Correlation Coefficients
If you decide to invest some money in stock indices then the first problem you should think about is the index correlation. If two indices are highly correlated there is no sense to buy both of them. It is equivalent buying one index. Diversification effect will be very small.

We calculated the correlation coefficients for 11 US stock indices:
Dow Jones Industrial Average
Dow Jones Transportation Average
Dow Jones Utility Average
NASDAQ 100
SP 500
SP MidCap
Silver and Gold Index
Oil Index
Semiconductor Index
Russell 2000
Biotech Index

The definitions of the average returns, risk, and correlation coefficients one can find in the e-book How to win the stock market game. We have studied the index histories for the period from 1985 to 2001. For some indices (semi and biotech) the history is shorter. The figure on the left shows the correlation coefficients between the Dow Jones Industrial Average and other indices. Correlation coefficients have been calculated for various returns: from 1 to 64 days.

We define m day return as
return = (P(i+m)/P(i) -1)*100%
where P(i) is the closing index price at day i.

One can see that Gold and Biotech indices correlate very little with the Dow. For very short term returns the Gold-Dow correlation coefficient is close to zero. The highest correlation coefficients have been found for Dow-SP 500, Dow-DJTA, Dow-Russell 2000, and Dow-SP MidCap. It was a surprise that the Dow correlates relatively highly with the NASDAQ 100 index.

There is a strong time dependence of the Dow - Biotech correlation coefficient. For 20 day returns (and longer) there is very little correlation between these indices.

The next table shows the correlation coefficients between all indices for 20 day returns.

One can see that the Dow Jones Utility Average (DJUA) and the Gold/Silver Index have very little correlation with other indices. Utilities are even negatively correlated with the semiconductor index. The highest correlation is between DJIA and SP 500. One can conclude that it is not a good idea to have these two indices in the portfolio together. They are practically moving together.

Efficient Two Index Portfolio
Now we consider how to build the efficient investment portfolio using two indices. We will suppose that an investor wants to buy two stock indices to have minimal risk/return ratio. In other words an investor wants to have optimal risk with a good return.

Suppose an investor spends X part of his/her capital to buy index #1 and (1-X) part of his capital to buy index #2. Example:

X = 0.2  (20 %)
1-X = 0.8 (80 %)

For calculation we will use 20 day return data. Using other time scale does not change the results significantly. Let us introduce some parameters:

r1 is the average monthly return of index 1
r2 is the average monthly return of index 2
s1 is risk (the standard deviation of the returns) of index 1
s2 is risk (the standard deviation of the returns) of index 2
c12 is the correlation coefficient for the returns of these indices
r is the average return of investment portfolio
s is risk (the standard deviation of the returns r) of investment portfolio

For calculations of r and s one can use simple equations:

r =  X * r1  +  (1-X) * r2

s ² = X ² * r1 ²  +  (1-X) ² * r2 ²  +  2 * X * (1-X) * r1 * r2 * c12

The next table shows the average returns, risk and risk to return ratios for the studied indices. We select the time period form 1994 to December of 2001 when all indices exist: the biotech and semiconductor indices were introduced recently. Due to dividends we also added 0.35% to 20 day returns of DJUA.

It is interesting to note that for this period of time the best risk/return ratio was for the SP MidCap and DJUA indices.  The highest return was for Biotech index but risk of using this index is high. The worst risk/return ratio was for the Dow Jones Transportation Average.

Using equations showed above we have calculated the risk/return ratios for all index pairs for different values of X from 0 to 1. The figure in the left shows example of calculations. It presents dependencies of the risk/return ratios for various combinations of DJIA and other indices. The best combinations (lowest risk/return ratios) for this period of time were observed for:

50% DJIA and 50% DJUA  (20 day return = 1.1%)
75% DJIA and 25% Biotech (20 day return = 1.5%)

In the table below we show some index combinations with low values of risk/return ratios.

Notes
We have shown the method of quick building the efficient investment portfolio. We must mention that the results depend on  the input parameters: average returns, risk, and correlation coefficients. The last two parameters are rather stable and do not depend much on the period of time considered. The returns are function of time. If an investor wants to make similar calculations he must select the time frame as large as possible.