Dollar cost averaging, explained

Dollar cost averaging fixes two things and lets the third do what it likes. The amount is fixed, the schedule is fixed, and the price is whatever the market is asking on the day. Because the amount is fixed and the price is not, the units each payment buys move inversely with the price: cheap months buy more, expensive months buy fewer.

Pay 200 dollars a month into something whose price falls and then recovers. Prices below are in dollars.

Five payments of 200 dollars is 1,000 dollars, and it bought 26 units, so the average cost was 38.46 per unit. The simple average of the five prices was 41.00. The schedule paid less per unit than the average price without anybody deciding anything.

That is not luck. Fixing the amount makes the average cost the harmonic mean of the prices:

$$\bar{P} = \frac{n}{\sum_{k=1}^{n} 1/P_k}$$

and the harmonic mean of a set of positive numbers is always at or below their simple average, with equality only when every price is identical. Fixing the quantity instead gives the simple average exactly: five units a month across those same prices costs 1,025 dollars for 25 units, which is 41.00 each.

So the averaging effect is real, and it is guaranteed rather than lucky. What it is not is a comparison with any other way of investing the same money. That is where most of the confusion starts.

Almost every argument about dollar cost averaging is two arguments wearing one label, and they have different answers.

The first is averaging by circumstance. Income arrives monthly, part of it goes into a fund every month, and prices are whatever they are on the day the transfer clears. Anyone paying into a workplace retirement plan is doing this whether they call it that or not. There is no alternative on offer, because there is no lump sum: money not yet earned cannot be invested early. Nothing is held back, so nothing is given up. The first worked example below is this case, and for anyone whose investing runs out of a salary it is the only one that applies.

The second is averaging by choice. A sum already exists, from an inheritance, a bonus, a house sale, a maturing deposit or a transfer between accounts, and you decide to feed it in over several months rather than invest it now. That is a real decision with a real price, because the part not yet invested is sitting in cash while you wait.

Keeping the two apart matters, because the evidence in the next section is entirely about the second one. It says nothing at all about paying into a plan out of each paycheque, which is what most people mean when they say they are dollar cost averaging. Judging monthly investing against a lump sum study is measuring it against an option that never existed.

The compound interest calculator at the top of this page is built for the first case: a starting amount, a fixed contribution and a rate. The savings goal calculator runs the same shape backwards, from a target to the monthly amount it needs.

A lower cost per unit than the average price sounds like an edge until you ask what the alternative was. The comparison that decides anything is not the schedule against the prices it met. It is the schedule against putting the same sum in on day one.

On a rising path the schedule loses, and the harmonic mean arithmetic still holds while it does:

The schedule paid 59.16 per unit against an average price of 60.00, so it won its own comparison and lost the one that counts. Its 1,000 dollars bought about 16.90 units. The same 1,000 dollars spent in month 1 bought 20. Valued at the month 5 price, and with the money still waiting earning nothing, the single purchase finishes 18.3 percent ahead.

Money waiting to be invested earns the cash rate rather than the market return, so every month of waiting gives up the difference, and share markets have returned more than cash over most historical stretches. Vanguard's rolling window study of United States, United Kingdom and Australian history put investing at once ahead of a twelve month schedule in roughly two thirds of periods, for that reason.

The other third of that sentence matters as much. Spreading won in about one period in three, those are three markets over particular runs of history rather than a law, and nothing tells you in advance which kind of period you are standing in. Two thirds is a frequency counted across many windows, not a prediction about yours.

The worked examples below put a number on the cost of waiting with the return held fixed and positive. That is the average case written out rather than evidence gathered: assume a market that rises steadily and investing at once has to win, so what those examples measure is the size of the gap, not its sign.

A narrower range of outcomes. That is the whole of it, and it is worth something.

What decides the winner is the average price the schedule pays inside the buying window against the price on day one. Where the market finishes drops out of both sides, because on the closing date both paths hold units worth the same price. So a window in which prices mostly sit below where they started favours the schedule, and one in which they mostly sit above favours the single purchase.

