## How Diversification Reduces Risk

Note: Please read the disclaimer. The author is not providing professional investing advice.

Most of us have probably heard that the way to reduce risk is to diversify. And we know that risk is another term for volatility, or how much a security or fund’s rate of return (ROR) might vary in any given year, quarter, etc. from the average.

We also know that the reason high variance is a problem – even if you’re investing for the long term – is because it can take many years of good investments to recover from only one double-digit loss year.

If you haven’t thought too deeply about the topic, you may think that diversification works its magic purely through averaging. After all, if instead of investing in one mutual fund in a given year you instead split your money equally between two funds, you’ll earn the average ROR of the two.

Is the relationship with risk the same? That is, is the effective risk when your money is split over two funds just the average of the two individual funds’ risks (quantified via standard deviation)?

Happily, it’s not!

Let’s look at a simple example. Suppose we have 3 investors who put \$100,000 into the market from January 1, 1999 until December 31, 2005.

Investor A puts all of his money into Fund #1 tracking the utilities sector.

Investor B puts all of her money into Fund #2 following the consumer discretionary sector.

Investor C diversifies by dividing his money equally between the two above funds. At the end of each year, he re-balances his investment so he always has equal dollar amounts in each stock going into the following year.

Let’s look at the annual performance:

Annual RORs

Fund 1999 2000 2001 2002 2003 2004 2005 ave std
dev
#1 -4.5% 21.6% -13.6% -28.8% 26.5% 23.6% 16.4% 3.8% 20.0%
#2 19.2% -17.1% 12.6% -18.6% 37.2% 12.9% -6.6% 3.9% 19.0%
#1&2 7.4% 2.3% -0.5% -23.7% 31.9% 18.3% 4.9% 4.5% 15.9%

Funds #1 and #2 both delivered about the same average (geometric) ROR, that is 3.8% versus 3.9%. The two funds also have roughly the same standard deviation (proportional to risk), 20% vs. 19%.

So should investor C, who split his money between two equally performing funds, have had any particular advantage? Shouldn’t his average ROR and risk (standard deviation) just be the average of that provided by Funds #1 and #2?

Checking the above table we see that his average annual ROR is not 3.85% (average of 3.8% and 3.9%) but a partially higher 4.5%! Likewise his standard deviation is not 19.5% (average of 19% and 20%) but a significantly lower 15.9%!

This is because both the geometric average and standard deviation operations involves nth power and nth root operations.

So we started with 2 funds that delivered about the same ROR at the same level of risk. Intuition might say that splitting your money equally between them would offer no real benefit.

The key is looking at the individual yearly returns of the two funds. They are not highly correlated as in 4 of the 7 years, when one fund was down, the other was up. This translates into a smaller yearly swing when you’re invested in both funds.

And again, your intuition might say that by reducing your yearly swing, you’ll avoid having really bad years but you may also avoid having really good years! True but don’t forget that + and – swings are not created equal. If you have a -50% return one year and a +50% return the next, you don’t break even, you lose 25%! Do the math…

If we examine our 3 investing options, we see that the individual funds both had 2 years of double digit losses, whereas if you had diversified across both, there was only 1 double digit loss year.

This turns out to really help in our example.

Investor A, completely invested in Fund #1, ended up with:
\$100,000 x (1-.045) x (1+.216) x (1-.136) x (1-.288) x (1+.265) x (1+.236) x (1+.164) = \$130,015

Investor B, completely invested in Fund #2, ended up with:
\$100,000 x (1+.192) x (1-.171) x (1+.126) x (1-.186) x (1+.372) x (1+.129) x (1-.066) = \$131,035

Investor C, who split his money each year equally between the two funds, ended up with:
\$100,000 x (1+.074) x (1+.023) x (1-.005) x (1-.237) x (1+.319) x (1+.183) x (1+.049) = \$136,531

Roughly \$5000 more just by diversifying across what may have initially appeared to be equivalent funds. Not bad…

Of course, this is just an example and the diversified return could have turned out lower, but I did use real historical data for two real ETF funds. In practice you may not end up with a higher average return but will almost always* experience lower risk.

* “almost always?” The exception is the rare case in which each stock or fund comprising the portfolio performs exactly the same – in which case you will indeed get the average standard deviation.