Is a big tax refund a good thing?
The big refund ($119.23 less in each of 26 paychecks, $3,100 back at the end of the year) minus the same money in each paycheck. From the refund layer.
Artificial Wasteland · a combine across five layers · commons
The Number Has No Date
A $1.25 billion jackpot, a $3,100 tax refund, buying against renting, starting to save at 25. Add up the money in each as if the dates did not matter, which is an interest rate of 0%, and you get a verdict: sometimes the familiar one, sometimes its opposite, once a tie. Turn one rate dial and watch each flip at a rate of its own (0%, 1.99%, 4.90%, 6.28%), and a fifth question, easing a windfall into the market, flip at the cash rate, most of the time.
green: the familiar answer holds at this rate · red: it flips · blue: the 0% reading, or a tie
Is renting throwing money away?
Buying the $500,000 house (20% down, a 6.5% mortgage, the home gaining 3.5% a year) minus renting at $2,083.33 a month and investing what the owner would have spent. From the renting layer.
stay
Is the jackpot $1.25 billion?
Powerball, December 17, 2025, as advertised before the drawing: the 30-payment annuity (each payment 5% larger) minus the $572.1 million cash option. From the lottery layer.
Does starting at 25 beat starting at 35?
$300 a month from 25 to 35 and then nothing, minus $300 a month from 35 to 65, compared at 65. Inflation-adjusted dollars, so on this row the dial is a real rate. From the starting-early layer.
Should you ease a windfall into the market?
All of it now, against twelve monthly slices with the rest in cash at 3.40%. On this row the dial is what the market earns while you wait. From the easing-in layer.
Adding is discounting at zero
Every one of these questions compares two ways of getting or paying money on different dates. Take the difference between them, payment by payment, and you have a list of dated amounts. To compare dates you need a rate: a dollar t years away is worth 1/(1+r)t of a dollar now, where r is what money earns in the meantime. Discount every amount that way and add. That total, the net present value, is the verdict: positive and the first option wins, negative and the second does.
Now set r to zero. Every factor becomes 1 and the calculation collapses into plain addition. That is all "the date erased" means. Two of the familiar answers below are exactly this reading; one is its opposite, which holds only above a rate; and one is a feeling the 0% reading does not even support. James Lorie and Leonard Savage noticed the same thing about investment projects in 1955:
As the cost of capital of the firm approaches zero, the present value of the investment proposal approaches the algebraic sum of net cash flow and will be negative if this sum is negative.Lorie and Savage, "Three Problems in Rationing Capital," The Journal of Business 28, no. 4 (1955); read from a retyped copy
Somewhere each list's total changes sign, for most of them above zero. That rate is the crossover (finance calls it the internal rate of return of the difference), and it is the one number each argument is secretly about. The five crossovers are spread from zero to a little over six percent: 0.00%, 1.99%, 3.40%, 4.90% and 6.28%.
Five readings at zero, and where each flips
The jackpot. "$1.25 billion" is the sum of the 30 annuity payments, the 0% reading. At 0% the annuity is worth $677.9 million more than the $572.1 million cash. (Both are the figures advertised before the drawing; Powerball's draw record gives the final ones as $1.27 billion and $579.7 million, which move the rate to about 4.92%.) The cash option is what the lottery would spend on Treasury securities to fund the payments, so the two options are worth the same at the single rate that price implies, 4.90%. If you can earn more than that, the cash is worth more; if less, the annuity is. The billboard number is not a lie about the payments. It is a present value with the rate set to zero.
The refund. Here the 0% reading is a tie, exactly: $3,100 taken in 26 equal slices (about $119.23 each) and $3,100 comes back. So the crossover is 0.00%, and at every positive rate the big refund is the losing side, because it is a loan to the government at no interest. The loss is small (at 5%, $73.89 on the day the refund arrives; the refund layer's simple-interest method gives $74.52), which is why the layer's real subject is the $119.23 a paycheck you could have had instead. The "windfall" feeling is not even the 0% reading. It forgets the outgoing half of the list.
