Correlation
Also written Correlation coefficient · Coefficient of correlation
A measure of the strength and direction of the relationship between two variables, running from -1 to +1, and the single factor that determines how much risk diversification actually removes.
In plain language
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.
How it works
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.
The formula
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
A worked example
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.
Why NISM asks about it
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.
Common exam traps
- 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.
Where this is taught
Free preparation for NISM Series X-ARelated terms
- Standard deviationA measure of how far returns typically stray from their own average — the standard statistic for total risk, counting company-specific and market-wide causes alike.
- DiversificationSpreading an exposure across holdings that do not move together, so that total risk falls by more than total return does — minimising risk per unit of return.
- Asset allocationThe decision on how to distribute a client's wealth across asset classes — the first decision in building a portfolio, and the one that explains most of what the portfolio then does.
- CovarianceA measure of the degree to which the returns of two assets move together relative to their own averages — the building block of portfolio risk, standardised into the correlation coefficient.
- Efficient frontierThe umbrella-shaped curve of portfolios offering the highest expected return for each level of risk, or the lowest risk for each level of return — any portfolio below it is dominated.
- Modern Portfolio TheoryMarkowitz's framework for building portfolios on expected return and risk together, in which the co-movement between holdings — not their individual riskiness — decides the risk of the whole.
- Utility scoreA single number that scores an investment for one particular investor: expected return minus a penalty for risk, scaled by that investor's own risk aversion.
- Arbitrage Pricing TheoryA multifactor alternative to CAPM, developed by Stephen Ross, in which expected return depends on several risk factors — each with its own beta — instead of the market alone.
- Information CoefficientThe correlation between a manager's forecasts and what actually happened — a measure of forecasting skill from −1 (always wrong) through 0 (coin toss) to +1 (always right).
- Perfect Positive CorrelationA correlation coefficient of +1: two assets' returns move together in a completely linear way, so combining them gives no diversification benefit at all.
- Perfect Negative CorrelationA correlation coefficient of −1: two assets' returns move in opposite directions, delivering the complete benefit of diversification and making a zero-risk combination possible.
- Variance-covariance matrixThe grid of every asset's variance and every pair's covariance that is needed to compute the risk of a portfolio holding more than two securities.
- Probability distribution of expected returnsThe list of returns an investment might deliver, each with the chance of it happening — the form in which Modern Portfolio Theory assumes an investor sees every investment choice.
- Expected Portfolio ReturnThe weighted average of the expected returns of the holdings in a portfolio, each weight being that asset's share of portfolio value — the return half of Modern Portfolio Theory.
- Variance of a SecurityThe probability-weighted average of the squared gaps between a security's possible returns and its expected return — the forward-looking measure of a single asset's risk in Modern Portfolio Theory.
- PortfolioUnder the SEBI PM Regulations, the total holdings of securities and goods belonging to any person — the basket whose return is a weighted average but whose risk also depends on correlation.
- Cross sectional diversificationHolding equities across different industries, sectors and geographies at a given time, so that sector- and company-specific risks offset. It cannot remove market risk.
- Real InvestmentsNon-financial investments such as real estate, gold and commodities — physical assets, as opposed to financial investments like equity and bonds, which are claims on future cash flows.
- Time diversificationReducing the risk of an equity investment by holding it across many time periods rather than across many securities, on the belief that bad periods are cancelled out by good ones.
- Counter-cyclical businessA business whose products stay in demand, or even do better, when the broader economy is in a downturn — the opposite of one that rises and falls with the business cycle.