Rebalancing Bonus II

the Rebalancing Bonus: Part II a continuation of Part I

We're following Bernstein and considering a portfolio with two assets, A and B, with annual returns a_k and b_k respectively
(for k = 1, 2, 3, ... N). We devote a fraction x to asset A and the balance, y = 1 - x to asset B and rebalance yearly to maintain this x:y allocation.

If the Annualized Gain Factor is G for our portfolio With rebalancing, then

(I) G^N = Π (1+x a_k+y b_k) gives 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:
(II) 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.

With Bernstein, we define:

Rebalancing Bonus = { G - (x P + y Q) }

and ask: How large is this Rebalancing Bonus?

To compare these two numbers, G and (x P + y Q), we note that each of G^N, P^N and Q^N have the form:
S^N = Π (1+s_k) = (1+s₁)(1+s₂)...(1+s_N).
Taking logs we get:
N log(S) = Σ log(1+s_k) = Σ { s_k - (1/2)s_k² + ...} = Σ s_k - (1/2)Σs_k² + ...
where we've used the fact that, for small values of z,
log(1+z) = z - (1/2) z² + (1/3) z³ - (1/4) z⁴ + - ...
>What!?
Pay attention. If you don't understand, just look at the picture
For values of z less than, say, 0.25 (meaning, in our case, a 25% annual return) we can replace log(1+z) by z - z²/2.
>Approximately!
Yes. Okay. Then: log(S) = (1/N)Σ s_k - (1/2N)Σs_k² approximately.
Now we use another magic formula, namely:
e^z = exp(z) = 1 + z + z²/2! + ... = 1 + z + z²/2 approximately
and, since exp(log(S)) = S, we get:

S = 1 + log(S) + (1/2)log²(S)
= 1 + { (1/N)Σ s_k - (1/2N)Σs_k² } + (1/2) { (1/N)Σ s_k - (1/2N)Σs_k² }² approximately

... and we recognize (1/N)Σ s_k as the average of the N values of s_k and, as we'll see, the sum Σs_k² involves the Standard Deviation of the numbers s_k.
Figure 1

Figure 2

>zzzZZZ
Okay, so we'll simplify the notation a bit.
Let M = (1/N)Σ s_k = the Mean of the numbers s_k.
Let SS = (1/N)Σs_k² = the average Square of the numbers s_k.
Then, as noted here:
SD² = the Variance, or (Standard Deviation)² = SS - M² so SS = M² + SD²
and the expresssion for S becomes:

S = 1 + {M - (1/2)SS} + (1/2){M - (1/2)SS}²
= 1 + {M - (1/2)M² - (1/2)SD²} + (1/2){M - (1/2)M² - (1/2)SD²}²
= 1 + {M - (1/2)M² - (1/2)SD²} + (1/2){M² + ...}
= 1 + M - (1/2)SD² approximately!

Do you remember what we're doing?
>zzzZZZ
We're investigating the expressions for the annualized Gain Factors G, P and Q.
They all have the form of our S, namely: S = 1 + M - (1/2)SD².
>zzz ... uh? Approximately!
Yes ... approximately!. Magic Formula

If S^N = Π (1+s_k) = (1+s₁)(1+s₂)...(1+s_N)

then S = 1 + Mean(s_k) - (1/2)Variance(s_k)
... approximately

... so we get our expressions for the annualized returns G, P and Q ... like so:

>Wait! Is that approximation any good?
Good question. If we look at the annual returns for, say, the S&P 500 and calculate the annualized returns from 1950 to 1951, then from 1950 to 1952 then from 1950 to 1953 etc. and see how they compare to the Mean - (1/2)Variance, 1950 to 1951 then ...
>Yeah, I get the idea.
Not bad, eh?
And below, for each of five decades ...
>With all those approximations? Amazing!

Figure 3

>So, have we finished with the rebalancing bonus?
Is it positive, this bonus?
>I have no idea.
Then we're not finished.

