Invest all at once or gradually: Risk, random walks and "dollar cost averaging"

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.

Buy stock all at once or over time?

Questions involving finance and financial markets can be tough.  Often one's intuition is misleading, and all kinds of priors and biases can get tangled up in decision-making.

With defined contribution pension plans, college tuition plans, annuities, health savings accounts, etc. it is all the more important for everyone to be able to make good financial decisions.

Here is a very practical issue, with a question that I am pondering.  I have intuition on it, but some lingering concerns as well.  I will put it out there for others to think about.

So suppose you all of a sudden come upon $1 million.  You would like to invest this in the stock market, using just one low cost index fund.  Your time horizon is long, say 15 years at least.

Should you put all the $1 million in at once, or should you do something like spread it out over 12 months, putting an equal amount in each month?

Climate Hypocrisy

It was -14 F this morning in Hanover, -26 C.

Why is it me, rather skeptical of much of climate policy, who has to go around Tuck getting storm windows to be shut for winter?  The heat is just pouring out of the windows while the steam plant burns #2 fuel oil to compensate.

Or why is it me who in the little burb of Hanover immediately goes to the block-away parking lot where a spot is 99% probability rather than drive around and around waiting for someone to leave?  A town of liberal climate-change believers and you would not believe how everyone wastes fuel looking for the closest spot.

Ah, the hypocrisy in this world will kill you if you let it.

Je Suis Charlie!

These words express my opinion quite well.

What a travesty.  Let's see what develops over the next few days...there are a lot of things to investigate here.

My thoughts are with all the French people tonight, especially the family members of those who died.

No Free Health Care in Germany!

One thing that really drives me nuts is when someone gets hurt or sick in some other country, they go to the doctor or hospital...and don't get a bill!  "It was so nice" they typically say..."all my care was free!"

Geez, that is what I want for the new year, free health care.

But now a colleague of mine reports that he had a health care problem in Germany...and he is getting medical bills sent to him!  Achtung! In German!  What efficiency...they are sending them to his US address!

I asked to see them, just to compare them to the bills that we get in this country ("your charges were $10,000 but your health insurer cut out all the nonsense and got them down to just $1,000, saving you $9,000.") Developing...

My colleague also reports that

By the way, the food in the hospital was awful.
Breakfast…one roll & tea OR coffee. Lunch..soup & a small yogurt + tea OR coffee. Dinner 2 pieces of cold cut, one pice of cheese, two slices of bread + tea OR coffee. Not only that, I shared a room with 2 other guys, this was OK, but no curtains between beds.

Germany by the way spends about 11% of its GDP on health care, much less than the US...and they don't have a single payer system.

Medicaid Reimbursement Rates Revisited

One of the lesser-known rules in the Affordable Care Act is that it requires states to reimburse providers at Medicare rates, not at prevailing state-level Medicaid rates.  This is just for "primary care" but that is a pretty large category of care.

I wrote about this back in March 2013 -- see here -- and made this prediction:

Ah, but here is the kicker and relation to the title of my post:  this requirement and in particular that the Federal government will pay for the higher rates only applies for two years!

The phrase "doc fix" refers to a law about ten years ago that was supposed to cut Medicare reimbursement rates to providers by a certain amount each year that the rate of increase in total Medicare expenses was too high.  Starting immediately, Congress overrode the mandated increase.  By now, there is around a 30% cumulative cut that is due, and each year Congress has to pass a law (the "doc fix") that keeps that cut from going into force.

Anyone besides me worry that we are going to get into a "Medicaid doc fix" situation? 

Look forward:  For two years, any Medicaid service that can be legally lumped into the "primary care" category is going to be paid at the relatively lucrative Medicare rates.  But in two years, states like NH are going to go back to the old rate schedule.  Really?

So the time has come for those higher rates to expire, and just like clockwork, here come the calls for extending them.  This NYT article titled "As Medicaid Rolls Swell, Cuts in Payments to Doctors Threaten Access to Care" is pretty good*; it definitely notes that the higher rates were meant to be temporary.  Given the results of the election this fall, it now appears that the higher rates will indeed expire (that should have happened yesterday!)  As the article notes, President Obama did propose an extension of the higher rates, but it was not accepted by Congress.

So my prediction was wrong, but I can live with that!  It should have been a conditional forecast:  IF the 2014 election does not bring a Republican-ruled Congress, then there will be a successful push to extend the Medicaid reimbursement rates.


*My caveat here is that I do think the article was slanted in favor of extending the higher rates.  Start with the title! And it focuses too much -- exclusively? -- on the negative aspects of ending the program.  Are there no positive aspects to leaving determination of Medicaid rates in the hands of the states?