Portfolio expected return, variance and the role of correlation

How do you find a portfolio’s expected return?

The expected portfolio return is the weighted average of the expected returns of its assets. The weights are each holding’s share of the portfolio’s value at the start of the period, and they add to 1 (the first lesson of this module shows how they drift).

$$E[r_P] = w_1 E[r_1] + w_2 E[r_2] + \cdots + w_N E[r_N] = \sum_{i=1}^{N} w_i\, E[r_i]$$

Everything in this lesson uses one running example. An investor puts 60% in an equity fund (expected return 9%, standard deviation 18%) and 40% in a bond fund (expected return 4%, standard deviation 6%). The correlation between the two funds is 0.2.

Questions also ask for the weight that reaches a target return.

Expected return is the easy part: the weights tell you the whole story, and how the assets move together plays no role. Risk is different.

How do two returns move together?

Covariance is the probability-weighted average of the product of the two returns’ deviations from their own expected values.

$$\sigma_{i,j} = E\big[(r_i - E[r_i])(r_j - E[r_j])\big]$$

Read the sign like this:

• Positive: when one return is above its expected value, the other tends to be above its own too (and below together). Average positive relationship.
• Negative: when one is above, the other tends to be below. Average inverse relationship.
• Zero: the returns are unrelated in this linear sense.

$\sigma_{i,i} = \sigma_i^2$: the covariance of a return with itself is its variance. That is why variances sit on the diagonal of the variance–covariance matrix later in this lesson.

Covariance is hard to read because its size depends on how volatile the assets are. Correlation removes that by dividing by both standard deviations, so it has no units and lies between −1 and +1.

$$\rho_{1,2} = \frac{\sigma_{1,2}}{\sigma_1\, \sigma_2}$$

Most exam questions give you the correlation and ask you to rebuild the covariance:

$$\sigma_{1,2} = \rho_{1,2}\, \sigma_1\, \sigma_2$$

Exam tip Pick one unit system and stay in it. Decimals (0.18, 0.0324) or percent (18, 324), never a mixture.

How do you calculate the risk of a two-asset portfolio?

Portfolio variance has three parts: each asset’s own variance scaled by its squared weight, and a cross term that counts the covariance twice.

$$\sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2\, w_1 w_2\, \rho_{1,2}\, \sigma_1 \sigma_2$$

The standard deviation is the square root, and it is the number that carries the same units as the returns.

$$\sigma_P = \sqrt{\sigma_P^2}$$

Why does a lower correlation reduce risk?

Only the cross term depends on the correlation. Hold the weights and the two standard deviations fixed, move ρ, and the expected return stays at 7.0% while the standard deviation changes.

Correlation	Portfolio standard deviation	Expected return
+1.0	13.20% (the weighted average)	7.0%
+0.5	12.18%	7.0%
+0.2	11.52%	7.0%
0.0	11.06%	7.0%
−0.5	9.82%	7.0%
−1.0	8.40%	7.0%

At ρ = +1 the formula is a perfect square, so $\sigma_P = w_1\sigma_1 + w_2\sigma_2$ and there is no gain. Any lower correlation pulls $\sigma_P$ below that line. The risk reduction from holding assets that do not move in step is the diversification benefit , and it grows as the correlation falls.

Portfolio varianceweights, own risks and how the assets move together

• Own-variance terms Memorize wᵢ²σᵢ², always positive
• Covariance terms Memorize 2wᵢwⱼσᵢ,ⱼ for every pair
  • Positive covariance adds risk; returns move together
  • Zero covariance adds nothing; returns unrelated
  • Negative covariance subtracts risk; this is the diversification benefit

Correlation between the two assetsweights and standard deviations held fixed; expected return never changes

• ρ = +1 Memorize σ_P is the weighted average of the two σs; no diversification benefit
• ρ between −1 and +1 Memorize σ_P below the weighted average; the lower ρ, the lower σ_P
• ρ = −1 Use it σ_P = |w₁σ₁ − w₂σ₂|; can reach zero

Notice that the ρ = 0 row equals what you get by forgetting the cross term altogether. Dropping it is the same as assuming the funds are unrelated, which is rarely true and never the right answer when the question gives you a covariance.

Go deeper Removing all risk with perfectly negative correlation

At ρ = −1 the variance is a perfect square again: $\sigma_P = |w_1\sigma_1 - w_2\sigma_2|$. It reaches zero when w₁σ₁ = w₂σ₂, which gives w₁ = σ₂ ÷ (σ₁ + σ₂).

