Magic Formulas ... and reBalancing

I often need certain magic formulas, so I thought I'd ...

>You thought you'd stick 'em here, right?
Right.

Suppose your portfolio has annual returns of r1, r2, ... rn, over n years ... where r = 0.123 corresponds to a 12.3% return
Then the total gain is:
[1]       G = (1+r1)(1+r2)...(1+rn)     ... meaning that \$1 grows to \$G in n years

Okay, let's take the log of this guy, giving:
[2]       logG = log(1+r1) + log(1+r2) +...+ log(1+rn)     ... since the log of a product is the sum of the logs
 But, for small values of a number z: [3]       log(1+z) = z - (1/2)z2 + (1/3)z3 - ... In fact, if the zs are "usual" returns ... hence pretty small numbers ... we can ignore (1/3)z3 and higher order terms, and take:   log(1+z) = z - (1/2)z2
Then [2] becomes:
[4]       logG = r1 + r2 + ... + rn - (1/2)(r12 + r22 + ... +rn2)
 Aah, but elogG = G and we know that: [5]       ez = 1 + z + (1/2!)z2 + (1/3!)z3 + ... Again, we ignore powers of z greater than 2 and take: ez = 1 + z + (1/2)z2
That allows us to get, from [4]:
 [6]       G = elogG = EXP[r1 + r2 + ... + rn - (1/2)(r12 + r22 + ... +rn2)] = 1 + [r1 + r2 + ... + rn - (1/2)(r12 + r22 + ... +rn2)] + (1/2)[r1 + r2 + ... + rn - (1/2)(r12 + r22 + ... +rn2)]2   ... approximately = 1 + [r1 + r2 + ... + rn - (1/2)(r12 + r22 + ... +rn2)] + (1/2)[r1 + r2 + ... + rn]2   ... ignoring higher order terms
We'll write this up fancy:

### n-year Gains

 (1+r1)(1+r2)...(1+rn) = Π (1+rk)   ... where Π means the product = 1 + Σrk + (1/2)[Σrk]2 - (1/2)Σrk2   ... where Σ means the sum

... approximately, for small numbers rk

Now we'll look again at [1], but this time we'll consider the annualized gain factor:
[1A]       GN = (1+r1)(1+r2)...(1+rn)     ... meaning that an annual gain of G gives the final value of a \$1.00 portfolio, over n years

Continuing, we get (taking logs):
[2A]       N logG = log(1+r1) + log(1+r2) +...+ log(1+rn) = r1 + r2 + ... + rn - (1/2)(r12 + r22 + ... +rn2) = Σrk - (1/2)Σrk2   ... as we did above

So
[2B]       logG = (1/N) Σrk - (1/2N)Σrk2
so that
[2C]       G = exp[log(G)] = 1 + [(1/N) Σrk - (1/2N)Σrk2] + (1/2)[ (1/N) Σrk - (1/2N)Σrk2]2
and, since we're neglecting powers of the rk higher than "2", we get:

### Annualized Returns

 [(1+r1)(1+r2)...(1+rn)]1/N = 1 + (1/N) Σrk - (1/2N)Σrk2 + (1/2N2) [Σrk]2

... approximately, for small numbers rk

Note:
Since the Variance of a set of returns is given by:
Variance = (average of the squares) - (the square of the average)
we can replace the average of the squares, namely (1/N)Σrk2, by   Variance + (the square of the average).
That is, we can put: (1/2N)Σrk2 - (1/2)[(1/N)Σrk]2 = (1/2)Variance

That gives:

### Annualized Returns

 [(1+r1)(1+r2)...(1+rn)]1/N = 1 + (1/N) Σrk - (1/2)Variance = 1 + Mean - (1/2) Variance

... approximately, for small numbers rk

Note: This gives an approximation to the Compound Annual Growth Rate, namely:

CAGR = Mean - (1/2) Variance

>Approximately.

>So you're talking about a portfolio with a single stock, right?
Wrong. I'm talking about a series of annual (or weekly or monthly) returns: r1, r2, r3, ... ...
If your portfolio is made of, say, two assets with returns a1, a2, a3, ... ... and b1, b2, b3, ...
and you rebalance annually to maintain a fraction x of your money in the first asset and y = 1 - x in the second asset,
then your annual portfolio returns would be: r1 = xa1+yb1, r2 = xa2+yb2, r3 = xa3+yb3, ...