That is not the same question as whether the market rose. Prices can spike through the whole window and still end below where they started, and the schedule loses anyway, because it spent the year buying the spike. Direction between the first day and the last is not what it pays for; the path in between is.

Both effects come out of the same held back cash, so the two tails move together. Spreading hands back the top of the range of outcomes to lift the bottom of it: a smaller best case bought with a less bad worst case. Vanguard's paper on the point is titled "Dollar-cost averaging just means taking risk later", which is the mechanism in one line.

Notice what that does and does not remove. During the buying window your exposure is smaller than it would have been, so a fall costs less. Once the last instalment goes in, both paths hold exactly the same thing and carry identical volatility from that day on. Spreading changes the path, not the risk you are left holding.

The case for paying that price is behavioural rather than mathematical. Committing a large sum days before a sharp fall is the kind of experience people give as their reason for stopping, and a rule set once in advance is easier to hold than one renegotiated every month against the news. If a schedule is what gets the money invested at all, then the comparison that applies is against leaving it in cash, not against investing at once. Which of those two comparisons is the real one is a question about the person holding the money, and no page can answer it from the arithmetic.

The lump sum against averaging explorer lets you drag the market trend and the buying window and watch both ends of the range move.

Four things, and none is the shape of the market, which nobody knows in advance.

• How long the window is. The cost is roughly the return given up times the average time the money spends out of the market. Twelve equal monthly instalments leave the average dollar uninvested for 5.5 of the 12 months, so at a 7 point gap the expected shortfall is about 3.2 percent, which is what the worked examples find. Double the window and the average dollar waits 11.5 months, and the shortfall a little more than doubles.
• What the waiting cash earns. The gap that matters is between the expected market return and the rate paid on the cash, not the market return alone. If cash pays 4 percent while the market is expected to return 7, the gap is 3 points rather than 7 and the expected cost falls by about 57 percent. When cash rates sit near zero, the whole expected return is on the line, which is the case the worked examples below set up, so they show the widest version of the gap rather than a typical one. Where cash is expected to beat the market over the window, the sign flips and spreading is the higher-return choice.
• How big the sum is next to what you already hold. Spreading an amount that is small next to an existing portfolio barely changes total exposure, so it buys little protection for its cost.
• What each purchase costs to make. Where a broker charges per trade, twelve small purchases pay twelve commissions instead of one, and fractional shares matter when a single unit is expensive.

Tax treatment is jurisdictional, and it is more than paperwork. It leaves the pre-tax arithmetic above untouched, but it decides what record each purchase creates and it can change what is owed on a sale, which is part of what the money actually returns. In the United States each purchase is a separate tax lot with its own cost basis and holding period, which allows particular lots to be identified at sale. In the United Kingdom, shares of the same class are pooled at an average cost, with same day and thirty day matching rules applied ahead of the pool. Neither of those is the general case, and local rules decide which applies.

Averaging into a single company is a different act. A schedule pointed at a broad fund keeps buying a slice of a whole market. A schedule pointed at one falling company keeps adding money to a position the market is repricing downward, and it raises that company's share of a portfolio at exactly the moment the market has marked the company down. Averaging down is a concentration decision wearing a discipline's clothes, and the protection people believe they are buying comes from diversification instead.

Stopping when prices fall removes the point. The cheap purchases are the ones that pull the average cost down. A schedule kept only while prices rise buys expensively and skips the cheap months, which is a common way the arithmetic gets undone.

Waiting for a better entry is not a schedule. Cash held back for a price that may never arrive carries the cost of averaging with none of its discipline, because there is no date on which the money definitely goes in.

A lower average cost is not a return. The cost per unit says what you paid. It says nothing about whether the holding was worth owning, and a schedule that faithfully averages into a losing position loses faithfully.

Compare like for like or not at all. Two paths can only be judged over the same period, valued on the same date, with the same money. A schedule that started at a different time is a different bet, and a chart that begins at a market peak is an illustration rather than evidence.

This page is educational material about how the arithmetic works, not financial advice about what to do with a particular sum.