Correlation

Diversification is not achieved by owning many things. It is achieved by owning things that do not move together.

Correlation is the number that says whether they do. It runs from -1 to +1. A value of +1 means the two move perfectly together, in the same direction, up or down. A value of -1 means they move perfectly in opposite directions. The size of the number says how strong the relationship is; the sign says which way it runs.

That is why the workbook calls correlation the most relevant factor in reaping the benefits of risk diversification. Two shares of the same bank are both equities, both Indian, both sensitive to the same policy rate — and owning both is scarcely safer than owning one.

Investments in the same asset class are sensitive to the same major economic and investment factors, so correlation within an asset class is expected to be high, and correlation between two different asset classes is expected to be low. This is the entire analytical basis of asset allocation: you get more risk reduction from moving across asset classes than from adding names inside one.

The mechanism shows up in the portfolio variance formula. For two assets the variance has three terms — the weighted variance of each asset, plus twice the weighted covariance between them, and it is that third term that correlation controls. Push correlation down and the third term shrinks; portfolio risk falls below the weighted average of the individual risks.

At perfect positive correlation there is no diversification benefit at all. Both the expected return and the standard deviation are simple linear combinations, so the risk-return opportunity set is a straight line joining the two securities.

One practical warning the workbook is careful to give: correlations change. They move over time and differ across economic situations, and future correlations may differ from those observed in the past because of changing economic and market regimes. Investors should not rely solely on past correlation metrics.

Portfolio variance, two assets:

  sigma_p^2 = w1^2 x sigma1^2 + w2^2 x sigma2^2 + 2 x w1 x w2 x r(1,2) x sigma1 x sigma2

  w      = weight,  sigma = standard deviation,  r(1,2) = correlation coefficient

Number of covariance terms for n securities:

  (n^2 - n) / 2        e.g. n = 50  ->  (2500 - 50)/2 = 1,225

The workbook's own computation. Two securities, each with an expected return of 15% and a standard deviation of 5%, held in equal weights, with a correlation coefficient of 0.5.

sigma_p^2 = (0.50^2 x 0.05^2) + (0.50^2 x 0.05^2)
          + (2 x 0.50 x 0.50 x 0.5 x 0.05 x 0.05)

          = 0.000625 + 0.000625 + 0.000625
          = 0.001875

sigma_p   = 0.0433, i.e. 4.33%

Each security on its own carries 5% risk. Together they carry 4.33% — and the expected return is still 15%, because a weighted average of 15% and 15% is 15%.

Risk fell by 13.4% and return did not fall at all. Nothing was given up. That free lunch is what correlation below +1 buys.

In rupees. On a Rs 40,00,000 portfolio, a one-standard-deviation year is a swing of Rs 2,00,000 in either security alone, but Rs 1,73,200 for the pair — roughly Rs 26,800 less volatility for the same expected Rs 6,00,000 of return.

Now break it. Set the correlation to +1 and the third term becomes 2 x 0.25 x 1 x 0.0025 = 0.00125:

sigma_p^2 = 0.000625 + 0.000625 + 0.00125 = 0.0025
sigma_p   = 0.05, i.e. 5%

The benefit vanishes entirely. Two perfectly correlated securities are, for risk purposes, one security bought twice.

Chapter 14 (Introduction to Modern Portfolio Theory) derives portfolio variance from covariance and correlation, gives the two-security computation above, and uses perfect correlation to show why the opportunity set becomes a straight line. Chapter 15 (Portfolio Construction Process), section 15.2, is titled "Understanding correlation across asset classes and securities" and supplies the -1 to +1 range and the within-class versus across-class rule. Expect a plug-in-the-numbers portfolio variance question and a conceptual question on which correlation gives the greatest diversification benefit.

• The greatest diversification benefit is at -1, not at 0. Zero correlation helps; perfect negative correlation helps most. Questions offer both as options.
• At +1 there is no benefit whatsoever — portfolio standard deviation is exactly the weighted average of the individual standard deviations.
• Standard deviation is not additive, but at perfect correlation it behaves as if it were. That special case is the only time you may average the risks.
• Square the weights in the first two terms, and do not square them in the third. The cross term is 2 x w1 x w2, not 2 x w1^2 x w2^2.
• Correlation is unit-free and bounded; covariance is neither. Correlation = covariance / (sigma1 x sigma2). A question giving you covariance expects that conversion.
• The covariance count grows with the square of the number of securities. For 50 securities there are 50 variances and 1,225 covariances, by (n^2 - n)/2 — which is the workbook's point about how heavy the estimation burden becomes.
• Past correlation is not future correlation. The workbook warns explicitly that correlations shift with economic and market regimes — typically rising in a crisis, exactly when the diversification was needed.