Dollar-cost averaging calculator

Why is your average cost lower than the average price?

When the amount stays fixed and the price moves, the number of units you get moves the other way. A cheap month buys more units and an expensive month buys fewer, so the low prices carry more weight in your cost than in a plain average of the prices. The result is that your average cost per unit is the harmonic mean of the prices, not their arithmetic mean.

The smallest example shows it. Put $100 in at a price of $100 and $100 in at $50. The first buys 1 unit and the second buys 2, so $200 buys 3 units at $66.67 each. The average of the two prices is $75, yet nobody holding these units paid $75 for them. The harmonic mean, 2 / (1/100 + 1/50), is exactly $66.67.

For any list of prices that are not all the same, the harmonic mean sits below the arithmetic mean. So with equal amounts, whole money invested and no charges, the cost per unit comes in under the average price you would read off a chart. That is the whole effect, and it is arithmetic, not a forecast.

How do you calculate dollar-cost averaging?

Three steps. Divide the amount by each price to get the units bought that time. Add the units. Then divide the total invested by the total units: average cost = (amount x number of purchases) / sum of (amount / price). Because the amount cancels, this is the same as number of purchases / sum of (1 / price), the harmonic mean.

The default inputs invest $500 six times at prices of 100, 90, 75, 80, 95 and 110:

The $3,000 buys 33.2808 units, so the average cost is $90.14. The six prices add up to 550, an average of $91.67. Your cost is $1.52 a unit, or 1.66%, below that average. Valued at the last price of $110, the holding is worth $3,660.89, a gain of $660.89 or 22.03% on the money invested.

How much do price swings widen the gap?

The gap between the two averages depends on how far the prices move around, not on whether they rise or fall overall. Here are four lists of prices that all average exactly $100, with $100 invested at each:

Swings of 10% either side cut the cost by 1%; swings of 50% either side cut it by a quarter. When prices alternate evenly around a level, the saving is the square of the swing: 0.10 squared is 1%, 0.25 squared is 6.25%. A steady price gives no saving at all. The panel chart under the result shows the same thing as bars: the lower the price, the taller the bar of units bought.

Dollar-cost averaging or a lump sum: what changes the answer?

A lower cost per unit than the average price is not the same as beating a lump sum. The calculator puts the whole total in at your first price and values both at the same price, so you can see each case.

• Prices dip and recover (the default). $3,000 at the first price of $100 buys 30 units, worth $3,300 at $110. The six smaller purchases hold 33.28 units worth $3,660.89, because half the money went in below $100.
• The same dip, valued lower. Enter 85 as the valuation price and the regular purchases are worth $2,828.87, a loss of $171.13 or 5.70%. The lump sum would be worth $2,550. Both lost money; the staged purchases lost less.
• A steady rise. €300 at €20, €22, €24 and €26 buys 52.67 units at an average cost of €22.78, worth €1,369.55 at €26. Investing €1,200 at €20 would have bought 60 units, worth €1,560.00.

In a market that only rises, money invested earlier buys more units, so the lump sum ends ahead. The harmonic-mean effect still holds in that case: €22.78 is below the €23.00 average of the four prices. It just cannot make up for buying later at higher prices.

Whole shares, fractional units and commission

Funds and some platforms sell fractions of a unit, so every dollar is invested. Some shares can only be bought whole. Switch on Whole shares only and each purchase buys as many whole shares as the money allows, then carries the remainder to the next one.

On the default list, $500 buys 5 shares at $100, then 5 at $90 with $50 left, then 7 at $75 using that $50, then 6 at $80, 5 at $95 and 5 at $110. That is 33 shares and $20 of cash that never found a share. The cost per share is ($3,000 - $20) / 33 = $90.30, a little above the fractional $90.14 because the rounding steals some of the cheap months' extra units. The value at $110 is $3,650 including the $20.

A flat commission works against the averaging. A $5 charge on each of the six $500 purchases costs $30 and lifts the fractional cost to $91.05, still below the $91.67 average price. Make the commission large compared with the amount and it can push your cost above the average price; the calculator says so when that happens. The lump-sum line pays the commission once, so it ends at $3,294.50 against $3,624.28 for the regular purchases.

Worked example in pounds: a monthly fund purchase

The arithmetic does not care about the currency. A saver puts £200 a month into a fund priced at £4.00, £3.20, £2.50, £3.20 and £4.00, paying £1.50 commission each time. Each purchase invests £198.50, so the units are 198.50 x (1/4.00 + 1/3.20 + 1/2.50 + 1/3.20 + 1/4.00) = 302.7125.

The £1,000 put in therefore costs £3.30 a unit including commission. Before commission it is £3.28, which is the harmonic mean of the five prices, while their simple average is £3.38. The price ends where it started, at £4.00, yet the holding is worth £1,210.85, because the months at £2.50 and £3.20 bought more units. One £1,000 purchase at £4.00 with a single commission would be worth £998.50 at the same price.

Switch the currency selector to GBP, EUR, AED or any of 24 currencies and the formatting follows. Prices you type in the list stay exactly as typed, so enter them in the currency you chose.

How to calculate dollar-cost averaging in Excel or Google Sheets

Both spreadsheets have a harmonic mean function, which gives the cost per unit directly when the amount is the same each time. Put the prices in B2:B7 and the amount in E1. Then:

• =HARMEAN(B2:B7) returns the average cost per unit: 90.14 on the default list.
• =AVERAGE(B2:B7) returns the simple average price, 91.67, for comparison.
• =E1*SUMPRODUCT(1/B2:B7) returns the units held, 33.28.

Microsoft's documentation notes that HARMEAN returns the #NUM! error if any value is zero or negative, so leave empty months out of the range rather than typing 0. With a commission, multiply the units by (amount - commission) / amount and divide the total invested by the result instead. If the amounts differ from month to month, HARMEAN no longer applies: use =SUM(amounts)/SUMPRODUCT(amounts/prices).

Dollar-cost averaging in the UK, US and elsewhere

The method is the same in every currency. It is also called pound-cost averaging, and the same idea applies to euros or dirhams. A regular monthly investment into a fund or a workplace plan that takes a set sum from each paycheck follows exactly the arithmetic on this page, whether or not the plan uses the name.

Tax is where countries differ. The average cost shown here is the plain figure across your purchases. Tax offices have their own rules for which cost applies when you sell part of a holding, and those rules can give a different number. The stock average calculator sets out how the UK, the US and Canada treat the average cost of shares, and the capital gains tax calculator estimates the tax on a gain by country.

Common mistakes and what this calculator leaves out

• Quoting the average price as your cost. With equal amounts your cost is the harmonic mean, which is lower whenever prices moved.
• Reading the gap as a return. A cost 1.66% below the average price says nothing about whether the holding is worth more than you paid.
• Mixing in different amounts. HARMEAN only fits when every amount is equal; otherwise divide total money by total units.
• Typing 0 for a skipped month. A zero price is left out here and breaks HARMEAN in a spreadsheet.
• Ignoring charges on small amounts. A flat commission is a bigger share of $100 than of $1,000.

The calculator holds no prices of its own and makes no forecast. It does not add dividends, fund running costs, currency conversion or tax, and it assumes every purchase happens at the price you list. For the effect of a fund's yearly charge over time, see the fund fee calculator.

Glossary

Dollar-cost averaging: investing a fixed amount at regular intervals whatever the price. Arithmetic mean: the sum of the prices divided by their count. Harmonic mean: the count divided by the sum of 1 / price; the cost per unit of equal purchases. Average cost per unit: money invested, commission included, divided by units held. Fractional unit: part of a share or fund unit. Lump sum: the whole amount invested at once.

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