Integrated Portfolio Return And Risk Workshop

Introduction

A portfolio decision is not fully evaluated by looking at each asset separately. The investor needs to know the portfolio’s expected return and the risk created by the assets’ variances and their relationships with one another. A complete analysis also asks which holdings contribute most to that total risk, and how the results change when the allocation is revised.

This workshop brings those calculations together. It uses a hypothetical one-period setting: asset returns are measured over the same period, portfolio weights are fixed at the start of that period, and no contributions or withdrawals occur during it. Expected returns, variances, and covariances are treated as inputs. No particular probability distribution is required to calculate expected return or variance; standard deviation describes dispersion, not every possible feature of the return distribution.

Definition: For a portfolio with $n$ assets, $w_i$ is the fraction of portfolio value invested in asset $i$, and $R_i$ is its one-period return. The portfolio return is $R_p=\sum_{i=1}^{n}w_iR_i$. In a fully invested portfolio without borrowing, $w_i\geq 0$ and $\sum_{i=1}^{n}w_i=1$. The expected return is $E[R_p]$, and the portfolio standard deviation is $\sigma_p=\sqrt{\operatorname{Var}(R_p)}$.

The Integrated Calculation

The portfolio’s expected return is the weighted average of the assets’ expected returns. Portfolio variance also uses the weights, but it must account for every pair of assets through covariance. If standard deviations and correlations are supplied instead, the covariance of assets $i$ and $j$ is $\operatorname{Cov}(R_i,R_j)=\rho_{ij}\sigma_i\sigma_j$. A covariance matrix $\Sigma$ places each variance on its diagonal and each pairwise covariance in the corresponding off-diagonal positions.

E[R_p]=\sum_{i=1}^{n}w_iE[R_i], \qquad \operatorname{Var}(R_p)=\mathbf{w}^{\mathsf T}\Sigma\mathbf{w} =\sum_{i=1}^{n}\sum_{j=1}^{n}w_iw_j\operatorname{Cov}(R_i,R_j)

The diagonal terms in the double sum are the weighted variance terms. Each covariance between two distinct assets appears twice: once with $i,j$ and once with $j,i$. Since covariance is symmetric, a useful equivalent expression is the sum of the weighted asset variances plus twice the weighted covariances for each distinct pair.

Derivation: From the definition of portfolio return, $R_p-E[R_p]=\sum_i w_i(R_i-E[R_i])$. Squaring this expression gives $\sum_i\sum_j w_iw_j(R_i-E[R_i])(R_j-E[R_j])$. Taking expectations changes each product’s expectation to $\operatorname{Cov}(R_i,R_j)$, which gives $\operatorname{Var}(R_p)=\sum_i\sum_j w_iw_j\operatorname{Cov}(R_i,R_j)=\mathbf{w}^{\mathsf T}\Sigma\mathbf{w}$. This explains why portfolio risk depends on both individual variances and the relationships between returns.

A useful way to attribute the calculated variance is to assign asset $i$ the contribution $c_i=w_i(\Sigma\mathbf{w})_i$, where $(\Sigma\mathbf{w})_i$ is row $i$ of the covariance matrix multiplied by the weight vector. These contributions add exactly to portfolio variance. Dividing each contribution by total variance gives its share of variance. For standard deviation, the corresponding component contribution is $c_i/\sigma_p$; those component contributions add to $\sigma_p$. An individual contribution can be negative if that asset’s covariance pattern reduces overall portfolio risk.

c_i=w_i(\Sigma\mathbf{w})_i, \qquad \sum_{i=1}^{n}c_i=\operatorname{Var}(R_p), \qquad \text{standard-deviation component}_i=\frac{c_i}{\sigma_p}

Worked Portfolio Analysis

A reliable order of operations keeps the calculations consistent: first establish weights, then expected return, then the covariance inputs, then total risk, and finally risk attribution. When evaluating a new allocation, repeat the risk calculation with the new weights rather than assuming risk changes in proportion to expected return.

Establish the allocation.
Divide each asset’s market value by the total portfolio value and confirm that the weights sum to one.

Calculate expected return.
Multiply each asset’s expected return by its weight and add the results.

Calculate variance and standard deviation.
Use either the covariance matrix or the equivalent diagonal-plus-pairwise-covariance expression.

Attribute risk and compare allocations.
Calculate $c_i=w_i(\Sigma\mathbf{w})_i$, check that the contributions sum to variance, and repeat for any revised weights.

Worked Example: Three-Asset Return, Risk, and Contributions

A hypothetical portfolio holds $40,000 in Asset A, $35,000 in Asset B, and $25,000 in Asset C. For the same one-year period, their expected returns are 6%, 10%, and 8%; their standard deviations are 12%, 20%, and 15%; and their pairwise correlations are $\rho_{AB}=0.25$, $\rho_{AC}=0.10$, and $\rho_{BC}=0.40$. Treat these estimates as known inputs for the calculation. Find the weights, expected return, portfolio standard deviation, and each asset’s contribution to variance.

