Investing a sum at once or over twelve months, priced in expectation and on three price paths, with the argument about average cost taken apart.
On expected returns, investing a lump sum at once beats dollar-cost averaging, because averaging keeps half the money in cash for half the period. Putting 12,000 into an index fund on day one at an illustrative 7 per cent expected return ends the year at 12,840; feeding it in at 1,000 a month, with the waiting cash at 4 per cent, ends at 12,676. The lump sum leads by 164, or 1.37 per cent, and averaging comes out ahead only if prices fall, stay flat or rise by less than about 4 per cent over the twelve months.
Worked in full in Stock Market Investing for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Dollar-cost averaging (pound-cost averaging in the UK) means splitting a sum into equal amounts invested at regular intervals. It is distinct from investing part of every pay cheque, which is simply investing money as it arrives and has no lump-sum alternative. The question here is narrower: you already hold 12,000. Do you invest it today or over a year?
| Input | Value |
|---|---|
| Sum available today | 12,000 |
| Averaging period | 12 months |
| Monthly tranche | 1,000 |
| Interest on cash waiting to be invested | 4% a year (0.3333% a month) |
| Expected return on the index fund | 7% a year (0.5654% a month) |
| Starting fund price | 100 |
Lump sum. Buy 12,000 ÷ 100 = 120.00 units on day one. Value after a year = 120.00 × price at month 12.
Averaging. At the start of each month k (0 to 11) buy 1,000 ÷ pricek units. The cash not yet invested earns 0.3333 per cent a month. Value after a year = total units × price at month 12, plus the interest the waiting cash earned, which stays in cash.
Expected case. If the price grows at 0.5654 per cent a month, (1.07)1/12 − 1, the lump sum ends at 12,000 × 1.07 = 12,840. Each averaging tranche earns 7 per cent only for the months it is invested and 4 per cent before; the twelve together end at 12,676.
The gap has a simple source. On average an averaging plan holds each pound in cash for 5.5 months before investing it. Half the money spends half the year earning the cash rate instead of the expected market return, and 12,000 × (7% − 4%) × 5.5 ÷ 12 is about 165, almost exactly the 164 the full calculation gives. The lump sum's advantage is the equity risk premium on the money averaging leaves waiting.
Expected values are not what anyone experiences. Here are three illustrative twelve-month paths, all starting at a price of 100.
| Path | Price at month 12 | Lump sum | Averaging | Averaging less lump sum |
|---|---|---|---|---|
| Steady rise, 100 to 112 | 112 | 13,440 | 12,979 | −461 |
| Fall to 80, recovery to 100 | 100 | 12,000 | 13,617 | 1,617 |
| Steady fall, 100 to 85 | 85 | 10,200 | 11,202 | 1,002 |
Averaging wins two of the three, but that is a property of the paths chosen, not of markets: two of them go down. In the fall-and-recovery case averaging buys 133.92 units against the lump sum's 120.00, because half its purchases happen near the low of 80, and it ends 1,617 ahead though the price finishes exactly where it started.
On a straight-line path the two methods tie when the price ends the year at 104.05, a rise of 4.05 per cent. Any year in which the index rises by more than that, on a smooth path, favours the lump sum. Whenever the expected return on the fund exceeds the cash rate, the expected result favours investing at once; how often it actually wins depends on how volatile the year is, an assumption this calculation does not supply.
The classic argument for averaging is that it buys more units when prices are low, so the average cost per unit is below the average price. That is true. On the rising path the average price paid over the twelve months is 105.50 and the averaging plan's cost per unit is 105.39. In the fall-and-recovery path it is 89.61 against an average price of 90.00.
But the comparison that matters is not average cost against average price. It is averaging against the lump sum, and the lump sum bought every unit at 100 on day one. On the rising path the averaging plan's cost of 105.39 a unit is worse than the lump sum's 100, and it ends 461 behind. A lower average cost than the average price says nothing about whether the money would have done better invested at once.
Averaging is insurance against regret: it reduces the chance of investing everything the week before a fall. On these figures the premium is about 164 a year in expectation on 12,000, 1.37 per cent, in exchange for a smaller worst case. In the steady-fall path the lump sum loses 1,800 while averaging loses 798, since the cash interest of 225.69 and the cheaper later purchases cushion it.
That trade can be worth paying for someone who would otherwise not invest at all, or who would sell in a panic after an early fall. Selling after a fall is far more expensive than either method: the book's companion workbook prices one such sale over a working life. If averaging is what makes the money go in and stay in, it has done its job. Shortening the period, three or six months instead of twelve, cuts the expected cost roughly in proportion to the months spent in cash.
With a lump sum already in hand and a long horizon, investing it at once has the higher expected result, because money in an index fund is expected to earn more than money in cash. Averaging gives up that difference, about the equity premium on half the money for half the period, to reduce the regret of bad timing. Know the price before choosing it, and remember that the fee on the fund will cost far more over the years than this decision; see what one percentage point of fees actually costs and what 500 a month grows to.
The free workbook for this book includes the rule for a falling market, written before the fall. All figures are illustrative; investing carries the risk of loss, and this is not advice.
In expectation, the lump sum, because money in an index fund is expected to earn more than cash. On 12,000 at an illustrative 7 per cent against 4 per cent on cash, investing at once ends the year 164 ahead of twelve monthly tranches. Averaging is ahead only in years when prices fall or rise less than about 4.05 per cent.
It lowers average cost per unit below the average price over the period, because a fixed amount buys more units when prices are low: 105.39 against 105.50 on a rising path. But a lump sum invested on day one bought every unit at 100, so on that path averaging finished 461 behind. Average cost is the wrong comparison.
The shorter the period, the smaller the expected cost, since the cost is the return given up on cash waiting to be invested. Over twelve months the average pound waits 5.5 months; over six months it waits 2.5 months, less than half as long, and the expected shortfall falls roughly in proportion. Pick the period before starting and do not extend it.
This article is one calculation from Stock Market Investing for Beginners. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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