Dollar-Cost Averaging: What It Actually Buys You

The identity almost nobody states: your average cost is a harmonic mean

Suppose you spend a fixed amount C in each of n periods, at prices p₁ … pₙ. The units bought in period i are C ÷ pᵢ, so the total units are C·Σ(1 ÷ pᵢ) and the total spent is nC. Your average cost per unit is the second divided by the first: nC ÷ [C·Σ(1 ÷ pᵢ)] = n ÷ Σ(1 ÷ pᵢ). That expression is the definition of the harmonic mean of the prices. The C cancels, so the result does not depend on how much you invest — only on the fact that the amount is constant.

Now change the rule. Buy a fixed number of units U each period instead of spending a fixed amount. The total spent is U·Σpᵢ and the total units are nU, so the average cost is Σpᵢ ÷ n — the arithmetic mean of the prices, which is also just the average price a passive observer would quote. The two rules produce two different averages of exactly the same price series, and that is the entire mathematical content of dollar-cost averaging.

Take a concrete path of twelve monthly prices: 100, 84, 72, 95, 120, 140, 105, 88, 76, 92, 130, 118. The arithmetic mean is 101.6667. Spending $200 a month buys 24.575589 units for $2,400, which is an average cost of exactly 97.6579 — and the harmonic mean of those twelve prices is 97.6579 to every decimal place a computer will print. Buying 2 units a month instead costs $2,440.00 for 24 units, an average of exactly 101.6667. The gap is 4.0088, or 4.10 percent of the harmonic mean, and it appeared for no reason other than the shape of the spending rule.

Why the harmonic mean is always the lower of the two

The inequality is not an empirical regularity, it is a theorem. For any positive numbers, the Cauchy-Schwarz inequality gives (Σpᵢ)·(Σ1÷pᵢ) ≥ n². Divide both sides by n²: (Σpᵢ ÷ n) · (1 ÷ [n ÷ Σ(1÷pᵢ)]) ≥ 1, which says arithmetic mean ÷ harmonic mean ≥ 1. Equality holds only when every price is identical. So the harmonic mean is at most the arithmetic mean, always, for every price path that ever existed or will exist, with no assumption about trend, distribution or market direction.

The size of the gap is governed by how dispersed the prices are. To a good approximation the difference between the two means is the variance of the prices divided by their arithmetic mean. On the path above the population standard deviation of the prices is 20.544 — a coefficient of variation of 20.21 percent — and the approximation gives 20.544² ÷ 101.6667 = 4.151 against an actual gap of 4.009. On a calmer path of 100, 102, 98, 101, 103, 99, 100, 102, 101, 99, 100, 101, the arithmetic mean is 100.5000, the harmonic mean is 100.4809, and the gap collapses to 0.019 percent. The benefit is real, it is proportional to price dispersion, and on a quiet asset it rounds to nothing.

What the harmonic mean does not prove

The comparison the theorem makes is between a fixed-amount plan and a fixed-quantity plan over the same prices. It is not a comparison with investing everything on day one, and treating it as one is where most of the confusion in this subject lives. Beating the average price is easy and slightly beside the point: the lump-sum investor did not pay the average price either. They paid the first price, 100, which on this path is below both means.

On this particular path the plan does win, and it is worth seeing why that is not evidence of anything. Twelve payments of $200 buy 24.575589 units, worth $2,899.92 at the closing price of 118 — a gain of 20.83 percent on $2,400 spent. The same $2,400 committed at the opening price of 100 buys 24 units worth $2,832.00, a gain of 18.00 percent. The plan is ahead by $67.92 because this path fell hard in the middle, and the monthly buyer's money arrived while prices were at 72, 76 and 88. Reverse the path so it rises first and falls later and the ordering reverses with it. One path proves nothing; a distribution does.

The honest comparison: 200,000 simulated paths

The parameters have to be stated, because they are assumptions and not observations: a twelve-month horizon, $12,000 to invest, lognormal monthly returns with an arithmetic mean of 8 percent a year and an annualised volatility of 20 percent, and 2 percent a year on cash not yet invested. The lump-sum investor puts $12,000 in on day one. The plan investor puts in $1,000 at the start of each of twelve months and earns the cash rate on the balance still waiting. Both are valued at the end of month twelve. No fees, no taxes, no spread — adding them does not change the direction of the result.

The simulation is not needed to get the expected values, which is a useful check on it. The plan's expected invested wealth is Σ 1,000·(1.08)^((12−t)÷12) over t = 0 … 11, which is $12,513.89, plus $110.39 of cash interest, giving $12,624.28. The lump sum's expectation is simply $12,000 × 1.08 = $12,960.00. The gap is $335.72 and the simulated means, $12,623 and $12,960, land on it. The reason is not subtle: each euro in the plan is invested for an average of 6.5 months rather than 12, so it earns roughly half the expected return, and half the cash sits in a low-yielding account meanwhile.

The distribution is where the plan earns its keep, and the table above gives it in full. The plan's standard deviation of terminal wealth is $1,559 against $2,625 — 41 percent lower. Its fifth percentile is $10,262 against $9,142, so in the worst one path in twenty the plan is $1,120 better off. Its ninety-fifth percentile is $15,352 against $17,661, so in the best one in twenty it is $2,309 worse off. Both statements are the same statement. The plan gave up $2,309 of upside to buy $1,120 of downside protection, and in 44.43 percent of paths it finished ahead.

