To show this, I put together a few comparisons. All are using monthly S&P 500 values for ~100 years. I then just use both strategies for every historical period, and compare the results.
First, you can simply look at the distributions of values resulting from each strategy. Here is that plot for a 2-year investing period using $10,000:
Investing all at once has a much higher spread in results as you'd expect. It also has a much higher median. Sure, you'll lose more every once in a while as you're more exposed to crashes, but those periods appear pretty rare. How rare?
Here's the next plot. This shows the difference in outcome for each strategy at every historical point:
Some wild swings there, but it's clear that investing all at once is better in more of the periods.
A final presentation of the results is simply to see what percentage of historical periods were better with each strategy...
- all at once wins 72% of the time
- dollar cost averaging wins 28% of the time
An obvious question you might have is if the current market value influences this. As a check, we can do the same thing except modify the strategy slightly:
- If market is more than 5% off the all-time high, invest lump
- Else, dollar cost average
Comparing that strategy (call it caution) with the all at once one from earlier:
- all at once wins 59% of the time
- caution wins 21% of the time
- they tie 20% of the time
5% off the all-time high is arbitrary, so what about 2%?
- all at once wins 59% of the time
- caution wins 20% of the time
- they tie 21% of the time
This isn't 100% thorough, but it's enough to push me more towards 'invest all at once' than I was in the past.
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