Of course, if you just invest \$x in the first asset and \$y in the second (with \$x+\$y=\$1.00), and you did NOT rebalance,
then, after n years, your \$1.00 portfolio would be worth   x(1+a1)(1+a2)...(1+an) + y(1+b1)(1+b2)...(1+bn)

 reBalancing ... yes or no?
>Shall I ask the burning question?

>Which is larger?

A fair question. If
Pn = (1+xa1+yb1)(1+xa2+yb2)...(1+xan+ybn) = Π (1+x ak+y bk)   ... with rebalancing
and
If Qn = x(1+a1)(1+a2)...(1+an) + y(1+b1)(1+b2)...(1+bn) = xΠ (1+ak) + yΠ (1+bk)   ... withOUT rebalancing
which is larger? P or Q?

It depends upon the returns, eh?
If I select 50 random returns for each of assets A and B, and pick x-fractions (to devote to asset A), we can get
better CAGR with rebalancing, or sometimes withOUT rebalancing, or sometimes it depends upon how much is devoted to asset A.

 sometimes reBalancing is better sometimes reBalancing is worser sometimes it depends upon x

Notice that, when x = 0% or 100%, the two CAGRs are the same. You're investing in just one asset.
These charts are available in the spreadsheet noted above
The two sets of 50 returns are randomly chosen from Normal distributions to (sorta) mimic actual returns.

>Okay, but maybe rebalancing reduces volatility.
And if you regard volatility as "risk" (heaven forbid!), then it'd reduce risk, eh?
Let's try that again ... with our 50 randomly selected returns for each of assets A and B.
As before, we consider varying the fraction of asset A (that's x).

 sometimes reBalancing is better sometimes reBalancing is worser sometimes it depends upon x

These charts are available in the spreadsheet noted above

Note:
In
another tutorial we talked about Bernstein's so-called Rebalancing Bonus.
In fact, Bernstein's "bonus" is never negative **, implying that rebalancing is good ... indeed, it's always good.
Alas, that ain't (always) the case

**
Bernstein's "bonus" = (1/2) x y ( VAR[A] + VAR(B) - 2 COVAR(A,B) )
and, if U and V are the Standard Deviations of assets A and B, and r is their correlation coefficient, then this can be written:
Bernstein's "bonus" = (1/2) x y (U2 + V2 - 2 r U V ) ≥ (1/2) x y (U2 + V2 - 2 U V ) = (1/2) x y (U - V)2 ≥ 0

 the Shape of Curves

Note that, over n years, the non-reBalanced portfolio has an annualized growth rate according to Q, where:
If Qn = x(1+a1)(1+a2)...(1+an) + y(1+b1)(1+b2)...(1+bn)
As a function of x, that right-side is a straight line.

In contrast, a rebalanced portfolio has an annualized growth rate according to P, where:
Pn = (1+xa1+yb1)(1+xa2+yb2)...(1+xan+ybn)
and that right-side ain't hardly a straight line!
However, the P-chart starts and ends at the same point as the Q-chart ... when x = 0% and x = 100%

So what does the P-chart look like:
We'll do a wee bit o' calculus here:

1. n log P = log(1+xa1+yb1) + log(1+xa2+yb2) + ...
2. n d/dx (log P) = (n/P) dP/dx = (a1-b1)/(1+xa1+yb1) + (a2-b2)/(1+xa2+yb2) + ... = Σ(ak-bk)/(1+xak+ybk)
3. So dP/dx = (1/n) P Σ(ak-bk)/(1+xak+ybk) = (1/n) P ΣWk ... where we put Wk = (ak - bk)/(1 + xak + ybk)
4. Then d2P/dx2 = (1/n) dP/dx ΣWk - (1/n) P ΣWk2 ... since dWk / dx = - Wk2
5. So (using 3) d2P/dx2 = P [(1/n)ΣWk]2 - P (1/n) ΣWk2
6. We can then write:
 [A]       (1/P) dP/dx = (1/n) ΣWk   ... from 3 [B]       (1/P) d2P/dx2= [(1/n)ΣWk]2 - (1/n) ΣWk2   ... from 5
>Wake me when you're finished ...
Let M = (1/n)ΣWk be the mean of the W's.
Then we can rewite things like so:
 (1/n) ΣWk2 = (1/n) Σ(Wk - M + M)2 = (1/n) Σ(Wk - M)2 + (1/n) 2M Σ(Wk - M) + (1/n) ΣM2 = (1/n) Σ(Wk - M)2 + (1/n) 2M 0 + (1/n) (n M2)   ... noting that Σ(Wk - M) = 0 since M is the mean of the Ws. = VAR[Wk] + M2   ...where VAR is the Variance (or StandardDeviation2) of the Wk
Then we can rewrite [B] above, like so:

