There's been plenty of discussion about how often to rebalance in order to maintain some fixed asset allocation. >Like 60% Stocks and 40% Bonds?
Okay. We start with a portfolio of $ - During the next m months, the monthly
**Stock**returns are s_{1}, s_{2}, ... s_{m} - ... and the monthly
**Bond**returns are b_{1}, b_{2}, ... b_{m}. - The
**Stock**component of our Portfolio, namely**x A**, will have grown by a factor given by our**Magic Formula**, becoming:**x A**[ 1 +**Mean**(stocks) - (1/2)**Variance**(stocks)]^{m}, or, approximately:**x A**[ 1 + m{**Mean**(stocks) - (1/2)**Variance**(stocks)}] using*******, above. - The
**Bond**component of our Portfolio, namely**y A**, will have grown by a factor given by our**Magic Formula**, becoming:**y A**[ 1 +**Mean**(bonds) - (1/2)**Variance**(bonds)]^{m}, or, approximately:**y A**[ 1 + m{**Mean**(bonds) - (1/2)**Variance**(bonds)}] using*******, above. - Adding, we get our Total Portfolio after
**m**months, namely:**A**{1 + m[**x****Mean**(stocks) +**y****Mean**(bonds) - (**x**/2)**Variance**(stocks) - (**y**/2)**Variance**(bonds)]} where we have used the fact that x + y = 1, so xA + yA = A - We conclude that, after
**m**months, our Portfolio has grown by a Gain Factor:**G**(m) = 1 + m[**x****Mean**(stocks) +**y****Mean**(bonds) - (**x**/2)**Variance**(stocks) - (**y**/2)**Variance**(bonds)]
"approximately"?
Sorry, I forgot. Approximately! Since our Portfolio has grown to G(m)A after m months,
we rebalance so that our Stock component becomes a fraction x of this new Portfolio and our Bond component
a fraction y: that's xG(m)A and yG(m)A, respectively.
But now we can repeat the above scheme umpteen times, meaning that we're letting our Stock and Bond components sit for m months, then we rebalance, then they sit for another m months, then we rebalance, then ...
>Yeah, yeah. I get it!
>If we rebalance every >And we pick
It's r = xMean(stocks) + yMean(bonds) -
(x/2)Variance(stocks) - (y/2)Variance(bonds).
It's like an annualized return, except here we're talking monthly, so it's the ... uh, monthlyized ...
Our $1.00 changes to (1+ r m) after m months, then we rebalance and that costs
us c cents so we have (1+ r m)-c.
That gets multipied by our m-month Gain Factor again, namely (1+r m), giving us
(1+ r m){(1+r m)-c} = (1+r m)^{2} - c(1+r m).
We rebalance again, subtracting our fixed cost of c cents, leaving us
(1+ r m)^{2} - c(1+r m) - c.
Over the next m months, that gets multiplied by (1+r m) again, then we subtract our c cents ...
To confuse you with mathematical hocus pocus
Okay, here's what we'll do: - We'll split our Portfolio between two asset classes, like Stocks and Bonds.
- Each month we select a random monthly return for the Stock and the Bond component.
- We rebalance every
**m**= 1 month to maintain, say, 60% Stocks and 40% Bonds. - We continue for
**N**= 100 months ... and look at our final Portfolio. - Then we repeat steps 1, 2, 3 and 4 with
**m**= 2 months (so now there are just**N/m**= 50 rebalancings, over the 100 months). - Then we repeat for
**m**= 3, 4, 5, ... up to a rebalancing period of**m**= 30 months.
>And the monthly returns for the two assets are random?
>And?
>Those thirty graphs, in Figure 4, they're pretty wild. I mean, just about any one could be the winner at Indeed, so we shouldn't take it too seriously because ... >But >I assume you did this with some spreadsheet.
>But that picture of the spreadsheet has Here's a couple more for your entertainment ... and edification: >Can I play with the spreadsheet?
>So what's your conclusion, about when to rebalance?
>Yes, yes! That's what I mean!
- Look a number of years into the future.
- Identify the monthly returns for your two (or more) asset classes.
- Run the numbers through a spreadsheet, using various
**m**values. - Pick the best rebalancing period.
>Very funny. >Of course, you've just considered an initial Portfolio without additional investments and ...
See also a Strategy (maybe:^) and perhaps Stocks & Bonds and maybe Stocks & Bonds & Frontiers and maybe ... >zzzZZZ |