Renting. At 0% the owner of the $500,000 house is $37,206 ahead of the renter after nine years: the slogan is right if the renter's savings earn nothing. The owner's advantage is what buying earns on the money put into it, counting the rent it saves, and for this house over nine years that is 1.99% a year. Any renter whose savings earn more than that comes out ahead. The crossover depends heavily on the length of the stay, because buying and selling cost about 10% of the price between them, paid once: -7.27% for three years (the owner loses even against money under a mattress, and would only win if money shrank faster than that), -1.51% for five, 0.81% for seven, 1.99% for nine, 3.41% for fifteen and 4.19% for thirty. None reaches the 6.5% the renting layer assumes a portfolio earns, which is why that layer finds the renter ahead at every stay with its defaults.
Starting at 25. At 0% the late saver wins, because they put in $108,000 to the early saver's $36,000: $72,000 ahead. The famous result (the early saver who stops at 35 still ends with more) needs a real return above 6.28% a year, which is 6.109% compounded monthly, the floor the starting-early layer found. Here the myth runs the other way: the 0% reading is the intuitive one, and the popular claim is the one that holds only above a rate.
Easing in. This row is different, and the section below says how.
The fifth row is different
Easing a windfall in is not a list of dated payments. Both choices end up fully in the market. The difference is only what each dollar does before its slice goes in: in the lump it is already in the market, in the slices it waits in cash. After the last slice both portfolios hold only the market, so everything that happens afterwards multiplies both by the same factor and cancels. The verdict is settled in the months you spend easing in.
If the market grew at a steady rate, the rule would be exact: the lump wins when the market beats cash, easing in wins when it does not, and they tie when the two rates are equal. That is the 3.40% on the ruler. It is the average 3-month Treasury bill rate from January 1934 to June 2024 in the easing-in layer's data, so on this row the crossover is the cash rate itself and not a number the question produces.
Markets do not grow at a steady rate, so the easing-in layer answers with a frequency: over 1,722 ten-year windows starting each month from January 1871 to June 2014 (cash earning the Treasury bill rate from 1934 and the ten-year yield before it, as that layer does), the lump beat twelve monthly slices in 63.9% of them. How well does the one-rate rule predict each window? Ask only whether the market beat cash over the twelve months of easing in. It did in 64.6% of windows, close to the lump's win rate, but the rule gets the individual window right only 86.8% of the time. With three slices it is right 81.2% of the time and with thirty-six 81.9%. The rest are windows where the path mattered: a market that ended the year ahead of cash but fell in the middle can reward the slices bought in the dip. The rate decides most of these races and not all of them: in the rest, the path the market took decided, and a rate says nothing about a path.
The ten-year horizon played no part, and cannot: everything after the last slice multiplies both sides by the same factor. The check confirms it anyway by stopping all 1,722 races right after the last slice and getting the same verdicts.
When there are two crossovers
Why should each list have only one crossover? Because each changes sign only once: the refund list is withdrawals then one repayment (the last withdrawal falls on the refund date, so they net), the lottery list one big negative then 29 positives, the renting list outflows then a sale, the saving list ten years of plus then thirty of minus. Put x = 1/(1+r), taken per payment period, and the total is a polynomial in x; Descartes' rule of signs says the number of positive roots is at most the number of sign changes, and has the same parity. One sign change, exactly one crossover. The check counts them for every list here.
Lorie and Savage's example of when this fails is an oil pump that gets the same oil out of the ground faster: money out for the pump, more oil for a while, then less oil at the end than the old pump would still have been giving. Out, in, out: two sign changes.
Investments of this nature are rare, but they do occur, especially in the extractive industries.Lorie and Savage, 1955
Their paper gives no numbers. The ones finance textbooks conventionally use for the pump (often traced to Ezra Solomon's reply in the same journal the next year, which this page did not read, so the attribution is unchecked) are: pay $1,600 now, receive $10,000 in a year, pay $10,000 in two. At 0% it is a loss of $1,600, at 100% a gain of $900, and it is zero at both 25% and 400%. "Is this worth doing at my rate?" still has an answer. "What is its rate of return?" has two.
