From Portfolio Weights to a Risk-Return Set
A two-asset portfolio does not have just one risk-return outcome. Its expected return and standard deviation depend on how its value is divided between the assets. Varying the allocation traces an opportunity set: the collection of attainable expected-return and risk pairs under the permitted weight rules. Locating the minimum-variance point identifies the lowest-risk portfolio in that set; the remaining combinations show how risk and expected return change as the allocation moves away from it.
Assume a single investment period, with asset returns \(R_A\) and \(R_B\) having finite means and variances. Let \(w\) be the portfolio weight in asset A, so the weight in asset B is \(1-w\). The portfolio is fully invested. Unless stated otherwise, returns and risk are measured over the same period. A long-only portfolio requires \(0\leq w\leq1\); allowing short sales permits weights outside that interval.
The expected return changes linearly with \(w\). Risk generally does not: portfolio variance includes a covariance term. The standard deviation plotted against expected return therefore traces a curved locus in the usual case, rather than a straight line between the assets’ standalone risk-return points.
How the Minimum-Variance Point Shapes the Set
Let \(C=\operatorname{Cov}(R_A,R_B)\). Using the two-asset variance formula, portfolio variance at weight \(w\) is
This is a quadratic function of \(w\). To see why its minimum matters, define \(D=\sigma_A^2+\sigma_B^2-2C\). This is \(\operatorname{Var}(R_A-R_B)\), so \(D\geq0\). In the non-degenerate case \(D>0\), completing the square gives the minimum-variance weight and the variance at that weight:
This identity is also a short justification of the minimum-variance formula: \(D(w-w^*)^2\) cannot be negative, and it equals zero at \(w=w^*\). Thus the smallest variance over unrestricted weights is \(\sigma_{\min}^2\). If only long-only weights are allowed, \(w^*\) is the minimum provided it lies between 0 and 1. If it falls outside that interval, the constrained minimum is at the nearer endpoint of the interval.
The expected return at any weight can be written \(\mu_p=\mu_B+w(\mu_A-\mu_B)\). When \(\mu_A\ne\mu_B\), the two weights that produce the same variance, if both are permitted, are on opposite sides of \(w^*\). In fact, for a standard deviation \(\sigma_p\geq\sigma_{\min}\), they satisfy
These two allocations usually have different expected returns. The lower-return branch means the lower expected return among portfolios at the same standard deviation. It is not always the branch with \(w<w^*\): that depends on which asset has the greater expected return. Also, both weights must meet the portfolio’s constraints. In a long-only set, one of the two weights may be outside \([0,1]\), leaving only one feasible portfolio at that risk level.
If \(\mu_A=\mu_B\), portfolios at the two weights have the same expected return, so there is no distinct lower-return branch. If \(D=0\), the variance is constant in \(w\) and the usual unique minimum-weight formula does not apply. The examples below use \(D>0\).
Worked Example: Tracing a Long-Only Set to Its Minimum
A hypothetical portfolio combines asset A, with expected return 6% and standard deviation 10%, and asset B, with expected return 8% and standard deviation 20%. Their correlation is 0.25. Find the expected return and standard deviation at \(w=0.5\), then find the long-only minimum-variance portfolio and check the endpoints.
Method. Convert correlation to covariance, use the quadratic variance formula, and then locate its minimum. This method avoids treating correlation itself as covariance.
First, the variances are \(\sigma_A^2=0.10^2=0.01\) and \(\sigma_B^2=0.20^2=0.04\). The covariance is
At \(w=0.5\), the expected return is
The variance and standard deviation at this weight are
Next, \(D=0.01+0.04-2(0.005)=0.04\). Therefore,
This is long-only feasible. The weight in B is \(1-0.875=0.125\), and the expected return at the minimum is
Using the completed-square result, the minimum variance is
Thus \(\sigma_{\min}=\sqrt{0.009375}=0.096824\ldots=9.68\%\). As an endpoint check, \(w=0\) invests entirely in B and gives an 8% expected return and 20% standard deviation; \(w=1\) invests entirely in A and gives a 6% expected return and 10% standard deviation. The minimum is below both endpoint risks, as required. Rounded only for the final results, to two decimal places, the \(w=0.5\) portfolio has a 7.00% expected return and 12.25% standard deviation, and the minimum-variance portfolio has a 6.25% expected return and 9.68% standard deviation.
Common mistake avoided: Using 0.25 directly as the covariance would produce an invalid variance calculation. Correlation is unit-free; covariance must be calculated as correlation times both standard deviations.
Reading the Two Branches
The minimum-variance point is the leftmost point of the risk-return set when standard deviation is on the horizontal axis. Moving the weight away from \(w^*\) increases variance on either side, because the increase is \(D(w-w^*)^2\). The expected return moves according to the sign and size of \(\mu_A-\mu_B\). Consequently, a given risk above the minimum can correspond to one portfolio on each side of the minimum, but not necessarily two portfolios within the chosen weight limits.
When both such portfolios are feasible, compare their expected returns rather than assuming that the lower-return branch always corresponds to the smaller weight. At equal risk, a portfolio with lower expected return is less attractive under a mean-variance comparison, all else equal. This comparison identifies the lower-return branch; it does not imply that every risk level has two feasible allocations.
Worked Example: Identifying the Lower-Return Branch
A hypothetical asset A has expected return 12% and standard deviation 20%; asset B has expected return 6% and standard deviation 10%. Their covariance is 0.005. Restrict weights to \(0\leq w\leq1\). Find the minimum-variance weight, then identify two portfolios with the same 10% standard deviation and determine which is on the lower-return branch.
Method. Find the minimum first, then check weights on either side of it using the variance formula. This avoids identifying a branch from its weight alone.