Two variations sharpen the picture. Raise the volatility to 40 percent and the plan wins on 53.39 percent of paths and on the median — $12,187 against $11,958 — while the mean gap stays at $340, because the mean is governed by time in the market and the median by the shape of the distribution. Set the expected return to zero and the plan wins on 57.47 percent of paths and on the mean too, at $12,109 against $12,000. That is the general rule stated cleanly: the plan's cost is the expected return it forgoes by holding cash, so where there is no expected return there is no cost. Remove the 2 percent cash rate and the plan's mean falls to $12,513 and its win rate to 41.15 percent, which is a reminder that where the money waits is part of the decision.

The real reason people use it, and it is a good one

The expected-wealth argument assumes an investor who executes whatever plan the arithmetic recommends, calmly, once. Real investors face a decision they have to keep making, and the phased plan is very often the one they will actually follow. Putting a full year's savings into a market on a single morning requires a tolerance for regret that many people discover, late, that they do not have; a plan that has already been set up and automated requires nothing at all in the month when prices are falling, which is exactly the month when a discretionary investor stops.

That is a real advantage and it is not measured anywhere in the table above, because the simulation compares two plans that are both carried out. The comparison an individual actually faces is often between a phased plan carried out and a lump-sum plan abandoned — and against a plan that never happens, $12,623 is a very good number. The honest framing is therefore: the expected cost of phasing in, on these assumptions, is about $337 on $12,000 over a year, or 2.8 percent of the sum, and that is the price of a plan you will keep. Whether it is worth paying is a question about you, not about the arithmetic.

The case where it genuinely dominates

Everything above compares a plan with a lump sum. That comparison presupposes a lump sum exists. For most people saving out of income, it does not: the money arrives monthly, and each month's contribution is invested as soon as it exists. There is no alternative strategy being forgone, no cash sitting idle, and no expected return being given up. Investing each month's savings on the month it arrives is simply investing immediately — it is the lump-sum strategy, applied twelve times to twelve small sums.

The distinction matters because the two situations get called by the same name and only one of them carries a cost. If you have inherited, sold something or received a bonus, and the money is sitting in an account, phasing it in has a measurable expected cost and buys a measurable reduction in dispersion, and you can weigh the two against each other with the figures above. If you are investing a share of each month's salary, there is nothing to weigh. The harmonic-mean effect is a genuine and free bonus in that case: your average cost across the year will still come out below the average price, at no expense whatsoever, purely because a fixed contribution buys more units when prices are low.

Frequently asked questions

Does dollar-cost averaging beat investing a lump sum?

Not on expected terminal wealth, on the assumptions used here. Over twelve months with $12,000, an assumed 8 percent average return, 20 percent volatility and 2 percent on waiting cash, the lump sum averages $12,960 and the monthly plan $12,623 — a shortfall of $337, or 2.8 percent of the sum. The reason is time in the market: each euro in the plan is invested for an average of 6.5 months rather than 12. The plan wins on 44.43 percent of the 200,000 simulated paths. What it does buy is a 41 percent lower standard deviation and a fifth-percentile outcome $1,120 better. Change the assumption to a zero expected return and the plan wins on the mean too, at $12,109 against $12,000 — which shows the cost is exactly the return forgone, nothing more.

Is the harmonic-mean advantage real, or a presentational trick?

It is real and it is provable, but it answers a narrower question than it appears to. It is a theorem that spending a fixed amount produces an average cost equal to the harmonic mean of the prices, which is never above the arithmetic mean — 97.66 against 101.67 on the path in this article, a 4.10 percent advantage. That is a true statement about your cost relative to the average price over the same window. It is not a statement about your cost relative to any other strategy, and in particular the lump-sum buyer paid the first price, not the average one, so the theorem says nothing about which of you does better. The gap also shrinks to almost nothing when prices are stable: on a calm path in this article it was 0.019 percent.

Over how many periods should I spread the money?

The two effects pull in opposite directions and both scale with the length of the schedule. A longer schedule leaves more money in cash for longer, so the expected cost rises roughly in proportion; on the assumptions here, spreading $12,000 over twelve months costs about $337 in expectation, and a schedule twice as long would leave each euro invested for an average of 12.5 months out of 24 instead of 6.5 out of 12 and cost correspondingly more. A longer schedule also reduces dispersion further, though with diminishing returns, since much of the variance reduction is already achieved in the first few instalments. There is no optimum that holds for everyone, because the trade is between a quantity you can compute and a tolerance only you can assess. What is worth avoiding is a schedule so long that the cash allocation, rather than the investment, dominates the outcome.

Does a monthly plan out of salary have the same drawback?

No, and this is the most important distinction in the subject. The expected cost of a phased plan is the return given up by leaving money in cash instead of investing it. If the money does not exist yet — because it arrives with each month's salary — there is no cash sitting idle and nothing is being given up. Investing each month's contribution as soon as it arrives is investing immediately, not phasing in. In that situation the harmonic-mean effect is a free bonus: your average cost over the year comes out below the average price purely because a fixed contribution buys more units when prices are low. The lump-sum comparison simply does not apply, because there was never a lump sum to deploy.

Do transaction fees change the conclusion?

They push in the same direction as the expected-return effect, so they widen rather than close the gap. A plan makes twelve purchases where a lump sum makes one, so any flat charge per order is paid twelve times; a flat $2 per order costs $24 against $2, and on $12,000 that is a further 0.18 percent handed away. Percentage-based commissions and spreads are broadly neutral, since they scale with the amount traded rather than the number of trades. The practical consequences are to prefer a provider with no per-order charge for a regular plan, and to be sceptical of very frequent schedules — a weekly plan pays four times the fixed costs of a monthly one for a further reduction in dispersion that is small.