How many times can it flip?
Descartes' rule caps the crossovers at the number of times the list changes sign, and when it changes sign once, that settles it. The pump changes sign twice, as can any list with money going out, coming in and going out again, and then the rule allows two crossovers or none. Solving finds the crossovers it finds. It does not say whether those were all of them.
The rows at the top of this page already draw the instrument that does. Discount every amount at some rate, call it the anchor, and keep a running total in date order: that is the curve on each of the first four rows, with the dial as the anchor. The number of times the running total changes sign is a ceiling on the crossovers above the anchor. Read the same total backwards, from the last amount to the first, and its sign changes are a ceiling on the crossovers below. Edmond Laguerre stated the rule in 1883. The statement this page uses is G. J. O. Jameson's, who proved it for amounts on any dates, daily ones included (Theorem 4.7 and Corollary 4.8 of the paper cited below, moved to any anchor by the scaling remark that follows them). At an anchor of 0%, a running total that changes sign once is what finance calls Norstrøm's criterion (1972) for a single positive rate of return; this page did not read that note.
At 0% the pump's running total reads -$1,600, +$8,400, -$1,600: two sign changes, so at most two crossovers above 0%. Backwards it reads -$10,000, $0, -$1,600 and never changes sign, so there are none below. The value itself changes sign at 25% and again at 400%. Two ceilings, met by the two crossovers you can see: there are exactly two, and no third is hiding at 900%. That is a proof, and at an anchor of 0% its ceilings are nothing but addition.
a list
running total of
A ceiling is not a count, and it can sit too high. At 450%, with nothing left above it, the pump's forward running total still changes sign 2 times. When that happens, take the running total of the running total, and keep it running past the last amount, where each further period adds the whole total again; a change of sign out there counts too. Each further round of adding can only lower the ceiling or leave it where it was, never raise it, and never take it below the truth. At 450% the second round changes sign 0 times.
Some lists need it. -1,000, +500, +500, +500, -600 changes sign twice and never crosses: its value is at best -$52.69, at -12.80%. Its running total agrees that nothing crosses above an anchor of 23.3752% or more, and nothing below one of -39.2729% or less, but at every anchor from -39.2728% to 23.3751% it allows two each way, and no choice of anchors gets rid of them. The running total of the running total, anchored at -14%, changes sign neither forwards nor backwards: no crossover at any rate. Why every rung is still a ceiling is shown below, from the same paper, and the check tests every rung against an exact count.
Why every rung is a ceiling
Take rates above the anchor a and write y = (1+a)/(1+r), which lies between 0 and 1 there. With bk the amounts discounted at the anchor, the list is worth Σ bk yk. Divide that by (1−y)m, which is positive there, so no zero moves. Dividing by 1−y once turns the coefficients of a power series in y into their running totals, so the result is the power series whose coefficients are the m-th running totals of the bk, kept running past the last amount, forever. Descartes' rule holds for a power series where it converges, here 0 < y < 1, as long as its coefficients change sign only finitely often (Jameson's remark (1) after his Theorem 3.1), and these do: past the last amount the m-th totals follow a polynomial in the period number. So on every rung, the sign changes, continuation included, are a ceiling on the crossovers above the anchor; the list read backwards gives the same below it. Each rung's totals are the running totals of the rung below, and a sequence's running totals change sign no more often than the sequence does (Jameson's Lemma 4.1), so a higher rung can only lower the count.
