Present value and discounting: what a future payment is worth today
Discounting is compounding run backwards. It converts money due on different dates into one comparable number, and it is how companies record leases and how investors value anything that pays over time. This lesson builds the formula, shows why the discount rate matters so much, and rebuilds Apple's lease liability from its own disclosures.
- Level
- Beginner
- Reading time
- About 26 minutes
- Know this first
- Compound interest and future value
By the end you will be able to:
- discount a future amount, or a stream of amounts, to present value
- explain why a small change in the rate moves long-dated values a great deal
- find the break-even rate between two offers
- rebuild a lease liability from its payment schedule and discount rate
Compounding in reverse
The previous lesson asked what a sum today will grow to. This one asks the opposite question: what is a sum promised for the future worth today? The answer is the present value, and finding it is called discounting. It uses the same arithmetic as compounding, run backwards. If $1,000 left at 5% grows to $1,050 in a year, then $1,050 due in a year is worth $1,000 today at 5%.
Almost every financial decision compares amounts that arrive on different dates: a loan repaid over years, a bond that pays interest twice a year, a machine that earns for a decade, a lease paid monthly. Those amounts cannot be added up as they stand, because a dollar in 2035 is not the same as a dollar now. Discounting converts each to its value on one date, today, so they can be added and compared.
See what waiting costs
Set a sum due in the future, how many years away it is, and the discount rate. The left page shows what it is worth today and how much of its face value is taken away by waiting. The right page shows the discount factor, the share lost, and what the same sum would be worth if the rate were two points higher or lower. The sentence below the ledger reads it in plain words.
$10,000 received in 10 years is worth $6,139 today at a 5.0% discount rate. Each dollar due then counts for 0.614 dollars now, so $3,861 of the face value is the cost of waiting. If the rate were 7.0% the value would be $5,083; at 3.0% it would be $7,441.
Four experiments. First, push the years to 30 and leave the rate at 5%. The $10,000 is now worth $2,314: a promise far enough away is worth very little today. Second, return to 10 years and raise the rate to 10%. The value falls from $6,139 to $3,855, so doubling the rate takes 37% off the value. Third, set the rate to 0%. The present value equals the face value, because with no return on offer there is no cost to waiting. Fourth, set 30 years and compare the two sensitivity lines. A two-point rise in the rate cuts the 30-year value from $2,314 to $1,314, a fall of 43%, while for a sum due in one year the same rise changes the value by only about 2%. The longer the wait, the more the answer depends on the rate.
Discounting a stream of payments
Discounting one amount is simple. Real contracts pay several, and the method is the same: discount each payment by the number of years until it is paid, then add. The sum is the present value of the whole stream.
Suppose you are offered a settlement. You can take $60,000 now, or $15,000 at the end of each of the next five years. The instalments total $75,000, which looks better. Whether it is depends on the discount rate.
| Offer | At 4% a year | At 7% a year | At 10% a year |
|---|---|---|---|
| Lump sum today | $60,000 | $60,000 | $60,000 |
| Year 1: $15,000 | $14,423 | $14,019 | $13,636 |
| Year 2: $15,000 | $13,868 | $13,102 | $12,397 |
| Year 3: $15,000 | $13,335 | $12,244 | $11,270 |
| Year 4: $15,000 | $12,822 | $11,443 | $10,245 |
| Year 5: $15,000 | $12,329 | $10,695 | $9,314 |
| Present value of the five instalments | $66,777 | $61,503 | $56,862 |
At 4%, the instalments are worth $66,777 today, more than the $60,000, so waiting pays. At 10% they are worth $56,862, less than the lump sum. The break-even rate, the one at which the two offers are worth the same, is 7.9%. If you could earn more than that on the lump sum, take it. If not, take the instalments. This is the pattern that returns in every lesson of this module: the same cash flows can be good or bad depending on the rate they are measured against.
Where the discount rate comes from
No law of nature sets it. In practice the rate is built from the following parts, and different users stress different ones.
- The return on a safe alternative. A government bond is the usual yardstick for money you are sure to receive. This is the base.
- A premium for risk. A cash flow that might not arrive, or might arrive late, is discounted at a higher rate. The more uncertain, the higher the rate.
