Present Value and the Interest Rate

Almost every decision worth making spans time. You spend money now to earn a degree that pays off for decades. A firm builds a factory this year to sell goods for years. A government borrows today and taxes tomorrow. In all of these, costs and benefits land at different dates, which raises a problem we have so far ignored: how do you compare a dollar today with a dollar next year, or thirty years from now? You cannot just add them up. A thousand dollars promised long ago and a thousand dollars in your pocket today are not the same thing, and treating them as equal is one of the most common and expensive mistakes people make.

This topic gives you the tool that solves it. It is called present value, and it puts amounts arriving at different times onto a single ruler so they can be compared. This is the capitalization tool the course has been promising since T7, when future profit first got baked into today’s asset price. Out of that one idea comes a surprising amount: what interest really is and where it comes from, how to price a loan, why a future payment is worth less than a present one, and why the interest rate you pick can swing the answer to a public debate. We build the arithmetic first in this post, then point it at durable assets, your own earnings, profit, and politics in the next. Everything here is one tool used over and over; state it clearly now and the rest of the topic falls out of it.

A Future Dollar Is Worth Less Than a Present Dollar

Start with the most basic move. If you put $100 in an account paying 6 percent a year, in a year you have $106. The thing that turned $100 into $106 is the rate of interest, the rate at which a present amount grows into a future one. Write it as a formula and the whole topic unfolds from it:

P × (1 + r) = F

Here P is the present amount, r is the interest rate, and F is the future amount. With P = $100 and r = 0.06, F comes out to $106. This forward direction, present amount to future amount, is compounding, and over many years it does more than beginners expect, because each year’s interest itself earns interest. The table below shows what $1 grows to over time at various rates. Read it as a multiplier: at 7 percent for ten years, $1 becomes $1.97, so $350 becomes about $690.

Future value of $1 (what a present $1 grows to). Selected rates and years.

Years	3%	4%	5%	6%	7%	8%	10%	12%	15%	20%
1	1.03	1.04	1.05	1.06	1.07	1.08	1.10	1.12	1.15	1.20
3	1.09	1.13	1.16	1.19	1.23	1.26	1.33	1.41	1.52	1.73
5	1.16	1.22	1.28	1.34	1.40	1.47	1.61	1.76	2.01	2.49
10	1.34	1.48	1.63	1.79	1.97	2.16	2.59	3.11	4.05	6.19
15	1.56	1.80	2.08	2.40	2.76	3.17	4.18	5.47	8.14	15.41
20	1.81	2.19	2.65	3.21	3.87	4.66	6.73	9.65	16.37	38.34
30	2.43	3.24	4.32	5.74	7.61	10.0	17.4	30.0	66.21	237.4

A couple of the entries are worth pausing on, because they show up in questions you will be asked. Put $250 in at 7 percent for three years and you have about $306 ($250 × 1.23). And notice how fast money doubles: at 5 percent, $1 grows past $2 in about fourteen years; at 10 percent, in about seven. That regularity has a handy shortcut called the Rule of 72: divide 72 by the interest rate and you get the rough number of years it takes money to double. At 7 percent, 72 ÷ 7 is about ten years. At 12 percent, about six. It is an approximation, but a good enough one to do in your head.

Compounding also explains how modest-looking sums become large fortunes. A piece of land bought for $5,000 that is worth $85,000 thirty years later has grown seventeenfold; scan the thirty-year row and you find that a seventeenfold gain corresponds to about 10 percent a year, the kind of return a broad stock-market investment has historically delivered. Compounding is quiet but relentless. The same arithmetic runs in a barnyard: a population of rabbits, or a stored stock of grain that is planted and harvested, grows by a net percentage each year, and that physical growth rate is just an interest rate wearing a different coat. Whenever something you hold compounds at a steady percentage, you are looking at the rate at which a present amount grows into a future one.

Present Value Reverses the Arithmetic

Now run the formula backward. If a future amount F is worth P × (1 + r) when it arrives, then an amount arriving in the future is worth less today, and we recover today’s value by dividing instead of multiplying:

P = F ÷ (1 + r)

Take $220 due one year from now at 10 percent. Its present value is $220 ÷ 1.10 = $200. We say the future $220 has been discounted back to a present value of $200. The number you multiply by, 1 ÷ (1 + r), is the discount factor; here it is 0.909, so $220 × 0.909 = $200. This is the single most useful operation in finance, because it lets you take any amount arriving at any future date and express it in today’s dollars, where it can be compared with anything else.

