Over at Marginal Revolution, Tyler Cowen puzzles over Ian Ayres‘s take on investment strategy:
In our risk-reducing implementation, we want people to borrow to invest more when young and then invest less when older. The lifetime exposure to stocks is held constant. Compare the following two investment paths:
Option 1:
- Year 1 Invest $1
- Year 2 Invest $2
- Year 3 Invest $3
Option 2:
- Year 1 Invest $2
- Year 2 Invest $2
- Year 3 Invest $2
Our view is that option 2 is the safer bet.
(Note that when Ayres says “invest $2” he does not mean “Add $2 to your investment”. He means “Have a total of $2 invested.” So under Option 1 you add a dollar a year to your investment. Under Option 2 you do all your investment up front and then scoop out all the profits every year (or scoop replacement funds in if you’ve taken losses). Option 1, in other words, is the widely touted but thoroughly ridiculous strategy often called Dollar Cost Averaging.
I’m surprised that Tyler is unsure about this. Here’s the clear intuition: Under Option 1 (the dollar cost averaging strategy) you’ve got $1 riding on the first year’s market performance, $2 riding on the second year’s, and $3 riding on the third year’s. Under Option 2, you’ve got equal amounts riding on each year’s performance. Therefore you’re more diversified (and hence safer) under Option 2. (This ignores a world of practical difficulties, like getting your hands on enough borrowed funds to make your entire lifetime investment up front. In the real world, you’ll never manage to implement Option 2—but you should still strive to get as close to it as possible.)
I first realized this about fifteen years ago when I was writing a chapter on investment strategy for The Armchair Economist. But of course I didn’t want to go to press without confirming my intuition, so I did some calculations to make sure I was right. Through the miracle of modern technology, I still have every note I’ve ever made to myself since I got my first computer, and through a greater miracle I was actually able to find this one. It’s a little technical, and a little terse because it was never meant to be read by anyone but me, but I think it answers Tyler’s question unambiguously. Here it is.
I think you’ve made a typo:
“In the real world, you’ll never manage to implement Option A—but you should still strive to get as close to it as possible.”
Should it not read Option B here? Option A is the dollar cost averaging option.
SL, I agree with what you said here. I’m also very disappointed in the comments over at MR on this topic. Most of them seem a) thoroughly confused about dollar cost averaging, b) thoroughly confused about the fallacy of time diversification, or just c) vague and aimless. I think you’ve managed to attract the smarter group of commenters to your blog than MR has! That’s actually quite a feat.
SL: I just quickly looked over your calculations, but did not spend as long as I perhaps should have. Is there anything restricting x and y from being less than $1? Or more than $0? In other words, is the exercise of starting with a dollar a binding constraint or does that not really matter?
tbedifferentphil: I think there’s no reason x and y can’t be negative.
Al: Good catch! I’m fixing this!
Seems silly.
First of all I don’t know why you assume A has anything to do with some model no average joe has ever heard of. It’s simply the result of accumulating savings over time.
More importantly though, in A, if the market tanks in year 3 (which we will use as a proxy for “near retirement”) it’s a major bummer but not necessarily catastrophic. In B, if the market tanks in year 1 (proxy for “shortly after entering the workforce”), and you’ve borrowed up to the eyeballs to invest, you are completely screwed. You will be forced into bankruptcy and your credit will be ruined for precisely the years of life you needed it most.
SL,
I have occasionally thought about DCA as well, but rather than coming at it from the perspective of “this is definitely wrong, I must find the simplest possible model that proves it so,” I believe you should take the opposite approach. That is, say “so many people believe this, there must be something to it, I must find the simplest possible model that makes it rational and see whether I believe it.” Towards that effort, I believe one possible approach is to assume that market prices are mean-reverting (i.e. not unit root) over relatively short periods (over very long periods, I would imagine the diversification benefit would win), but you cannot know whether prices are currently high or low.
Here is a simple model: Suppose you have $100 you can invest over the next 10 days. You can invest all $100 on day 1 (lump sum), or $10/day over each day (DCA). Also suppose that over this relatively short period stocks have 0 expected return and the only risk is from a pure mean-reverting process, prices are either $1 or $2, alternating each day. Under the LS strategy, you will end up with 100 shares with p=.5 and 50 shares with p=.5. Under DCA, you will end up with 75 shares with p=1.
