Portfolio and Allocation | Malone Math | Malone Global University

The Central Idea

An allocation problem asks how a total resource should be divided among several alternatives.

If $w_i$ is the fraction assigned to alternative $i$, a fully allocated portfolio usually satisfies

\[ \boxed{\sum_{i=1}^{n}w_i=1}. \]

The weights determine how the individual outcomes combine into one portfolio outcome.

Portfolio

A portfolio is a collection of positions, projects, assets, or activities considered together.

The term is common in finance, but the mathematics applies more broadly to:

project funding;
production capacity;
energy sources;
research programs;
transportation routes;
inventory;
staffing;
other resource allocations.

Allocation Weights

Suppose a total resource $B$ is allocated across $n$ alternatives.

If alternative $i$ receives amount $x_i$, define its weight as

\[ \boxed{w_i=\frac{x_i}{B}}. \]

If the entire resource is allocated,

\[ \sum_i x_i=B \]

and therefore

\[ \sum_iw_i=1. \]

Example: Converting Amounts to Weights

Suppose $100,000 is divided as follows:

$50,000 to A;
$30,000 to B;
$20,000 to C.

The weights are

\[ w_A=\frac{50{,}000}{100{,}000}=0.50, \] \[ w_B=0.30, \] \[ w_C=0.20. \]

Check:

\[ 0.50+0.30+0.20=\boxed{1}. \]

Weighted Portfolio Outcome

If alternative $i$ produces outcome $R_i$, then the portfolio outcome is

\[ \boxed{ R_P=\sum_{i=1}^{n}w_iR_i }. \]

This is a weighted average when the weights are nonnegative and sum to one.

Example: Portfolio Return

Suppose

\[ w_A=0.60,\qquad w_B=0.40 \]

and realized returns are

\[ R_A=10\%,\qquad R_B=4\%. \]

Then

\[ R_P=0.60(10\%)+0.40(4\%). \] \[ =6\%+1.6\%. \] \[ \boxed{R_P=7.6\%}. \]

Expected Portfolio Return

Expected value is linear, so

\[ \boxed{ E[R_P] = \sum_{i=1}^{n}w_iE[R_i] }. \]

Expected portfolio return is the weighted average of the individual expected returns.

Example: Expected Return

Suppose:

\[ E[R_A]=8\%,\qquad E[R_B]=5\% \]

and

\[ w_A=0.70,\qquad w_B=0.30. \]

Then

\[ E[R_P] =0.70(8\%)+0.30(5\%). \] \[ =5.6\%+1.5\%. \] \[ \boxed{E[R_P]=7.1\%}. \]

Expected Return Is Easy; Risk Is More Subtle

Expected portfolio return depends only on the weights and individual expected returns.

Portfolio risk also depends on how the components move together.

This introduces covariance and correlation.

Two-Asset Portfolio Variance

For two random returns $R_A$ and $R_B$,

\[ \boxed{ \operatorname{Var}(R_P) = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\operatorname{Cov}(R_A,R_B) }. \]

Using Correlation

Since

\[ \operatorname{Cov}(R_A,R_B) = \rho_{AB}\sigma_A\sigma_B, \]

portfolio variance can be written

\[ \boxed{ \sigma_P^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho_{AB}\sigma_A\sigma_B }. \]

Portfolio Standard Deviation

Once portfolio variance is known,

\[ \boxed{ \sigma_P=\sqrt{\sigma_P^2} }. \]

Standard deviation is often used as a measure of portfolio variability.

Diversification

Diversification spreads exposure among alternatives whose outcomes are not perfectly aligned.

The main idea is not merely to own or fund many things.

The benefit comes from combining components whose unfavorable outcomes do not occur together perfectly.

Perfect Positive Correlation

\[ \rho_{AB}=1, \]

the two returns move together perfectly in standardized terms.

Diversification provides the least variance reduction in this extreme case.

Zero Correlation

\[ \rho_{AB}=0, \]

the covariance term disappears:

\[ \sigma_P^2 = w_A^2\sigma_A^2+w_B^2\sigma_B^2. \]

Combining the assets can reduce variability relative to simple concentration.

Negative Correlation

\[ \rho_{AB}<0, \]

the covariance term reduces portfolio variance.

Negative dependence can therefore provide especially strong diversification benefits.

Perfect Negative Correlation

\[ \rho_{AB}=-1, \]

certain weight combinations can theoretically eliminate variance in a two-asset model when the component standard deviations permit exact offsetting.

Example: Two-Asset Variance

Suppose

\[ w_A=0.6,\qquad w_B=0.4, \] \[ \sigma_A=0.10,\qquad \sigma_B=0.05, \]

and

\[ \rho_{AB}=0. \]

Then

\[ \sigma_P^2 = (0.6)^2(0.10)^2 + (0.4)^2(0.05)^2. \] \[ =0.36(0.01)+0.16(0.0025). \] \[ =0.0036+0.0004. \] \[ \boxed{\sigma_P^2=0.004}. \]

Complete the Standard Deviation

From

\[ \sigma_P^2=0.004, \]

we obtain

\[ \sigma_P=\sqrt{0.004} \] \[ \boxed{\approx0.0632}. \]

That is approximately $6.32\%$.

General Portfolio Variance

For $n$ components,

\[ \boxed{ \operatorname{Var}(R_P) = \sum_{i=1}^{n} \sum_{j=1}^{n} w_iw_j\operatorname{Cov}(R_i,R_j) }. \]

Every pairwise covariance can affect total portfolio risk.

Matrix Form

If $\mathbf{w}$ is the vector of portfolio weights and $\Sigma$ is the covariance matrix, then

\[ \boxed{ \sigma_P^2 = \mathbf{w}^{T}\Sigma\mathbf{w} }. \]

This compact form becomes especially useful for large portfolios.

Concentration Risk

Concentration risk occurs when too much exposure is placed in one asset, project, sector, supplier, region, or other shared source of uncertainty.

A portfolio can contain many positions and still be concentrated if those positions depend on the same underlying risk.

Diversification Does Not Eliminate All Risk

Diversification can reduce component-specific risk.

It cannot automatically remove risks shared across the entire system.

Common macroeconomic, technological, regulatory, environmental, or network shocks may affect many portfolio components together.

Allocation Constraints

Real allocation problems include constraints.

Common constraints include:

\[ \sum_iw_i=1, \] \[ w_i\ge0, \]

and upper or lower bounds such as

\[ 0.10\le w_i\le0.40. \]

No-Shorting Constraint

In financial language,

\[ w_i\ge0 \]

is often called a no-short-selling constraint.

More generally, it means the allocation cannot assign a negative amount to an alternative.

Budget Constraint

If allocations are expressed in dollars rather than weights,

\[ \boxed{ x_1+x_2+\cdots+x_n=B }. \]

The budget $B$ limits the total amount that can be assigned.

Capacity Constraints

An alternative may have a maximum feasible allocation:

\[ x_i\le c_i. \]

Capacity constraints arise when a project, supplier, asset, or facility cannot absorb unlimited resources.

Minimum Allocation Constraints

Strategic or operational requirements may impose

\[ x_i\ge m_i. \]

For example, a portfolio may require a minimum reserve, minimum service level, or minimum allocation to a mandatory activity.

Expected-Return Maximization

A simple allocation objective is

\[ \boxed{ \max_{\mathbf{w}} \sum_iw_i\mu_i } \]

subject to the relevant allocation constraints.

If risk is ignored, this can push the solution toward the alternative with the largest expected return.

Risk Minimization

Another objective is to minimize portfolio variance:

\[ \boxed{ \min_{\mathbf{w}} \mathbf{w}^{T}\Sigma\mathbf{w} }. \]

Constraints can be added to require a minimum expected return or other performance target.

Mean-Variance Tradeoff

Many portfolio models balance expected return and variance.

A stylized objective is

\[ \boxed{ \max_{\mathbf{w}} \left( \mathbf{w}^{T}\boldsymbol{\mu} - \lambda\mathbf{w}^{T}\Sigma\mathbf{w} \right) }. \]

Here $\lambda>0$ controls how strongly variance is penalized.

Risk-Preference Parameter

In the preceding objective, a larger

\[ \lambda \]

places more emphasis on reducing variance.

A smaller $\lambda$ places relatively more emphasis on expected return.

The parameter encodes a tradeoff rather than a universal truth.

Efficient Portfolios

A portfolio is mean-variance efficient if no other feasible portfolio has:

higher expected return with no more variance; or
lower variance with no less expected return.

Efficient Frontier

The set of mean-variance efficient portfolios forms an efficient frontier.

Portfolios below the frontier are inefficient because another feasible portfolio offers a better risk-return combination.

Portfolio Choice Still Requires Preferences

The efficient frontier identifies attractive tradeoffs.

It does not by itself tell every decision maker which portfolio to choose.

Final selection depends on:

risk tolerance;
objectives;
constraints;
time horizon;
loss functions or utility.

Minimum-Variance Portfolio

Among feasible portfolios, the minimum-variance portfolio has the smallest possible variance.

It is the leftmost point in a conventional risk-return representation of the feasible set.

