Portfolio variance

For fixed weights collected in a column vector $\mathbf{w}$ and a return covariance matrix $\Sigma$,

The transpose symbol turns the weight column into a row so that this matrix product produces a single number. What matters economically is the interaction it retains: each position's return is paired with every other position's return.

A 60/40 allocation with two sources of risk

Its diagonal holds variances, so the 15% input enters as $0.15^2=0.0225$. Substituting correlation 0.2 directly into the off-diagonal cells would mix different units and produce an invalid risk calculation.

Portfolio return is $R_p=0.6R_A+0.4R_B$. Its mean is $0.6(8\%)+0.4(12\%)=9.6\%$. Its variance requires squared weights and covariance:

Why covariance appears twice

Let $X=R_A-\mathbb{E}[R_A]$ and $Y=R_B-\mathbb{E}[R_B]$. Squaring the portfolio's deviation gives

Taking expectations turns the squared deviations into variances and each product into covariance. The two cross terms are equal. This is why the two-asset formula contains $2w_Aw_B\text{Cov}(R_A,R_B)$.

With many assets, either count every ordered pair once, or count each distinct pair once and multiply it by two:

Check it by constructing the actual outcomes

A covariance model should agree with the variance of the portfolio returns it generates. Build two independent factors $U$ and $V$, each equally likely to be $+1$ or $-1$, and define

There are four equally likely joint states. A depends on the common factor $U$; B has a smaller loading on $U$ plus its own independent factor $V$.

Both factors have mean zero and variance one. Consequently A's variance is 0.0225, B's is $0.05^2+0.06=0.0625$, and their covariance is $0.15(0.05)=0.0075$.

More directly, the portfolio return simplifies to

Its centered squared outcomes average to

What changing correlation really changes

Keeping these weights and individual volatilities fixed gives

At correlation zero, volatility is about 13.45%. At correlation one, it is 19%. At correlation minus one, it is 1%: the weighted volatility exposures, 9% and 10%, nearly offset but do not cancel completely.

Negative correlation is therefore unnecessary for a diversification benefit relative to perfect positive correlation. But the direction depends on the exposure signs: with one long and one short position, $w_Aw_B<0$, so higher covariance reduces this pair's variance, holding everything else fixed. Shorting also introduces financing, borrowing and liquidity considerations outside this covariance calculation.

From a clean identity to a credible risk number

The formula is exact for the stated model. Whether its inputs describe the portfolio is a separate question.

• Align the return definition and horizon. Use common observation dates, currency treatment and price-versus-total-return conventions. The one-year inputs above need no annualisation. Multiplying a daily covariance matrix by a trading-day count concerns sums of daily returns under dependence assumptions; it does not generally give the exact covariance of compounded annual simple returns.
• Separate a population from an estimate. For $T$ aligned observations, the usual sample matrix has entries $\widehat\Sigma_{ij}=\frac{1}{T-1}\sum_{t=1}^{T}(R_{i,t}-\bar R_i)(R_{j,t}-\bar R_j)$. With the same rows, centering, divisor and fixed weights, $\mathbf{w}^{\top}\widehat\Sigma\mathbf{w}$ equals the sample variance of the weighted return series. The four states above describe an entire stipulated probability distribution, so their population calculation divides by four. Historical estimates remain uncertain forecasts of future co-movement.
• Require a coherent matrix. A covariance matrix is symmetric and positive semidefinite: $\mathbf{x}^{\top}\Sigma\mathbf{x}\ge0$ for every real exposure vector $\mathbf{x}$. Estimating different pairs on different date sets can violate that requirement. Valid pairwise correlations alone do not guarantee a valid many-asset matrix. A singular matrix can still calculate variance; inversion is unnecessary here.
• Match the exposure model to the holdings. Beginning-of-period wealth weights give the simple-return identity for the held portfolio. Log returns do not aggregate this way. Rebalancing requires a return calculation for each holding period. Signed exposures can instead be measured relative to stated reference capital, with financing included explicitly. For options and other nonlinear positions, full revaluation or a justified local risk approximation is needed before using underlying-factor covariances.

Variance treats upside and downside deviations alike. It is useful for comparing spread around a mean, but cannot by itself describe tail shape, funding pressure or the probability of a damaging drawdown. Choosing weights using expected return and this risk measure belongs to mean–variance optimisation; calculating variance for the weights already held is the step developed here.

Sources

• Elton, E. J., Gruber, M. J., Brown, S. J. and Goetzmann, W. N., Modern Portfolio Theory and Investment Analysis, 9th edition, Chapter 4, “Characteristics of Portfolios in General,” pp. 50–54: portfolio-return aggregation, two-asset variance and covariance, and direct outcome verification.
• Eric Zivot, University of Washington, Portfolio Theory with Matrix Algebra, Spring 2015, slides 6–10: return and covariance-matrix notation. The slides also impose a normal-return model; the variance identity used here follows without that assumption, as the discrete construction demonstrates.