The table below gives the discount factors directly. Each entry is the present value of $1 received after the listed number of years. (These are simply the reciprocals of the future-value table above.)

Present value of $1 (today’s value of $1 received in a future year). Selected rates and years.

Years	3%	4%	5%	6%	7%	8%	10%	12%	15%	20%
1	0.97	0.96	0.95	0.94	0.94	0.93	0.91	0.89	0.87	0.83
3	0.92	0.89	0.86	0.84	0.82	0.79	0.75	0.71	0.66	0.58
5	0.86	0.82	0.78	0.75	0.71	0.68	0.62	0.57	0.50	0.40
10	0.74	0.68	0.61	0.56	0.51	0.46	0.39	0.32	0.25	0.16
15	0.64	0.56	0.48	0.42	0.36	0.32	0.24	0.18	0.12	0.07
20	0.55	0.46	0.38	0.31	0.26	0.22	0.15	0.10	0.06	0.03

Two patterns in this table are themselves economic lessons. First, hold the rate fixed and read down a column: the farther into the future an amount lies, the smaller its present value. At 4 percent, $2,500 due in five years is worth $2,500 × 0.82 = $2,050, but the same $2,500 due in ten years is worth only $2,500 × 0.68 = $1,700. Distance in time eats into value, because over a longer wait, a smaller present sum could have grown to the target on its own. Second, hold the year fixed and read across a row: the higher the interest rate, the smaller the present value. That same $2,500 in five years is worth $1,950 at 5 percent but only $1,700 at 8 percent. A higher rate is a stronger pull, discounting future amounts more heavily.

Once you can do this, comparing options that pay off at different times becomes routine. Suppose someone offers you either $30,000 in ten years or $40,000 in fifteen years. Which is better? It depends on the rate. Discount each back to today and compare. At 6 percent, $30,000 in ten years is worth about $16,800 (0.56) and $40,000 in fifteen years about $16,800 (0.42), a near tie; at 4 percent the longer, larger prize pulls ahead. The lesson is not the particular answer but the method: reduce every alternative to its present value, then they sit on the same ruler. A subsidized loan reveals the same logic from the other side. A college “loan” of $1,000 at no interest, repaid years later, is partly a gift: the present value of the repayment is less than $1,000, and the difference is what the lender has handed you. Whenever a future obligation is fixed in dollars but those dollars are discounted, the gap between the face amount and the present value is the real subsidy.

Interest Is the Price of Current Income, Set Where Saving Meets Investment

We have used the interest rate as a given. Where does it come from, and why is it positive at all? Why will people pay extra to have things now rather than later?

There are two reasons, and you should keep them distinct. The first is the productivity of investment. Resources set aside and put to work can yield more later: grapes left to ferment become wine worth more than the juice; grain stored and planted yields a larger harvest; grapes dried into raisins are worth more than the fresh fruit. (Here “more” means more value, not necessarily more physical stuff, a pile of raisins weighs less than the grapes it came from.) Investing simply means giving up some consumption now to get something more valuable later. Painting your own house instead of spending the weekend gaming is an investment, because you sacrifice leisure now for a preserved, more valuable house later. Because such opportunities exist, current resources are worth more than future ones: with current resources you can start the process and reap the growth.

But not every roundabout, time-consuming method pays off. The claim that “more roundabout production is always more productive” is false. Only the right methods in the right amounts add to wealth; pour resources into the wrong long projects and you get less, not more. So the productivity of investment is real but not automatic.

The second reason is time preference. People generally prefer goods sooner rather than later, partly from simple impatience, partly because the future is uncertain and life is finite. Given the choice between a feast today and the same feast a year from now, most take it today. That preference, too, makes current income command a premium over future income.

Put these together in a market and the interest rate is a price like any other, set by supply and demand, the price of current income in terms of future income. People who want current income they don’t yet have, to invest or to consume, demand it; people willing to part with current income in exchange for more future income supply it. The buyers and sellers meet in what we can call the loan market, and the price that clears it is the interest rate. If $100 today trades for a promise of $105 next year, the rate is 5 percent: r = (F − P) ÷ P = (105 − 100) ÷ 100.

