For example, we may decide to invest 60% in asset class A, 30% in class B and the remaining 10% in C, in which case
x = 0.6, y = 0.3 and z = 0.1 and, of course, x+y+z=1.

Further, we label things like so:

1. At the end of year n the value of our portfolio is P_n.
2. The number of units in each class is A_n, B_n and C_n.
3. The price of each unit is a_n, b_n and c_n.
4. Hence, we have P_n = A_na_n+B_nb_n +C_nc_n.
5. Now we re-balance (and hold these units for all of year n+1). We assume there are no costs associated with this ritual and that the market remains calm and unchanged while we buy and sell (!), hence the value of our portfolio doesn't change, and we have:
   (1)    P_n = A_n+1a_n+B_n+1b_n +C_n+1c_n where A_n, B_n and C_n are the new number of units of each, after re-balancing to maintain the fractions x, y and z, hence
   (2)    A_n+1a_n = x P_n and B_n+1b_n = y P_n and C_n+1c_n = z P_n.
6. At the end of year n+1 (just before re-balancing), we have:
   (3)    P_n+1 = A_n+1a_n+1+B_n+1b_n+1 +C_n+1c_n+1 where a_n+1, b_n+1 and c_n+1 are the prices at the end of year n+1.

(4)    P_n+1-P_n = A_n+1(a_n+1-a_n) + B_n+1(b_n+1-b_n) + C_n+1(c_n+1-c_n)

If we call the changes in portfolio value and prices ΔP_n, etc., we get:

(5)    ΔP_n = A_n+1Δa_n + B_n+1Δb_n + C_n+1Δc_n

and, dividing each side by P_n gives:

(6)    ΔP_n/P_n = A_n+1Δa_n/P_n + B_n+1Δb_n/P_n + C_n+1Δc_n/P_n

However, our annual re-balancing results in correcting the fractions devoted to A, B and C. In particular, after re-balancing at the end of year n, we identify the amounts of money invested in each class as: Shares * Price = A_n+1a_n, etc. according to Equation (2), hence:

• A_n+1/P_n = x/a_n
• B_n+1/P_n = y/b_n
• C_n+1/P_n = z/c_n

(7)    ΔP_n/P_n = xΔa_n/a_n + yΔb_n/b_n + zΔc_n/c_n

Conclusion? The annual gain in our portfolio, each year, is a weighted sum of the individual gains in each asset class, the weights being identical to the fractions devoted to each class.

Now we add 1 to the left side of Equation (7) and x+y+z = 1 to the right side, getting:

(8)    G_n(P) = xg_n(A) + yg_n(B) + zg_n(C)

where 1+ΔP_n/P_n = G_n(P), the annual Gain Factor for the Portfolio, and
1+Δa_n/a_n = g_n(A), the annual Gain Factor for asset class A, and ... etc. etc.

>Gain Factor? What's that?
If $1.00 grows to $2.34 the Gain Factor is 2.34 so the Gain Factor is just the Factor by which the stock grows and if it's less than "1" it means the stock price has decreased but if it's greater than "1" then ...

>Okay, I get it.
Our conclusion? To get the Gain Factor over n years, we multiply together the n annual Gain Factors getting:

(9)    G₁(P)G₂(P)...G_n(P) = {xg₁(A)+yg₁(B)+zg₁(C)} {xg₂(A)+yg₂(B)+zg₂(C)} ...{xg_n(A)+yg_n(B)+zg_n(C)} = G₁G₂G₃...G_n

where we write G_n as a simpler label for G_n(P) = xg_n(A)+yg_n(B)+zg_n(C).

Note that if we did not rebalance, but started with a portfolio which had a fraction x devoted to asset A and a fraction y devoted to asset B and a fraction z devoted to asset C ... and we just ignored our portfolio (no rebalancing), then the gain after n years would be: (9a)    x {g1(A)g2(A)...gn(A)} + y {g1(B)g2(B)...gn(B)} + z {g1(C)g2(C)...gn(C)}

Now it would be interesting to know the best choice for ...

>Wait! Is it better to rebalance every year ... or just ignore your portfolio?
It depends. Here's some Pictures.

Anyway, it would be interesting to know the best choice for the fractions x, y and z (with annual rebalancing) if we are given the Gain Factors for each asset class. Of course, it means we're looking at historical values and we'll get what would have been the best choice and that doesn't mean, of course, that it'll be the best choice for the future evolution of the asset classes so ...

Are you still awake?
>zzzzzz ... huh? It seems to me that the best choice would be putting all your money into the stock with the greatest gain.

