# Mathematics of Rebalancing

Last Updated Sep 20, 2010 8:26 PM EDT

The need to rebalance a portfolio to its target allocation is something most investors understand. Yet understanding it, and actually doing it, don't apparently go hand in hand. Though rebalancing is often regarded as contrarian, and going against the herd, there's far more to it. To coin a much used phrase, let's "do the math."

Let's start by looking at what I call the moderate Second Grader Portfolio during the ten years of the last decade. It's a 60 percent equity and 40 percent bond portfolio comprised of:

Vanguard Total Stock Market Index (VTSMX) - 40%

Vanguard Total International Stock Index (VGTSX) - 20%

Vanguard Total Bond Index (VBMFX) - 40%

During the "lost decade," only the bond fund had a high return. The simple average annual return over the decade was a mere 2.72 percent.

The power of rebalancing
This portfolio, rebalanced only once a year at year-end, clocked in a 3.54 percent annual return, or 0.82% more per year than the 2.72 percent in the table. That may not seem like much, but \$10,000 growing at 2.72% would be worth \$13,078, while the extra 0.82% annually was worth \$14,161. I happen to think that the difference of \$1,087 is real money.

The math behind the extra dough
To better understand where this extra grand came from let's take two examples. In the first, assume you make one investment that earns 30 percent the first year, and loses 10 percent in year two. The simple average annual return is 10 percent, meaning your \$10,000 investment would grow to \$11,700 (\$10,000 x 1.3 x .9).

In the second example, your investment earns 10 percent each of the two years. The simple annual average return is also 10 percent, but the \$10,000 investment is now worth \$12,100 (\$10,000 x 1.1 x 1.1). That's \$400 more than in the first example, even though both earned a simple average annual return of 10 percent.

What matters is the geometric average annual return, and here's why. The first example turned in an 8.17 percent annual geometric return, while the second earned a 10.0 percent geometric return. The more the variation in the annual return, the more the geometric return will lag the simple average return.

Rebalancing lowers variation
Because there will always be periods when each of the three funds are hot, rebalancing into funds that are not, reduces variation. That reduction of variation helps to boost your return. It won't work every year, as it didn't work every year of the last decade. It is, however, likely to work in the long run.