In that market, the words run backward from ordinary usage. A lender is buying a future income (you hand over money now to buy the borrower’s promise to pay later), and a borrower is selling future income (selling a claim on next year’s money to get cash now). The demand for current income slopes down: at a lower interest rate, more investment projects clear the bar and more current income is demanded. The supply slopes up: a higher rate coaxes out more saving.

It does not matter whether the current income is wanted for consumption or for investment; both wants press on the same market and help set the same rate. And though we often say that saving equals investment by definition, the economy still has to coordinate the people who save with the people who invest, because they are usually different people acting for different reasons. The saver setting aside money for retirement and the entrepreneur building a plant never meet, yet the interest rate brings their plans into line, rationing current income to its most valued uses, exactly as any price coordinates strangers.

One warning about what interest is not: it is not “the price of money.” Printing more money does not push the interest rate down for long; mostly it raises the price level. The interest rate is the price of current versus future income, a real thing, and we are deliberately leaving the money-supply machinery to a later course.

That last point has a long history. Interest has for centuries been condemned as usury, a slippery word for either a rate judged too high or any interest at all. Early Christian teaching called it sin, a strict reading of the Koran forbids it outright, Aristotle called money sterile, and Communist doctrine branded it exploitation. Yet the people who condemned interest kept collecting it under another name: medieval Christians let Jewish lenders charge it and then borrowed from them, the Church booked it as a fee or a discount, and Communist planners revived it as an “efficiency index.” The reason is the one we just gave: at a zero rate far more people want present goods than will supply them, so a positive price reasserts itself whatever a statute forbids.

Early New England shows the pattern. Its first legal code, the Massachusetts Body of Liberties of 1641, capped interest at “eight pounds in the hundred for one yeare” while warning that even this must not become “a coulour or countenance to allow any usurie amongst us contrarie to the law of god.” Such caps have never held for long. When the Supreme Court ruled in 1978 that a national bank could charge out-of-state customers the rate allowed in its own home state, South Dakota abolished its usury ceiling in 1980 to attract the credit-card business, and the ceilings other states kept on their books became a dead letter.

An Annuity Is a Stream, and a Perpetuity Lasts Forever

Most real decisions involve not a single future amount but a stream of them: a pension that pays every year, a loan repaid in installments, a building that throws off rent for decades. A level stream of equal annual payments is an annuity. Its present value is just the sum of the discounted individual payments, but adding up a column of discount factors every time is tedious, so we tabulate the totals. The table below gives the present value of $1 received at the end of each year for the listed number of years, an annuity factor: multiply it by the annual payment to get the whole stream’s present value.

Present value of $1 received at the end of each year (annuity factors). Selected rates and years.

Years	3%	4%	5%	6%	7%	8%	10%	12%	15%	20%
1	0.97	0.96	0.95	0.94	0.94	0.93	0.91	0.89	0.87	0.83
3	2.83	2.78	2.72	2.67	2.62	2.58	2.49	2.40	2.28	2.11
5	4.58	4.45	4.33	4.21	4.10	3.99	3.79	3.61	3.35	2.99
10	8.53	8.11	7.72	7.36	7.02	6.71	6.15	5.65	5.02	4.19
15	11.94	11.12	10.38	9.71	9.11	8.56	7.61	6.81	5.85	4.68
20	14.88	13.59	12.46	11.47	10.59	9.82	8.51	7.47	6.26	4.87
30	19.60	17.29	15.37	13.77	12.41	11.26	9.43	8.06	6.57	4.98

This one table does enormous work. A three-year annuity of $1,000 at 8 percent is worth $1,000 × 2.58 = $2,580. If your rich uncle buys you fifteen years of $50,000 a year, beginning next year, at a 10 percent rate the gift is worth $50,000 × 7.61 = $380,500 today, not the $750,000 you will eventually collect; that gap between the headline sum and the present value is exactly the point of the exercise, and it is the same gap that turns a no-interest “loan” or a below-market employee-stock loan into a partial gift. The factor also runs in reverse to find an implied rate. If $500,000 today buys you a fifteen-year stream of equal payments and you want to know the built-in interest rate, divide and search the table: $500,000 ÷ 7.61 ≈ $65,700 a year at 10 percent. The same reverse lookup answers questions like “my grandparents sold their house for a stream of payments; what interest rate is built in?”, divide the price by the per-dollar annuity factor and read the rate off the table.

