This is the third of three articles about Diversification and Risk Reduction. In this article we examine monthly and daily correlations between Canadian, US and EAFE indexes and ETF's. We find that indeed recent levels of correlation have been relatively high, and then we explore the impact this has had on overall risk level

We continue our exploration of risk by returning to a point made in Part 1: the risk-reducing aspect of diversification is largely a function of imperfect correlations between components of a portfolio. We argued that the recent and continuing market crisis was not “a failure of diversification”, since the crisis was essentially a global market event and hence not diversifiable by definition. However, we left open the possibility that correlations were indeed very high, which might be a contributing factor to higher than normal levels of risk.

__Monthly and Daily Correlations__

With our focus on investment opportunities for retail and smaller institutional investors, we will look only at correlations between Canadian stocks, US stocks and EAFE (developed markets ex North American) stocks. These correspond to the three ETF’s we followed in Part 2, XIU (Canadian SP/TSX 60), XSP (S&P500 with currency hedged into $CN), and XIN (EAFE, with currencies hedged into $CN).(1)

We have monthly price data for all three underlying market indexes back to December 1969, and in Figure 1 we plot the rolling 12-month, 24-month and 60-month average correlations between the SPTSX, S&P500, and EAFE, all in local currencies.(2)

While Figure 1 shows that correlations for all three time-spans are currently relatively high, only the longest 60-month measure (the blue line) is actually at its highest point since the beginning of 1970. The 24-month measure (green line) has recently peaked to roughly match its all-time highs in 2003, but the shortest 12-month measure (red line) has not matched its all-time high posted in 2002.

Figure 2 shows the three ETF’s investors could have used. Over this short history since 2000,(3) the 60-month average correlation (blue line) is at its highest point, but the 24-month and 12-month correlations are very high but not at their absolute peaks.

Daily correlations show much the same results. We have common daily history for the three market indexes underlying the ETF's back to 1987, and show average correlations for 20 days and 60 days in Figure 3. Once again we see that average correlations are very high, but have not reached their respective peaks --- this time in 2001.

Again this is confirmed by looking directly at the ETF’s in Figure 4 below, although the data there only begin in December of 2005.(4)

Recent levels of 20-day average correlations have been high, but slightly lower than peak levels that occurred several times during 2007. The peak in 60-day correlation from November 2008 roughly matches the peak in December 2007.

__The Implications of Changing Levels of Correlation__

Of all five monthly and daily correlation measures tracked in the previous section, only the longest monthly measure (60-month correlations) has made new highs in the past 12 months. Generally correlations have been very high, but not at unprecedented levels. So how do we evaluate these findings relative to risk?

We can see the effect of correlation changes with a simple example. We build a simple case with three assets, such as the three ETF’s we have used above. First, we assume that they all have the same expected returns and risks. For example, we might believe that all three equity markets should generate returns of 4.0% per year after inflation, and that their expected risk is 16%. No matter what combination of the three assets we were to hold, our expected return would remain at 4.0%. However, as long as they are not perfectly correlated, there will be a risk reduction benefit by holding all three assets.

We will vary the correlations, but we make the simplifying assumption that in each case the correlations between each pair are the same. Then if we “optimized” with no further constraints, our lowest risk position would be to hold all three assets in equal portions. Varying the correlation level, we would have the following expected risks:

- If correlations are 1.0 then expected portfolio risk is 16.00% ---no reduction in risk
- If correlations are 0.9 then expected portfolio risk is 15.46% ---a reduction of 3.37%
- If correlations are 0.8 then expected portfolio risk is 14.90% ---a reduction of 6.88%
- If correlations are 0.7 then expected portfolio risk is 14.31% ---a reduction of 10.56%
- If correlations are 0.6 then expected portfolio risk is 13.70% ---a reduction of 14.38%
- If correlations are 0.5 then expected portfolio risk is 13.06% ---a reduction of 18.38%(5)

To put this in perspective, using the 60-month average correlation measure, we can see from Figure 1 that until 1994, the average correlation was mostly between .6 and .7, with a few dips below to as low as .53. From 1995 to mid 1997 the measure took a prolonged dip down to about .5, before starting a multi-year climb to .8 and above. So for the last ten years, from late 1998 to the present, correlations have generally held between .7 and .8, with several extended periods above and one spike below. Its all-time high was .84 in October 2008.

With many investors believing in the continuing convergence of global equity markets, it may be reasonable to assume that 60-month correlation levels will stay at higher levels such as between .7 and .8, with brief spikes up or down. If this is the case, then the risk-reducing effects of correlation will be less going forward — between 6% and 11% on average — rather than 11% to 15% at lower levels of correlation witnessed from the 1970’s to mid 1990’s.

