Dollar Cost Average with MSCI World

I recently wondered whether it made sense to invest a larger sum of money in one go (lump sum) or spread it out over multiple months. The basic idea is the Dollar cost average (DCA) effect where by regularly buying for a fixed amount of money, one ends up getting more shares than regularly buying a fixed amount of shares.

Say that the price of something is 10 EUR per unit on average, with a standard deviation of 1 EUR. There is no long-term trend, it is just fluctuating with a normal distribution whenever you buy it. Take 100 random points in time where you buy. The mean price in my example is 9.992 EUR. But when I buy for a fixed amount of EUR each time and look at the effective average price, it is just 9.888 EUR. So I have effectively saved 1 %.

This only works when there is no long-term trend, or the long-term trend is much smaller than the current fluctuations. The latter is quantified in the Sharpe ratio. For something like Bitcoin, it is rather low, and the DCA will give you nice gains there. But what about an ETF based on the MSCI World index, which has relatively low fluctuation and steady long-term gains; is the DCA stronger than the overal gains? Does it make sense to hold back money and buy over time, or invest all right now?

I just wanted to test this empirically. So I downloaded the historical data for the MSCI World index. One can see that it grows rather steadily, with a few dips during crises:

One can then just simulate investing from any starting point and spread out over 1 month (investing all at once) and 12 months (investing a twelth each month). I compare the two strategies and see how much I would gain by spreading out my investment. Taking the data since 2000 only, we see this:

If one just looks at the median line in each box, one can see that the median goes down as one increases the period. So in at least half of starting months it has not payed out to wait. And the longer the wait, the worse it gots. However, there are some outliers going up to above 30 %. So there are a few months where it would have been better to invest over a long period instead of going all in at the same time.

To get a better idea for this, I have made a heatmap showing the starting month on the horizontal axis, the length of the investment period on the vertical axis. The advantage in percent is visualized in color. I have only taken the months since 2015, otherwise the graphic would be too large.

One can see that most of the chart is pinkish, so one is at a loss when spreading out the investment. So although the index goes up and down, it usually goes upwards such that the price never dips below the starting point.

It is green around 2015, but the months alternate. So one would have to have really clear timing to get it right. And the other one is in the beginning of 2020. Because there was a drop March 2020, it would be bad to have bought the month before that and better to have spread it out a bit longer. However, if you started in March, it would have been best to go all in during the local minimum.

Since I don't want to time the market, I want a strategy which does the right thing in most cases. And apparently with a relatively stable index like MSCI World, the winning strategy is to invest whenever there is extra money, and not try to use DCA.