The same machinery prices an ordinary loan, and it clears up something most borrowers never notice. When you take out a mortgage or a car loan and repay it in equal yearly (or monthly) installments, you are selling the lender an annuity: the loan amount is the present value, and the level payment is whatever annual figure makes that stream’s present value equal the sum you borrowed. Run the annuity table in reverse to find it. Borrow $80,000 over twenty years at 5 percent, and the twenty-year, 5 percent annuity factor (about 12.46) says the level payment is roughly $80,000 ÷ 12.46, about $6,400 a year.

Here is the part worth slowing down for: every one of those level payments splits into two pieces, interest on the balance still owed plus a repayment of principal, and the mix shifts over the life of the loan. Early on, you owe a lot, so most of the payment is interest and little goes to principal; as the balance falls, the interest piece shrinks and a growing share of the same fixed payment chips away at principal. Your equity, the part of the asset you actually own free and clear, therefore builds slowly at first and then accelerates. The table below traces the first two years of that $80,000 loan.

How a level loan payment splits into interest and principal (first two years). $80,000 borrowed at 5%, level payment $6,400/year.

Year	Balance owed at start	Interest (5% of balance)	Principal repaid (payment − interest)	Equity built this year
1	$80,000	$4,000	$2,400	$2,400
2	$77,600	$3,880	$2,520	$2,520

In the first year, $4,000 of the $6,400 payment just covers interest, leaving $2,400 to build equity. In the second year, you owe less ($77,600), so interest falls to $3,880 and $2,520 of the same payment goes to principal. The payment never changes, but its composition does, which is why your equity grows faster and faster as the years pass and why paying a mortgage down feels glacial at the start. It is the present-value arithmetic of an annuity, read one year at a time.

Streams need not be level. When the yearly amounts vary, you discount each separately and add. The next table works a six-year stream at 8 percent. Multiply each future amount by its discount factor, then sum the present-value column.

Present value of a six-year varying annuity at 8%.

Year	Future amount	Discount factor (8%)	Present value
1	$1,000	0.926	$926
2	$2,000	0.857	$1,714
3	$3,000	0.794	$2,382
4	$3,000	0.735	$2,205
5	$2,000	0.681	$1,362
6	$1,000	0.630	$630
Total			$9,219

This little table also delivers a deeper lesson, the one this topic keeps returning to: a present value changes the instant expectations about the future change, not when the future arrives. Suppose news today doubles the expected receipts in years five and six to $4,000 and $2,000. The present value jumps immediately from $9,219 to $11,211. Nothing has actually happened yet in years five or six; only the anticipation changed. The same is true of an anticipated future tax cut or any other foreseen change: today’s value moves the moment the expectation forms. We will come back to this in Part B, when we talk about politics and about profit.

That arithmetic makes the chosen discount rate a political lever. Advocates of California’s Feather River water project could make the long-lived project look worthwhile by discounting its future benefits at about 2 percent. At a higher market rate, the same resources were worth more in other uses for the time being, so postponing the project could dominate building it immediately. The public debate largely skipped the choice of rate even though that choice could reverse the verdict. Whenever a public project “pays for itself,” ask which discount rate made the statement true and what alternative return was set aside.

Now stretch the stream out forever. An annuity that pays a constant amount in every future year, without end, is a perpetuity, and it has a beautifully simple value:

P = A ÷ r

The present value of a perpetuity is just the annual amount divided by the interest rate. A perpetuity of $1 a year at 5 percent is worth $1 ÷ 0.05 = $20. That seems impossibly small for an endless stream, until you see why: invest $20 at 5 percent, collect the $1 of interest each year, spend it, and the $20 is still there to repeat the trick forever. The distant future barely adds anything, because those far-off dollars are discounted to almost nothing. The next table makes this concrete by splitting a perpetuity into its first fifty years and everything after.

How much of a perpetuity’s value comes from the distant future.