Uh ... yeah, I guess you're right, so we'll consider adding to our portfolio each year, some fixed amount, say $D, of which xD will be invested in asset A, yD in asset B and zD in asset C. If we start with P₀, our annual portfolios will look like this at the end of each year:

Year	Portfolio
0	P0
1	P1 = G1P0 + D
2	P2 = G2P1 + D = G1G2P0 + (G2+1)D
3	P3 = G3P2 + D = G1G2G3P0 + (G2G3+G3+1)D
...	...
n	Pn = GnPn-1 + D = G1G2...GnP0 + (G2G3...Gn+G3G4...Gn +...+Gn+1)D

Note that guy in front of D, namely (G₂G₃...G_n+G₃G₄...G_n +...+G_n+1):

• The first $D, invested at the end of year 1, grows by a factor G₂G₃...G_n through years 2 to n.
• The second $D, invested at the end of year 2, grows by a factor G₃G₄...G_n through years 3 to n.
• The third $D, invested at the end of year 3, grows by a factor G₄G₅...G_n through years 4 to n.
• ...
• The last $D, invested at the end of year n, grows by a factor 1 ('cause it don't hardly have no time to grow none).

From the last equation in the above table, we can write:

>It'd make things simpler if you just invest $D every year, so the initial portfolio is just a $D investment then D/P₀ = 1 and ...

Hey! If I put P₀ = D the problem simplifies to:

>Brilliant ...

This isn't a simple problem since f(x,y) is a polynomial of degree n, in x and y, but we can make a few simplifying assumptions:

• We'll let r_k(A) be the annual gain for asset A, for year k. That is, 100 r_k(A) would be the percentage gain for the k^th year.
• The annual gain Factor is then g_k(A) = 1 + r_k(A)   (for asset A and year k).
• The annual Gain Factor for our Portfolio, for year k, is then
  G_k(P) = x{1+r_k(A)} +y{1+r_k(B)} +z{1+r_k(C)} = 1 + x r_k(A) + y r_k(B) + z r_k(C)   (where x + y + z = 1)
  and this has the form 1 + R_k(P) where R_k = x r_k(A) + y r_k(B) + z r_k(C) is the annual Portfolio gain.
• A product such as G₃G₄G₅...G_n now has the form:   (1+R₃)(1+R₄)(1+R₅)...(1+R_n) and, if the annual gains R₃, R₃, etc. are reasonably small, this product is approximately 1 + R₃ + R₄ + R₅ + ... + R_n.

>What?!
Well, it's like simple interest, without compounding. It's like taking the gains out of your Portfolio each year and ...

>Okay, I get it.
Our "Problem" (now simplified!) is approximated (!) by the solution to the problem:

Note:
A thing which looks like r₁+2r₂+3r₃+...+nr_n is a weighted sum of the n annual gains, with the most recent gain, r_n, being weighted most heavily and the earliest gain, r₁, being weighted the least ... but to get a proper weighted sum we divide these things by K = 1+2+3+...+n = n(n+1)/2. (See Weighted Moving Averages, where we discuss this type of thing ... but for prices, not gains.)

Further, the thing we want to maximize has the form:

F(x,y) = Ax + By + Cz + D = maximum with x + y + z = 1, x>0, y>0, z>0

>zzzZZZ ...

Look, there's a picture a-coming ... with two asset classes: a Large Cap (the DOW) and Hi Tech (the Nasdaq).

Here's what we do:

1. At the end of each week we calculate the weighted moving average (WMA) of prices, for each class, using the 1, 2, 3, ... weights, going back, say, eight weeks like so: p₁+2p₂+3p₃+...+8p₈ where p₈ is this week's closing price and p₇ is last week's closing price, etc. (and, after summing, we divide by K = 8(9)/2 = 36).
2. In order change these prices into Gain Factors, we divide each WMA by the Index price in Jan, 1985 ... the day we started.
3. For the following week we invest only in that asset class which has the larger WMA.
4. At the end of next week, we repeat this ritual, selling all of one stock, if necessary, to buy the other.

>Pretty lousy strategy if you ask me. After ten years you only managed a 10.7% annual gain and after some 15 or 16 years a gain of 10.0% so you'd be better off just buying the DOW or the NAZ and ...
Aah, I knew that'd wake you up. But pay attention. Look at the WMA for each Index. What we'd really like is to invest in that Index whose rate of increase is largest. Forget the values of the red and green WMA graphs. Look carefully at their slopes. We want to always switch to the Index which has the larger slope, the one rising most rapidly, the one with the largest upward velocity, the ...

>Wait! How many times did you have to switch investments? Every week?
Good question. Here's the chart. When I'm invested in the Nasdaq it's red and when I'm in the DOW, it's green.

>It's still a lousy strategy because ...
Pay attention. We now modify the strategy so we keep track, at the end of each week, of the change in the WMA:
{this week's WMA - last week's WMA}, except, of course, we're really talking about Gain Factors, not Prices, because we divide each Price by the starting Price when we began our investments: Jan 2, 1985.