Method and weights. Total value is $40{,}000+35{,}000+25{,}000=\$100{,}000$. Therefore, $w_A=40{,}000/100{,}000=0.40$, $w_B=0.35$, and $w_C=0.25$. The check is $0.40+0.35+0.25=1.00$.

Expected return. The weighted return is $(0.40)(0.06)+(0.35)(0.10)+(0.25)(0.08)=0.024+0.035+0.020=0.079$, or 7.90%.

Covariances. Convert each correlation into covariance using $\operatorname{Cov}(R_i,R_j)=\rho_{ij}\sigma_i\sigma_j$. Thus, $\operatorname{Cov}(R_A,R_B)=(0.25)(0.12)(0.20)=0.006$; $\operatorname{Cov}(R_A,R_C)=(0.10)(0.12)(0.15)=0.0018$; and $\operatorname{Cov}(R_B,R_C)=(0.40)(0.20)(0.15)=0.012$. The variances are $0.12^2=0.0144$, $0.20^2=0.0400$, and $0.15^2=0.0225$.

Portfolio variance. The diagonal terms are $(0.40^2)(0.0144)=0.002304$, $(0.35^2)(0.0400)=0.004900$, and $(0.25^2)(0.0225)=0.00140625$. The three distinct pair terms, each counted twice, are $2(0.40)(0.35)(0.006)=0.001680$, $2(0.40)(0.25)(0.0018)=0.000360$, and $2(0.35)(0.25)(0.012)=0.002100$. Adding all six terms gives $\operatorname{Var}(R_p)=0.01275025$. Therefore, $\sigma_p=\sqrt{0.01275025}=0.112917$, or 11.29% when rounded to two decimal places.

Asset contributions. The A row product is $(\Sigma\mathbf{w})_A=(0.0144)(0.40)+(0.006)(0.35)+(0.0018)(0.25)=0.008310$, so $c_A=(0.40)(0.008310)=0.003324$. For B, $(\Sigma\mathbf{w})_B=(0.006)(0.40)+(0.0400)(0.35)+(0.012)(0.25)=0.019400$, so $c_B=(0.35)(0.019400)=0.006790$. For C, $(\Sigma\mathbf{w})_C=(0.0018)(0.40)+(0.012)(0.35)+(0.0225)(0.25)=0.010545$, so $c_C=(0.25)(0.010545)=0.00263625$.

The check is $0.003324+0.006790+0.00263625=0.01275025$, equal to portfolio variance. The variance shares are $0.003324/0.01275025=26.07\%$, $0.006790/0.01275025=53.25\%$, and $0.00263625/0.01275025=20.68\%$ (rounded to two decimal places). The method avoids treating each asset’s standalone risk as its portfolio contribution; covariance with the other holdings matters.

Assessing a Revised Allocation

A change in weights affects both the weighted average return and every weighted covariance term. As a result, an allocation that raises expected return may increase or decrease portfolio standard deviation depending on the assets’ covariance structure. To make a valid comparison, keep the return and covariance assumptions fixed and change only the weights.

Worked Example: Comparing a Rebalanced Portfolio

Use the same hypothetical three-asset return and covariance inputs as in the previous calculation. A revised allocation is 55% in A, 25% in B, and 20% in C. Find its expected return and standard deviation, then compare them with the original allocation. Treat all figures as one-year estimates.

Method and expected return. The revised weights sum to $0.55+0.25+0.20=1.00$. The expected return is $(0.55)(0.06)+(0.25)(0.10)+(0.20)(0.08)=0.033+0.025+0.016=0.074$, or 7.40%. This is 0.50 percentage points below the original 7.90%.

Variance. Using the same variances and covariances, the diagonal terms are $(0.55^2)(0.0144)=0.004356$, $(0.25^2)(0.0400)=0.002500$, and $(0.20^2)(0.0225)=0.000900$. The pair terms are $2(0.55)(0.25)(0.006)=0.001650$, $2(0.55)(0.20)(0.0018)=0.000396$, and $2(0.25)(0.20)(0.012)=0.001200$. Their sum is $\operatorname{Var}(R_p)=0.011002$, giving $\sigma_p=\sqrt{0.011002}=0.104890$, or 10.49% to two decimal places.

Comparison. The revised portfolio’s standard deviation is 10.49%, compared with 11.29% for the original allocation, a decrease of about 0.80 percentage points. Its expected return also falls by 0.50 percentage points. This is a comparison of the stated estimates, not a guarantee about realized returns. The calculation avoids assuming that the portfolio’s risk moves in the same proportion as its expected return.