Interest rate	Present value of first 50 years	Present value of year 51 onward	Present value of entire perpetuity	Share beyond year 50
3%	$25.70	$7.30	$33.33	about 22%
5%	$18.30	$1.70	$20.00	about 8.5%
10%	$9.91	$0.09	$10.00	under 1%

At 10 percent, everything beyond the first fifty years is worth nine cents on a ten-dollar claim. This is why a thousand-year stream is worth barely more than a perpetuity, and it is the arithmetic behind a real bias in public debates. Advocates of enormous long-lived projects, a bullet train, say, like to use a low interest rate, because a low rate inflates the present value of distant benefits. The same trick appears in lawsuits over a person’s lost future earnings: a plaintiff wants a low discount rate to make the present value of those lost earnings look large, while the defense wants a high one. The rate you choose is not a technicality; it can swing the answer by a lot.

Interest Is a Cost Even When You Never Borrow

Here is a mistake that wrecks business decisions: thinking that if you don’t borrow, interest costs you nothing. Interest is the cost of waiting: the cost of having something later rather than sooner, and that cost is there whether or not a loan is involved.

Suppose a venture spends $1,000 at the start of year one and collects $1,300 at the end of year two, with nothing left over. Is it profitable? You cannot just subtract $1,000 from $1,300 and declare a $300 profit, because those amounts sit two years apart. You have to capitalize both to a common date using the interest rate. The table below does it two ways, valuing everything at the start and again at the end, and it shows the same conclusion either way.

Capitalizing costs and receipts to a common date, at a 7% interest rate.

	Capitalized to start of year 1	Capitalized to end of year 2
Outlay ($1,000 at start of year 1)	$1,000	$1,000 × 1.07 × 1.07 = $1,145
Receipts ($1,300 at end of year 2)	$1,300 ÷ (1.07 × 1.07) = $1,136	$1,300
Result	$1,136 − $1,000 = $136 profit	$1,300 − $1,145 = $155 profit

At 7 percent the venture earns a profit (the two figures, $136 and $155, are the same profit measured at different dates). But the answer depends on the interest rate. Redo it at 15 percent and the project turns into a loss:

The same venture at a 15% interest rate.

	Capitalized to start of year 1	Capitalized to end of year 2
Outlay ($1,000 at start of year 1)	$1,000	$1,000 × 1.15 × 1.15 = $1,322
Receipts ($1,300 at end of year 2)	$1,300 ÷ (1.15 × 1.15) = $983	$1,300
Result	$983 − $1,000 = $17 loss	$1,300 − $1,322 = $22 loss

No money was borrowed in either case, yet the interest rate decides whether the venture makes money. This is why a firm flush with its own cash is not immune to interest rates: it can always lend its funds out at the market rate instead of tying them up in a project, so every internal project competes against that alternative, and the market rate still binds. Ignore interest and you will count phantom profits by adding up dollars from different years as if they were the same dollars. That is also why a lottery paying “$1,000,000” as $100,000 a year for ten years is not really a million-dollar prize: its present value is smaller, because the later payments are worth less today. Capital value, costs and receipts discounted to one date, is therefore the honest measure of cost in any decision that spans time. (Why this is also the honest measure of profit, and what profit precisely means, is a thread we pick up in Part B.)

The same insight settles a question businesses face constantly: should you lease an asset or borrow and buy it? A leasing pitch promises you avoid “tying up capital” and avoid “the loss of depreciation.” Both claims dissolve once you think in present value. The leasing company is simply lending you the asset and charging rent that already builds in the depreciation it bears and the interest on the money it has sunk into the asset; you could instead borrow the purchase price, buy the asset, and pay interest on the loan. At the same implied interest rate, the two paths cost the same, because the underlying real cost (the asset’s services over its life, discounted) is identical no matter how you finance it. The honest way to compare is to set the purchase price against the present value of the lease payments and see which is cheaper at the going rate. What genuinely tips a real lease-versus-buy decision is usually not the financing romance but the tax code: in the United States, lease payments, loan interest, and (when you own) depreciation can all be deductible, and which deductions a buyer can claim is what actually swings the after-tax cost one way or the other. Strip out the tax differences and financing form changes nothing real; that is the lesson capital value keeps teaching.