For the two funds that is 6 ÷ (18 + 6) = 25% in equity and 75% in bonds. Check: 0.25 × 18% = 4.5% and 0.75 × 6% = 4.5%, so the two terms cancel. In practice no two real assets are perfectly negatively correlated, so risk is reduced but not removed.

Go deeper Where the cross term comes from

The portfolio return is w₁r₁ + w₂r₂. Its variance is the expected squared deviation from its mean. Multiplying out (w₁(r₁ − E[r₁]) + w₂(r₂ − E[r₂]))² gives w₁² times the first variance, w₂² times the second, and two copies of w₁w₂ times (r₁ − E[r₁])(r₂ − E[r₂]). Taking expectations turns that last product into the covariance. The factor of 2 appears because the cross product occurs twice.

How do covariance and correlation matrices hold the inputs?

The same logic extends to any number of assets. The variance–covariance matrix puts the variances on the diagonal and the covariances off it, and each cell is weighted by the product of its two weights. The matrix is symmetric, a mirror image across the diagonal, because $\sigma_{i,j} = \sigma_{j,i}$.

$$\sigma_P^2 = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i\, w_j\, \sigma_{i,j}$$

The same inputs can be shown as a correlation matrix , which has 1s on the diagonal. To build the variance–covariance matrix from a correlation matrix, multiply each correlation by the two standard deviations: $\sigma_{i,j} = \rho_{i,j}\,\sigma_i\,\sigma_j$.

N assets need N variances but only (N² − N) ÷ 2 distinct covariances.

$$\frac{N^2 - N}{2} = \frac{N(N-1)}{2}$$

As N grows, covariance terms outnumber variance terms: ten assets give 10 variance cells and 90 off-diagonal cells, which hold 45 distinct covariances, and 100 assets need 4,950. In a large portfolio, how the assets move together matters more than how risky each one is alone. Lesson 3 turns this into a formula and asks how many assets you need.

Go deeper Why a bigger covariance is not a higher correlation

Take two equity funds with covariance 0.0018 and standard deviations 5% and 4.5%: ρ = 0.0018 ÷ (0.05 × 0.045) = 0.80. Take two bond funds with covariance 0.00028 and standard deviations 1.5% and 2.0%: ρ = 0.00028 ÷ (0.015 × 0.02) = 0.93.

The equity pair has a covariance more than six times larger, yet the bond pair is more tightly correlated. The equity funds are simply more volatile. To compare how closely pairs move together, compare correlations. To compute portfolio variance, you still need the covariances.

Go deeper Forecast versus sample covariance

The covariance in this lesson is forward-looking: it is an expected value, so there is no n − 1. When you estimate it from n past observations of two return series, the sample covariance sums the products of deviations from the sample means and divides by n − 1, as in the statistical measures lesson (Learning Module 5).

Go deeper Expected covariances are not built from expected correlations

A forecast of portfolio variance uses expected covariances: $E[\sigma_P^2] = \sum\sum w_i w_j\,E[\sigma_{i,j}]$. Since $\sigma_{i,j} = \rho_{i,j}\sigma_i\sigma_j$, it is tempting to multiply an expected correlation by two expected standard deviations. In general that product is not the expected covariance, because correlations and volatilities move together (both tend to rise in a crisis). Exam questions give you the covariance, or say the inputs are fixed.

Go deeper Matrix shorthand (not examinable)

With a weight vector w, expected returns μ and variance–covariance matrix V, the expected return is $E[r_P] = w^{\top}\mu$ and the variance is $\sigma_P^2 = w^{\top}Vw$. These are the same sums written compactly, and spreadsheets and optimizers use this form. You will not be asked to do matrix algebra.

How do you get portfolio risk from scenarios?

When the forecast is a set of scenarios with probabilities, find the portfolio’s return in each scenario first, then its probability-weighted mean and variance. That gives the scenario-based portfolio return and variance .

$$r_{P,s} = \sum_{i=1}^{N} w_i\, r_{i,s}$$

$$E[r_P] = \sum_{s} p_s\, r_{P,s}$$

$$\sigma_P^2 = \sum_{s} p_s \left(r_{P,s} - E[r_P]\right)^2$$

Go deeper Mean–variance analysis

Mean–variance analysis evaluates investments using only expected return (reward) and variance of return (risk). It needs just the inputs of this lesson: expected returns, variances and covariances. It holds exactly when investors are risk averse and either returns are normally distributed or investors have quadratic utility. It remains a useful approximation when those conditions fail, which is why it underlies the minimum-variance frontier in lesson 4 of this module.