Constantinides's Argument

Another day, another article about dollar-cost averaging. I just finished skimming through "A Note on the Suboptimality of Dollar-Cost Averaging as an Investment Policy" by George Constantinides. The core argument there is quite devastating, I feel, and can be explained at an intuitive level without any of the mathematical apparatus from the paper.

To set the stage first note that although we've talked about dollar-cost averaging as a strategy for shifting money from a safe investment (cash) to a risky investment (stocks), an adherent of dollar-cost averaging should equally well recommend the same approach when moving from stocks to cash (or, in general, from any asset class to any other asset class). The core argument still applies: you want to enhance your chances of getting a "fair" price for the stock and not run the risk of a bad price by happening to be very unlucky in your choice of the day for the lump sum transition. Whether you happen to be buying or selling the stock, the logic is the same.

Second, note that we've implicitly been assuming all along that there are no transactions costs of any sort. We can instantly and without cost shift any part of our portfolio from one asset to another without paying commissions, capital gains tax, or anything like that. Of course, this is an idealization, but it seems like a harmless one. Certainly the proponents of dollar-cost averaging don't argue that the reason the approach works is because of transaction costs. (If anything, transactions costs hurt the case for dollar-cost averaging since you have more transactions.)

Now consider two investors John and Mary. John has inherited $1,000,000 in cash while Mary has inherited $1,000,000 in stock. Let's call that point in time t0. They have identical preferences as far as risk and return and each feel their optimal long-term allocation is 50% stocks and 50% cash. If we subscribe to dollar-cost averaging, at t0 John should move a little money into stocks and Mary should move a little money into cash. But clearly they will still have different portfolios: John will still be mostly in cash, and Mary will still be mostly in stocks.

Now here's where the trap closes on the hapless DCA proponent. Observe that John and Mary are really in equivalent positions at t0. By our assumption above of no transaction costs, Mary can move all her money from stocks to cash at t0 without any cost (and vice versa for John). If Mary can transition freely and instantaneously to John's portfolio at t0, and John can transition freely and instantaneously to Mary's portfolio at t0, how can they possibly have different optimal investment strategies to pursue at t0?

To put it another way, suppose we can quantify John's utility at t0 (after he readjusts his portfolio as per DCA) as Uj,0 and Mary's utility at t0 as Um,0. This utility will be a function of the expected return and volatility of their portfolio going forward from t0. If Uj,0 > Um,0 then you have to wonder why Mary doesn't just shift all her money to John's portfolio at t0 which would give her utility Uj,0. Likewise if Um,0 > Uj,0. The only way we don't get a contradiction is if Uj,0 = Um,0 but in general that is not going to be the case - a 95% allocation to stocks and a 95% allocation to cash will not offer identical utility.

A Better Demonstration

One of these days I'll write about a topic other than dollar-cost averaging. Today is not that day.

I think I've come up with a better way to demonstrate the futility of dollar-cost averaging that sidesteps some of the complexity we got mired in back in my earlier posts. It relies on computer simulation to compare a hypothetical dollar-cost-averaging approach to a "lump sum investment" approach.

Let's say we have a portfolio that is at this moment 100% cash and 0% stocks. You want to use dollar-cost averaging to transition your portfolio gradually to 100% stocks over some initial period of time. After that you are going to hold the all-stock portfolio for some period of time. You can pick the initial duration (the length of the "ramp up" period) and the "steady state" duration to be whatever you like.

My claim is that whatever values you select, I can construct a superior portfolio that takes a "lump sum investment" approach. This portfolio will start with X% of its funds allocated to stock and maintain exactly that percentage for the full duration - i.e., we have a lump sum investment at the very beginning of the experiment. This portfolio will be superior to the DCA portfolio because it will have an identical expected return, but the volatility - measured as the standard deviation of the observed returns - will be lower.

As you might guess, X will be chosen so that the average allocation to stocks - across the entire time period - is the same for the lump sum investment approach as for the DCA approach.

My main assumption is that the return of stocks over a single unit of time is drawn from a normal distribution and is independent of the return of stocks over any other unit of time.

I ran the simulation with these parameters, but, again, the assertion is that I could choose essentially any set of values:

• Expected stock return: 0.01 (meaning expected appreciation of stocks over one unit of time is 1%)
• Standard deviation of stock returns: 0.03
• Return on cash: 0 (with no variability)
  
• Length of "ramp up" period: 10 time units
• Length of "steady state" period: 100 time units
• Starting value of portfolio: $1000

For a given run, I simulated each portfolio 1,000,000 times. I calculated the means of the final portfolio values and the standard deviations. Over thirty runs, I observed the following range of values:

Dollar-cost averaging:
Means: 2833.938-2836.723
Stddevs: 879.314-883.154
Lump-sum investment:
Means: 2834.745-2837.861
Stddevs: 863.836-866.589

As you can see, the means fall very close to each other, but the standard deviations of the lump-sum approach are lower. Perhaps not hugely lower, but clearly there is a difference that is not random noise.

Response to Comments

I thought I'd respond to some of the comments in the main blog rather than limit all of the interesting discussion to the comments section.

