Monday, 5 May 2008

Correlation & Standard Deviation: Worthless Risk Tools for the Investor?

Retired mathematician and investment aficionado Gummy (see link to his great website in my Resources sidebar on the right) wrote a provocative entry (see the lower part titled Correlation means ... what?) on his website in which he casts doubt on the value of standard deviation for risk measurement and correlation for risk control.

He shows, and I am sure his math is impeccable, that an example portfolio with standard deviation of zero as a result of holding two perfectly negatively correlated assets, can lose a lot of money. Huh? Standard deviation as the accepted measure of risk is one of the bedrocks of finance theory. Negatively correlated assets are the foundation of hedging and building portfolios more protected against loss. What gives?

The basic reason I believe that we are ok continuing to believe in standard deviation and negative correlation as essential investment principles is that reality has not ever worked and by all logic should never work the way his example does. For one asset or asset class to produce negative returns in seven out of ten years and to have an average negative return of over 6% per year for a decade is very unlikely. For an uncorrelated second asset to exhibit the same steadily downward behaviour at the same time (Gummy's second asset goes down an average of 3.7% per year) is even more improbable. I went back and scanned Roger Gibson's book on Asset Allocation (my review here) for his many charts and tables of investment returns. Even in the worst period of the 1930s, while stocks were going down year after year, other asset classes like bonds were still producing positive returns. Over ten years, almost no asset class will average negative returns. For example, Seeking Alpha has just published charts of the latest Major Asset Class 1, 3, 5, 10 & 15 Year Returns. Not one had negative annualized ten year returns.

Any investment or asset class must ultimately be expected to have positive returns. Otherwise, why would anyone invest? The beforehand expectation is of course not always what happens, especially where individual companies are concerned. That's why I believe that I am better off with funds and ETFs that spread the individual company risk over many companies so that the expected pattern of positive asset class returns emerges.

With respect to correlation, the patterns are variable according to the choice of interval, as the upper part of Gummy's post shows in his analysis of correlation according to the number of days, and they are also unstable over time even for a set interval (which he does not explore). Even with these considerable imperfections, non- or negative correlation between asset classes yields diversification benefits in a portfolio in the form of a reduction of risk, or chance of loss.

To illustrate, I took Gummy's data and graphed the cumulative value of each asset X and Y, separately and as a 50-50 portfolio. The results are in the chart you see. Would you rather have owned asset X or Y or the portfolio of the two? Suppose along the way in some random year, unknown in advance, you had needed to sell, what would be you be better off with, one or the other asset or the portfolio? I think I'd rather have the portfolio despite the steady 5% loss year after year.

Just for fun, I also tried out another portfolio best practise, which is to periodically rebalance the portfolio back to the original 50-50 allocation to each asset. Lo and behold, rebalancing once a year after each year's return produces a portfolio whose individual assets vary much less in dollar value year to year and whose end value is considerably higher - 60% of the starting investment vs only 52% - than the non-rebalanced portfolio. To be more realistic, the cost of trades would have to be subtracted from the rebalanced portfolio but the impact would depend on the size of the portfolio - as the amount invested got very large the impact of that trading cost would become very small. In other words, even with trading costs, the rebalanced portfolio would probably come out ahead.

From a practical point of view, the only type of investment asset that one is likely to find with perfect and predictable negative correlation to another is insurance. e.g. through the use of options. But insurance comes at a net cost that will reduce overall returns. Conversely, no asset classes have perfect positive correlation, which means that at least some diversification benefit can always be obtained with different assets.

In the real world it worth remembering that there is no such thing as a risk-free asset. Even T-bills, though they are free of default risk, are subject to inflation and taxes that have at times in the past resulted in net losses in real purchasing power terms.

The practical world is messy and imperfect as Gummy found but the principle of using standard deviation as a measure of risk is still very useful to the investor, as is seeking out asset class combinations in a portfolio with positive long term returns as negatively correlated as possible.


Michael James said...

I agree with you that correlation and standard deviation are not worthless. But, their importance can be overstated. If two investments have the same expected return, but one has a higher standard deviation, the other portfolio with the lower standard deviation will have better long-term returns.

If investment B is negatively correlated with investment A, then owning some of B with your A will lower the standard deviation of your portfolio. This would seem to improve long-term returns. But, we haven't taken into account B's expected return yet. If B's expected return is less than A's expected return, then this will drag on the mixed portfolio's returns. Reducing standard deviation has to be balanced against the effect on expected return.

A good example of this is buying put options on a stock index ETF you own. The put option helps a little in that it reduces the portfolio's standard deviation. But, the expected return of the option is negative. The overall strategy is expected to give lower long-term returns than simply owning the ETF, despite the reduced volatility.

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