Note: this point only applies to a choice over how to invest over short periods of time when you have all the money available up front, which seems to be the point addressed by your pdf notes. The choice over how to invest over your lifetime is driven primarily by costs of leverage, and in this case I agree with you that the LS strategy reduces risk in the abstract, but I agree with Noah that there are real-world risks you are ignoring that make that result irrelevant.
Tal F: If prices are either $1 or $2, alternating each day, you should make all your purchases on the days when the price is $1. I do not think this kind of assumption is going to come even close to justifying dollar cost averaging.
Prof. Landsburg writes: “(Note that when Ayres says “invest $2″ he does not mean “Add $2 to your investment”. He means “Have a total of $2 invested.” So under Option 1 you add a dollar a year to your investment. Under Option 2 you do all your investment up front and then scoop out all the profits every year (or scoop replacement funds in if you’ve taken losses). Option 1, in other words, is the widely touted but thoroughly ridiculous strategy often called Dollar Cost Averaging.”
Unless “invest $X” means something different with respect to Option 1 than with respect to Option 2, that doesn’t sound quite right. I (and other comments) understood DCA to mean adding the same dollar amount to an investment in every period. However, that is not what Option 1 means if “invest $X” is understood to mean have $X invested. For example, if an investment of $1 at the beginning of period 1 had grown to $1.50, under Option 1, one would only add $0.50 at the beginning of period 2, while under DCA, one would add $1, regardless of performance. That may make Option 1 and even worse strategy than DCA (I have not thought carefully enough to make sure), but it certainly makes it different.
Aeternitas: You are right; Option 1 is not exactly dollar cost averaging, though it is close to it in spirit, and a bad strategy for exactly the same reason that DCA is.
Steve,
I was afraid you would dismiss the argument by making that point. The example is clearly very stylized. Perhaps you can make it more realistic by adding a unit root process, so that you cannot tell for sure if the underlying price is high or low. The optimal strategy would not be LS or DCA, but I speculate it would be closer to DCA, so DCA could be thought of as a convenient rule of thumb. Or maybe I’m wrong and the intuition does not survive the addition of these complications, but that is not the point of my comment. My point was really that you should strive to find a model which will support DCA then see if you believe its assumptions are realistic. For my model, I think the question is whether the short-term mean-reversion is significant enough to justify DCA, and at least in backtests, I have yet to find a situation where it is.
Tal F: I thought about this back in 1994 and was not able to construct a model that justified anything like DCA. Of course there might be something I haven’t thought of.
I find it hard to believe that so many people can believe so strongly in DCA yet there is no simple model that will justify it. After all, DCA is an idea that has somehow survived the test of time. Is it not our job as economists to figure out why? I would agree that the logic of lump sum investing is very compelling, but I just can’t dismiss DCA until I at least understand where it comes from. Maybe there is something irrationally seductive about DCA that makes people believe it, but I just don’t see it.
A separate but related point: option 2 is not really feasible unless one sets aside a portion of their total wealth at the outset to cover potential losses. In that case, one may have no choice but to increase their investments over time (although perhaps not to the degree commonly done today), as the risk of having to add funds to make up for losses decreases.
Tal F:
Maybe there is something irrationally seductive about DCA that makes people believe it
This, I think, is dead on right.
Steve: Sorry, I can’t quite assimilate the math in the analysis. Maybe this will help a bit, if you could answer: since DCA seems to be based on the expectation that the market in general will rise over time, is there anything built into your model that gives some upward bias to the variation (or ‘risk’) that occurs in the market?
DCA is implemented, essentially without any choice, in my company’s 401k plan, since every two weeks a purchase is made, adding to the investment. I could allocate it all to a money market fund, but that seems sub-par. There are rules against a lot of trading that would preclude the Option 2 strategy, I think.
