Expected Return And Variance For Multi-Asset Portfolios

From Two Assets to a Multi-Asset Portfolio

With three or more assets, portfolio risk depends not only on each asset’s standalone variance but also on how every pair of asset returns moves together. A covariance between one pair cannot stand in for the covariances of the other pairs. The calculation therefore requires a separate term for each distinct pair.

This tutorial considers a single holding period and fixed portfolio weights set at the beginning of that period. Returns are random variables with finite expected values and variances. No normality assumption is needed. Unless stated otherwise, returns and weights are expressed as decimals, and the weights sum to 1. The methods are aligned with the SOA VEE Accounting and Finance syllabus.

Definition: Let \(R_i\) be the return on asset \(i\) over the holding period, and let \(w_i\) be its portfolio weight. The portfolio return is \(R_p=\sum_{i=1}^{n}w_iR_i\). Its expected return is \(E[R_p]\), and its variance is \(\operatorname{Var}(R_p)\). For assets \(i\) and \(j\), \(\operatorname{Cov}(R_i,R_j)\) is their return covariance. A covariance is positive when the returns tend to vary in the same direction and negative when they tend to vary in opposite directions.

For weights that sum to 1, the portfolio return is the weighted sum of the asset returns. If the weights do not sum to 1, the same weighted-sum formulas still calculate the return on those positions, but the result need not describe a fully invested portfolio; the interpretation depends on what the weights represent. A negative weight represents a short position when short selling is permitted. The formulas below require that the portfolio return is represented by the stated weighted sum and that the relevant second moments exist.

Expected Return and Portfolio Variance

Linearity of expectation makes the expected return a weighted average of the asset expected returns. Variance, in contrast, includes both the individual variances and the joint variation between each pair.

E[R_p]=\sum_{i=1}^{n}w_iE[R_i]

\operatorname{Var}(R_p) = \sum_{i=1}^{n}w_i^2\operatorname{Var}(R_i) + 2\sum_{1\leq i<j\leq n}w_iw_j\operatorname{Cov}(R_i,R_j)

The first sum has one term per asset. The second sum has one term per distinct pair of assets. It is multiplied by 2 because the full expansion of the variance contains both the \(i,j\) term and the \(j,i\) term. If a problem gives standard deviations \(\sigma_i\), first use \(\operatorname{Var}(R_i)=\sigma_i^2\). Covariance is already a variance-scale quantity; do not square a covariance.

Why Every Pairwise Covariance Appears

The variance formula follows by subtracting the portfolio’s expected return from its return, squaring, and taking the expectation. Write \(E[R_i]=\mu_i\). Then \(R_p-E[R_p]=\sum_i w_i(R_i-\mu_i)\). Expanding the square produces one squared deviation for each asset and two cross-products for each distinct pair:

\begin{aligned} \operatorname{Var}(R_p) &=E\left[\left(\sum_{i=1}^{n}w_i(R_i-\mu_i)\right)^2\right]\\ &=\sum_{i=1}^{n}w_i^2E[(R_i-\mu_i)^2] +\sum_{i<j}2w_iw_jE[(R_i-\mu_i)(R_j-\mu_j)]\\ &=\sum_{i=1}^{n}w_i^2\operatorname{Var}(R_i) +2\sum_{i<j}w_iw_j\operatorname{Cov}(R_i,R_j). \end{aligned}

The middle line uses the fact that the terms for a pair are equal: multiplying the deviations for \(i\) and \(j\) gives the same product as multiplying the deviations for \(j\) and \(i\). This derivation also explains why the covariance term is weighted by \(w_iw_j\), not by \(w_i^2\) or \(w_j^2\).

Record the inputs.
List every weight and expected return. Record either each variance or each standard deviation, and list the covariance for every distinct pair.

Calculate expected return.
Multiply each weight by its asset’s expected return, then add the products.

Calculate the variance terms.
Multiply each asset variance by its weight squared. For each distinct pair, multiply its covariance by twice the product of the two weights.

Check and interpret.
Add all terms, including negative covariance contributions. If a standard deviation is requested, take the square root of the final variance.

Worked Calculations

Worked Example: Three Long Positions With Mixed Covariances

A hypothetical portfolio invests 50% in Asset A, 30% in Asset B, and 20% in Asset C. Their expected returns are 8%, 6%, and 10%. Their standard deviations are 20%, 15%, and 25%. The pairwise covariances are \(\operatorname{Cov}(R_A,R_B)=0.006\), \(\operatorname{Cov}(R_A,R_C)=0.005\), and \(\operatorname{Cov}(R_B,R_C)=-0.0075\). Calculate expected return, variance, and standard deviation. Use decimal returns throughout.

Method. Apply the weighted expected-return formula and the expanded variance formula, including all three distinct pairs. The stated weights sum to \(0.50+0.30+0.20=1.00\).

Expected return. Convert the expected returns to decimals and weight each one:

$$ E[R_p]=(0.50)(0.08)+(0.30)(0.06)+(0.20)(0.10) =0.040+0.018+0.020=0.078. $$

The expected return is \(7.8\%\).

Individual variance terms. Square each standard deviation to obtain its variance, then multiply by the squared weight:

\begin{aligned} (0.50)^2(0.20)^2&=0.25(0.0400)=0.010000,\\ (0.30)^2(0.15)^2&=0.09(0.0225)=0.002025,\\ (0.20)^2(0.25)^2&=0.04(0.0625)=0.002500. \end{aligned}

The individual terms total \(0.010000+0.002025+0.002500=0.014525\).

Pairwise covariance terms. Include each of the three distinct pairs once in the sum, with the factor of 2:

\begin{aligned} 2(0.50)(0.30)(0.006)&=0.001800,\\ 2(0.50)(0.20)(0.005)&=0.001000,\\ 2(0.30)(0.20)(-0.0075)&=-0.000900. \end{aligned}

The covariance terms total \(0.001800+0.001000-0.000900=0.001900\). Therefore,

\operatorname{Var}(R_p)=0.014525+0.001900=0.016425, \qquad \operatorname{SD}(R_p)=\sqrt{0.016425}\approx0.1282.

The portfolio standard deviation is \(12.82\%\), rounded to two decimal places. The calculation checks: all three asset variances and all three distinct pairwise covariances are included, and the variance is positive.

Worked Example: Four Assets and Six Covariances

A hypothetical portfolio has weights \(0.40, 0.30, 0.20,\) and \(0.10\) in Assets A, B, C, and D. Expected returns are \(7\%,9\%,5\%,\) and \(11\%\), respectively. Standard deviations are \(10\%,20\%,15\%,\) and \(25\%\). The covariances are: AB \(=0.005\), AC \(=0.0015\), AD \(=-0.005\), BC \(=0.003\), BD \(=0.005\), and CD \(=0.00375\). Find the expected return and standard deviation.

Method. There are four individual variance terms and \(4(4-1)/2=6\) distinct pairs. The weights sum to \(1.00\).

Expected return.

$$ (0.40)(0.07)+(0.30)(0.09)+(0.20)(0.05)+(0.10)(0.11) =0.028+0.027+0.010+0.011=0.076. $$

The expected return is \(7.6\%\).

Individual variance terms. Squaring the standard deviations gives \(0.0100, 0.0400, 0.0225,\) and \(0.0625\). After weighting by squared weights, the terms are:

\begin{aligned} (0.40)^2(0.0100)&=0.001600,\\ (0.30)^2(0.0400)&=0.003600,\\ (0.20)^2(0.0225)&=0.000900,\\ (0.10)^2(0.0625)&=0.000625. \end{aligned}

Their total is \(0.001600+0.003600+0.000900+0.000625=0.006725\).

Six pairwise terms. For each pair, multiply its covariance by twice the product of its weights:

\begin{aligned} AB &: 2(0.40)(0.30)(0.005)=0.001200,\\ AC &: 2(0.40)(0.20)(0.0015)=0.000240,\\ AD &: 2(0.40)(0.10)(-0.005)=-0.000400,\\ BC &: 2(0.30)(0.20)(0.003)=0.000360,\\ BD &: 2(0.30)(0.10)(0.005)=0.000300,\\ CD &: 2(0.20)(0.10)(0.00375)=0.000150. \end{aligned}

The six terms sum to \(0.001200+0.000240-0.000400+0.000360+0.000300+0.000150=0.001850\). Thus,

\operatorname{Var}(R_p)=0.006725+0.001850=0.008575, \qquad \operatorname{SD}(R_p)=\sqrt{0.008575}\approx0.0926.

The standard deviation is \(9.26\%\), rounded to two decimal places. The negative AD covariance reduces the variance, while the positive covariances increase it. This method avoids overlooking a pair simply because its covariance has a different sign.

Worked Example: A Portfolio With a Short Position

A hypothetical portfolio has weights \(0.60,0.50,\) and \(-0.10\) in Assets A, B, and C. The negative weight represents a short position, and the weights sum to 1. Expected returns are \(6\%,10\%,\) and \(4\%\). Standard deviations are \(12\%,18\%,\) and \(10\%\). Covariances are AB \(=0.0108\), AC \(=-0.0036\), and BC \(=-0.0036\). Calculate the portfolio expected return and standard deviation.

Method. Use the same weighted-sum formulas, retaining the negative weight in every term that contains Asset C.

Expected return.

$$ (0.60)(0.06)+(0.50)(0.10)+(-0.10)(0.04) =0.036+0.050-0.004=0.082. $$

The expected return is \(8.2\%\). The short position’s expected contribution is negative because its weight is negative.

