Dollar-Cost Averaging vs Lump Sum: What the Arithmetic Actually Says

The severance cheque cleared on March 3. Twelve thousand dollars, sitting in a savings account, and you have already decided it belongs in a broad index fund. What you have not decided is whether to buy the whole thing Monday morning or split it into six monthly pieces and buy a little at a time until September.

Here is the answer, before the caveats. If the fund’s price is higher in September than it is Monday, buying everything Monday wins. If the price sags for a few months and then comes back, spreading the purchases wins. That is the whole comparison. Neither approach is better in the abstract — the winner is decided by the price path, which nobody has yet.

What follows is the arithmetic that makes that true, run on your $12,000, with every assumption written down so you can check it.

The two mechanics, stated precisely

Lump sum means converting cash to shares in one transaction, on one day, at one price. Monday, $12,000, done.

Dollar-cost averaging means splitting the same cash into equal instalments and buying at fixed intervals regardless of price. The SEC’s investor glossary defines it exactly that way: a fixed dollar amount, on a schedule, whatever the price happens to be. Your version is six payments of $2,000, one on the third of each month, March through August.

Both paths end with you owning shares of the same fund. The only difference is the sequence of prices you paid. Everything people argue about downstream — risk, regret, timing, discipline — flows out of that one difference.

Why the expected-value argument favours buying Monday

Start with an assumption, because the whole argument rests on it: you believe the fund has a positive expected return. If you did not, you would not be buying it.

Grant that, and the rest is bookkeeping. Money sitting in the savings account is not earning the fund’s return. Under the six-month schedule, only $2,000 of your $12,000 is exposed for the full six months. The last $2,000 is exposed for zero months. On average, your money spends about two and a half months in the market instead of six. You have voluntarily switched off a large share of the return you said you expected to get.

That is not a claim about history, and it is worth being clear that it is not. Plenty of writing on this subject leans on backtests showing lump sum winning in most historical windows. Those backtests exist, but this article does not cite them, because the point stands without them: if the expected return is positive, delaying exposure lowers expected value. Arithmetic, not evidence.

Two things follow. First, “higher expected value” describes an average over many possible futures, and you get exactly one. Second, the size of the effect scales with how long you stretch the schedule. Six months costs you a little. Three years costs you a lot.

Your $12,000, two price paths

Assumptions, all of them: no fees, no dividends, fractional shares allowed, price starts at $100 on March 3, six instalments of $2,000. The lump-sum version buys 120 shares on March 3 and never trades again.

The first path rises steadily to $120. The second sags to $82, then claws back to exactly $100 — six months of drama that ends where it started.

Rising path. The lump sum holds 120 shares bought at $100. At August’s price of $120 that stake is worth $14,400. Dollar-cost averaging accumulates 109.51 shares at an average cost of $109.57, worth $13,142 at the same price. Lump sum is ahead by $1,258, or a bit over ten percent of the original stake, and the reason is visible in the table: every dollar you held back bought fewer shares than the dollar before it.

Dip path. The lump sum still holds its 120 shares at $100, worth $12,000 in August. Flat — six months of movement, nothing to show. Dollar-cost averaging accumulates 131.88 shares at an average cost of $90.99, worth $13,188. Spreading the purchases wins by $1,188, because $2,000 buys 24.39 shares at $82 and only 20 at $100.

Same investor, same $12,000, same six months, opposite answers. The strategy did not change. The price path did.

The average-cost effect is real, but smaller than it sounds

There is a genuine mathematical property hiding in that table, and it is usually oversold.

Look at the rising path. The six prices average $110. Your average cost was $109.57 — below the average price. Now the dip path: the six prices average $91.50, and your average cost was $90.99. Below again. This is not luck. Buying a fixed dollar amount means you automatically pick up more shares when the price is low and fewer when it is high, so your average cost always comes in at or below the plain average of the prices you paid across.

That property holds every time, on any price path. It is also not what you actually care about. Your average cost beats the average of the prices during the buying window, but the lump sum paid the first price. When prices rise, the first price is the lowest one in the window, and beating the average of a rising series is no consolation. The average-cost advantage is real; it just is not the benchmark that decides who ends up with more money.

What the expected-value argument leaves out

You do not experience an average. You experience one Monday.

Put the whole $12,000 in on March 3 and watch it fall twelve percent by Friday, and you are down $1,440 in four days. The arithmetic above still says the decision was the favoured one. That will not help. Some investors abandon the plan at exactly this point, sell into the decline, and lock in a loss that the expected-value calculation never contemplated, because the calculation quietly assumed you would hold.