The second rung also has a picture in time. Take rates above the anchor again, write s = ln((1+r)/(1+a)), which is positive there, and let bk be the amounts discounted at the anchor, so the list is worth Σ bk e−ks. Let B(τ) be the running total as a step function of time: b0 + … + bk from period k until period k+1, and the whole total after the last amount. Summing by parts, the list is worth s ∫ B(τ) e−sτ dτ over τ from 0 to infinity, which is Jameson's route to his Theorem 4.7. Integrate by parts once more, with C(τ) the integral of B from 0 to τ: the list is worth s2 ∫ C(τ) e−sτ dτ, because C grows at most in proportion to τ and the end terms vanish when s is positive. His Lemma 4.5 says the number of zeros of such an integral is at most the number of sign changes of what is being integrated; he states it on a bounded interval, and the same induction (Rolle's theorem, one sign change at a time) runs on the half-line when s is positive. At whole periods C is the running total of the running total, straight between them, and past the last amount it grows at the rate of the whole total, so its sign changes are those of the second rung with its continuation counted.
What no ceiling can see is a touch: a value that comes down to zero at one rate and turns back without crossing. The ceiling still reserves two crossovers for it, the instrument above will say the list is not settled, and it will not pretend to know whether that means none or two.
What the functions say
A spreadsheet's IRR function does not count crossovers. Most start from a guess and step toward a rate where the value is zero, and each call hands back a single number or an error, never a list (or, three times in the bench below, never returns at all). A bench run for this page handed 310 lists whose crossovers were known exactly to 15 IRR and XIRR functions from ten libraries and programs (LibreOffice Calc, Gnumeric, numpy-financial, pyxirr, formula.js, HyperFormula, the npm package financial, two R packages and Apache POI), each called with its default and, where it takes one, with each of seven fixed starting guesses from -50% to 300%: 32,860 calls, and not one of them said a list had more than one answer. LibreOffice Calc 24.2.7.2 answers the pump with 25% from its default guess and 400% from a guess of 3. On a list that crosses at 10%, 20% and 30%, no function answered 20%, from its default or from any of the seven guesses, and nothing about an answer of 10% or 30% says there is another crossover to look for. And formula.js 4.6.1 answers -100, +250, -160, a list that never crosses, with 36%. The bench is published, with every answer at every guess, and so is its report; none of it is about Excel or Google Sheets, which were not run.
What this does not show
- Which rate is yours. The page puts every row on one dial to show where each flips. It does not say what you earn, and the right rate differs by row: a Treasury yield for the lottery, a savings rate for the refund, a portfolio's expected return for the renter and the saver. A rate you might earn is not a rate you will earn.
- Which convention. The dial is an effective annual rate. Two members quote monthly-compounded rates: the saver's 6.109% is 6.28% effective, and the renting layer's slider reads the 1.99% crossover as 1.976%. The easing-in layer compounds cash monthly too; the steady-growth tie at the cash rate holds in either convention.
- Nominal and real. The starting-early row is in inflation-adjusted dollars, as its layer runs it, so its 6.28% is a real rate. The other rows are in ordinary dollars. One dial carries two kinds of rate, and the page says which is which rather than converting with an inflation figure it would have to invent.
- Taxes, risk, and people. Every row inherits its layer's simplifications: no taxes (a flat tax rate applied to both lottery options scales both by the same factor and does not move the crossover at all), constant rent, a steady home appreciation, a smooth return. The members each name theirs. And a rate does not describe a path: on the easing-in row the path, not the rate, decided about 13% of twelve-slice windows (about 19% with three slices and 18% with thirty-six).
- That the myths are stupid. Each 0% reading is correct arithmetic about the amounts. What it leaves out is the dates.
