Lump sum vs dollar-cost averaging: which the math favors

Lump sum versus dollar-cost averaging is a cash-deployment rule, not a personality test. You have $120,000. You can buy equity tonight, or you can buy 1/12th a month for a year and leave the rest in cash. If expected equity returns beat cash, the math of waiting is a drag. If the first month is a crash, the math of waiting is a gift. This page runs both stories: 1,000 orderings from the regime engine, and every 12-month window on a 1928+ monthly tape.

Treatment: lump at t=0 versus 12 equal monthly purchases. After month 12 both hold equity to month 120 as a companion. Primary is month 12. Cash in the DCA sleeve earns the engine's cash yield (Q1) and earns 0 extra on the historical tape (Q3). No spending, no tax lots, no look-ahead.

Lump sum vs dollar-cost averaging: which the math favors

On Q1, lump finishes ahead in 56.3% of 1,000 shared worlds at month 12, and 55.9% at month 120. The median ratio at month 12 is 1.016. That is not a blowout. It is a sign. A 56.3% win share with a 1.016 median ratio is "more often than not, by a little," which is what a positive equity premium plus a year of cash drag should look like. A 90% win share would have been a different generator.

Dollar-cost averaging is often sold as risk reduction. It does reduce the share of the pot that takes the first month's return. It also reduces the share that earns the next eleven months. Whether that is cheaper depends on whether those months have a positive premium over cash. On this engine they usually do. That is not bravery. It is the sign of the premium.

The Q1 median paths (not the historical chart below) stay close: lump's median is above DCA's through the year. Close medians and a win share a bit over half can live together. Means and win fractions are different meters, the same lesson as success versus balance.

The tape, with n named honestly

Overlapping 12-month windows on the 1928+ monthly series: lump wins 67.0% of 1161 starts. Independent non-overlapping 12-month blocks: 62.9% of 97. January-only starts: 62.9% of 97. Overlapping windows share crashes. They are not Bernoulli n = 1,161. The independent count is the one to put next to a probability-sounding sentence, and even 97 is a small n for a century of one country.

The line on the chart is a rolling 10-year share of January starts where lump won the next 12 months. It spends long stretches above 50% and ducks under in the early 1930s, which is the example you want if you are arguing for DCA as crash insurance. Insurance has a premium. On this tape the premium is paid in the other decades. The historical veto did not overturn lump's lead. It did show that some windows reverse it.

Q2: stock mu minus 0.02 on the Monte Carlo. Win fraction at month 12 falls to 52.8%, still above 50%. Month 120: 50.8%. The card said a flip to 50% or below would falsify the "more often than not" sentence. It did not flip.

What this is not

It is not tax. Realizing a lump in a taxable account, or harvesting lots along a DCA, can dominate the 1.6% median gap. We did not model lots. Takeaway: pre-tax.

It is not a cash-need study. If the $120,000 might be a house down payment in nine months, the object is not month-12 equity wealth. It is the chance the house fund is intact. Different primary. Different paper.

It is not advice to lump a bonus tonight. The math on these tapes favors lump as a 12-month equity-versus-cash deployment rule when the premium is positive. Households have other constraints. The freeze does not see them.

Scope

Q1 run. Q2 run. Q3 run. Q4 not run as costs: no trading bps, no bid-ask. Information set: each month's purchase uses that month's return only. The DCA schedule is calendar, not a crash rule. A rule that waits for a −20% month would be a different roster and would be search if we invented the −20% after seeing 2008.

What a 56.3% win share is, and is not

It is not "always lump." 43.7% of the 1,000 worlds still finish the year with DCA ahead. Those are the worlds where the first months are kind to cash and cruel to equity. A 12-month DCA is crash insurance with a premium. The premium is the cash drag in the worlds where equity rose. On this engine the premium is worth paying less often than not, which is the same sentence as a positive premium.

It is not a 67.0% probability that lump will win your bonus. That 67.0% is overlapping windows on one country's monthly tape. Independent n is 97. If you must sound like a probability, use 62.9% of 97 non-overlapping 12-month blocks, and then admit that 97 blocks from 1928 to now are still one history. Q1's 56.3% is the number that is allowed to sound like a share of modelled orderings.

