity formula assumes the payments come at the end of each period, which is called an ordinary annuity. When payments come at the start, as with rent paid in advance, the series is an annuity due, and every payment is received one period earlier. The present value rises by exactly one period of interest, which is a factor of one plus the rate. The table shows the effect for a payment of 1,000 a year at 6%.
| Payments of 1,000 a year at 6% | Present value, paid at year end | Present value, paid at start of year | Difference |
|---|---|---|---|
| 5 years | 4,212.36 | 4,465.11 | 252.74 |
| 10 years | 7,360.09 | 7,801.69 | 441.61 |
| 20 years | 11,469.92 | 12,158.12 | 688.20 |
Over ten years, 1,000 a year is worth 7,360.09 if paid at year end and 7,801.69 if paid at the start of each year. The same contract with a different payment date is a different amount, so check the date before applying the formula. Lease contracts, insurance premiums and pensions all differ in whether they pay in advance or in arrears.
Monthly or yearly payments
Take a loan of 200,000 at a nominal rate of 5% over twenty years. If payments are made monthly, each is 1,319.91, so the borrower pays 15,838.94 a year. If they are made once a year, each is 16,048.52. Because monthly payments reduce the balance sooner, the monthly loan costs less interest in total: 116,779 against 120,970, a difference of 4,192. Conventions vary by country and by product: payments may be monthly, quarterly or yearly, and interest may be added at a different frequency from the one at which payments are made. The first question about any loan is therefore how often payments are made and how often interest is added.
The flat rate trap
Some lenders in some markets quote a flat rate, which charges interest on the original amount for the whole term and not on the balance that is left. A 200,000 loan at a flat 5% over five years carries 50,000 of interest in total. Spread with the principal over 60 months, each payment is 4,166.67. A normal reducing-balance loan at 5% over the same term has a monthly payment of 3,774.25. Solving for the rate that makes the flat-rate payments equal to a reducing-balance loan gives an annual rate of 9.15%, nearly double the quote. The difference arises because the borrower repays principal every month but is charged as if none had been repaid.
This is why regulators in many countries require lenders to disclose an annual percentage rate or an effective rate computed on the reducing balance. Use this standard when comparing offers, and when a quote does not say how interest is calculated, ask.
Common mistakes
Using yearly rates with monthly payments
For monthly payments divide the yearly rate by 12 and count the months. Using 6% with 360 payments would make the loan far cheaper than it is.
Comparing loans by payment alone
A lower payment over a longer term usually means much more interest in total. Compare total interest and the length of the loan as well.
Forgetting that the balance falls slowly at first
After a third of a 30-year loan, only 16.3% of it has been repaid. Selling or refinancing early means owing most of the original amount.
Treating the perpetuity formula as exact
It is a limiting case. Use it with a rate comfortably above the growth rate, and show how the answer changes if you shift either.
Ignoring payment timing
An annuity due is worth more than an ordinary annuity by one period's interest, because each payment comes a period earlier. For the settlement example that would be a factor of 1 + r.
Try it yourself
1. What is the present value of $1,000 paid at the end of each year for 10 years, discounted at 6%?
Show the working
1,000 × (1 − 1.06−10) ÷ 0.06 = 1,000 × 7.3601 = $7,360.09. It is less than $10,000 because the later payments are discounted more.
2. You borrow $20,000 at 5% a year for 5 years, with monthly payments. What is the payment?
Show the working
Monthly rate = 0.05 ÷ 12 = 0.4167%. Number of payments = 60. Payment = 20,000 × 0.004167 ÷ (1 − 1.004167−60) = $377.42. Total paid is $22,645, so the interest is $2,645.
3. A share is expected to pay $3 next year, with the payment growing 2% a year after that. If you require 8%, what is it worth?
Show the working
3 ÷ (0.08 − 0.02) = $50.00. If you required 7%, it would be worth $60.00, a rise of 20% for one point.
4. Apple's lease liability is $12,430 million at 3.1%. Roughly how much interest does it carry in the first year, and what does that show about the speed of repayment?
Show the working
12,430 × 0.031 = $385 million. Against a first-year payment of $1,991 million, only about 19% is interest and the rest repays principal, a much faster repayment than a 30-year mortgage.
Where this is used
- A bank prices a mortgage, a car loan or a business loan by solving the annuity formula for the payment.
- A saver or planner works out how much a regular contribution grows to, or how much is needed each month to reach a target.
- An accountant records a lease or a long-term obligation as the present value of fixed payments.
- An analyst values a mature, steadily growing stream of payments using a growing perpetuity.
Key terms
- Annuity
- A stream of equal payments at regular intervals.
- Ordinary annuity
- An annuity whose payments are made at the end of each period.
- Annuity due
- An annuity whose payments are made at the start of each period.
- Perpetuity
- A stream of payments that continues forever.
- Amortization
- Repaying a loan through payments that cover both interest and part of the principal.
- Amortization schedule
- A table showing, for each payment, the interest, the principal and the balance left.
- Principal
- The part of a loan payment that reduces the amount owed.
Questions people ask
Why does a 15-year loan cost so much less in total?
Because the balance falls faster, so less interest is charged for fewer years. The payment is higher, but the total paid is much lower.
Is it better to pay extra on a loan or invest the money?
It depends on how the loan rate compares with what you can earn after tax and risk. Repaying a loan is a certain saving at the loan rate, which makes it a useful benchmark.
What is the difference between an annuity in finance and an insurance annuity?
In finance an annuity is any level stream. An insurance annuity is a product that sells you such a stream, usually for life, in exchange for a lump sum.
Why does the perpetuity formula use the next payment?
Because the first payment in the formula arrives one period from now. Using this year's payment would count a payment that has already been made.
Sources and review
- Apple Inc., Form 10-K for the fiscal year ended 28 September 2024, filed with the US Securities and Exchange Commission. Source of the lease payments by year, the lease liabilities and the weighted-average term and discount rate.
- EDGAR Tools, Apple lease disclosures, a data site that reproduces the lease note, used to read the note tables.
- AccountingTools, How to calculate present value, for the discounting basis of the annuity formulas.
- Brilliant, Time value of money, for annuities and perpetuities.
- US Securities and Exchange Commission, Compound Interest Calculator, for checking the saving examples.
How the numbers were checked: the loan calculator was tested against separate arithmetic for more than three hundred combinations of amount, rate, term, extra payment and check year. Total interest was shown to equal the total paid less the amount borrowed, and the first payment's interest and principal to add to the payment. The balance was computed by a month-by-month loop and by the closed formula, and the extra-payment case by a full month-by-month schedule. The amortization table was recomputed by loop, and Apple's lease figures were tied out as in the lesson on present value.