The variances are \(0.20^2=0.04\) and \(0.10^2=0.01\). Hence \(D=0.04+0.01-2(0.005)=0.04\), giving
The minimum expected return is \(0.125(0.12)+0.875(0.06)=0.015+0.0525=0.0675=6.75\%\). The minimum variance is \(0.01-(0.005)^2/0.04=0.009375\), so the minimum standard deviation is \(9.68\%\) to two decimal places.
Consider \(w=0\). The portfolio consists entirely of B, so its standard deviation is 10% and its expected return is 6%. Now consider \(w=0.25\). Its variance is
Its standard deviation is therefore \(\sqrt{0.0100}=10\%\), the same as the all-B portfolio. Its expected return is \(0.25(0.12)+0.75(0.06)=0.03+0.045=7.5\%\). Both weights are long-only feasible. At 10% risk, the lower-return branch is the portfolio with \(w=0\), returning 6%; the portfolio with \(w=0.25\) is the higher-return branch, returning 7.5%. The weights are equally distant from \(w^*=0.125\), which explains their equal variances.
Common mistake avoided: Assuming that higher weight in A must always mean higher risk. Risk is minimized at \(w^*\); it rises as the weight moves away from that point in either direction.
Long-Only Constraints and Short Selling
The unrestricted minimum-variance weight is a mathematical result, not a guarantee that the portfolio can be held under the stated rules. If the allowed weights are long-only, an answer outside \([0,1]\) must be replaced by the feasible endpoint with the lower variance. If short sales are permitted, a weight below zero means a short position in A and a weight above one means a short position in B, financed by a larger position in the other asset.
The following example separates the unconstrained minimum from the long-only minimum. This distinction matters because the opportunity set changes when the permitted weights change.
Worked Example: An Unrestricted Minimum Outside the Long-Only Set
A hypothetical asset A has expected return 10% and standard deviation 20%; asset B has expected return 6% and standard deviation 10%. Their covariance is 0.015. Find the unrestricted minimum-variance weight and compare it with the long-only minimum.
Method. Calculate \(w^*\), then test its feasibility before reporting a constrained answer. This avoids presenting a short position as a long-only allocation.
The variances are \(0.04\) and \(0.01\), so \(D=0.04+0.01-2(0.015)=0.02\). Thus
The unrestricted minimum has a 25% short position in A and a 125% position in B. Its expected return is
Its variance, checked directly, is
Its standard deviation is \(\sqrt{0.00875}=0.093541\ldots=9.35\%\) to two decimal places. This is less than B’s 10% standard deviation, which is consistent with diversification through the short position. However, \(w^*=-0.25\) is not long-only feasible. Over \(0\leq w\leq1\), the variance is increasing from \(w=0\), since its minimum occurs to the left of that interval. The long-only minimum is therefore \(w=0\): 100% in B, with a 6% expected return and 10% standard deviation.
Common mistake avoided: Reporting the unrestricted minimum as the long-only answer. Always compare the calculated weight with the stated weight constraints.
Calculation Checks and Common Traps
A reliable opportunity-set calculation has three checks. First, the endpoint weights must reproduce the standalone assets: \(w=0\) gives B and \(w=1\) gives A. Second, the covariance term must use covariance, not correlation; if correlation is supplied, convert it using \(C=\rho_{AB}\sigma_A\sigma_B\). Third, the calculated minimum must be feasible under the weight rules. For \(D>0\), the variance cannot fall below \(\sigma_{\min}^2\), and any equal-variance pair must lie at equal distances from \(w^*\).
A frequent arithmetic trap is to reverse the endpoint expected returns. For example, if A has a 6% expected return and B has an 8% expected return, then \(\mu_p=0.06+w(0.08-0.06)\) would incorrectly assign the 8% return to \(w=1\). With \(w\) defined as the weight in A, the correct expression is
It gives 8% at \(w=0\) and 6% at \(w=1\), matching the asset identities. Writing down and checking both endpoints is a simple way to catch a reversed weight convention before calculating the rest of the set.
Check Your Understanding
For each problem, identify the relevant weights and use returns and risks measured over the same investment period.
- A hypothetical portfolio has asset A with expected return 4% and standard deviation 12%, and asset B with expected return 10% and standard deviation 18%. Their correlation is 0.25. At \(w=0.30\), calculate the covariance, expected return, and portfolio standard deviation. (Answer: covariance \(=0.0054\); expected return \(=8.20\%\); standard deviation \(=13.94\%\).)
- Asset A has expected return 7% and standard deviation 10%; asset B has expected return 11% and standard deviation 15%; their covariance is 0.006. Find the minimum-variance weight in A and the expected return and standard deviation there. (Answer: \(w^*=0.8049\); expected return \(=7.78\%\); standard deviation \(=9.60\%\).)
- Asset A has expected return 12% and standard deviation 20%; asset B has expected return 6% and standard deviation 10%; their covariance is 0.005. For long-only weights, identify the two portfolios with 10% standard deviation and name the lower-return branch. (Answer: \(w=0\) gives 6%; \(w=0.25\) gives 7.5%; the lower-return branch is \(w=0\).)
- Asset A has expected return 9% and standard deviation 18%; asset B has expected return 5% and standard deviation 12%; their covariance is 0.020. Compare the unrestricted minimum with the long-only minimum. (Answer: unrestricted \(w^*=-0.8235\), expected return \(=1.71\%\), standard deviation \(=9.89\%\); long-only minimum \(w=0\), expected return \(=5\%\), standard deviation \(=12\%\).)
- Asset A has expected return 8% and standard deviation 16%; asset B has expected return 5% and standard deviation 12%; their covariance is 0.0096. Find the weight in A needed for a 6.8% expected return and calculate the resulting standard deviation. (Answer: \(w=0.60\); standard deviation \(=12.70\%\).)