- The borrower's own cost of funds. A company often uses the rate at which it can borrow, or, for an investment, the return its investors require. The lesson on net present value returns to this.
- Inflation. If the cash flows are in nominal dollars, the rate should include expected inflation, as the previous lesson showed.
| Discount rate | $100 due in 10 years is worth | $100 due in 30 years is worth |
|---|---|---|
| 2% | $82 | $55 |
| 5% | $61 | $23 |
| 8% | $46 | $10 |
| 12% | $32 | $3 |
The table gives the value today of $100 due in 10 years and in 30 years at four different rates. At a low 2% rate the 30-year promise is worth $55.21; at 12% it is worth $3.34. A rate of 12% is six times the 2% rate, but it cuts the value by 94%. Valuation work on long-lived assets is sensitive to the rate for this reason, and an analyst who shows one answer without saying which rate was used has not shown the full picture.
Rebuilding Apple's lease liability
Apple rents much of what it uses: retail stores, offices, data centers. Under current US accounting rules, a company that signs such a lease records a lease liability on its balance sheet. The liability is the present value of the fixed payments it has promised, discounted at a rate set when the lease starts. The rate used is the one implied in the lease if it can be determined, and otherwise the company's own borrowing rate for a loan of similar term and security. In general it is not updated when market rates move later, unless the lease is changed.
Apple's annual report gives enough to test this. The table is from the lease note in the fiscal 2024 report (the year ended 28 September 2024), in millions of US dollars. It lists the payments due in each of the next five years, the total after that, and then takes off imputed interest, which is the discount, to arrive at the liability. Operating and finance leases are combined, because Apple states a single weighted-average discount rate of 3.1% for the total.
| Payments due in | Operating leases | Finance leases | Total |
|---|---|---|---|
| Fiscal 2025 | 1,820 | 171 | 1,991 |
| Fiscal 2026 | 1,914 | 131 | 2,045 |
| Fiscal 2027 | 1,674 | 59 | 1,733 |
| Fiscal 2028 | 1,360 | 38 | 1,398 |
| Fiscal 2029 | 1,187 | 36 | 1,223 |
| Thereafter | 5,563 | 837 | 6,400 |
| Total undiscounted payments | 13,518 | 1,272 | 14,790 |
| Less: imputed interest | (1,984) | (376) | (2,360) |
| Lease liabilities (present value) | 11,534 | 896 | 12,430 |
The undiscounted payments are $14,790 million. Apple's note takes off $2,360 million of imputed interest and arrives at a liability of $12,430 million. Now discount the payments ourselves at the disclosed 3.1%. For the five years with a stated amount, treat each payment as falling at the end of its year:
| Payment due | Years away | Payment | Discount factor at 3.1% | Present value |
|---|---|---|---|---|
| Fiscal 2025 | 1 | 1,991 | 0.9699 | 1,931 |
| Fiscal 2026 | 2 | 2,045 | 0.9408 | 1,924 |
| Fiscal 2027 | 3 | 1,733 | 0.9125 | 1,581 |
| Fiscal 2028 | 4 | 1,398 | 0.8850 | 1,237 |
| Fiscal 2029 | 5 | 1,223 | 0.8584 | 1,050 |
| First five years | 8,390 | 7,723 | ||
| Thereafter, spread over fiscal 2030 to 2038 | 6 to 14 | 6,400 | 4,731 | |
| Total, our rebuild | 14,790 | 12,454 | ||
| Reported by Apple | 14,790 | 12,430 |
The first five years are worth $7,723 million, which is 62% of the liability. The remaining $4,707 million of the liability must therefore be the value of the $6,400 million of payments after fiscal 2029, a discount factor of 0.735 on average. The note does not say when those payments fall. A test: spread them evenly over fiscal 2030 to 2038, nine years, giving $711 million a year. Their present value is $4,731 million. The total is $12,454 million, which is within 0.2% of Apple's reported $12,430 million. The imputed interest comes out at $2,336 million against Apple's $2,360 million.
