Covariance
A 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.
In plain language
Two shares can each be risky on their own and still make a calm portfolio together — if one tends to rise when the other falls. Covariance is the number that captures that tendency.
The workbook's definition (Box 3.1): covariance is a measure of the degree to which two variables "move together" relative to their individual mean values.
- Positive covariance: when one is above its average, the other tends to be above its average too.
- Negative covariance: when one is above, the other tends to be below.
- Near zero: no consistent pattern.
The trouble with covariance is that its size is hard to read. It is measured in squared return units, and it grows with how volatile each asset is. So the workbook immediately converts it into the correlation coefficient, which strips out the individual volatilities and always lies between −1 and +1.
How it works
Harry Markowitz used covariance to derive the risk of a portfolio. In the workbook's general formula, portfolio risk depends on three things only:
- the weights of the investments,
- the risk (standard deviation) of each, and
- the co-movement between each pair — the covariance.
The covariance between two assets is linked to correlation by Cov(ij) = r(ij) × σ(i) × σ(j). Put another way, correlation is covariance divided by the product of the two standard deviations.
Two useful facts the workbook adds:
- The covariance of an asset with itself is its variance. That is why the market portfolio's beta works out to exactly 1.
- Beta is a covariance ratio: the covariance of a security with the market divided by the variance of the market.
Table 3.4 applies this to two securities, each with expected return 15% and standard deviation 5%, held 50:50, with correlation 0.50. The covariance is 0.50 × 0.05 × 0.05 = 0.00125, and portfolio variance is:
0.5²(0.05²) + 0.5²(0.05²) + 2(0.5)(0.5)(0.00125)
= 0.000625 + 0.000625 + 0.000625 = 0.001875
σ(portfolio) = √0.001875 = 4.33%
Both securities have 5% risk; together they have 4.33%. The gap is the covariance at work.
The formula
Cov(i,j) = Σ Pₛ × [Rᵢ,ₛ − E(Rᵢ)] × [Rⱼ,ₛ − E(Rⱼ)]
r(i,j) = Cov(i,j) ÷ (σᵢ × σⱼ)
σ²(portfolio of two) = wᵢ²σᵢ² + wⱼ²σⱼ² + 2wᵢwⱼCov(i,j)
A worked example
The return forecasts for Stock Y are illustrative. Stock X uses the workbook's own Table 3.2 figures.
An investor splits Rs 10 lakh equally between two stocks. Four states of the economy are equally likely (probability 0.25 each):
| State | Stock X | Stock Y | X deviation | Y deviation | Product |
|---|---|---|---|---|---|
| 1 | 8% | 16% | −3 | +6 | −18 |
| 2 | 10% | 4% | −1 | −6 | +6 |
| 3 | 12% | 14% | +1 | +4 | +4 |
| 4 | 14% | 6% | +3 | −4 | −12 |
Expected returns: X = 11%, Y = 10%.
Covariance = 0.25 × (−18 + 6 + 4 − 12) = 0.25 × (−20) = −5 (in %²)
σX = √[0.25 × (9+1+1+9)] = √5 = 2.24%
σY = √[0.25 × (36+36+16+16)] = √26 = 5.10%
Correlation = −5 ÷ (2.24 × 5.10) = −0.44
The covariance is negative: Y tends to do badly when X does well. Now the portfolio:
σ²p = 0.25(5) + 0.25(26) + 2(0.5)(0.5)(−5) = 1.25 + 6.50 − 2.50 = 5.25
σp = √5.25 = 2.29%
Expected return is 10.5%, or Rs 1.05 lakh. And the portfolio's risk, 2.29%, is barely above the safer stock's 2.24% — even though half the money is in a stock more than twice as volatile. That is what a negative covariance buys.
Why NISM asks about it
Chapter 3, section 3.6 (Variance of return for a portfolio) and Box 3.1 define covariance and correlation; Table 3.4 is the two-security computation; section 3.13 reuses covariance to derive beta. The first sample question for Chapter 3 asks what happens to portfolio risk when the correlation between two stocks falls — it decreases. Expect that, the −1 to +1 range, and "the covariance of an asset with itself is its variance".
Common exam traps
- Covariance is not bounded; correlation is. Only the correlation coefficient runs from −1 to +1.
- Correlation = covariance ÷ (σᵢ × σⱼ), not covariance × the standard deviations.
- Lower covariance lowers portfolio risk, all else equal. It does not change expected return, which is a plain weighted average.
- The covariance of an asset with itself is its variance — the footnote that makes market beta equal 1.
- Negative covariance is not the same as a hedge. A correlation of −0.44 reduces risk; only −1 can, in theory, remove it.
- The risk-free asset has zero covariance with every risky asset, because its return does not vary at all.
