About Dollar Cost Averaging ... part I

I'm not a fan of Dollar Cost Averaging (DCA). Putting money into investments at the end of each month (without thought, synchronized with the phases of the moon) seems an admission of cerebral frailty. So I decided to prove that waiting for a correction would have been a better strategy (in the past, of course, since I can't predict what'll happen tomorrow).
PROLOGUE
Having found (with Bylo's help) a website which has monthly returns from Jan, 1975 to June, 1997, for various world markets, I set out to prove my thesis:
SAVE YOUR $100/month in Money Market (MM) UNTIL A CORRECTION,
then INVEST IT ALL AT ONCE.
(Of course, there are those who will argue that DCA is to reduce your risk, not to improve your return ... so I address this to those who want to maximize returns.)

I chose a very volatile market (annualized return = 14% over this period), namely Pacific Large Cap, with a Standard Deviation (SD) of 6.3% over this time period (compared to U.S. Large Caps with SD=4.1%).

I considered nine strategies:
Keep putting your $100 each month into MM (at 5%/annum) until there are FOUR successive drops in the (Pacific) market, then invest it all at once. If the market drops again the following month, invest your $100 again. Continue until the market goes up, then switch to putting your $100 into MM again ... waiting for the next drop for FOUR consecutive months. (Below, I call this strategy DOWN-4.)

I repeated this strategy, waiting for corrections over THREE, TWO, and ONE consecutive month(s), as well as waiting until the market has gone UP for ONE, TWO, THREE or FOUR months.
(These last strategies are for those who mumble, "Aah, now we're on a roll ... time to jump in.")

In the following table, the value of your investment (in some FUND which mimics the Pacific Index) and the amount in MM and your TOTAL portfolio is given as of June, 1997 (assuming you began these strategies in Jan, 1975):

PACIFIC LARGE CAP
STRATEGY FUND MM TOTAL
DOWN-4 $93,337 $7,037 $100,374
DOWN-3 $113,458 $7,037 $120,495
DOWN-2 $117,277 $504 $117,781
DOWN-1 $122,609 $301 $122,910
DCA $126,229 $0 $126,229
UP+1 $124,847 $0 $124,847
UP+2 $123,241 $0 $123,241
UP+3 $122,338 $0 $122,338
UP+4 $115,198 $5,411 $120,609

EPILOGUE
"SweetiePie," I groan, "Look at this! Must be something wrong here! Lemme check the calculations ... no ... they're okay ... so lemme try U.S Large Cap ... same thing!! ... Cdn Small Cap? ... Aaargh! ... DCA wins again!!!"
"That's okay HoneyBun, you're right sometimes," she sez. "You married me, didn't you?"

>So ... why does DCA work?
When the price goes down you buy more units with your monthly investment. Take a peek:

>So, you buy more units with your $100, at the lower prices?
Yes. This example is due to James Stowers, author of "Yes, You Can Achieve Financial Independence".

Of course, linear changes in returns are unusual, so consider this:

>So, will the green portfolio ever catch up to the blue?
No.
>Will it catch up if you change the percentage, from 1% to, say ...?
No.
>Can you prove that?
Yes. Want to see the proof?
>No. But if linear changes are unusual then so are smoooth percentage changes, eh?
Uh ... yes, so here's a more realistic scenario.

>So is DCA always best?
Of course, not. It depends upon the sequence of stock prices and the time interval and ...
>... and the weather in Bermuda?
Sure.
>So you're now a fan of Dollar Cost Averaging?
Nope. It requires a certain sociological profile, a certain genetic configuration, a certain aversion to volatility, a certain comfort in ritual, a certain ...
>None of which you have.
Right. Besides, DCA has a Dark Side.

You can peruse the Fund Library thread that started all this.

Also, you may want to consider Value Averaging.

>Wait! I assume that if DCAing into your investments is good, then ...
DCAing out is bad? Yes. If you think the market is going to go South, then you might consider taking ALL your money out rather than a piece at a time. Withdrawing, say, $100 a month from a $10K portfolio will guarantee that you sell more shares when the price is low and fewer shares when the price is high. Further ...

>A picture is worth a thousand ...
Here's some pictures. You're investing in the S&P 500. You expect a downturn. You withdraw $100 each month, putting it into Money Market at 3%. Compare your total monies (stock + money market) if you do this DCA-out thing ... or just hold onto the stock. Of course, if you sell everything and put the cash under your pillow - you keep the $10K intact:

>You're comparing holding onto your stock or DCAing out at $100 per month, eh?
Right.
>I'd say you're better off with the money under the pillow.
Notice that when the S&P goes down, you sell more of the low priced stock. That's like ...

>Buy high. Sell low.
Very clever.
>Thanks.
However, remember we're talking about getting out during a market downturn. You're better selling all-at-once. On the other hand, if you needed a bunch of money, in a year or so - and you're not expecting a downturn - you may be better off cashing in stock a bit at a time.

