# Proof showing that dollar cost averaging always worse than lump sum alternative

I am referring to the article here. In a nutshell the article says that using data based on S&P 500 index going back as far as to 1950, dollar cost averaging is performing worse than a lump sump investment.

My question is, assuming that the market works according to random walk hypothesis, is it possible to show mathematically that dollar cost averaging is always worse than lump sum investment, as far as the return on investment goes?

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You did not carefully read the article you yourself linked to. Dollar cost averaging is a generalized concept. What the author compares is a full-sized investment or time-specific partial investments. So, dca is a concept and you draw conclusions from one single approach to dca. There is no mathematical proof that dca works or not because it is one single concept that is only part of the overall investment process. Its as if I tell you that getting higher education does not pay off because those who undergo it and become philosophers do not recoup their investment. I did not include those who become lawyers, doctors, financiers,...the same applies to this article.

What if I told you that dca hugely outperforms a lump-sum investment approach if instead of investing the parts at time-shifted periods you instead disregards time and buy companies with great fundamentals but depressed stock prices because of overall market sentiment. Each time the stock price trades another 2% below your previous investment you invest another portion. It is one alternative approach from many but it clearly demonstrates that you are comparing apples and oranges here.

Another example to debunk the point made in the link is if I claimed trend following strategies do not work because the market is 70% of times range-bound and the strategy cannot make money in a range bound environment thus it has to be inferior to a range-bound strategy approach.

Hope you got my point. So, in short, there is no mathematical proof. I could specify a strategy approach that hugely under-performs when investing in a dca way and another strategy that hugely outperforms a lump sum investment approach.

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What if I told you that dca hugely outperforms a lump-sum investment approach if instead of investing the parts at time-shifted periods you instead disregards time and buy companies with great fundamentals but depressed stock prices because of overall market sentiment-- this is not what we generally mean by dollar cost averaging. This is fundamental analysis-- a popular strategy to pick stock. Dollar cost averaging is designed to avoid picking stocks/analyzing stocks specifically. – Graviton Mar 5 '14 at 4:29

There are a number of papers in the literature which show that Dollar Cost Averaging is suboptimal, in the sense that, given a DCA investment strategy, then there exists an alternative investment strategy which will be strictly preferred by a utility maximising agent. This preferred strategy may not necessarily be a "lump-sum" strategy, but a better strategy can be given that beats DCA for any utility maximising investor.

I think that the original paper on this was by Constantinides, "A note on the suboptimality of dollar-cost averaging as an investment policy"

http://www.jstor.org/stable/2330513

(edit: The class of models that Constantinides considers is large, and should include the ones you want.)

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this is total nonsense. There are some highly profitable professional hedge fund traders that dollar cost average every single day, they just called it "legging in and out of trades". I would caution to be careful to take trading related academic treatises too seriously. Academicians are excellent at researching things, but trade entries and exits is something you definitely do not learn in text books nor by performing academic research. Also note that the point of Malkil in your reference is again PERIODIC investments, hinting that the timing of the investments is not linked to a strategy – Matt Wolf May 28 '12 at 13:33
...which is completely unrelated to dca. All that dca is about is the fact that not all proceeds are invested at a single point but at several instances. There does not have to be any time-based relationship, dca does not hint at all at what the underlying motivation is of the specific timing of the sub-investments. Its a very trivial yet hugely misunderstood concept. – Matt Wolf May 28 '12 at 13:39
In the article: "Let’s look at an example of DCA. Lets say you just inherited 20,000 and you decided to invest it in the stock market. Once you have determined a proper asset allocation ... you could invest it all at once or do “dollar cost averaging” into the securities by purchasing a set dollar amount of securities at pre-determined intervals. An example would be investing 1666.67 on the first of each month for 12 months." It seems to me that the DCA described here and in the Constantides paper is exactly periodic investments. Perhaps you should define DCA to clarify your view. – amgc May 28 '12 at 13:52
The definition is clear: DCA simply means to average into trades/investments, nothing else. The function (whether its utility, a trading strategy, or fixed time intervals) does not have an impact on whether an investment is averaged into, price wise, or not. The academicians cited in your referenced paper seem to have a problem to distinguish between those two concepts. They compare two different things and conclude from that that DCA is suboptimal which is factually incorrect. – Matt Wolf May 29 '12 at 1:54
@Freddy The definition in the academic literature, the link cited, and even online dictionaries link excludes any strategy that changes due to additional information (e.g. price changes), and it's pretty much due to this that DCA is suboptimal. I don't think that anyone claims that splitting trades into small pieces can have benefits if it incorporates some of this extra information, but then it is not DCA by the commonly used definition. (I don't follow your definition - is the average fixed at the outset, and costed in cash) – amgc May 29 '12 at 7:33