Check yourself
1.If a portfolio comprises two stocks and the correlation coefficient between them were to decrease over time, everything else remaining constant, the portfolio's risk would:
- a)Decrease
- b)Remain constant
- c)Increase
- d)Fluctuate positively and negatively
Show the answer
Answer: (a) Decrease
Decrease. Portfolio variance contains a covariance term built from the correlation: Cov(ij) = r(ij) x sigma(i) x sigma(j). Lower the correlation and that term shrinks; make the correlation negative and the term turns negative, actually subtracting from total variance.
This is exactly why the workbook lists three variables which affect portfolio risk: (1) weights of investments, (2) risk of investments, and (3) CO-MOVEMENT between investments. The third is Markowitz's contribution, and it is the only one that can work in the investor's favour.
The workbook's own Table 3.4 demonstrates it numerically. Two securities, each with a standard deviation of 5 per cent, correlated at 0.50, combine in equal weights into a portfolio with a standard deviation of 4.33 per cent - with no loss of return, since both return 15 per cent.
Had the correlation been +1.0, the portfolio standard deviation would have been exactly 5 per cent - no benefit at all. Perfectly correlated assets are, for risk purposes, the same asset.
And it is the same fact that shapes the geometry: when the correlation between them is LESS THAN PERFECT 1, the various combinations get plotted on a CURVE rather than a straight line.
2.A completely diversified portfolio would have a correlation with the market portfolio that is:
- a)Equal to zero, because it has only unsystematic risk
- b)Equal to one, because it has only systematic risk
- c)Less than zero, because it has only systematic risk
- d)Less than one, because it has only unsystematic risk
Show the answer
Answer: (b) Equal to one, because it has only systematic risk
Equal to one, because it has only systematic risk.
A completely diversified portfolio has, by definition, eliminated all unsystematic risk. What remains is only the risk common to the whole market - so nothing is left to make the portfolio move differently from the market, and its correlation with the market must be +1, which the workbook defines as perfect positive correlation, meaning that returns for the two assets move together in a completely linear manner.
This is the fact on which CAPM rests. When investors invest in the portfolio M (market portfolio), THAT HAS ONLY SYSTEMATIC RISK, they need to be COMPENSATED FOR BEARING ONLY THE SYSTEMATIC RISK. BECAUSE THE UNSYSTEMATIC RISK IS DIVERSIFIED, THERE IS NO NEED FOR ANY COMPENSATION FOR THE SAME.
That is why the CAPM equation contains beta and not total standard deviation: beta is a good proxy of the NON-DIVERSIFIABLE portion of the risk.
And it explains why the market's own beta is 1. The risk of the market portfolio is the covariance of the market portfolio WITH ITSELF, which is the VARIANCE of the market portfolio - the workbook noting that the covariance of an asset with itself is its variance - so the beta formula's numerator and denominator are identical.
3.The correlation coefficient between the return on the market portfolio and the return on a risk-free asset is:
- a)+1
- b)-1
- c)Between 0 and +1
- d)Zero
Show the answer
Answer: (d) Zero
Zero. Since the return of a risk-free asset is CERTAIN, its STANDARD DEVIATION IS ZERO. Also, the COVARIANCE between the return of the risk-free asset and the risky asset IS ZERO TOO.
Box 3.2 lists four characteristics of a risk-free asset:
- An asset where its returns yield ZERO STANDARD DEVIATION
- ZERO CORRELATION with all other risky assets
- Provides the RISK-FREE RATE OF RETURN
- Will lie on the VERTICAL AXIS of a portfolio graph, as it has no risk
The intuition is simple: correlation measures how two things move together. A risk-free asset does not move at all - its return is certain - so it cannot move with anything. There is nothing to correlate.
And this fact has a large consequence. When a risk-free asset is combined with a risky portfolio, every term in the Markowitz risk formula involving the risk-free asset vanishes, leaving portfolio risk as simply the weight in the risky asset multiplied by the risk of the risky asset. Because that relationship is linear, such portfolios plot on a straight line - which is what makes the Capital Market Line straight, where Markowitz's frontier was curved.
Where this is taught
- Series XXI-B · Chapter 15: Introduction to Modern Portfolio Theoryintroduced here
- Series XIX-C · Chapter 4: Introduction to Modern Portfolio Theoryintroduced here
- Series X-A · Chapter 14: Introduction to Modern Portfolio Theoryintroduced here
- Series XIX-E · Chapter 3: Introduction to Modern Portfolio Theory and Capital Market Theoryintroduced here
Related terms
- BetaHow sharply a share moves relative to the market index — beta 1 moves with the index, above 1 amplifies it, below 1 dampens it. The standard measure of systematic risk.
- 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.
- CorrelationA 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.
- 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.
- 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.
- 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.
- Estimation riskThe potential for error that comes from having to estimate the returns, risks and correlations that feed portfolio optimisation — the output is only as accurate as those statistical inputs.
- Optimum portfolioThe combination of investments, chosen from the feasible portfolios, that has desirable risk-return characteristics within the investor's constraints and meets that investor's objectives.
- Risk-free AssetAn asset whose return is certain: zero standard deviation, zero correlation with every risky asset, earning the risk-free rate and plotting on the vertical axis of a risk-return graph.
- 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.