Consider the Magic Formula applied to G, our annualized portfolio Gain Factor:

G = 1 + Mean(x a_k+y b_k) - (1/2)Variance(x a_k+y b_k)

Remember what we're calling the Mean returns of assets A and B?
>Huh?
We're calling them a and b, so

G = 1 + x a +y b - (1/2)Variance(x a_k+y b_k)

Remember how to calculate the Variance? The (Standard Deviation)² ?
>You kidding?
The Variance is the average Squared Deviation from the Mean:

Variance(x a_k+y b_k) = (1/N) Σ{ x (a_k - a) +y (b_k - b) }²
= x² (1/N)Σ (a_k - a)² + y² (1/N)Σ (b_k - b)² + 2x y (1/N)Σ (a_k - a)(b_k - b)
= x² Variance(A) + y² Variance(B) + 2x y Covariance(A,B)

Altogether now:
G = 1 + x a +y b - (1/2){ x² Variance(A) + y² Variance(B) + 2x y Covariance(A,B) }

and, similarly for P and Q which involve a single asset, either A or B:

P = 1 + a - (1/2)Variance(A)
Q = 1 + b - (1/2)Variance(B)

>I've completely forgotten what we're ...
We're after the Rebalancing Bonus = G - (x P + y Q). If we use x + y = 1 and x - x² = x (1-x) = x y, we get:

Rebalancing Bonus = x y [ (1/2){Variance(A) + Variance(B)} - Covariance(A,B) ]

>Are we still working on "approximately"?
Yes ... approximately.

>Is that what Bernstein got?
Yes ... though I should point out that the term "rebalancing bonus" means different things to different people. Some might use the term to mean the difference in annualized returns for portfolios With and Without rebalancing, the latter being a Buy-and-Hold portfolio.

>So, what do we do with our portfolio in order to get an annualized Gain Factor of
x P + y Q? That's what we're comparing to, eh? We're comparing the annualized Gain Factor of our portfolio With rebalancing, that's our G, so is that x P + y Q a Gain Factor of some kind? For some kind of portfolio Without rebalancing?
I have no idea.
>So why are we doing all this?
Isn't it fun? Edifying?
>No! Are we finished?
Yes, for now ... except to notice that (1/2)[Var(A) + Var(B)] - Cov(A,B) is given in Figure 4, for A = S&P 500 and B = 5-year Treasuries so you can multiply by
x y = x - x² to see what you'd get for some x:(1-x) allocation. You may also want to note that the maximum of x - x², hence the maximum of the Rebalancing Bonus, occurs for x = y = 1/2 (when x y = 1/4).

Figure 4

If you're interested, you can figure out the Rebalancing Bonus (for the period 1975 - 2000) from the following numbers:

100*Covariance (1975-2000) LG LV SG SV T-bill 5-yr T Lng Bnd S&P EAFE
LG 2.520 1.487 2.603 1.109 -0.030 0.262 0.527 2.133 0.944
LV 1.487 2.365 2.053 2.469 0.014 0.538 1.030 1.650 0.794
SG 2.603 2.053 5.420 3.073 -0.008 -0.023 -0.054 2.323 1.462
SV 1.109 2.469 3.073 3.978 0.018 0.338 0.607 1.427 0.676
T-bill -0.030 0.014 -0.008 0.018 0.072 0.049 -0.001 -0.018 -0.066
5-yr T 0.262 0.538 -0.023 0.338 0.049 0.497 0.841 0.291 0.019
Lng Bnd 0.527 1.030 -0.054 0.607 -0.001 0.841 1.614 0.595 0.162
S&P 2.133 1.650 2.323 1.427 -0.018 0.291 0.595 1.979 1.016
EAFE 0.944 0.794 1.462 0.676 -0.066 0.019 0.162 1.016 4.109
Standard Deviations LG LV SG SV T-bill 5-yr T Lng Bnd S&P EAFE
(1975-2000) 15.88% 15.38% 23.28% 19.94% 2.68% 7.05% 12.70% 14.07% 20.27%

For example, if A = LG (meaning Large cap Growth) and B = Lng Bnd (meaning govt. Long term Bonds) then, since the Variance is (Standard Deviation)², we have

Var(A) = 0.1588² = 0.0252
Var(B) = 0.1270² = 0.0161
100*Cov(A,B) = 0.527 so Cov(A,B) = 0.00527
Rebalancing Bonus = xy [ (1/2)(0.0252 + 0.0161) - 0.00527 ] = xy [0.01538] and 0.01538 is about 1.5%

>zzzZZZ
One interesting thingy:
Since that [0.01538] factor has nothing to do with your allocation (defined by x and y = 1-x), you'd get the largest "bonus" by
maximizing xy = x(1-x) ... and that occurs for x = y = 0.5 so (for our example) that'd give a "bonus" of (0.5)(0.5)[0.01538] = 0.0038 or 0.38%

>zzz ... huh? So I should always choose 50% of each?
Why not?