In any case DCA worked quite well in the period of the past 2-3 years, since a lot of shares were purchased around March 2009, when prices were very low, and subsequently the market recovered a good bit. In my personal case, at least the total now exceeds the original investment, even though starting this investment strategy just about a year before the worst market period in decades. That’s anecdotal, I know, but I wonder if a similar experience has persuaded at least some people to think DCA is a good idea. Again, this was dependent on the market going up, over a long term.
ToddM: is there anything built into your model that gives some upward bias to the variation (or ‘risk’) that occurs in the market?
Yes, this is captured in the parametr mu (Greek letter mu), which can measures the upward drift, and can take any value without affecting the analysis.
If you believe the stock market is a better place to invest than a money fund then shouldn’t the typical person take his 401K money and invest as much as he can in the stock market?
Of course, it is important to be able to sleep at night but it seems how much risk you can handle should be the determining factor.
Therefore, in the real world, shouldn’t something like DCA be the best way to invest? Otherwise you are stuck with people chickening out at the bottom and selling.
The eggs pictured in the basket are from Araucana hens, who lay different colored eggs as pictured. They’re often called Easter hens. Thus, putting all your eggs in one basket still yields diversity.
I think that people decide they can afford to risk an amount each month. If you then retrospectively apply a DCA type reasoning to make it seem that it is a good way to invest anyway, then it makes you feel better.
The alternative for most people would be to borrow lots on money up front to invest. You could work out the amount you could risk each month, take out a fixed rate loan with repayments of this amount each month, and invest the lump sum at the beginning. This will obviously be a smaller amount than the total investment you would otherwise make by adding up all the monthly contributions. However, you would not do it at all if you thought the stock market would not perform better than an interest account, or at least be prepared to risk the monthly sum on that gamble. As far as I know, this is almost unknown, so why is this? Is it that people cannot get the loan for this purpose? I have never heard of anyone trying. The real cost is the difference between what I could get in interest and what I have to pay in interest, so the stock market needs to perform better than this amount for it to pay off.
As a practical matter what does Professor Landsburg comment “…but you should still strive to get as close to it as possible.” mean? Surely it’s not use whatever one’s borrowing capacity may be to fund an initial portfolio that risks the Noah Yetter outcome. Does this leave us with the “child’s birthday gift” solution—any unanticipated cash gets immediately invested to move us closer to our Option 2 portfolio with the anticipation that, say, after ten or twenty years of employment and, hopefully, a series of unanticipated favorable cash flows, e.g., promotions, job changes or bonuses, we will reach or near reach that nirvana of “as close to it [Option 2] as possible”?
Steve
Re: “If prices are either $1 or $2, alternating each day, you should make all your purchases on the days when the price is $1″.
Note that DCA is attractive to those with who value time more than mere money (consumer surplus of leisure!). If you add transaction costs to your model, the results will therefore change.
The cost of watching stocks and buying them only when they reach $1 (vs. simply ’set and forget’ DCA strategy) will add **considerable** costs to each stock I buy. More importantly, if the negative externality or stress of making such irritating decisions is added, then it is even more unattractive to watch the stock ticker.
Those with high (real or imagined) opportunity costs of time (my personal opportunity cost is close to infinity – I discount money almost entirely: anything beyond my minimum needs doesn’t add much value to me, but additional time adds enormous value) will avoid watching the stock market and picking stocks.
DCA strategies are therefore perfectly designed for people like me. In any event, stock picking is a dangerous game, and most such people under-perform ordinary index funds. I therefore regularly borrow to invest at automatic, regular intervals in an index fund. In addition, when I do accidentally find stocks at particularly low values, I shift money from low interest savings into stocks: a one shot investment. The combination of both (DCA and random picking of stocks when they fall to exceptionally low values) is time-convenient and, in my view, effective.
In brief, transaction costs (including perceived value of time) will show why the vast majority of common investors prefer ’set and forget’ strategies to active management. People are rational. Your model perhaps doesn’t adequately capture their rationality.
Sanjeev: Your story about transactions costs works only in the presence of the underlying assumption of mean reversion (i.e. “prices are either $1 or $2, alternating each day”, or something like it). If you believe the evidence that stock prices are random walks (or more generally, martingales), then your strategy still makes no sense.