[C]       (1/P) d2P/dx2 = M2 - [VAR[Wk] + M2] = - VAR[Wk]

Since Variance = StandardDeviation 2 is never negative, the P-curve has a negative second-derivative ... hence is concave down.
That means that, with rebalancing, the P-curve begins and ends at the points where the Q-curves begins and ends, but is concave down, so ...

>So the maximum, with rebalancing, is somewhere between x = 0% and x = 100%, right?
Sometimes ...

 the "REAL" Bonus

We saw in an earlier investigation here , that if:

[7]       Pn = Π (1+x ak+y bk)     gives the final value of a \$1.00 portfolio, With rebalancing.

then the annualized gain factor is (approximately) :

P = 1 + x a +y b - (1/2){ x2 VAR(A) + y2 VAR(B) + 2x y COVAR(A,B) }

Similarly, if the annualized Gain Factors for each of assets A and B are A and B respectively, then:
An = Π (1+ak)     and     Bn = Π (1+bk)

and (approximately) :

[7A.a]       A = 1 + a - (1/2)VAR(A)
[7A.b]       B = 1 + b - (1/2)VAR(B)

where a and b are the average (or mean) returns and (remember!) A and B are the annualized gain factors... for each asset.

Uh ... yeah, to keep you awake.
Anyway, it's not reasonable to compare P to x A + y B since the latter doesn't represent the gain factor of the nonreBalanced portfolio.
Indeed, the nonreBalanced portfolio has an n-year gain of x An + y Bn, hence an annualized gain factor of:

[8]       Q = [ x An + y Bn ] 1/n

If we look at the size of the terms in, say, A = 1 + a - (1/2)VAR(A) we might get something like:
1 + 0.08 - (1/2)(0.02) = 1 + 0.08 - 0.01 = 1 + 0.07
so it's very near the value 1.0

Then, as an approximation, let's use (1+z)n ≈ 1 + nz when z is small.(and n isn't too large!).
That'd give:

[8A.a]       An ≈ 1 + na - (n/2)VAR(A)
[8A.b]       Bn ≈ 1 + nb - (n/2)VAR(B)

and finally:

[8B]       Q = [ x An + y Bn ] 1/n = [ 1 + n (xA + yB) - (n/2) (x VAR(A) + y VAR(B) ] 1/n   noting that x + y = 1

Now let's use (1+z)1/n ≈ 1 + (1/n)z and we'll get:

[8C]       Q = 1 + (xA + yB) - (1/2) (x VAR(A) + y VAR(B))   approximately

And now, finally, we consider the difference:

[8D]       P - Q = (1/2) (x VAR(A) + y VAR(B)) - (1/2){ x2 VAR(A) + y2 VAR(B) + 2x y COVAR(A,B)}   approximately

Aah, but there are terms involving x - x2 = x (1-x) = xy and y - y2 = y (1-y) = xy so we get finally:

>How many times have you said "finally"?
Pay attention! We now have the "real" rebalancing bonus:

[9]       P - Q = (xy) (1/2){VAR(A) + VAR(B)) - 2 x y COVAR(A,B)}   approximately

>But that's the same as Bernstein's bonus ... isn't it?
Uh ... yes, so it is.
I guess that (1+z)n = 1 + nz isn't such a good approximation. Let's check:
 z = 0.1, n = 30 (1+z)n = 17.4 1 + nz = 4 z = 0.1, n = 10 (1+z)n = 2.6 1 + nz = 2 z = 0.1, n = 5 (1+z)n = 1.6 1 + nz = 1.5

