Variance-covariance matrix
Also written Covariance matrix
The 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.
In plain language
With two securities, portfolio risk needs three numbers you can hold in your head: two variances and one covariance. Add a third security and you need three variances and three covariances. At fifty securities the pairs run into the hundreds.
The variance-covariance matrix is simply the tidy way of laying all of this out. Put the assets along the top and down the side. Each diagonal cell holds an asset's variance; every other cell holds the covariance of the pair where that row and column meet.
The workbook's point is practical: calculating the risk of a portfolio of more than two securities works the same way as for two, but it requires the variance-covariance matrix. There are several methods for building it, and it can be easily calculated on a spreadsheet or in Visual Basic.
How it works
Markowitz's portfolio-risk formula (section 3.6) needs three kinds of input — weights, individual risks, and co-movement:
- the weights of each investment;
- each asset's variance (the diagonal of the matrix);
- the covariance between each pair (the off-diagonal cells), where
Cov_ij = r_ij × σ_i × σ_j.
The matrix is symmetric: the covariance of A with B is the same as B with A, so each pair appears twice.
How fast it grows. For n securities the number of distinct correlation estimates is:
(n² − n) ÷ 2
The workbook's figure: a portfolio of 50 securities needs 1,225 correlation estimates. The potential error arising from all these estimates is estimation risk — and the output of the portfolio allocation is only as accurate as these statistical inputs.
The formula
σ²_port = Σᵢ wᵢ² σᵢ² + Σᵢ Σⱼ (i ≠ j) wᵢ wⱼ Cov_ij
Cov_ij = r_ij × σᵢ × σⱼ
σ_port = √σ²_port
A worked example
The portfolio, risks and correlations are illustrative.
A family office holds Rs 5 crore across three funds: Rs 2.5 crore in a large-cap fund (w = 0.5, σ = 20%), Rs 1.5 crore in a mid-cap fund (w = 0.3, σ = 15%) and Rs 1 crore in a debt fund (w = 0.2, σ = 10%). Correlations: large/mid 0.6, large/debt 0.2, mid/debt 0.4.
Covariances:
Cov(L,M) = 0.6 × 0.20 × 0.15 = 0.0180
Cov(L,D) = 0.2 × 0.20 × 0.10 = 0.0040
Cov(M,D) = 0.4 × 0.15 × 0.10 = 0.0060
The matrix:
| Large | Mid | Debt | |
|---|---|---|---|
| Large | 0.0400 | 0.0180 | 0.0040 |
| Mid | 0.0180 | 0.0225 | 0.0060 |
| Debt | 0.0040 | 0.0060 | 0.0100 |
Portfolio variance:
Diagonal: 0.25(0.0400) + 0.09(0.0225) + 0.04(0.0100)
= 0.010000 + 0.002025 + 0.000400 = 0.012425
Pairs: 2[0.5×0.3×0.018 + 0.5×0.2×0.004 + 0.3×0.2×0.006]
= 2[0.00270 + 0.00040 + 0.00036] = 0.006920
Total variance = 0.019345
σ_port = √0.019345 ≈ 13.9%
The weighted average of the three standard deviations is 0.5(20) + 0.3(15) + 0.2(10) = 16.5%. The portfolio's risk is only about 13.9% — roughly Rs 69.5 lakh of one-standard-deviation movement on Rs 5 crore, instead of Rs 82.5 lakh. The gap is the diversification the off-diagonal cells capture.
Why NISM asks about it
Chapter 3, section 3.6 (Variance of return for a portfolio), gives Markowitz's formula, and section 3.9 (Estimation issues) introduces the variance-covariance matrix for portfolios of more than two securities along with the 1,225-estimates figure. Expect a question on what is needed beyond two securities, the (n² − n) ÷ 2 count, and the three drivers of portfolio risk.
Common exam traps
- The diagonal holds variances; the off-diagonal cells hold covariances.
- For 50 securities the count is 1,225, from (n² − n) ÷ 2 — not 2,500 and not 50.
- Each pair enters the variance twice, because the matrix is symmetric. Forgetting the factor of 2 understates risk.
- Portfolio risk is not the weighted average of individual risks unless every correlation is +1.
- Three variables drive portfolio risk: weights, individual risks and co-movement.
- The workbook does not teach a method for building the matrix — it says several exist and refers readers to financial modelling texts (Benninga). What it tests is why the matrix is needed and how many estimates it implies.
Check yourself
1.For a portfolio drawn from a universe of 50 securities, the number of correlation estimates required is:
- a)50
- b)100
- c)1,225
- d)2,500
Show the answer
Answer: (c) 1,225
For a portfolio of 50 securities, the number of correlation estimates is 1,225, computed from the workbook's formula (n² - n) / 2:
(50² - 50) / 2 = (2,500 - 50) / 2 = 1,225
The formula counts distinct pairs - each of 50 securities pairs with 49 others, giving 2,450 ordered pairs, halved because the correlation of A with B is the same as B with A.
Put this alongside the other two inputs the manager must estimate: the expected return of every asset (50) and the standard deviation of each asset's expected returns (50). Correlations outnumber everything else by more than twelve to one.
Why that matters, in the workbook's own words: "It is important to bear in mind that the OUTPUT of the portfolio allocation DEPEND ON THE ACCURACY OF THE STATISTICAL INPUTS." And the danger has a name: the potential source of error that arises from these estimations is referred to as ESTIMATION RISK.
The computation itself is not the problem. Portfolios with more securities require calculation of a VARIANCE-COVARIANCE MATRIX, for which there are several methods, and which can be easily calculated on a spreadsheet or visual basic application. The difficulty is that every entry is a forecast.
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 XIX-E · Chapter 3: Introduction to Modern Portfolio Theory and Capital Market Theoryintroduced here
- Series X-A · Chapter 14: Introduction to Modern Portfolio Theoryintroduced here
Related 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.
- VarianceA measure of how widely a scheme's periodic returns fluctuate around its own average return — the square of standard deviation, and a risk measure relevant to both debt and equity schemes.
- 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.
- 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.
- 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.
- 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.
- Monte Carlo simulationA VaR method that generates the portfolio's returns at random from chosen statistical parameters, repeats this about 10,000 times, sorts the outcomes and reads off the required percentile.
- 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.