> but it won't (statistically) deliver higher returns.

Would you take a coin-flip where heads you double your net worth and tails you lose all your money and assets?

(Statistically) the expected outcome is zero, so it really shouldn't bother you either way, you could take the bet or not, it's the same if you go for it or if you don't.

In the real world, though, lowering your risk means reducing the chance of both winning the bet and losing it. For many people it's worth more to have less risk of randomly timing the market poorly, even though they have less chance of randomly timing it well.

Yes, over all investors the two decisions cancel out, but to the individual DCA can be worth lowering their risk.

DCA'ing provides no mitigation of the risk of randomly timing the market poorly.

The only risk it mitigates is the risk of a sudden bankruptcy, or some other massive exogenous event which disrupts your future plans.

However even accounting for those events the expected value of DCA is still lower.

The point is to sacrifice some return for lower variance. The absolute return is lower in expectation, but the argument would be that the risk-adjusted return is better.

If you're committing to the strategy in advance, it delivers neither lower variance nor better risk-adjusted return.

It certainly does deliver lower variance, how could it not?

How could it?

So assume there's a normal distribution of price possibilities at each purchase point. You're going to end up with the _sum_ of those distributions, not the correlation or combination of them.

Presumably the deviations of those are growing more larger over time, but the are still the same underlying distribution curve as they sum up. The mean will trend steadily upwards.

The resulting price will be a normal distribution with a deviation of somewhere around the midpoint, and a mean of somewhere around the midpoint.

Now, with the mean increasing over time, your expected return will be decreasing the longer you wait. But what about the risk:return?

Unless you have outside knowledge about risk growing or lessening over time, it will remain constant over the DCA period, which means the risk factor of your investment (the standard deviation of the resulting DCA price) will be at exactly the midpoint of your purchase series.

So your expected return will be slightly lower, and your risk profile will be substantially larger, than just buying immediately.

DCA: math does not check out, not even close, _if you have all the money upfront_.