That is what dollar-cost averaging actually buys. Not lower risk in any general sense — you still end up fully invested, exposed to everything the market does afterwards. What it buys is a smaller chance that one bad entry date defines your whole experience of investing, and with it a better chance that you stay in the position at all. A strategy you abandon under stress does not earn its expected return. It earns whatever you did instead, which is usually cash, and occasionally something worse.

Whether that trade is worth roughly $1,258 in the rising case is a question about your own behaviour, not about which method is correct. If you have been through a drawdown before and held, you have data on yourself. If you have not, be careful about assuming you know.

Fees and taxes, briefly

Six trades used to cost six commissions, which mattered a great deal on $12,000. Most retail brokers now charge nothing per trade on standard US equity and ETF orders, so for most readers this line item has gone to zero. Check yours rather than assuming; the fee schedule is where the assumption gets tested.

Taxes are the more durable difference. Six purchases create six tax lots instead of one, each with its own cost basis and its own holding-period clock. That matters later, when you sell part of the position and have to identify which shares went. The IRS covers basis and holding periods in Publication 550. Inside a 401(k) or an IRA, none of it applies — the lots exist but produce no reporting for you.

What this does not tell you

It assumes the money is already earmarked for investing. Nothing above speaks to whether $12,000 of severance should go into an index fund at all rather than toward high-interest debt, an emergency fund, or six months of expenses while you look for work.

It uses two hand-built price paths, not a distribution of thousands of historical windows. Those two paths were chosen to show the mechanism cleanly — one where delay hurts, one where it helps. They are illustrations of how the arithmetic works, not evidence about how often each outcome occurs.

It assumes you set the schedule in advance and follow it. Real investors skip a month when the news is bad, or dump the remaining balance in early when the market rallies. Either move breaks the clean arithmetic and turns the plan into market timing wearing a schedule’s clothes.

It ignores dividends, which would be reinvested at whatever price prevailed and would nudge both share counts up slightly, in the same direction for both strategies.

And it says nothing about your particular fund, your particular horizon, or the level of prices today. A generic path starting at $100 is a teaching device, not a forecast.

FAQ

Is lump sum investing better than dollar-cost averaging?

For expected value, yes, if you believe the asset has a positive expected return — money held back earns nothing while it waits. For any single outcome, no strategy is better in advance, because the result depends entirely on what prices do after you commit. The gap between those two sentences is where the whole debate lives.

Does dollar-cost averaging lower my average cost per share?

It lowers it relative to the average price during your buying window, always, because fixed dollar amounts buy more shares at low prices than at high ones. In the rising example above, prices averaged $110 and the average cost came to $109.57. But that is not the number that decides the comparison, since a lump sum pays the first price rather than the average one.

How long should I spread out a lump sum?

There is no correct length, only a trade-off that gets steeper as you stretch it. A three-month schedule behaves almost like a lump sum and costs almost nothing in expected exposure. A three-year schedule leaves most of the money in cash for most of the period, which is a large give-up if the reason you are investing is that you expect returns to be positive.

Can I invest half now and spread the rest?

Yes, and it does exactly what it looks like: half the expected-value cost of full averaging, half the protection against a bad entry date. There is no formula that identifies the right split, because the split is a statement about how much short-term loss you want to be exposed to, not a quantity that can be solved for.

Does dollar-cost averaging create more work at tax time?

In a taxable account, six purchases mean six cost-basis lots and six holding-period clocks, which matters when you sell part of the position and have to decide which shares to sell. Brokers track basis for you on most covered securities, but you are the one choosing lots at sale time. IRS Publication 550 covers the rules. In an IRA or 401(k), there is nothing extra to do.

I invest from every paycheck. Does any of this apply to me?

Not really. Buying with each paycheck is dollar-cost averaging only in name — you are investing money as it arrives, because there is no lump sum to deploy. This comparison only bites when you are holding a pile of cash and choosing when to convert it.

Where to look next

The useful next step is not picking a side. It is filling in the table above with your own numbers: your amount, your fund’s current price, and two price paths you consider plausible — one that rises, one that dips and recovers. The arithmetic takes ten minutes in a spreadsheet and tells you the size of the stakes, which is usually smaller than the argument suggests.

Then check the two things that are actually knowable: whether your broker charges per trade, and whether the account is taxable. After that, the only remaining question is how you have behaved during past paper losses. That answer predicts which plan you will still be following in September better than any historical statistic can.

This article is general information, not financial advice. See our disclaimer.

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