I have been thinking a bit about my last post on whether if one has a bundle of cash and wants to put it into the stock market, you should invest all at once or more gradually. For instance, suppose you have $1 million in cash; you could put it into an equity index fund all at once, or do 1/12 each month for the next year. Bottom line for me is leaning very strongly towards all at once.

As to be expected, I was not the first one in history to pose this question and there is a fair amount written about it. The phrase "dollar cost averaging" or DCA is sometimes used to refer to the idea of investing a lump sum gradually, although DCA is also used for other investment policies (such as just investing a fixed amount each month).

Wikipedia has an entry, dollar cost averaging, which starts out good but doesn't really solve the problem. The references however are pretty good, including a 1979 paper by George Constantinides, "A Note on the Suboptimality of Dollar Cost Averaging as an Investment Policy."

The company Wealthfront, an online financial advisory service, has an

entry in their FAQ section that addresses the question and gives a link to a paper published by Vanguard. This is the same paper referenced by a commenter on my earlier post -- see

here.

Both Wealthfront and Vanguard give pretty good reasons for investing all at once. Vanguard even does a study using historical data to test DCA against all-at-once.

I liked the Constantinides paper because it is the most analytical and because it provides a reference to an even more analytical paper, a 1971 piece in Management Science by Gordon Pye, "Minimax Policies for Selling an Asset and Dollar Averaging."

The answer that many give to my question is along these lines: If you have decided your optimal asset allocation, 80/20 stocks/bonds or whatever, you should just get to it right away. If you like the risk/return profile of that asset allocation, then why would you not get to it right away? If that is your optimal allocation in 12 months, why isn't it your optimal allocation right now?

That is pretty well said and convincing, if I do say so myself.

However, I had the following idea that caused me to consider seriously the gradual policy. By investing gradually over a year, you end up investing at the average price during the year -- 1/12 each month at the price of that month is the same as putting all the money in at the average of the 12 months' prices.

Putting a statistical hat on, I then thought that the variance of that average price would be lower than the variance of any individual price, and therefore that going in gradually would get me to my optimal allocation in a less risky fashion.

Or put it this way. Suppose I am going to put my money into the stock market tomorrow. You give me a choice: I can put in my order and take whatever the market price of the index is at that time, or you will let me buy in at the average index level during the day. Again, I thought that I should prefer the average price by the compelling (seemingly) logic that the variance of an average is less than the variance of a single draw from a distribution.

There is a serious flaw in this line of reasoning, though, and it is the Constantinides paper that made me see it -- actually the reference to the Pye article because Pye deals with this problem.

My reasoning about the variance of the average being lower is true if the prices are independent random draws from a distribution. That was the model of stock prices I implicitly had in my mind.

But probably a better model of stock prices is that they are a random walk. That means, roughly, that the next price is the current price plus a random shock:

p(t) = p(t-1) + e

where e is a random variable with mean zero and some variance.

In this case, if you do a little math, you find out that the variance of prices increases over time, and the variance of the average price over a period is not less than the variance of any one price. Just to do a little math, suppose we have 5 periods. Then we have

p1 = e1

p2 = p1 + e2

p3 = p2 + e3

p4 = p3 + e4

p5 = p4 + e5

Then doing some substitutions, you can write

p5 = e1+e2+e3+e4+e5

So you can see what is going on -- the end of period price is the sum of all the random shocks to that point. Its variance is going to be higher than the variance of any of the previous prices -- the random walk is causing variance to increase with time. That is one of the key ideas of a random walk -- the meandering in the future can be pretty far off course!

And in this case, the variance of the average of the five prices is not lower than the variance of just p1. I will spare the math here, but it is pretty straightforward: write out the formula for the average price given the five equations above, and calculate its variance.

So my early intuition was based on one model of stock prices -- that prices are fluctuating randomly around a mean -- rather than what is a better model, that of a random walk.

Now there are some reasons why you might still reasonably want a gradual policy. One pretty good reason is based on a different kind of utility function, one that has a "regret" characteristic. You can read the Constantinides paper and see also that he comes up with some reasonable situations where a gradual investment policy does make sense. If you really think that the market is over-valued now, well then you should wait, but that is market timing which in practice is very difficult to do. My intuition on why gradualism might be good was not based on market timing, just on the idea that maybe I could reduce the volatility of my wealth (at an acceptable price of lower return).

But in most cases, all at once will be the rational, utility maximizing policy. Just be ready to face the prospect that you will see prices decline after you go all-in. Such is the world.