Target returns and the feasible frontier

What you will learn

Derive a piecewise target-return frontier over a three-sleeve capital simplex, certify the cash-exhaustion transition and distinguish domination, missed targets and infeasibility.

Prerequisites: Portfolio covariance and minimum variance · Comparing wealth distributions

Define a set of choices before ranking it

A minimum-variance answer can be useless for a task that also requires a return. A fictional reserve account has three sleeves: cash C, project A and project B. Write their initial capital fractions as c, a and b, with c+a+b=1 and every fraction nonnegative. Cash returns zero. Four equally probable joint states give A returns (0,0,4,4) and B returns (0,8,0,8), measured in percentage points. Each row describes simultaneous exposures over one period. These are stipulated population outcomes, not historical estimates, and four percent means a factor of 1.04 rather than a factor of four.

Assume divisible holdings, fixed initial weights, a common valuation clock, no borrowing, shorting, fees, taxes, flows, execution constraints or within-period trading. The population model is known exactly. Our objective is to minimize population variance subject to expected return at least t. A target is an expectation constraint, not a promise about every state. Indeed the first state pays zero on every allocation. Any positive target can therefore coexist with a realized return below that target. This lesson constructs the attainable mean-risk set under these assumptions; it does not infer real expected returns or select a suitable investment.

Map the allocation triangle into mean and risk

A has mean 2 and centered values (−2,−2,2,2); B has mean 4 and centered values (−4,4,−4,4). Their centered products are (8,−8,−8,8), averaging zero. Thus population variances are 4 and 16 and covariance is zero. For any allocation, m=2a+4b and V=4a²+16b². Cash enters the budget but adds neither mean nor variance. The allocation triangle has vertices all cash, all A and all B. Its image in the mean-variance plane is a region, not merely the line joining the three vertex summaries. Squared weights make risk nonlinear.

Two portfolios illustrate why feasibility and efficiency require separate checks. All A has mean 2 and variance 4. Half A, one quarter B and one quarter cash also has mean 2 but variance 2. The second portfolio strictly improves risk while preserving mean, so all A is dominated in the mean-variance sense. This does not assert first-order stochastic dominance: their cumulative distributions cross. Nor does it say every investor with an arbitrary utility function prefers the second. Here an efficient point means no feasible portfolio has at least its mean and at most its variance with one strict improvement.

Derive the cash-positive frontier with a certificate

For t>0, a minimum-risk allocation must have m=t. To prove this rather than assume it, take any allocation with m>t and multiply both risky weights by k=t/m, moving the released capital into cash. The budget remains feasible, mean becomes t and variance becomes k²V, strictly smaller because positive mean requires positive risk in this model. For t=0, all cash attains zero variance uniquely. A minimum-risk solution for an at-least target therefore lies on an exact-mean slice, although arbitrary feasible portfolios can exceed the target.

On the slice m=t, substitute a=t/2−2b. Then V=4(t/2−2b)²+16b²=t²−8tb+32b²=t²/2+32(b−t/8)². This identity is a global lower-bound certificate, not a grid-search guess. Its center gives b=t/8, a=t/4 and c=1−3t/8. All weights are nonnegative when 0≤t≤8/3. Consequently minimum variance is t²/2 on this segment. Increasing a target changes both the allocation and the best achievable risk; the unconstrained all-cash optimum alone cannot answer the constrained question. The ratio a:b remains two to one only while cash can finance that scaling.

Follow the active budget boundary rather than borrowing

Beyond t=8/3 the same center would require negative cash. The exact-mean slice also requires c=1−t/2+b≥0, hence b≥t/2−1. For t>8/3 this lower bound lies to the right of t/8. The completed square increases as b moves further right, so the feasible minimum occurs at b=t/2−1 and a=2−t/2, with c=0. This proves which constraint becomes active and why simply continuing the earlier ratio is invalid. The new minimum variance is (4−t)²+4(t−2)² for 8/3≤t≤4.

At t=3, the solution is a=b=1/2, c=0. State returns are (0,4,2,6), whose mean is 3 and whose centered squared deviations are (9,1,1,9). Their average is 5, independently checking the formula 1+4=5. At the join t=8/3 both expressions give 32/9 and the same weights (2/3,1/3,0). At t=4 only all B is feasible and variance is 16. For t>4 no allocation qualifies: a convex combination of sleeve means 0,2,4 cannot exceed 4. Reporting an enormous risk or a negative cash balance would conceal infeasibility rather than solve it.

Build the frontier and challenge the interpretation

In the workshop, move conserved one-twenty-fourth capital packets between the three sleeves. Every supported feasible target has its exact continuous optimum on this packet grid; the proof above still establishes continuous optimality between grid points. The state cards display your realized returns while you supply mean, variance and a regime certificate. Use cash for the zero target, mix for positive cash, boundary when cash is exhausted, and infeasible when no qualifying portfolio exists. An infeasibility certificate concerns the entire model, so no particular packet allocation needs to be constructed in that case.

Start at target 2. Compare all A with the allocation (a,b,c)=(1/2,1/4,1/4), then save the verified frontier point. Move to 8/3 and 3, retaining the certificates to see the boundary transition. At target 3 the tempting unconstrained weights (3/4,3/8) sum to more than one, demonstrating why the budget matters. Return to target zero and then try five. Distinguish a feasible allocation that misses the target, a target-meeting allocation that wastes risk, and a problem with no feasible allocation at all. A higher expected return requires higher minimum variance here, but this is a property of this stipulated frontier, not a universal law for every collection of assets. Estimation error or different constraints would require rebuilding the certificate.

Hands-on: Build a constrained frontier

Move conserved capital packets, certify target minima and retain frontier points across the cash boundary.

Common mistakes

• Calling an expectation target a guaranteed realized return.
• Continuing the cash-positive ratio into negative cash.
• Using weighted variances without squared weights, or dividing a specified population variance by n−1.
• Confusing a dominated portfolio or a missed target with an empty feasible set.
• Interpreting mean-variance domination as stochastic dominance or suitability.

Keep these ideas

• The allocation simplex maps to m=2a+4b and V=4a²+16b².
• A positive minimum-risk target binds: scale excess risky exposure into cash.
• Up to 8/3, weights are (a,b,c)=(t/4,t/8,1−3t/8) and V=t²/2.
• From 8/3 to 4, cash is zero and V=(4−t)²+4(t−2)². Targets above four are infeasible.

Further reading

• MOSEK Portfolio Optimization Cookbook: sections 2.1.1 and 2.2.1–2.2.3 ↗

  Read target-return formulation, efficient-frontier definition, cash-inclusive budget and long-only constraints. Supports modeling definitions. Our four-state law, piecewise proof, packet activity, examples and diagnostics are original. No numerical solver, source example, quote, credential or video review claimed. Checked 2026-10-25.