Invest all at once or spread it out?

Two paths for one sum. Investing all at once buys everything on day one at the starting price. Spreading buys an equal slice at the start of each month across the buying window you set, which is dollar cost averaging applied to money you already hold. Both are valued at the same date, 24 months in, so the comparison is like for like.

Both end up holding units worth the same price, so the whole answer collapses to a ratio of two purchase prices:

$$\frac{V_L}{V_S} = \frac{\bar{P}}{P_0}$$

$V_L$ is the value of investing at once, $V_S$ the value of spreading, $P_0$ the price on day one, and $\bar{P}$ the average price the spreading actually paid. That average is a harmonic mean, because a fixed sum buys more units when the price is low:

$$\bar{P} = \frac{N}{\sum_{k=0}^{N-1} 1/P_k}$$

Where the market finishes cancels out of both sides. Only the prices paid inside the buying window decide the winner.

On a steady upward trend every purchase after the first costs more than the first, so the average price paid lands above the day one price and spreading finishes behind. Reverse the trend and the same arithmetic runs the other way: each later purchase is cheaper, the average price paid falls below the day one price, and spreading finishes ahead.

Two settings tie exactly. A flat trend puts every purchase at the day one price, so both paths buy identical units. A buying window of one month is investing at once under another name.

The two are not symmetric in practice, because the two market directions are not equally likely. Share markets have risen across most long historical stretches, and studies of rolling historical windows put investing at once ahead of spreading in roughly two thirds of them. Money held back is money not earning the total return that made the market worth owning, and the longer the buying window the more of the sum sits out.

Not a higher expected outcome, but a narrower range of them. The lower bar in the tool shows it: as the buying window grows, the span of finishing values pulls in at both ends. Over a 12 month window the worst trend allowed here leaves the spread sum at 58 percent of what went in, against 49 percent for investing at once, and the best trend leaves it at 150 percent against 169 percent. Spreading hands back the top of the range to lift the bottom of it.

One thing the arithmetic misses is the behaviour. A single large purchase made days before a fall is the decision people abandon a plan over, and a schedule set once in advance removes the monthly argument with the news. Regular investing out of income is a different case again: no sum is sitting in cash waiting, so there is nothing to compare it against.

The tool holds waiting money at a zero return, which keeps the picture about the market path alone. Paying interest on that cash lifts the spreading side, and it flips the answer wherever the cash rate beats the market trend, so a flat market with cash earning something is a case for spreading.