Remco writes:

I think its purpose is exactly what its name implies, which is to purchase (or sell) stock at a short-term average price.... Dollar cost averaging tries to makes sure you get a "fair" price, i.e., that you don't buy stock at a local maximum or sell it at a local minimum. What you get is a reduction in risk (clearly), at the expense of reduced returns (because stock goes up on average and hence you end up paying more / getting less, on average).

Two responses. First, I agree and have already pointed out that dollar-cost averaging does reduce risk as compared to a lump-sum investment. But the reason it reduces risk is because you are holding a lot of cash during the transition period, not because that gradual transitioning process has some variance-reducing power over and above that. My assertion is that if it is suboptimal (because too risky) for you to hold the entire amount in stocks, then the rational thing to do is to simply reduce the amount you move into stocks. In general, to control variance, reduce the amount you assign to the riskiest asset class; but always make a lump-sum shift into your target allocation. If you want to argue for dollar-cost averaging, you should argue not just that it reduces variance, but that it reduces variance in a way superior to the alternatives (e.g., simply continuing to hold some cash).

Second, I would assert that the only rational way to measure risk or return to compute them with respect to your whole portfolio. It's true that dollar-cost averaging will ensure that you don't buy the stock at a local minimum or maximum. Over a sufficiently short time period, it will also reduce the variance in your return on that stock over that period. But so what? What you should really care about is your overall net worth, and the variance on that. Imagine that you keep your money in your left pocket and your right pocket. It would be irrational to make decisions based solely on the returns and volatility of your "right pocket" money. My earlier posts lay out the argument why dollar-cost averaging has no value when looking at your portfolio overall.

Dollar-cost averaging is clearly marketed to people as something that would be beneficial to them. It's possible that those marketing it only directly point out that they are going to get a "fairer" price for the stock. But it's implicit that this matters, that this is beneficial to their overall financial situation.

Dollar-Cost Averaging Revisited

I want to back up and try to restate my argument against dollar-cost averaging, and maybe make it a tad more explicit in the process. My original post didn't spell out a lot of the assumptions I was making.

The central point of my argument against dollar-cost averaging is that the (expected) risk-adjusted return of a portfolio is time-invariant. (Although I don't actually quite believe that - see below.) Implicitly, I am using "risk-adjusted return" as an overall goodness measure of a portfolio. In other words, your only objective as an investor is to select a portfolio that maximizes the risk-adjusted return. Once we accept that your risk-adjusted return is time-invariant, then it follows immediately that dollar-cost averaging is irrational, because dollar-cost averaging is a specific way of varying your portfolio over time which is alleged to be beneficial.

So what exactly is "risk-adjusted return"? Well, different people might define it in different ways. Obviously the expected return of the portfolio should play a role. (For example, a portfolio with an expected annual return of 10% would be preferable, all other things being equal, to a portfolio with an expected annual return of 5%.) Additionally, most people would agree that some measure of risk or volatility should contribute. Given a choice between a portfolio that always yields exactly 10% annually, and one that has an expected return of 10% but may be much higher or lower, it is natural to assume that a rational investor would choose the former. How do you measure risk or volatility? Typically people use the standard deviation of the return, but you could probably use a different measure without affecting the present argument. How do we combine the expected return of the portfolio with the risk measure (e.g., standard deviation)? Again, different people will do it differently and it ultimately may not matter for our purposes. But, for concreteness, we could use the Sharpe ratio.

Why then should we believe that risk-adjusted return is time-invariant? Well, there are good theoretical reasons to believe that expected returns are time-invariant, stemming from the idea that markets are reasonably efficient. Suppose expected returns were not time-invariant; in other words, we have reason to believe that the returns of some asset class will be higher or lower at some point in the future than the expected return over the long run. Well, we would presumably take advantage of that fact. For example, if we thought stocks were about to perform poorly in the near future we would sell them today; if we thought they were about to shoot up, we would buy them. Furthermore, the market is full of smart people looking for any edge; it would be unrealistic to expect ourselves to be the only people with this information. Enough people trying to exploit this "mispricing" will force prices down to their "true" value. So, contrary to assumption, in an efficient market we do not expect there to be mispricings of this sort. Or, at least, we do not expect them to be large and we do not expect them to persist long enough for an amateur investor to take advantage of.

Unfortunately, I don't think there are similar arguments that expected volatility must be time-invariant. As I write this in October of 2008, we are in the middle of one of the biggest market crashes of the century. I do not know whether the market will move up or down over the next few weeks, but I do have good reason to believe that volatility will be unusually high. Knowledge about future volatility cannot be arbitraged away (it seems to me), so there is no reason for it not to exist.

Still, this doesn't really constitute an argument for dollar-cost averaging. I can see some grounds for making some adjustments (probably minor) to your portfolio over time depending on your expectations of future volatility. But, as I mentioned in my original post, the kind of gradual portfolio shift entailed by dollar-cost averaging, would only be sensible in the highly unusual circumstance where you have information that volatility will be high in the near future, but will gradually decrease over time.