Diversification, Rebalancing and Different Means

Question

I have found many financial authors making generalizations about the geometric mean (GM) and arithmetic mean (AM) but they are wrong in certain circumstances. Could someone explain their reasoning?

My fact why they are wrong is based Jensen inequality:

$\sum^{n}_{i=1} p_{i} f(x_{i}) \geq f( \sum_{i=1}^{n} p_{i} x_{i})$

for concave up functions, verified here. Now the special case is:

$\sqrt[n]{x_{1}x_{2}...x_{n}} \leq \frac{x_{1}+x_{2}+...+x_{n}}{n}$

For concave down functions, the corresponding result can be obtained by reversing the inequality, more here.

Author 1

Rebalancing and diversification go hand in hand. There is no diversification benefit without rebalancing – otherwise the total return will simply be the weighted average of the long-term geometric returns. If you don’t rebalance to asset types, you will get no diversification benefit. If you can’t rebalance to an asset type, you cannot get diversification benefits.
Rebalancing benefits increase as volatility rises, and decreases in less volatile times. The benefit of rebalancing after a 10% movement is more than 10 times the benefit after a 1% movement, and the benefit from rebalancing after a 50% move is more than 5 times the benefit after a 10% move. The greatest benefit comes in times, like 2008-2009, when there are wild movements in portfolios. -- diversification does not assure profit or protect against loss in a declining that affects numerous asset types. Source.

Suppose we have a concave-down environment so

$\sqrt[n]{x_{1}x_{2}...x_{n}} \geq \frac{x_{1}+x_{2}+...+x_{n}}{n}$.

1. Now following the logic in the paragraph the total return is simply the weighted average of the long-term geometric returs. We know from the latter result that it is greater than or equal to the arithmetic mean. Is there a diversification benefit with rebalancing?
2. Then the numbers thrown in the next bolded sentence are a bit odd. Why would there be such a benefit if we just noticed that it is not necessarily so if our environment is concave-down environment (valuations going down)?
3. But the last sentence saves this writer! No error here.

Author 2.

William Bernstein here goes one step further, ignoring the Jensen:

The effective (geometric mean) return of a periodically rebalanced portfolio always exceeds the weighted sum of the component geometric means.

The implicit premise behind such statements probably is that the market in the long-run is rising, very well right to some extent. But with that assumption, the problem of asset allocation simplifies to the Jensen -- an even with such premise, it should be noticed to the reader (or such sentences are wrong).

Author 3.

Many works rely on the ambiguous assertions such as Ilmanen's Expected Return -book, page 485:

the arithmetic mean of a series is always higher than the geometric mean (AM > GM) except when there is a zero volatility (AM = GM). A simple Taylor series expansion show a good approximate is $GM \approx AM - \frac{Variance}{2}.$

...but the bolded sentence is wrong. You can make a function with concave values and $AM < GM$.

Questions

1. What are the authors meaning here?
2. Are they wrong or do they have some some hidden premises that I am missing?
3. Why are they stating such issues about AM and GM as they cannot always be true?

Answer 1

First of all, AM is always greater than or equal to GM

$$ x_1 + x_2 + ... + x_n \geq \sqrt[n]{x_1x_2...x_n}~\forall x_i \geq 0 $$

You can prove it by induction from $\frac{x_1 + x_2}{2} \geq \sqrt{x_1x_2}$ or put $f(x) = \ln(x), p_i = \frac{1}{n}$ to Jensen's inequality to get it. The equality holds when $x_1 = x_2 = ... = x_n$.

For author 1 and 2,

We want to diversify (put different weight) on $n$ different assets $p_i$ (including risk-free asset) with weight $w_i$ (assuming short-selling is possible and borrowing money is possible, $w_i \in \mathbb{R}$).

Assume those assets follow $n$ geometric brownian motion with drift $\mathbf{\mu} = (\mu_1,\mu_2,...,\mu_n)$ and the covariance matrix of their volatility term is $\mathbf{C}$. One can show that the optimal weighting vector $\mathbf{w} = (w_1,w_2,...,w_n)$ should be

$$ \mathbf{w} = \mathbf{C}^{-1} \mathbf{\mu} $$

One verify the above to the one-dimensional case $\displaystyle w = \frac{\mu}{\sigma^2}$.

It is known as volatility pumping in literature.

The result above is optimal, that's mean any other weighting vector will underperform the above (Of course, we have assumption that the asset is really following geometric brownian motion and our estimate of $\mu$ and $\mathbf{C}$ is correct). This answer the question of author 2.

As the asset move, the money you invested in the assets will change accordingly and move away from the correct weight you should put. Therefore you need to rebalance your portfolio to the optimal weight.

Since we cannot rebalancing continuously, there is an error with an upper bound of $\displaystyle O(\frac{1}{\sqrt{n}})$, where $n$ is the number of rebalancing.

Assume the transaction cost of rebalancing is zero, I don't think

The benefit of rebalancing after a 10% movement is more than 10 times the benefit after a 1% movement, and the benefit from rebalancing after a 50% move is more than 5 times the benefit after a 10% move.

is correct for author 1.

For author 3,

As I pointed out AM is always greater than or equal to GM with equality holds when $x_1 = x_2 = ... = x_n$. I think he is correct. When an random variable $X$, we have $\mathrm{Var}(X)$, it would implies $P(X=\mu) = 1$ for some $\mu$, or $X$ is constant almost surely. In finite case, it simply means $x_1 = x_2 = ... = x_n$.

For the Taylor's series expansion argument, it should be talking about the proof of Ito's lemma (See Informal derivation section). The idea behinds the proof is to make a taylor's expansion at $x$ and $t$.

Answer 2

I believe the assumption here in the first quote is that returns are either strictly positive or strictly negative and the authors are comparing the effect of volatility on geometric returns to arithmetic. The issue of diversification benefits have little to do with this difference as opposed to a time varying covariance matrix. The benefits of diversification in volatile times assumes low correlation and the assumption that asset correlations do not have a strong positive correlation with volatility. This is a common simplification for teaching basic finance. The second quote will be true when the ratio of return to risk is approximately equal across assets and assets are not perfectly positively correlated. The third quote is true on a long time horizon because real interest rates are positive. This follows from the fundamental assumption that a dollar today is worth more than a dollar tomorrow.