Investment Rituals a continuation of DCA: average price

We're going to invest every month (day? year? decade?) and we require that our portfolio increase according to some predetermined scheme.

>Huh?
I mean, if we start by investing, say, $A at the start of the first month, then, at the end of month 1 we want our portfolio to be
$f(1)A (so we buy or sell units to make this so) and, after 2 months it should be $f(2)A and after 3 months ...

>It should be $f(3)A. Yeah, I get it. So?
So, this prescription will generalize both Dollar Cost Averaging and Value Averaging.
To see this we consider the following (where the stock prices at the end of each month are P0, P1, P2 ... etc.):

  1. At the end of month n (after our stock purchase), our portfolio is worth f(n)A.
  2. At the end of month n the stock price is Pn (which is the price at the start of month n+1).
  3. We then have f(n)A/Pn units = (portfolio value)/(stock price).
  4. At the end of the next month, month n+1 (before our stock purchase), the stock price is Pn+1.
  5. Our f(n)A/Pn units are then worth [f(n)A/Pn] Pn+1 = (number of units) * (stock price).
  6. Since we're insisting that our portfolio be worth f(n+1)A, we buy or sell stock so as to make this so.
  7. We buy f(n+1)A - [f(n)A/Pn] Pn+1 dollars worth of stock (and, since this may be negative, we might have to sell stock).
  8. Our investment at the end of month n+1 (after our purchase) is then: f(n+1)A - f(n)A Gn+1
  9. Gn+1 = Pn+1 / Pn is the Gain Factor for month n.

>That's confusing. I mean ...
Okay, let's do it again:

  1. At the end of month 1 (after our stock purchase), our portfolio is worth f(1)A.
  2. At the end of month 1 the stock price is P1 (which is the price at the start of month 2).
  3. We then have f(1)A/P1 units = (portfolio value)/(stock price).
  4. At the end of the next month, month 2 (before our stock purchase), the stock price is P2.
  5. Our f(1)A/P1 units are then worth [f(1)A/P1] P2 = (number of units) * (stock price).
  6. Since we're insisting that our portfolio be worth f(2)A, we buy or sell stock so as to make this so.
  7. We buy f(2)A - [f(1)A/P1] P2 dollars worth of stock (and, since this may be negative, we might have to sell stock).
  8. Our investment at the end of month 2 (after our purchase) is then: f(2)A - f(1)A G2
  9. G2 = P2 / P1 is the Gain Factor for month 2.

>I get it. Please continue.
Remember our monthly investments?
After N months, how long have these amounts been invested?
I0 = f(0)A     ... invested for N months
I1 = f(1)A - f(0)A G1     ... invested for N-1 months
I2 = f(2)A - f(1)A G2     ... invested for N-2 months
I3 = f(3)A - f(2)A G3     ... invested for N-3 months
etc. etc.
IN = f(N)A - f(N-1)A GN     ... invested for 0 months

>Invested for 0 months?
Yes, that's our last investment, at the end of month N ... and our portfolio is now worth f(N)A.

>Dreamer.
Our compound return, R (expressed as a fraction, so R = 0.123 means 12.3%) is then calculated by solving the equation:

I0(1+R)N + I1(1+R)N-1 + I2(1+R)N-2 + ... + IN(1+R)N-N = f(N)A
or, if we put x = 1+R and cancel the factor A on each side of the equation, we get the Magic Equation:

We invest monthly to achieve a portfolio worth $f(n) in month n. Then:

Form I:       f(0)xN + [f(1) - f(0)G1]xN-1 + [f(2) - f(1)G2]xN-2 + ... + [f(N) - f(N-1)GN] = f(N)

which can also be written:

Form II:       f(0)xN-1 [x - G1] +f(1)xN-2 [x - G2] +f(2)xN-3 [x - G3] + ... +f(N-1) [x - GN] = 0

where R is the compound monthly Return achieved
and x = 1 + R
and Gn = Pn / Pn-1 is the stock Gain Factor for month n.

where, if we suppose the investments were ANNUAL instead of MONTHLY, then we'd get the ANNUALIZED return.

>The $A disappears? It doesn't matter if it's $100 or $1000? Are you saying ...?
I'm saying that the annualized return is the same whether the investments are in dollars, pounds, yen or gummybuks.

>gummybuks?
Yes. If you say A = $1000 then I say, Hey! That's 1 gummybuk!
So I replace your "A" dollars by my "1" gummybuk ... and that effectively makes "A" disappear from the equation. After all, if our currency were in yen we'd expect the same annualized gain as someone executing the same ritual in lira or francs.