In general, we'd want to ... uh ... Buy Low, Sell High.

That means that if we're buying stock, then we're accumulating units. The number of units purchased should be higher when the price, P, is lower. That means we'd buy a fixed number of dollars worth, say $A, so the number of units purchased is A/P and is inversely proportional to the stock price. When the stock price is low, we'd buy more units. That's Buy Low.

>Wrong colour.
That's Buy Low.

On the other hand, if we were selling stock, we're probably accumulating dollars, to pay for that trip to Hawaii. Then we'd might want to consider selling fewer dollars worth when the price is low and selling more dollars worth when the price is high. We could do that by selling a fixed number of units, say N units.

>Instead of a fixed number of dollars.
Exactly. If we sold N units at the price P, the dollar amount is then NP, so when the price is high we'd be selling more dollars worth. That's Sell High.

>Ain't there some formula, a mathematical gesticulation, a ...
Sure. You may (or may not) want to peek at Part II.
 for Part II: DCA for masochists

It has some math, and ...

>I'd rather not.

.... you may want to look at the thread.

Of course, we might examine the consequences of withdrawing with either a fixed DOLLAR amount or a fixed UNIT amount, assuming you start with N units of some stock (or mutual fund) and withdraw for m months ... like so:
Stock PricePortfolioUnits WithdrawnRemaining Units
P1N P1n1N - n1
P2(N - n1)P2n2N - n1 - n2
P3(N - n1 - n2)P3n3N - n1 - n2 - n3
etc. etc. etc. etc.
Pm(N - n1 - n2 - ... - nm-1)PmnmN - n1 - n2 - ... - nm
Our objective should be to end up with the maximum number of units, right?

>Uh ... if you say so.
Okay, then we want to maximize N - n1 - n2 - ... - nm
If we withdraw a FIXED number of UNITs, say U, then
n1 = n2 = ... = nm = U
leaving us with N - m U units.

On the other hand, if we withdraw a FIXED number of DOLLARs, say $D, then
n1 = D/P1, n2 = D/P2, .... nm = D/Pm.
leaving us with N - D (1/P1 + 1/P2 + ... + 1/Pm) units.

In order that our FIXED UNIT portfolio exceed our FIXED DOLLAR portfolio (after m withdrawals), we'd want to withdraw fewer total units. That is:

m U < D (1/P1 + 1/P2 + ... + 1/Pm)
or, to put it differently:

(1)             D / U > 1/{ (1/m) (1/P1 + 1/P2 + ... + 1/Pm) }
and we recognize (on the right-side) the reciprocal of the average reciprocal of the m stock prices.

>What!
You take the reciprocal of all the stock prices, then you take the average of all these reciprocals (call the average B) then you take the reciprocal of this average ... it's 1/B, see?

>No!
Well, here are some "sample" stock prices, over 10 months (or days or years) and the reciprocal of the average reciprocal that we're talking about:

>So?
Well ... since we don't know the average reciprocal - these are future prices, remember - then we'd probably say:
"I need $10,000 and the current stock price is $10 so I'll withdraw 100 units each month for 10 months"
That'd make U = 100 units, right?

On the other hand, we might say:
"I need $10,000 so I'll withdraw $1000 each month for 10 months"
That'd make D = $1000, right?

That makes D / U = $10.

>How do you know that 10 withdrawals, 100 units each time, will remove the same amount, $10K, from my portfolio?
Good question. If we withdraw D each month, for m months, we've withdrawn m D dollars.
To withdraw the same amount, by withdrawing U units each month, we need U (P1 + P2 + ... + Pm) = m D
and that means:

D / U = (1/m) (P1 + P2 + ... + Pm) = Average Stock Price
In order to satisfy Equation (1) (so our portfolio with FIXED UNIT withdrawals exceeds our portfolio with FIXED DOLLAR withdrawals), we'd need:
(1/m) (P1 + P2 + ... + Pm) = Average Stock Price > 1/{ (1/m) (1/P1 + 1/P2 + ... + 1/Pm) }
If course, we're talking about future prices ... so the question is: "Will they satisfy this inequality?"

>That's the question, but what's the answer?
I have no idea.

>A picture is worth a thousand ...

Okay, here's what we'll do:

  • We have a $100K portfolio and we want to withdraw $20K over about 10 months.
  • The current stock price is $10 ... and varies randomly!
  • We can either:
    1. Withdraw a fixed dollar amount of $2K per month for ten months, or
    2. Withdraw 200 units each month for ten or eleven months ... with a lesser amount for the last month so the total amount withdrawn is exactly $20K
  • Here are typical pictures, comparing Constant Unit withdrawals and Constant Dollar withdrawals:



>I don't see much difference ... so, what's best?
It depends upon the future ... wait'll I check.