The match is not a proof. We chose the nine-year spread because it fits, and payments are made throughout each year, not on its last day. What the exercise shows is that the machinery is simple and does reproduce a reported balance sheet figure: a liability of $12,430 million sits on Apple's balance sheet because $14,790 million of promised payments, many of them far in the future, are worth less than their face value today. It also shows the weight of the near years. Over 62% of the liability is explained by the first five years of payments alone.
The rate matters here too. Discount the same payments at 2.1% and the liability is about $13,135 million. At 4.1% it is about $11,832 million. A change of one point in either direction moves the balance sheet figure by roughly $652 million, and that is before any of the payments change. For the fiscal 2025 report, Apple states fixed lease payment obligations of $16.8 billion, of which $2.6 billion fall due within 12 months. That is the undiscounted figure, comparable to the $14,790 million above, and it has grown.
What discounting cannot tell you
- The answer is only as good as the rate. A small change in the rate changes a long-dated value a great deal, as the experiments and the Apple test both show.
- Timing is assumed. The formula puts each payment at one date. Real payments arrive through the year, so end-of-year discounting understates value slightly.
- It treats the cash flows as certain. Risk is dealt with only through the choice of rate, which is crude.
- A present value is not a price. Two parties with different rates will value the same payments differently, and a trade happens in the gap.
- Accounting rates are fixed in the past. A lease liability uses the rate at the time of signing, so it does not show what the same payments would be worth at today's rates.
What is a stream of dividends worth today?
The present value of a stream of payments is the sum of each payment divided by one plus the discount rate, raised to the number of years until it is received. The table applies this to a real set of payments: the dividends Apple, Microsoft and Toyota paid in their latest year, held flat for five years and discounted at three rates. It is an exercise and not a valuation, since dividends change and five years is not the whole life of a company.
| Latest year, millions of each currency | Dividends paid | Change on the year before | Worth today at 6% | Worth today at 8% | Worth today at 10% |
|---|---|---|---|---|---|
| Apple | $15,421 | 1.2% | $64,959 | $61,572 | $58,458 |
| Microsoft | $24,082 | 10.6% | $101,442 | $96,152 | $91,290 |
| Toyota | ¥1,132,329 | 28.6% | ¥4,769,782 | ¥4,521,061 | ¥4,292,418 |
Take Apple at 8%. Five payments of $15,421 million add up to $77,105 million, but they are worth $61,572 million today, because the later payments are discounted more. The same pattern appears in every row: a higher rate lowers the value, and the effect is bigger for payments further away. The three companies' numbers cannot be compared by size, since they are in different currencies, and the table makes the point that a yen stream is valued with a yen rate.
Match the currency of the cash flow and the rate
Discount rates differ by currency, because they contain the expected inflation and the interest rates of that currency. A cash flow in yen must be discounted with a yen rate, and a cash flow in dollars with a dollar rate. Mixing them produces an answer that is wrong by a predictable amount. The example is hypothetical: ¥1,500 will be received in one year, the yen interest rate is 1%, the dollar rate is 4%, and today's exchange rate is ¥150 per dollar.
- Method one, stay in yen: discount at the yen rate to get ¥1,485.15, then convert at today's rate to get $9.90.
- Method two, convert first: use the one-year forward exchange rate, which with these interest rates is ¥145.67 per dollar, to turn the payment into $10.30 next year, then discount at the dollar rate. The answer is $9.90 again.
- The mistake: discounting the yen amount at the dollar rate and converting at today's rate gives $9.62, which is −2.9% too low.
The two correct methods agree, because the forward rate is set so that the lower yen interest rate and the higher dollar interest rate offset each other. The wrong method matters most when the interest rate gap is large, which is why a company with operations in several currencies keeps a separate discount rate for each one.
Common mistakes
Adding payments from different years
The settlement example adds up to $75,000, but those dollars are spread over five years and are worth less than $75,000 now. Discount first, add second.
Discounting for the wrong number of periods
A payment at the end of year 3 is discounted three times, not two or four. Count from today to the date of the payment.
Mixing a monthly rate with annual periods
If payments are monthly and the rate is a yearly figure, convert one or the other. A 6% yearly rate is about 0.5% a month, and 120 months, not 10 years, is the number of periods.
Using one rate for everything
A safe government payment and a risky project's profit do not deserve the same rate.