>Why Jan, 1985?
We've got to start somewhere. Think of it like the Consumer Price Index. It's set at 100 at some arbitrary time in the past and goes from there. Here, we start with the value 1.00, because we're dividing each price by the Jan 2, 1985 price, so the ratio is "1" on Jan 2, 1985.

>We do this for both asset classes, right?
Right.
We have, at the end of this week:
(1) ...     K WMA(now) = G₁+2G₂+...+NG_N   the WMA of N Gain Factors
assuming we're doing an N-week average.

We have, at the end of next week:
(2) ...     K WMA(next) = G₂+2G₃+...+(N-1)G_N+NG_N+1 the next WMA of N Gain Factors

Subtracting, we get an expression for our slope, namely:
(3) ...     K{WMA(next) - WMA(now)} = - G₁ - G₂ -G₃ - ... -G_N+NG_N+1 = NG_N+1 - Σ G_k

and we now divide by N and get an expression which is proportional to the Rate of Change of our WMA ... the slope of the WMA graph ... namely:

>Why call it gMA?
You see, this is a strategy for attempting to predict the future evolution of the two asset classes so we can invest in the one with the greatest potential - for the next week or three - and no technical strategy is perfect ... in fact, some are pretty lousy ... so we needn't concern ourselves with trivial details and besides it looks nice to see the ordinary garden variety average of the previous N Prices ... uh, Gain Factors ... and besides it says we should look at how much this week's Price exceeds the previous N-week average but of course we'll divide all the prices by the price on Jan 2, 1985 so they're really cumulative Gain Factors because we don't want to be comparing a 10,000 DOW with a 3,000 Nasdaq and ...

>You didn't answer my question, but please continue.
Okay, here's what we'll do:

1. For each asset class we make a table of weekly closing prices and divide each price by the price at the start of our time period, so the modified numbers represent cumulative Gain Factors.
2. At the end of each week we calculate the average of these end-of-week prices, for each class, going back, say, thirty weeks like so: (p₁+p₂+p₃+...+p₃₀)/30 where p₃₀ is last week's closing price and p₂₉ is the closing price for the week before that, etc.. and we then divide this average by the price when we started our table (to get Gain Factors).
3. We then compare, for each asset class, this week's Gain Factor with this 30-week average, a la gMA.
4. For the following week we invest only in that asset class which has the larger gMA.
5. At the end of next week, we repeat this ritual, selling all of one stock, if necessary, to buy the other.

>Haven't you already done that?
Pay attention. For our 30-week averaging, it looks like so:

>Well, that's a bit better, but why 30 weeks?
Wait! I'm not finished! We can compare the two gMAs and require that, say, the Nasdaq gMA be 10% higher than the DOW gMA in order to justify switching from DOW to Nasdaq ... or even 20% or 30% higher ... then we ...

>Example?
Okay, here we do the 30-week gMA but won't switch to Nasdaq until its gMA is 10% greater than the DOW's gMA:

>And you switch when ...
When the Nasdaq gMA crosses the DOW from below and continues upward, so it's 10% greater than the DOW gMA. If that happens, then I switch to the Nasdaq and stay with the Nasdaq as long as it retains its 10% advantage.

>Lots of switching, I presume.
Yeah, I guess ... every three months or so.

Aah, but notice that, in 1999, we went up on the Nasdaq roller coaster and, when the dot.com bubble burst, we came down on the DOW. >You actually did that? Uh ... no, but I wish I had. >Why didn't you practise what you preach? That was then. This is now ... and I'm much smarter now. >You sure? Just wait till the next time! >Two questions that you didn't answer. First ... I chose 30 weeks because it worked quite well. It's sorta like the famous 200-day moving average. And gMA? I guess the MA is for Moving Average and the g could stand for gain.

>... or gummy? Or gummy. >So what does that UP on NAZ, DOWN on DOW look like, up close? You mean from 1999? Here's the close up: The percentages at the top are the annualized gains just for this period: Jan, 1999 to April, 2001 ... 115 weeks. Oh, and for a change of pace I've plotted the Gain Factors, namely the Prices, each divided by the Jan 2, 1985 Price. For example, notice that the Nasdaq reached nearly twenty times the Jan/85 value before it came crashing down to end this period with a negative gain, at about eight times the Jan/85 value. >Why do you call this rebalancing? It's more like sector rotation, where you switch everything from one sector to ... Good point. I began because I couldn't understand why rebalancing was something one did in sync with the months of a calendar instead of according to the dictates of the market. However, now that we're here, let's call it Sector Rotation.

for Sector Rotation.