Target Return and Risk Attribution in a Two-Asset Case

A target return can determine weights when there are two assets and the portfolio must be fully invested. It does not, by itself, determine whether those weights minimize risk: that requires comparing feasible allocations under a stated objective. The following example uses the target only to find a feasible allocation, then calculates its risk.

Worked Example: Finding Weights for a Target Return

A hypothetical fund invests only in Asset X and Asset Y for one year. Their expected returns are 5% and 11%, their standard deviations are 10% and 20%, and their correlation is 0.25. The fund is fully invested with no borrowing and seeks an expected return of 8%. Find the weights, portfolio standard deviation, and variance contributions.

Method and weights. Let $w_X+w_Y=1$. The target-return equation is $0.05w_X+0.11w_Y=0.08$. Substituting $w_Y=1-w_X$ gives $0.05w_X+0.11(1-w_X)=0.08$, so $0.11-0.06w_X=0.08$, $0.06w_X=0.03$, and $w_X=0.50$. Therefore, $w_Y=0.50$. Checking, $(0.50)(5\%)+(0.50)(11\%)=8\%$.

Covariance and variance. The asset covariance is $(0.25)(0.10)(0.20)=0.005$, and the variances are $0.0100$ and $0.0400$. Thus, $\operatorname{Var}(R_p)=(0.50^2)(0.0100)+(0.50^2)(0.0400)+2(0.50)(0.50)(0.005)=0.0025+0.0100+0.0025=0.0150$. The standard deviation is $\sqrt{0.0150}=0.122474$, or 12.25% to two decimal places.

Contributions and check. For X, $c_X=0.50[(0.0100)(0.50)+(0.005)(0.50)]=0.00375$. For Y, $c_Y=0.50[(0.005)(0.50)+(0.0400)(0.50)]=0.01125$. Their sum is $0.00375+0.01125=0.01500$, equal to the portfolio variance. The calculation meets the return target, but it does not establish that this is the lowest-risk allocation capable of doing so.

Common Traps and Interpretation

The most consequential arithmetic error is omitting or miscounting the covariance terms. In the first worked example, adding only the three weighted variance terms gives $0.002304+0.004900+0.00140625=0.00861025$, which would produce a misleading standard deviation of about 9.28%. The correct variance also includes the three pair terms, each counted twice, and is $0.01275025$, giving 11.29%. The correction is to use the full covariance matrix expression or explicitly include every distinct pair with its factor of two.

Confusing correlation with covariance. Correlation is unitless and ranges from -1 to 1; covariance is on the return scale. Convert using $\operatorname{Cov}(R_i,R_j)=\rho_{ij}\sigma_i\sigma_j$ before using a covariance-based formula.
Using percentages inconsistently. If returns are entered as decimals, 12% is $0.12$, and its variance is $0.12^2=0.0144$, not $12^2$. Keep all inputs on the same scale.
Forgetting that portfolio weights are squared in diagonal variance terms. The asset’s contribution to portfolio variance is not simply its weight times its standalone variance; covariance terms also affect the contribution.
Failing to check the result. Confirm that fully invested weights sum to one and that the asset variance contributions sum to portfolio variance. A failed check often reveals a missing pair or a weight error.

These calculations support allocation analysis by showing both the average return implied by the weights and the combined variability implied by the covariance structure. The risk contributions help identify which assets account for portfolio variance under the current allocation, while a revised-weight calculation shows whether a proposed change alters risk as intended. They remain estimates based on the inputs; they do not describe every possible outcome or guarantee a future return.

Check Your Understanding

For each problem, use returns and weights on a consistent decimal scale. Round final standard deviations to two decimal places.

A hypothetical portfolio holds $18,000 in Fund M, $12,000 in Fund N, and $30,000 in Fund P. Find each portfolio weight and verify that the weights sum to one. (Answer: $w_M=0.30,\ w_N=0.20,\ w_P=0.50$; total $=1.00$.)
The expected returns on M, N, and P are 4%, 9%, and 7%. Using the weights in question 1, find the expected portfolio return. (Answer: 6.50%.)
A portfolio invests 60% in Asset D and 40% in Asset E. Their variances are 0.0100 and 0.0225, and their covariance is 0.003. Find portfolio variance and standard deviation. (Answer: variance $=0.00864$; standard deviation $=9.30\%$.)
A two-asset portfolio has weights 0.40 and 0.60. Its covariance matrix has variances 0.040 and 0.090, and covariance 0.012. Find each asset’s contribution to variance and check the total. (Answer: contributions $=0.00928$ and $0.03528$; total variance $=0.04456$.)
Using the first worked example’s three-asset inputs, change the weights to 0.30 in A, 0.45 in B, and 0.25 in C. Find expected return and standard deviation. (Answer: expected return $=8.30\%$; variance $=0.01539225$; standard deviation $=12.41\%$.)

What You'll Learn