Dollar cost averaging and the harmonic mean

The DCA is able to lower the average purchase price of an investment. Some people find in this property of dollar cost averaging something magical.

However, the credit for this result comes from the properties of a particular type of less known average: the harmonic mean.

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals: given a set of n observations x1, x2, …, xn, the harmonic mean H is obtained by applying the formula in the image.

DCA is one of the best-known applications of harmonic mean, since it consists of periodically investing a fixed sum.

Let’s look at an example. Suppose you need to invest five times in a given fund. The prices of the investments are respectively: 51.88; 53.11; 49.99; 53.41 and 56.29.

Applying the harmonic mean, the average purchase price can be calculated to be 52.8562.

To verify the correctness of the formula, one simply needs to calculate how much the average price would be after 5 deposits of 1,000 dollars on an asset that has the prices listed above.

With the simple arithmetic mean, the average price is 5,000 (amount invested) / 94.5963 (number of stocks bought) = 52.8562.