Believing a precise present value is accurate
A result such as $12,454 million looks exact, but it depends on a rate and on timing assumptions. Quote the assumption beside the answer.
Try it yourself
1. What is $20,000 due in 6 years worth today at a discount rate of 5%?
Show the working
20,000 ÷ 1.056 = 20,000 ÷ 1.3401 = $14,924.
2. You are promised $5,000 at the end of each of the next three years. Discounting at 8%, what is the stream worth today?
Show the working
5,000 ÷ 1.08 = 4,629.63; 5,000 ÷ 1.082 = 4,286.69; 5,000 ÷ 1.083 = 3,969.16. The sum is $12,885, against $15,000 undiscounted.
3. A bond-like promise will pay $1,000 in 5 years. If you pay $780 for it today, what yearly return do you earn?
Show the working
Turn the formula round: (1,000 ÷ 780) to the power 1/5, minus 1, is 5.09%. At any discount rate below that, the promise is worth more than $780.
4. Apple's lease note shows undiscounted payments of $14,790 million and a liability of $12,430 million. What does the difference represent, and what share of the undiscounted total is it?
Show the working
The difference, $2,360 million, is the imputed interest: the cost of waiting for payments spread over many years. As a share of the payments it is 16.0%.
Where this is used
- A company records leases and long-term obligations at present value, as Apple does.
- An investor values a share, a bond or a business as the present value of what it is expected to pay.
- A court or insurer converts a stream of future losses into a lump sum for a settlement.
- A manager compares two projects by discounting each one's cash flows to today.
Key terms
- Present value
- What a future amount is worth today, at a stated discount rate.
- Discounting
- Converting a future amount into its value today by dividing by (1 + r) for each period of waiting.
- Discount rate
- The rate used to discount: the return available elsewhere, or required, for money of similar risk.
- Discount factor
- The number a future amount is multiplied by to give its present value: 1 ÷ (1 + r)^n.
- Lease liability
- The present value of the fixed payments a lessee has promised, recorded on its balance sheet.
- Imputed interest
- The difference between total lease payments and their present value: the discount.
- Break-even rate
- The discount rate at which two alternatives have the same present value.
Questions people ask
What is the difference between present value and future value?
They are the same relationship seen from two ends. Future value grows a sum forward at a rate. Present value brings a future sum back at a rate.
How do I choose a discount rate?
Start with the return on a safe alternative for the same term, then add a premium for the risk that the money does not arrive as promised. State the rate you used.
Why is a larger discount rate worse for long payments?
Because the rate is applied once for every year of waiting. A payment 30 years away is discounted 30 times, so a small rate change is multiplied many times over.
Does Apple really discount at 3.1%?
That is the weighted-average rate it reports across its leases at the end of fiscal 2024. Each lease has its own rate, set when the lease began, and the average moves as leases are added and end.
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 note: payments by year, imputed interest, lease liabilities and the weighted-average discount rate.
- Apple Inc., Form 10-K for the fiscal year ended 27 September 2025. Source of the fixed lease payment obligations of $16.8 billion stated in the discussion of liquidity and capital resources.
- 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 present value formula.
- Brilliant, Time value of money, for discounting and the discount factor.
- Apple Inc., Form 8-K, Exhibit 99.1: fourth quarter and full-year fiscal 2025 results, filed with the US Securities and Exchange Commission. Source of Apple's statements.
- Microsoft Corporation, Form 8-K, Exhibit 99.1: fourth quarter and fiscal year 2025 results, filed with the US Securities and Exchange Commission. Source of Microsoft's statements.
- Toyota Motor Corporation, FY2025 Financial Summary (IFRS, unaudited, in millions of yen), furnished to the US Securities and Exchange Commission on Form 6-K. Source of Toyota's statements.
How the numbers were checked: Apple's lease table was added up by code and matched its printed totals: the operating and finance payments sum to their totals, the imputed interest sums to the total, and the liability equals payments less imputed interest, both in total and for each type. Its current and non-current lines add up to the liabilities. The present value rebuild was computed twice, once with a loop and once with the closed formula for an annuity. The calculator was tested against separate arithmetic across hundreds of combinations, including that discounting and then compounding returns the original sum.