So while the risk-reducing benefits of imperfect correlations have been lower over the past ten years, and may continue at these lower levels going forward if correlations remain high, the impact of higher correlations on risk levels is generally quite small. For example, under the conditions of our example, the jump in risk is from 13.7% at a correlation of .6 to 14.9% at a correlation of .8. This change pales by comparison to the outright changes in risk we actually observed in Part 2 of this paper. Overall, we can conclude that recent higher levels of correlation have been a small contributor to higher levels of risk.

__Summary and Lessons Going Forward__

In Part 1 of "Diversification and Risk" we argued that diversification has not "failed". By diversifying equities globally, investors have obtained exposure to the global market, and hence are unprotected from global equity events that affect all global markets.

In Part 2 we examined measures of risk directly, and found that in most cases realized risks have been high, but not at unprecedented levels. The one exception was daily volatility, which was extremely high and matched only once since 1950 in value, but not in duration. While the daily risk levels are virtually unprecedented, they have not yet shown up in the longer term measure of risk.

In Part 3 we examined correlations between Canadian, US and EAFE markets, and again found high levels of correlation. However the impact of higher levels of correlation on risk is not very large in itself, and the outright levels of risks as found in Part 2 are much higher than would be generated simply by increases in correlations.

We believe that it is important to measure and evaluate realized market risks on an ongoing basis. In general, a better diagnosis of the properties of the assets in which you are investing should help you to make more insightful investment decisions. And when you have experienced problems, as many investors have in the past year, then a correct diagnosis of what has gone wrong in the past is more likely to help you better position your investments for the future.

In particular, if realised risk levels in equity markets have not been outside the bounds of reasonable expectations, then "unexpected levels of risk" is not a useful diagnosis of portfolio return problems in 2008. Investors should have been aware of ongoing risk levels, and should have been aware of the magnitude of any potential downturn. We tend to believe that a more useful diagnosis of the problem for most investors would point towards the implicit or explicit overconfidence in, or higher than reasonable forecasts for, the performance of equity markets towards the end of 2007. In the face of known risk levels, this overconfidence in equity prospects would have led to higher equity allocations than is justifiable.

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(1) Again we chose the XSP and XIN ETF’s with currencies hedged into $CN as their returns more closely approximate the returns of investments in the local markets in local currencies, with the effect of movements in the $CN relative to other currencies largely removed. (Return to text)

(2) Local currencies closely approximate the effect of holding the foreign ETF’s with currencies hedged back to $CN. With three assets, we have three sets of correlations: XIU vs. XSP, XIU vs. XIN, and XSP vs. XIN. The numbers shown on the graphs take the average of the three correlations. (Return to text)

(3) The XSP and XIN ETF’s only started trading at the end of 2005, but prior to trading we have monthly data for their benchmark indexes back to 2000 (See endnote 6 in Part 2.) (Return to text)

(4) For the XSP and XIN, we do not have daily benchmark data prior to the commencement of trading, and so we can only use data from the commencement of trading in December 2005. (Return to text)

(5) Our calculations assume equal correlations between all pairs, which won’t actually hold in practice, but the examples are meant to give readers a feel for the risk-reducing impact of changing correlations. The calculations themselves are based on the implied covariances between all assets, and are explained in most finance textbooks.

A second point is that there will often be other constraints in play. For example, as Canadians we acknowledge that the Canadian equity market is less than 3% of the “developed” equity market, while the combination of EAFE and the S&P500 approaches 90%, split roughly 50-50. If we are interested in more global equity exposure, we might want to hold a lower proportion in Canadian stocks. So suppose in the equity portion of your portfolio you held 15% XIU, 40% XSP and 45% XIN. With the previous parameters for risk and return, and equal correlations between all pairs, the risk reducing results would be:

- If correlations are 1.0 then expected portfolio risk is 16.00%
- If correlations are 0.9 then expected portfolio risk is 15.50%
- If correlations are 0.8 then expected portfolio risk is 14.98%
- If correlations are 0.7 then expected portfolio risk is 14.45%
- If correlations are 0.6 then expected portfolio risk is 13.89%
- If correlations are 0.5 then expected portfolio risk is 13.31%

You can see that the effects of risk reduction are slightly less than in the unconstrained case. However the difference is not dramatic. (Return to text)

Structured Capital presents a book by Tim Appelt:

Learning and Playing the Long-term Investment Game

Historical analysis of long-term global equity and bond returns is used to develop an analytical framework for a historical attribution of returns. In turn this attribution approach is used to develop expectations of future returns that acknowledge the past but take into account current market conditions.

Further information: