Deriving Capital-Constrained Portfolio Rules

Introduction

A portfolio can contain opportunities that are individually profitable but cannot all be accepted because the organization has limited capital. The decision is therefore not simply whether each opportunity earns a positive profit. It is how to select a feasible combination that produces the greatest total profit, given the value of the resources each opportunity consumes.

A constrained optimization model makes that trade-off explicit. Its shadow prices measure the local value of relaxing a binding constraint. They also give a precise acceptance rule when business volume is divisible. The rule needs more care when opportunities are indivisible: a shadow-price ranking can help explain the trade-off, but it does not necessarily identify the best discrete portfolio.

Definition: Let $x_j$ denote the amount of opportunity $j$ selected, $p_j$ its profit per unit, and $k_j$ the capital required per unit. In a simple additive-capital model, the portfolio-selection problem is to maximize $\sum_j p_j x_j$, subject to $\sum_j k_j x_j \le K$ and any other resource constraints. Here, $K$ is the available capital. The model assumes that profits and capital requirements are measured over a consistent horizon, that the stated contributions are appropriate for the decisions being modeled, and that the constraints represent the relevant capacity limits.

From a Capital Budget to an Acceptance Rule

First consider divisible opportunities: each $x_j$ can take any value between zero and a maximum amount $u_j$. For example, a business may be scalable by adjusting its volume. With only a capital constraint, the model is

\max_{0\le x_j\le u_j}\ \sum_j p_jx_j \qquad\text{subject to}\qquad \sum_j k_jx_j\le K.

The multiplier on the capital constraint, denoted by $\lambda$, is the shadow price of capital. Under the usual regularity conditions for this linear program, it is the marginal increase in the optimal objective from a small increase in $K$, as long as the active constraints and optimal basis remain unchanged. Its units are profit per unit of capital. It is not automatically an interest rate or a universal charge: it is specific to this objective, constraint, and local solution.

Key result: For a divisible opportunity with no other binding opportunity-specific constraint, its contribution after pricing capital at shadow price $\lambda$ is $p_j-\lambda k_j$ per unit. Select up to its maximum when this amount is positive, select none when it is negative, and be locally indifferent when it is zero. This marginal rule presumes the stated linear, additive model; it is not, by itself, a solution method for indivisible opportunities.

The rule follows directly from the Lagrangian. With the upper bounds on $x_j$ set aside temporarily, the Lagrangian for the capital constraint is

L(x,\lambda)=\sum_j p_jx_j+\lambda\left(K-\sum_j k_jx_j\right) =\lambda K+\sum_j(p_j-\lambda k_j)x_j,\qquad \lambda\ge 0.

For a given $\lambda$, each unit of $x_j$ changes the Lagrangian by $p_j-\lambda k_j$. A positive change favors increasing that activity; a negative change favors reducing it. At an interior amount, the change must be zero. The multiplier is chosen to satisfy the capital constraint and the complementary-slackness condition: $\lambda(K-\sum_j k_jx_j)=0$. Thus, if capital is slack, its multiplier is zero; if capital is binding, its multiplier may be positive. The upper bounds matter at endpoints: a positive net contribution can justify selecting an opportunity only up to its permitted maximum.

Worked Example: Finding the Marginal Opportunity Value of Capital

A hypothetical insurer can scale three portfolios continuously. Their annual profit contributions and capital requirements are: A, $21 thousand profit per unit and $30 thousand capital per unit; B, $30 thousand and $50 thousand; and C, $20 thousand and $40 thousand. Each can be selected up to one unit. Available capital is $70 thousand. Find the optimal selection and the local shadow price of capital.

Method. Compare profit per unit of capital, $p_j/k_j$, to identify the order in which divisible opportunities are used. A has $21/30=0.70$, B has $30/50=0.60$, and C has $20/40=0.50$, all in thousands of profit per thousand of capital.

Select A first: it uses $30 thousand and earns $21 thousand, leaving $70-30=40$ thousand of capital. Use the remaining $40 thousand for $40/50=0.8$ unit of B. That earns $0.8(30)=$24$ thousand. C is not selected. Total capital is $30+0.8(50)=70$ thousand, and total profit is $21+24=$45$ thousand.

The marginal selected opportunity is B, so the shadow price is $30/50=0.60$ thousand of annual profit per thousand of capital over the range where B remains the marginal opportunity. Equivalently, at $\lambda=0.60$, B has $p_B-\lambda k_B=30-0.60(50)=0$. A has $21-0.60(30)=3$ thousand per unit, and C has $20-0.60(40)=-4$ thousand per unit. These signs agree with the selection. The computed capital use equals the $70 thousand limit, as required.

Common mistake avoided: Selecting only whole units would incorrectly rule out the feasible partial selection of B. This example assumes opportunities are divisible; its acceptance calculation should not be transferred uncritically to a yes-or-no decision.

Additional Constraints and Multiple Shadow Prices

Capital is not always the only scarce resource. A portfolio may also be limited by underwriting capacity, concentration limits, operational capacity, or another explicitly modeled resource. In a linear model, associate a nonnegative multiplier with each upper-bound constraint. An opportunity’s adjusted contribution then subtracts the shadow-priced use of every constrained resource. For example, if opportunity $j$ uses $a_{rj}$ units of resource $r$, and that constraint has multiplier $\lambda_r$, the adjusted contribution per unit is $p_j-\sum_r\lambda_r a_{rj}$.

This is a local decision rule for a continuous linear model. A positive multiplier indicates that the constraint is valuable to relax at the current solution, within the range over which the current marginal value remains valid. A zero multiplier indicates that relaxing that constraint alone does not improve the objective locally. Multipliers can change when the optimal mix changes, so they should not be treated as fixed across large capacity changes.

Worked Example: Pricing Two Binding Capacity Limits

A hypothetical portfolio can choose divisible amounts $x_A$ and $x_B$, each nonnegative. Per unit, A earns $40 thousand and uses 50 units of capital and one unit of underwriting capacity. B earns $22 thousand and uses 20 units of capital and one unit of underwriting capacity. The limits are 100 units of capital and three units of underwriting capacity. Find the optimum and the two shadow prices.

The model is to maximize $40x_A+22x_B$, subject to $50x_A+20x_B\le100$ and $x_A+x_B\le3$. At the intersection of the two constraints, subtracting twice the second equation from the first gives $30x_A=40$, so $x_A=4/3$ and $x_B=3-4/3=5/3$. Capital use is $50(4/3)+20(5/3)=200/3+100/3=100$. Underwriting capacity use is $4/3+5/3=3$. Profit is $40(4/3)+22(5/3)=160/3+110/3=90$ thousand.

To find the multipliers, let $\lambda$ price capital and $\mu$ price underwriting capacity. Since both activities are positive at the interior solution, their adjusted marginal profits are zero:

50\lambda+\mu=40,\qquad 20\lambda+\mu=22.

Subtracting the second equation from the first gives $30\lambda=18$, so $\lambda=0.60$ thousand dollars of profit per capital unit. Then $20(0.60)+\mu=22$, giving $\mu=10$ thousand dollars of profit per underwriting-capacity unit. Both are positive, consistent with both limits binding. The corner solutions on the axes are $x_A=2$, with profit $80$ thousand, and $x_B=3$, with profit $66$ thousand. The intersection’s $90 thousand profit is greater, confirming the optimum among the feasible vertices.

Interpretation. Locally, one additional capital unit is worth $0.60 thousand in optimal profit if the current basis remains valid. One additional unit of underwriting capacity is worth $10 thousand under the same qualification. These marginal values are different because the constraints limit different resources.

Common mistake avoided: A binding constraint does not necessarily have the largest shadow price, and its multiplier is not its dollar limit. The multipliers describe marginal objective changes, not the amount of resource available.

Indivisible Opportunities Require a Discrete Decision

Many portfolio choices are indivisible: an entire contract, facility, or transaction is either accepted or rejected. Then $x_j$ is binary, taking only zero or one, and the model becomes an integer optimization problem. The Lagrangian still helps explain the cost of scarce capital, but ranking opportunities by profit per capital unit is not guaranteed to maximize total profit. A high-ratio opportunity may leave unusable capacity, while a combination of lower-ratio opportunities may produce more profit.

Worked Example: Why a Capital-Ratio Ranking Can Miss the Best Portfolio

A hypothetical insurer must choose among four indivisible opportunities under a $100 thousand capital limit. Their annual profit and capital requirements, both in thousands of dollars, are: A, profit 52 and capital 70; B, 39 and 50; C, 31 and 40; D, 22 and 30. Find the profit-maximizing portfolio.

Method. Check feasible combinations because each opportunity is indivisible. The profit-to-capital ratios are A, $52/70\approx0.7429$; B, $39/50=0.78$; C, $31/40=0.775$; and D, $22/30\approx0.7333$. A ratio-first greedy choice selects B, then C, using $50+40=90$ thousand of capital and earning $39+31=70$ thousand. No further opportunity fits.

Now compare the feasible combinations. A with D uses $70+30=100$ thousand and earns $52+22=74$ thousand. B with C uses 90 thousand and earns 70 thousand; B with D uses 80 thousand and earns 61 thousand; C with D uses 70 thousand and earns 53 thousand. The individual choices earn 52, 39, 31, and 22 thousand, respectively. A cannot be combined with B or C because their capital totals would be 120 and 110 thousand. The three-opportunity combination B+C+D uses 120 thousand, and any combination containing A plus two other opportunities exceeds the limit. Therefore A and D is the best feasible portfolio, with $74 thousand of profit.

Common mistake and correction. The wrong calculation is to treat the ratio ranking as an optimality rule and stop at B+C with $70 thousand of profit. The correction is to respect the binary decision and compare feasible combinations; A+D earns $4 thousand more. Shadow-price or ratio reasoning can support analysis, but discrete choices may require enumeration or an integer optimization method.

Applying the Rule Responsibly

A capital-constrained acceptance rule depends on the model’s assumptions. It is most direct when contributions are linear, capital is additive, volumes can be varied continuously, and all relevant constraints are represented. If capital depends on the combined portfolio rather than adding across opportunities, the constraint must use the appropriate portfolio capital function instead of $\sum_j k_jx_j$. In that case, local marginal capital use, rather than a fixed $k_j$, is needed for a local shadow-price comparison. A simple linear rule may not capture the resulting portfolio effects.

For a practical decision, state the objective and constraints first, use consistent profit and capital units, and identify whether choices are divisible. Then find the feasible optimum and interpret multipliers at that solution. The method is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus: it connects a capital budget to a defensible profit-maximizing decision without treating a single performance ratio as a universal ranking device.

Use a marginal acceptance cutoff for divisible business only under the stated model assumptions.
Interpret a shadow price as a local value, not as an invariant charge for all future decisions.
For indivisible choices, check feasible combinations or use an appropriate integer model.
Confirm that a proposed portfolio satisfies every constraint, not just the capital limit.

Check Your Understanding

For each problem, identify the relevant model before applying a capital or shadow-price calculation.

A divisible opportunity earns $18 thousand per unit and requires 30 units of capital. The current capital shadow price is $0.50 thousand of profit per capital unit. Calculate its adjusted contribution per unit and state whether the marginal rule favors accepting more. (Answer: $18-0.50(30)=3$ thousand; accept more, subject to its upper bound and other constraints.)
Two divisible opportunities use only capital. A earns $24 thousand on 40 capital units, and B earns $27 thousand on 60 capital units. Capital available is 70 units and each opportunity is capped at one unit. Find the profit-maximizing amounts and the shadow price. (Answer: Take A fully and $30/60=0.5$ of B; profit is $24+13.5=$37.5$ thousand, and the local shadow price is $27/60=0.45$ thousand per capital unit.)
Two indivisible opportunities require 45 and 35 capital units and earn $34 thousand and $25 thousand, respectively. The limit is 80 units. Compare the joint choice with each single choice. (Answer: Accept both; they use $45+35=80$ units and earn $34+25=$59$ thousand, more than either single choice.)
A divisible solution has positive amounts of A and B, and both capital and exposure constraints bind. Their resource use per unit is A: 40 capital and one exposure unit; B: 20 capital and one exposure unit. Their profits are $32 thousand and $20 thousand per unit. Solve for the two constraint shadow prices from the interior conditions. (Answer: $40\lambda+\mu=32$, $20\lambda+\mu=20$, so $\lambda=0.60$ thousand per capital unit and $\mu=8$ thousand per exposure unit.)
An indivisible portfolio has two feasible choices: X uses 60 capital units and earns $45 thousand; Y uses 40 units and earns $32 thousand. The capital limit is 60 units. Which choice maximizes profit, and why is comparing ratios alone unnecessary here? (Answer: Choose X for $45 thousand; it is feasible and earns more than Y’s $32 thousand, so direct comparison of feasible choices settles the decision.)

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