# It’s time to perform a wave of magic~

We make some basic assumptions:

1. Suppose you hold a bunch of uncorrelated assets with a compound annualized return of 0.

2. Create a portfolio that includes these assets and rebalances regularly.

3. Do some magic.

4. Now, the compound annualized return of the portfolio is magically greater than 0.

Why is this happening?

The magic secret is actually very simple. If an asset has a compound annualized return of 0%, then it must have a positive arithmetic return.

As for what is the arithmetic rate of return and what is the compound rate of return, I have written it in the previous article, so I will not repeat it here.

Likewise, I’ve written before about what is called “fluctuation drag”

Its formula is: g ≈ μ – 0.5 * σ^2 (I will not deduce the formula, please check Baidu if you are interested)

g is the compound rate of return, if g=0, then μ (arithmetic rate of return) is equal to 0.5 * σ^2. Here σ stands for volatility.

Now, let’s assume an asset with CAGR (compound annualized rate of return) = 0% and volatility σ = 10%.

Then set the above formula, we can calculate the arithmetic rate of return μ = 0.5%.

What happens if we have two independent assets, each with CAGR=0% and σ=10%, and then the two assets are equally divided?

The arithmetic return of this portfolio P μ_P = 0.5 * (0.5 * σ^2) + 0.5 * (0.5 * σ^2) = 0.5%

The volatility of this portfolio P is σ_P = √(0.5*0.5*σ^2+ 0.5*0.5*σ^2) = 7.01%

A magical scene occurs, the g of this combination is no longer 0%. Instead 0.25% (0.5% – 0.5 * 7.01% * 7.01%).

Although neither of these 2 assets has a return, only one combination generates a return of 0.25%.

As the number of such assets increases, the volatility σ _p of the entire portfolio approaches 0% indefinitely, (basically, the formula can be simply written as √(σ²/N)), and ultimately, the portfolio g equals 0.5%.

This is the meaning of pluralism, told through mathematical language. The increase in the compounded annualized return of your portfolio comes primarily from a reduction in portfolio volatility.

That’s why the last post came up. When you hold 50% TSM and 50% LMT, the CAGR of LMT is obviously a serious drag, but the combined CAGR is higher than 100% TSM. Because the volatility of the portfolio is directly suppressed.

And why should diversification balance your portfolio on a regular basis?

Because the volatility of the portfolio is minimized by equal weighting of each asset.

If we allow the weights to float, then the volatility of the portfolio goes up.

For example, an extreme assumption is that the other assets of the portfolio fall to negligible levels, and when one asset 99% dominates the portfolio, σ_P will reach approximately 10%, and g will return to zero.

Of course, the above is completely explained based on mathematics, and the default is to know the volatility and correlation in advance. The reality will be relatively more complex.

But the good thing about math is that it can give you a positive expectation of what you might get with an accurate model.

Even if you hold a lot of unprofitable assets, with reasonable diversification and rebalancing, you can still generate real money.

This is probably the power of magic!

As a digression, the above should be regarded as an introductory course for wanting to become a qualified investor, not the metaphysics of chicken soup. This content seems extremely boring and far from touching the love-hate story of individual stocks, but these are indeed the only way for you to become an investor without relying on luck.

This topic has 21 discussions in Snowball, click to view.
Snowball is an investor’s social network, and smart investors are here.