Individual variance terms.

\begin{aligned} (0.60)^2(0.12)^2&=0.36(0.0144)=0.005184,\\ (0.50)^2(0.18)^2&=0.25(0.0324)=0.008100,\\ (-0.10)^2(0.10)^2&=0.01(0.0100)=0.000100. \end{aligned}

These terms total \(0.005184+0.008100+0.000100=0.013384\). The short position’s weight is squared in its individual variance term, so that term is positive.

Pairwise terms.

\begin{aligned} AB &: 2(0.60)(0.50)(0.0108)=0.006480,\\ AC &: 2(0.60)(-0.10)(-0.0036)=0.000432,\\ BC &: 2(0.50)(-0.10)(-0.0036)=0.000360. \end{aligned}

The pairwise terms total \(0.006480+0.000432+0.000360=0.007272\). Hence,

\operatorname{Var}(R_p)=0.013384+0.007272=0.020656, \qquad \operatorname{SD}(R_p)=\sqrt{0.020656}\approx0.1437.

The standard deviation is \(14.37\%\), rounded to two decimal places. The sign of each cross-term comes from the product of both weights and the covariance; it cannot be inferred from the covariance sign alone.

Common Mistake: Counting Each Pair Only Once Without Doubling

In the first worked example, suppose the covariance contributions are calculated as \(w_iw_j\operatorname{Cov}(R_i,R_j)\), but the factor of 2 is omitted. That incorrect calculation gives half the correct pairwise total: \(0.000900+0.000500-0.000450=0.000950\). Adding the individual terms would then give an incorrect variance of \(0.014525+0.000950=0.015475\).

The correction is to use \(2w_iw_j\operatorname{Cov}(R_i,R_j)\) for every distinct pair, yielding the correct variance \(0.016425\). Alternatively, one may explicitly list both ordered cross-products, but must not do both that and multiply by 2. Another frequent error is using a percentage such as \(20\) where the decimal standard deviation \(0.20\) is required; squaring \(20\) instead of \(0.20\) changes a variance by a factor of \(10{,}000\).

Interpretation and Checks

A positive covariance contribution adds to portfolio variance when the product of the two weights is positive. A negative covariance contribution subtracts from variance in that case. If one weight is negative, the weight product changes sign, so the covariance contribution may change direction. The entire calculation must be considered; one negative covariance by itself does not establish that the portfolio variance is lower than a particular comparison portfolio’s variance.

Before accepting a result, check that each standard deviation was squared, each distinct pair was included exactly once with the factor of 2, and all inputs use consistent return units. Portfolio standard deviation is the square root of portfolio variance, not the weighted average of the asset standard deviations. A valid set of joint return assumptions produces a nonnegative portfolio variance; a negative result signals inconsistent inputs or an arithmetic error.

Key takeaway: For \(n\) assets, calculate \(n\) individual variance terms and \(n(n-1)/2\) distinct pairwise covariance terms. Weight each individual variance by the squared asset weight, and each pairwise covariance by twice the product of its two weights.

Check Your Understanding

For each problem, use decimal returns and include every pairwise covariance required.

A hypothetical portfolio has weights \(0.50,0.30,0.20\), expected returns \(6\%,8\%,10\%\), and standard deviations \(10\%,20\%,15\%\). Its covariances are AB \(=0.006\), AC \(=-0.003\), and BC \(=0.009\). Find expected return, variance, and standard deviation. (Answer: expected return \(7.4\%\); variance \(0.00928\); standard deviation \(9.63\%\).)
A portfolio has weights \(0.40,0.35,0.25\), expected returns \(5\%,9\%,7\%\), and standard deviations \(10\%,16\%,12\%\). Covariances are AB \(=-0.0048\), AC \(=0.0036\), and BC \(=0.0096\). Calculate its expected return and standard deviation. (Answer: expected return \(6.9\%\); variance \(0.006692\); standard deviation \(8.18\%\).)
A four-asset portfolio has weights \(0.40,0.30,0.20,0.10\), expected returns \(8\%,5\%,11\%,6\%\), and standard deviations \(10\%,15\%,20\%,10\%\). Pairwise correlations are AB \(=0.2\), AC \(=-0.1\), AD \(=0.5\), BC \(=0.3\), BD \(=0\), and CD \(=-0.2\). Find expected return and standard deviation. (Answer: expected return \(7.5\%\); variance \(0.007045\); standard deviation \(8.39\%\).)
A portfolio has weights \(0.70,0.50,-0.20\), expected returns \(4\%,9\%,6\%\), and standard deviations \(10\%,20\%,15\%\). Covariances are AB \(=0.010\), AC \(=-0.003\), and BC \(=0.006\). Calculate its expected return, variance, and standard deviation. (Answer: expected return \(6.1\%\); variance \(0.02244\); standard deviation \(14.98\%\).)

What You'll Learn