a topic suggested by Chuck W.
Once upon a time, Bernstein
considered the effect of rebalancing a portfolio consisting of two assets, A and B.
>Like Stocks and Bonds?
Yes, or S&P 500 and Foreign or Large and Small Cap or ...
>Okay, okay. I get it.
Fine. Suppose that, over the past umpteen years, the average annual return on asset A is
a and the average annual return on asset B is b and we devote a fraction
x of our portfolio to asset A (example: x = 0.70) and a fraction
y = 1  x to asset B (example: y = 0.30), and we rebalance each year in
order to maintain this x:y ratio ... then what's the average annual return of our portfolio?
>Give me an example.
Okay, suppose we devote 70% of our portfolio to A and 30% of B and their average returns
are 10% and 8% respectively.
>Our average portfolio return would be 70% of 10% plus 30% of 8% which is ... uh ...
That's 9.4%, namely the weighted average of the individual average returns. In general, it'd
be x a + y b.
>You can prove that, I suppose.
Yes. Suppose that, over N years (months? days?) the returns ...
>No, I don't need to see any proof! I'll take your word ...
... the returns for asset A are
a_{1}, a_{2}, ... a_{N}
with average a = (1/N){a_{1}+a_{2}+ ... +a_{N}}
= (1/N)Σa_{k}
where Σ means we add them all together,
and the returns for asset B are
b_{1}, b_{2}, ... b_{N}
with average b = (1/N){b_{1}+b_{2}+ ... +b_{N}}
= (1/N)Σb_{k}
then our annual portfolio returns, each year, are
(x a_{1}+y b_{1}), (x a_{2}+y b_{2}), ...
(x a_{N}+y b_{N})
or, to put it more simply,
(x a_{k}+y b_{k}) in year k (where k = 1, 2, 3, ... N).
>Are you're rebalancing each year, to maintain the x and y fractions?
Yes, of course. Didn't I say that already? Pay attention.
The average return, for our portfolio, is then the average of all those
annual portfolio returns, namely
(1/N)Σ(x a_{k}+y b_{k})
= x {
(1/N)Σa_{k}}
+ y {
(1/N)Σb_{k}}
= x a + y b.
>So that's our portfolio return, eh?
That's our AVERAGE annual return. If you got this average annual
return every year you'd be one lucky fella. (See
Average & Annualized.)
Figure 1 shows the actual S&P 500 and a portfolio which got, each year, the
Annualized Return (annualized over 19702000) and one which got the Average Return
(averaged over 19702000).
>I like the Average!
Yes, you would. It's always larger.
 Figure 1

What you REALLY want to know, in order that your portfolio end up after
umpteen years with the correct value, is the
Annualized return.
>I assume the annualized portfolio return is the weighted average of the annualized returns for A and B.
Wrong! That's okay with the Average Returns, but not the Annualized.
That's where our Rebalancing Bonus comes in.
>I was wondering when you'd get around to that.
Our portfolio Gain Factor, each year, is 1+Annual Return and since our annual returns are
(x a_{k}+y b_{k}), our Gain Factor (during year k) is:
(1+x a_{k}+y b_{k}) so, to get the Total Gain over N years,
namely (FinalPortfolio)/(Initial Portfolio), we multipy these annual Gain Factors together,
getting:
(1+x a_{1}+y b_{1})
(1+x a_{2}+y b_{2})
(1+x a_{3}+y b_{3})
... (1+x a_{N}+y b_{N})
= Π (1+x a_{k}+y b_{k})
where Π means we multiply them all together.
If we rebalance each year to maintain the x:y allocation
then our portfolio, after N years, would have grown by a Gain Factor
Π (1+x a_{k}+y b_{k})
giving the final value of a $1.00 portfolio, With rebalancing.
On the other hand, the price of asset A would have grown by a Gain Factor
Π (1+a_{k})
the final value of a $1.00 portfolio devoted to asset A
and the price of
asset B would have grown by a Gain Factor
Π (1+b_{k})
the final value of a $1.00 portfolio devoted to asset B.