Month-12 median wealth on Q1 is $129,607 for lump and $127,296 for DCA. The gap is small in the middle and still signed. Sign plus a 56.3% win share is the finding. A marketing chart that only showed the median gap without the win share would have understated how often the sign flips.

See both rules as a number

The lump sum vs dollar-cost averaging calculator is the matching URL. It will not know your tax lot. It will show the identity this freeze measures. If it only compounds a single 5% rate, it is the deterministic article again, not this one. This one needs a win share and an independent n. Q1 win share 56.3%. Independent tape n 97 at 62.9%. Quote both, or you are selling a round percentage with the wrong denominator. Overlapping windows feel like a lot of evidence. They share crashes. Shared crashes are not extra luck. They are extra copies of the same luck.

What this page refuses to say

It refuses "always lump." 43.7% of Q1 worlds still finish the year with DCA ahead. It refuses "DCA reduces risk" without naming the cash drag. It refuses to treat overlapping n = 1161 as independent trials. It refuses tax advice. A 1.6% median gap can vanish in a lot. It refuses a house down-payment study. Different primary. It refuses a crash-timing DCA that waits for −20%. That rule was not on the card.

It refuses to drop the 1930s from the tape to make lump look cleaner. The rolling line is allowed to duck under 50%. That is the insurance working in the decade people cite. The rest of the tape is the premium. It refuses a T-bill sleeve added after seeing 1931. Cash yield is in Q1 from the engine and is 0 extra on the tape, as registered.

It refuses a personality test. Impatient people do not "deserve" lump. Patient people do not "deserve" DCA. The math on these tapes favors lump as a 12-month equity-versus-cash deploy more often than not. Households have payroll dates, tax lots, and cash needs this freeze cannot see. The calculator is an identity. The person is not.

Q2 stayed above 50%. That keeps the "more often than not" sentence. It does not make 52.8% a slogan. Small leads are allowed to be small. They are not allowed to be rounded to always.

Replication spec

Q1: 1,000 episodes from the regime environment at seed 20260622. $120,000. Lump rides the stock lane from month 0. DCA moves 10,000 into stocks at the start of each of the first 12 months; cash earns the step cash yield. Primary comparison at the end of month 12. Companion at month 120. Win fraction 12: 56.3%. Win fraction 120: 55.9%. Median ratio 12: 1.0164. Median series length 13 (month 0 through 12). Month-12 medians: lump $129,607, DCA $127,296.

Q2: same deploy after stock mu −0.02. Win 12 52.8%, win 120 50.8%. Still above 50%. Q3: public/sp500_monthly_returns.csv, dates on or after 1928-01-01. Overlapping n 1161, win 67.0%. Independent n 97, win 62.9%. January n 97, win 62.9%. The line chart is a 10-year rolling January win share, not the overlapping 1,161 as if they were independent.

Cash on the tape earns 0 extra. A T-bill sleeve would have narrowed lump's lead in some decades and widened it in others. We did not add it after seeing 1931. That would have been search. CSV: /data/studies/lump-sum-vs-dollar-cost-averaging.csv. Pre-tax. No trading bps on this custom loop.

What would falsify: Q1 win fraction at or below 50%, or independent windows showing DCA ahead, or the mu-shave flipping Q1 below 50%. None of those three landed. The 1930s dip on the rolling line is a window reversal, not a global reversal.

Last recap: lump versus 12-month DCA on $120,000. Q1 win share 56.3% of 1,000 orderings. Q2 still 52.8%. Independent tape windows 62.9% of 97. Overlapping 67.0% of 1161 is the noisy cousin, not the n you quote as a probability. Pre-tax. Not a house fund. Not a crash-timing rule. More often than not, by a little, on these tapes.

Notes. B4_LUMP: $120,000, 12-month DCA versus lump, optional 10-year hold. Q1: 1,000 regime paths, seed 20260622, cash yield from the engine. Q2: stock mu −0.02. Q3: public/sp500_monthly_returns.csv from 1928-01, overlapping and independent 12-month windows. Independent n = floor(months/12). Q4: pre-tax, 0 trading bps on this custom deploy. CSV: /data/studies/lump-sum-vs-dollar-cost-averaging.csv. Card dated 2026-08-28. This article is educational analysis, not investment advice, and does not recommend any security or strategy.