In fact, I think it's clear that the annualized return, after N years, will be a function of the sequence of stock prices, expressed in dollars:
      R = F(P0,P1,P2, ... PN)
where F is some function. Now, if there are 100 yen to the dollar, then, in terms of yen, the return must be the same.
That is:
      R = F(100P0,100P1,100P2, ... 100PN) = F(P0,P1,P2, ... PN)

In general we must expect that the formula for annualized return satisfies:
      R = F(P0,P1,P2, ... PN) = F(c P0,c P1,c P2, ... c PN)     for any positive constant c.
In particular, if we set c = 1/P0 we get a formula:
      R = F(1,P1/P0,P2/P0, ... PN/P0)
hence involves the various Gain Factors (which, of course, are independent of the currency used).

>So what do we do now?
We solve that equation.
>Dreamer.


If all the investments were equal, then f(0) = 1, f(1)-f(0)G1 = 1 and f(2)-f(1)G2 = 1 and f(3)-f(2)G3 = 1 etc.
and the Magic Equation becomes: xN + xN-1 + xN-2 + ... + 1 = f(N)
where the portfolio values are generated sequentially according to the scheme:

(1)         f(0) = 1, f(1) = 1 + f(0)G1, f(2) = 1 + f(1)G2, etc. etc.

>All investments are equal to 1 gummybuk, right? Isn't that DCA?
Yes, and after you've discovered the formula for f(N), by using (1), you'll recognize the Magic Equation as the same one we generated here.

>The formula for f(N)? I can't remember ...
It's f(N) = PN [ 1/P1 + 1/P2 + ... + 1/PN]
>The prices are expressed in ... what? Dollars?
It doesn't matter since f(N) involves the ratio of prices ... the Gain Factors.
>So I could use gummybits?
Be my guest ... but it's gummybuks.

Further, if we set f(n) = n (so our portfolio increases by 1 gummybuk each month meaning that, after n months, it has the
value n) then we'd get the Value Averaging equation.

>What else could we ...?
We might ask that our portfolio increase by a constant FACTOR (instead of a constant dollar amount).
That means that f(n) = (1 + r)n, for example, where r is some constant like 0.01 ... so we want a 1% increase every month.

>What if f(n) is ... uh, $1,000,000?
Meaning that you start with f(0) = $1M and want to keep it there? Sure.
As it happens, $1,000,000 is 1 Gummybuk, so The Magic Equation would be:
      xN + [1 - G1]xN-1 + [1 - G2]xN-2 + ... + [1 - GN] = 1

and if we set Gn = 1 + gn where gn is a monthly gain (as opposed to a monthly gain FACTOR), the equation becomes:
      xN - g1xN-1 - g2xN-2 - ... - gN = 1

If f(n) = n Pn it means we're buying 1 unit each month (or year) at the current price so by month n (or year n) we've got n units and they're each worth Pn so ...
>What if f(n) = n2 or maybe f(n) = log(n) or maybe ...
Go back to sleep.
>Do you realize that you don't have a single picture?
Go back to sleep.
>zzzZZZ
So can we find a best ritual?
>zzzZZZ
Here's a "typical" picture with f(n) = 10 (1+r)n with r = 0.08 (or 8% annual portfolio increases) showing your portfolio, the number of units you hold U(n), the annual investments I(n), the prices P(n), your out-of-pocket cost ... as the years go by:

Figure 1

With f(n) = 10 (1.08)n, it means that, at n = 0, you invest f(0) = $10 (buying 1 unit @ $10 per unit).
Thereafter, you insist that your portfolio increase at 8% per year. This means that sometimes you buy units ... and sometimes you sell units. After 20 years, your annualized return (for this particular example) is 7.2%

>What! 7.2%? Isn't it supposed to be 8%!
It depends. Sometimes you get your 8%, sometimes you don't. After all, if the unit prices are continually decreasing there's no ritual which will give you an 8% return.

For example, suppose the annual gain in the unit price is constant. For example, 10% so G1=G2=...=GN=1.10.
Further, suppose we have some initial portfolio and we simply want to maintain that original value, say f(n) = 1 Gummybuk.
Then the Magic Equation, namely:
      xN + [1 - G1]xN-1 + [1 - G2]xN-2 + ... + [1 - GN] = 1
becomes (in Form II)
      [x - G]{ xN-1 + xN-2 + xN-3 + ... + 1} = 0 with G = constant.
The solution is x = G, so your annualized return will be that of the stock in which you invest ... even though you are maintaining a constant portfolio value. It doesn't increase at all!

>Picture?
Okay, here's a picture where we start with a $10 portfolio and we maintain that value even though the unit stock prices increase 10% each year (so we have to withdraw units each year):


Figure 2

Here's another, quite palatable example, where the stock gains are random (selected from a Normal distribution with
Mean = 10% and SD = 25%) and you insist that your portfolio increase 8% per year. Your out-of-pocket cost ends up being negative (so you have all your money back again ... and MORE) and your cost per unit is then negative


Figure 3

>zzzZZZ
Don't you see? The annualized return is 21.7%. Good, eh?
>zzzZZZ