The check
Every curve and verdict above is computed live by engine.mjs, and the check, research/the-number-has-no-date/verify.mjs, imports that same file. It rebuilds each row against an independent copy of its member layer's own engine (copied from the member's verifier, not from this page): the lottery's implied rate, the refund's installment interest, the renting layer's month-by-month race in 21 cases, the saver's 6.109% floor and $55,461 gap at 7%. It counts the sign changes of every list, confirms that moving every date by the same amount (paying at the start of each month instead of the end) moves no crossover, and scans each total across −50% to 200% for its zero crossings; runs the easing-in layer's own 1871 to 2024 series (read out of that layer's page, where it is inlined verbatim) window by window; and reads every number printed on this page in a data-v attribute and requires it to equal what it computed. The section on counting crossovers is checked by the same file against its own module, ladder.mjs: an independent exact count of every list's crossovers, touches counted twice (Sturm's theorem on a chain of greatest common divisors, written into the check and sharing no code with the module), runs over 1,500 seeded random lists and 47 built to be hard (crossovers as little as 0.0001% apart, touches, triple crossings, and values that come close to zero and turn back without reaching it). It requires that no ceiling ever falls below the true count, no higher rung ever sits above the one below it, every bracket, read back from the text the instrument prints, has opposite exact signs at its two ends and proves each digit it prints, and no list is called settled unless its brackets are all its crossovers (the ladder settles 1,499 of the random lists and 20 of the hard ones). The figures quoted from the bench are checked against bench.json, which research/the-number-has-no-date/irr-bench/summarise.py builds from the raw answers; the check cannot rerun fifteen programs, so what it checks there is the quotation. --mutate plants twelve faults and each must turn something red: six in a copy of the engine (a 4% lottery step, a refund a year late, selling costs dropped, an early saver who stops a year sooner, an annual rate used as a monthly one, cash that earns nothing) and six in a copy of the ladder (running totals that forget where they head after the last amount, a backwards reading at the wrong anchor, a list settled on one direction's proof, a bracket taken on trust, a crossover printed rounded instead of to the digits its bracket proves, a zero counted as a change of sign).
You can run it. In an empty folder, with Node 22, these lines fetch the check and the files it reads to the paths they have in the repository, then run it:
curl -L --create-dirs -o research/the-number-has-no-date/verify.mjs https://artwaste.land/checks/research/the-number-has-no-date/verify.mjs curl -L --create-dirs -o public/strata/the-number-has-no-date/engine.mjs https://artwaste.land/strata/the-number-has-no-date/engine.mjs curl -L --create-dirs -o public/strata/the-number-has-no-date/ladder.mjs https://artwaste.land/strata/the-number-has-no-date/ladder.mjs curl -L --create-dirs -o public/strata/the-number-has-no-date/bench.json https://artwaste.land/strata/the-number-has-no-date/bench.json curl -L --create-dirs -o public/strata/the-number-has-no-date/index.html https://artwaste.land/strata/the-number-has-no-date/ curl -L --create-dirs -o public/strata/dollar-cost-averaging-vs-lump-sum/index.html https://artwaste.land/strata/dollar-cost-averaging-vs-lump-sum/ node research/the-number-has-no-date/verify.mjs
What it does not check: the sources of the member layers' inputs (the Powerball figures, the 5% rule, the return conventions, Shiller's data), which each member checked and cites; the two quotations, which were read, not computed; and the bench's answers themselves, which come from running LibreOffice, Gnumeric, R, Java and three JavaScript and two Python libraries (the harness that reproduces them, and every raw answer, is published with the bench).
Sources, as read for this page. J. H. Lorie and L. J. Savage, "Three Problems in Rationing Capital," The Journal of Business 28(4), October 1955, read from a retyped copy hosted by Montclair State University (msuweb.montclair.edu/~lebelp/), which carries no original page numbers; both quotations are from its section on the rate-of-return rule. G. J. O. Jameson, "Counting zeros of generalised polynomials: Descartes' rule of signs and Laguerre's extensions," The Mathematical Gazette 90(518), 2006, 223–234, read for this page: Theorem 4.7, Corollary 4.8 and the scaling remark after them for the running-total ceiling, remark (1) after Theorem 3.1 and Lemma 4.1 for every rung above it, and Lemma 4.5 for the picture of the second. E. N. Laguerre's 1883 paper and C. J. Norstrøm's 1972 note (Journal of Financial and Quantitative Analysis 7(3), 1835–1839) were not read. Everything else is inherited from the five member layers and cited there.