>Pretty lousy, I'd say.
Yeah, that approximation is like using simple interest rather than compounding the returns.
I guess the moral is that you may get Bernstein's rebalancing bonus for the short term, but ...
>And the long term?
Okay, let's investigate. First, we note that:
 z = 0.1, n = 30 (1+z)1/n = 1.0032 1 + (1/n)z = 1.0033 z = 0.1, n = 10 (1+z)1/n = 1.0096 1 + (1/n)z = 1.010 z = 0.1, n = 5 (1+z)1/n = 1.019 1 + (1/n)z = 1.020

>So using (1+z)1/n = 1 + (1/n)z is good, eh?
Yeah, so now we'll use an improved approximation for (1+z)n, namely (1+z)n ≈ 1 + n z + (1/2)n(n-1)z2

Look again at [7A.a] and [7A.b], namely
[7A.a]       A = 1 + u ≈ 1 + a - (1/2)VAR(A)   where u is the annualized return
[7A.a]       B = 1 + v ≈ 1 + b - (1/2)VAR(B)   where v is the annualized return

We'll use our new, improved approximation to get:
[8A.a]       An ≈ 1 + n u + (1/2)n(n-1)u2
[8A.b]       Bn ≈ 1 + n v + (1/2)n(n-1)v2

Then
xAn + yBn = 1 + n (xu+yv) + (1/2)n(n-1) (xu2+yv2)   putting x + y = 1

and, using (1+z)1/n = 1 + (1/n)z, we get:

[10]       Q = [ x An + y Bn ] 1/n = 1 + (xu+yv) + (1/2)(n-1) (xu2+yv2)

Now consider P again:
 P = 1 + x a +y b - (1/2){ x2VAR(A) + y2VAR(B) + 2xy COVAR(A,B) } = 1 + x [ u + (1/2) VAR(A) ] + y [ v + (1/2) VAR(B) ] - (1/2){ x2VAR(A) + y2VAR(B) + 2xy COVAR(A,B)}       putting a = u + (1/2)VAR(A) and b = v + (1/2) VAR(B) = 1 + x u + y v + (1/2) x y { VAR(A) + VAR(B) - 2 COVAR(A,B) }       putting x - x2 = y - y2 = xy

If we subtract from Q from P we get, finally, the difference between the reBalanced and nonreBalanced gains:

P - Q = (xy) (1/2){VAR(A) + VAR(B)) - 2 COVAR(A,B)} - (1/2)(n-1) (xu2+yv2)

>I recognize that first piece. Isn't he ...?
Bernstein's Rebalancing Bonus? Yes. Interesting, eh?
But now we have an extra "correction", namely (1/2)(n -1) (xu2+yv2)

>Uh ... what's that u and v, again?
They're:
u = a - (1/2)VAR(A)
v = b - (1/2)VAR(B)
which are the annualized returns for assets A and B ... approximately

>Everything is "approximately", right?
Yes.

Okay, how big is this correction? Remember that the Bernstein "Bonus" is typically less than 1%.
If the annualized returns u and v are, say about 8% (or 0.08) then the correction is about (1/2)(n -1) (x 0.082+y 0.082)
and, for a 50-50 allocation (so x = y = 0.5) and n = 10 years, we'd get something like 3%.

>Wow! That's a huge"correction" !
Ain't it though ... and maybe the next correction would be even larger

>And the next correction term is ... what?
I think it's (1/3!) (n-1) (n-2) (xu3+yv3) so, for n = 10 and x = y = 0.5, this is about 1%.

In general, I guess we should write the rebalancing bonus as:
 Rebalancing Bonus = (xy) (1/2){VAR(A) + VAR(B)) - 2 COVAR(A,B)} - (1/2!)(n-1) (xu2+yv2) - (1/3!) (n-1) (n-2) (xu3+yv3) + ...
Note that the terms get smaller when (1/m!) nm-1 um and (1/m!) nm-1 vm are small.
For example, for n = 10 and u = 0.08, then
(1/m!) 10m-1 0.08m = 0.0032 when m = 2
and (1/m!) 10m-1 0.08m = 0.0085 when m = 3
and (1/m!) 10m-1 0.08m = 0.0017 when m = 4
and ...

>Enough! We're talking approximations, approximations, approximations ...
Uh ... yes, but I think we can conclude that the "bonus" is reduced for assets that are highly correlated (so COVAR(A,B) is large) and whose annualized returns are large (so u and v are large).