# Dollar cost averaging

It is often recommended  that when investing money in a mutual fund or security to use a strategy called  dollar cost averaging.  This simply means you invest a fixed amount of money at periodic intervals rather than say buy a fixed number of shares at periodic intervals or invest as a lump sum.  The argument is that when the price of the fund is low you buy more shares and when it is high you buy fewer shares.  In this way you can ride out the fluctuations in the stock price.

Evaluating which strategy is best is a great example of how concepts in calculus and in particular Jensen’s inequality enter every day life and more reasons for why an average person should learn more math.  The question is how do the investment strategies compare over time. Let $p_t$ be the price of the fund at time $t$.  In dollar cost averaging, you would invest $s$ dollars at periodic intervals (say each week).  So the number of shares you would buy each week is $n_t=s/p_t$.  Over a time period $T$ you will have purchased $\sum_t s/p_t = s T *(1/T)\sum_t 1/p_t$  shares, which can be rewritten as $sT E[1/p_t]$, where $E$ is the expectation value or average.  Since you have invested $s T$ dollars, you have paid $\tilde p_t=1/E[1/p_t]$ per share on average, which is also called the harmonic mean.

If on the other hand you decided to buy a fixed number of shares $n$ per week  then it would cost $n p_t$.   Over the time period $T$ you will have spend $\sum_t n p_t$ and have $n T$ shares.  Thus you have paid $\bar{p_t}= E[p_t]$ on average.    Now the question is how does the mean $\bar{p_t}$ compare to the harmonic mean $\tilde p_t$.  This is where Jensen’s inequality comes in, which says that for a convex function $f(x)$ $E[f(x)]\ge f(E[x])$.  A convex function is a function where a straight line joining any two points on the function is greater than or equal to the function.  Since $1/x$ is a convex function, this means that $E[1/p_t] \ge 1/E[p_t]$ which can be rearrange to show that $\bar{p}\ge \tilde{p}$.  So dollar cost averaging is always better than or as good as buying the same number of shares each week.  Now of course if you buy on a day when the price is below the harmonic mean of a time horizon you are looking at you will always do better .

## 7 thoughts on “Dollar cost averaging”

1. Pretty math with a tidy result. I haven’t seen this before and find it elegantly appealing.

DCA is an optimal strategy, but the assumptions for the game are interesting. DCA is the best answer to How do I optimize my investment in a fluctuating asset with an upward trend?

Under these optimistic assumptions, DCA is the equivalent to the idiom: make hay while the sun shines. Downward price fluctuations are viewed as opportunities to use the same resources to make even more of a good investment. Under less certainty, this is what gamblers call doubling down–when you are loosing, bet more so when you win, you make up your previous losses.

(My protestant parents don’t like it when I make work-ethic slogans equivalent to gambling folly.)

If you want to minimize losses when there is a good chance the trend is down, your outcome is improved if you buy the same number of shares every month. When prices fall, you use the left over money to diversify by buying the same number of shares of another company or fund, doing your best to avoid correlations.

DCA is only an optimization under optimistic assumptions–certainty that you chose the asset with the maximum assured appreciation within your time horizon.

DCA runs counter to the efficient market hypothesis as it depends on confidence in a final, better outcome in the face of what the EMH considers the discouraging evidence of currently falling prices. (This is ironic as many investment advice memes contain both DCA which is information agnostic and diversity, which is EMH compliant.)

DCA assumes proficiency in choosing assets (so called value investing). But if you can do that once, you can do it a dozen times. Then, the strategy is to accumulate cash for a year and watch your dozen surely appreciating assets. At the end of the year, buy the biggest looser with all the money you can find.

As you can see, I am torn on DCA. On one hand, it is clearly the optimal strategy under the optimistic assumptions. On the other, optimizing strategy under the optimistic assumptions may make me feel good when it is more like good interior design work on a sinking ship.

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2. […] April 23, 2010 by Dr. Skippy Over at Scientific Clearing house, Carson posted a pretty proof of the how dollar cost averaging is an optimal investment strategy.  Unfortunately, I think the […]

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3. Hi Scott,

I actually don’t think DCA is an optimal result. It’s just better than simple averaging. In an upward or downward trend DCA is not optimal. Buying earlier, which is lower, is better if prices are rising and buying later is better if prices are falling. In a fluctuating trend it is a little more interesting in that you would then balance the volatility with the trend in determining when to buy.

It may be in the spirit of but it is not equivalent to a doubling down (i.e. a Martingale) strategy because your expected value is not your current value.

Generally, DCA is applied to a single fund or security so it is independent of portfolio optimization. I think it just says that if you want to invest regularly then DCA is better than buying the same number of stocks each time or random numbers of stock each time.

cc

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5. […] is beneficial when compared to buying a fixed number of shares. Here’s an introductory analysis. In summary, cost averaging yields the harmonic mean of the entry prices. The resulting cost basis […]

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