Conclusion? Had we NOT rebalanced, but put $x into asset A and $y into
asset B and left them there, our portfolio would have grown
by a Gain Factor:
xΠ (1+a_{k}) +
yΠ (1+b_{k})
the final value of a $1.00 portfolio Without rebalancing.
>Put $x into asset A? What does that mean?
If our portfolio starts with $1.00 and x = 0.70 and y = 0.30 then we'd put
$0.70 into asset A, that's $x, and ...
>And $0.30 into asset B, right?
Right. So now we want to compare the Total Gain Factor With and
Without rebalancing.
>Remind me. What are the ...?
We ask: Which is larger?
(1)
With rebalancing: Π (1+x a_{k}+y b_{k})
giving the final value of a $1.00 portfolio, With rebalancing
or
(2)
Without rebalancing: xΠ (1+a_{k}) +
yΠ (1+b_{k})
giving the final value of a $1.00 portfolio, Without rebalancing
>Don't tell me! The Rebalancing Bonus is the difference, eh?
Uh ... not exactly.
>Well, what's the rebalancing bonus?
You may think that the difference in these returns (With and Without
rebalancing, annualized) would be the bonus, but remember, we're following Bernstein.
In fact, these two returns (annualized) wouldn't provide a bonus but may generate a
... what's the opposite of "bonus"?
>Deficit?
Thank you. A rebalancing deficit.
Look at Figure 2. Here we have a portfolio with 70% devoted
to the S&P 500 and 30% to 5year Treasuries, from 1950 to 2000. If we rebalance we get an
annualized return of 11.3%, but if we just stick 70% of our money into the S&P 500 and the rest
into the Treasuries ... and do NOT rebalance ... we'd get an annualized return of 12.4%, so ...
>Do not rebalance, meaning we just leave the monies there?
Exactly. You'll notice that there's no bonus here. Rebalancing generates a ... uh ...
>Deficit?
Yes, a deficit of 12.4  11.3 = 1.1% in annualized return.
>But the volatility is reduced, eh?
Yes, the Standard Deviation is smaller. Maybe that's the "bonus".
 Figure 2

>So I'll ask again. What's the rebalancing bonus?
We compare the annualized return when we rebalance with the weighted average of the two
annualized returns for assets A and B.
>Huh?
If our annualized Gain Factor With rebalancing is G, then, applying this
annualized Gain Factor to a $1.00 portfolio N times gives the final portfolio, namely
G^{N} which is given by (1), above. Hence:
(3)
G^{N} = Π (1+x a_{k}+y b_{k})
giving the final value of a $1.00 portfolio, With rebalancing.
On the other hand, if the annualized Gain Factors for each of assets A and B are
P and Q respectively, then:
(4)
P^{N} = Π (1+a_{k})
and
Q^{N} = Π (1+b_{k})
and the weighted average of these annualized Gain Factors is:
x P + y Q.
>Don't tell me! We want to compare G with x P + y Q, right?
Right.
>And the rebalancing bonus is {
G  (x P + y Q) }, right?
Right again.
>So, is this really a bonus or is it a ... uh ...
Deficit?
>Thank you.
For the 70% (S&P) + 30% (5yr T) example, the annualized returns for each (over the period
from 1950 to 2000) are 13.1% and 6.2% respectively so the weighted average (with the 70/30 split)
is 70% x (13.1) + 30% x (6.2) = 11.0% whereas the actual annualized return, With
rebalancing, is ...
>I get 12.4%, from Figure 2.
No, that's Without rebalancing. With rebalancing, it's 11.3%, the red curve from Figure 2.
>So the rebalancing bonus is 11.3%  11.0% = 0.3%, right?
Right.
>So it IS a bonus, for that example, but is it ALWAYS a bonus?
We'll see ...
for PART II.
