Dollar Cost Averaging, myth or fact?

Almost all banks prompt the “Dollar Cost Averaging” strategy to their mutual fund customers, e.g., http://helpmeinvest.scotiabank.com/knowhow/win-dollar-cost-averaging

The idea is that a customer should invest in a mutual fund on a regular basis, instead of in one shot. The reason is “Because the stock market goes up and down, your unit price may average out to be lower than it would be if you invested an equivalent lump sum at onetime at the beginning of the year.”

Is the above claim a myth or fact?

Assume the fund unit price is a random variable X and a custom has Ydollars to invest. If the customer invests all Y dollars in one shot, the average number of units he obtains is E[Y/X]. If the customer invests N times and each time invests Y/N dollars, the average number of units he has will be NE[Y/(NX)] = E[Y/X].

Myth buster!

The mistake from the above reasoning is: it assumes that when a customer invests Y dollars, the average number of units obtained by him is Y/E[X] (which is incorrect as E[Y/X]=YE[1/X] which is not equal to Y/E[X]); when invests N times and Y/N dollars each time, the average number of units obtained is Y/N/X_1 + Y/N/X_2 +… Y/N/X_N = Y E[1/X], and as1/E[X] < E[1/X], so dollar cost averaging is desirable (which is a myth).

Here is an example: the mutual fund price is uniformly distributed between $10 and $20 per unit, and the average price is $15/unit. Assume that the unit price is $10, $20, and $15 at t1, t2, and t3, respectively. If we invest $10 at t1, t2, and t3, we get 10/10+10/20+10/15 = 2.1667 units (which is larger than 30/15=2 units). But if we randomly choose a time instance to invest the $30, on average, we can obtain (30/10)/3 + (30/20)/3 + (30/15)/3 = 2.1667 units, the same as the first equal installment investment.

Among the 5 advantages listed in the bank’s website, the only point makes sense to me is ” Smaller investments on a regular basis can be easier than one large lump sum.” Therefore, dollar cost averaging is desirable for lazy investors who do not have time to manage the money.