Two-Asset Minimum-Variance Weight

For two assets A and B with

\[ w_B=1-w_A, \]

the minimum-variance weight in A, when unconstrained beyond full investment, is

\[ \boxed{ w_A^* = \frac{\sigma_B^2-\operatorname{Cov}(A,B)} {\sigma_A^2+\sigma_B^2-2\operatorname{Cov}(A,B)} }. \]

The remaining weight is

\[ w_B^*=1-w_A^*. \]

Allocation by Expected Utility

Instead of optimizing expected return and variance separately, a decision maker can evaluate final outcomes through a utility function.

Then the objective is

\[ \boxed{ \max E[U(W)] }, \]

where $W$ is the uncertain final portfolio outcome or wealth.

Allocation by Expected Loss

If outcomes are represented through a loss function, the allocation problem may instead be

\[ \boxed{ \min E[L(\mathbf{w},X)] }. \]

This allows downside costs, shortages, tracking error, or other decision objectives to be modeled directly.

Allocation Across Projects

Suppose three projects have expected returns per dollar:

\[ 0.08,\qquad0.06,\qquad0.04. \]

If no other constraints or risks exist, maximizing expected return would place all available resources in the first project.

Real allocation problems become more interesting when risk, capacity, diversification, timing, or strategic constraints are included.

Marginal Allocation

A useful question is:

Where should the next unit of resource go?

The answer depends on the marginal benefit, marginal risk, and constraints associated with increasing each allocation.

Marginal Return

If total benefit from allocation $x$ is $B(x)$, marginal benefit is

\[ \boxed{B'(x)} \]

when the derivative exists.

Diminishing returns mean this marginal benefit decreases as allocation increases.

Diminishing Returns Encourage Diversification

Even without uncertainty, diminishing marginal returns can make spreading resources optimal.

If one project becomes progressively less productive as more resources are assigned to it, another project may eventually offer the larger marginal benefit.

Equal-Marginal Principle

In a smooth constrained allocation problem, an interior optimum often occurs when marginal benefits are equal across alternatives:

\[ \boxed{ B_1'(x_1)=B_2'(x_2)=\cdots }. \]

If one activity had a larger marginal benefit, shifting a small amount toward it could improve the objective.

Allocation under Uncertain Scenarios

An allocation can be tested across several states of the world.

Allocation	Weak Scenario	Base Scenario	Strong Scenario
Concentrated	-10	50	100
Diversified	20	45	70

The concentrated allocation has more upside but more downside.

Robust Allocation

A robust allocation performs acceptably across a broad range of plausible assumptions.

Robustness can be useful when expected returns, probabilities, or correlations are estimated imprecisely.

Sensitivity of Portfolio Decisions

Portfolio recommendations can be sensitive to small changes in:

expected returns;
volatility estimates;
correlations;
constraints;
risk-preference parameters.

Sensitivity analysis is therefore a natural part of allocation modeling.

Estimation Risk

The inputs to a portfolio model are often estimated from data.

The optimizer may respond strongly to small estimation errors, especially in expected returns.

An mathematically precise allocation can therefore be unreliable if its inputs are poorly known.

Rebalancing

Portfolio weights change when component values change.

Rebalancing restores or revises target weights.

Rebalancing decisions may consider:

transaction costs;
new information;
changed constraints;
changed risk;
changed objectives.

Allocation Costs

Moving resources between alternatives may itself create costs.

A more complete optimization problem can include:

\[ \text{Net Objective} = \text{Portfolio Benefit} - \text{Allocation Costs}. \]

Ignoring switching or transaction costs can lead to excessive reallocation.

Liquidity and Flexibility

Some allocations are easier to reverse than others.

Flexibility can have value when future information is expected.

A slightly lower expected-return allocation may be preferred if it preserves the ability to respond later.

Portfolio and Value of Information

Better estimates of expected returns, correlations, demand, or failure risk can improve allocation decisions.

The value of that information depends on whether it changes the optimal allocation enough to improve expected outcomes.

Portfolio and Strategic Decisions

Allocations may also affect competitors or other decision makers.

For example, investing heavily in one capacity may change the incentives of rivals.

In such cases, portfolio optimization and game theory can interact.

Common Error: Forgetting That Weights Must Be Consistent

If weights are intended to represent a complete allocation,

\[ \sum_iw_i=1. \]

A set of weights summing to $1.2$ or $0.8$ requires explanation or correction.

Common Error: Averaging Standard Deviations

Portfolio standard deviation is generally not

\[ w_A\sigma_A+w_B\sigma_B. \]

Covariance must be incorporated through the variance formula.

Common Error: Ignoring Correlation

Two alternatives can have identical individual variances but very different diversification value depending on how their outcomes co-move.

Common Error: Assuming More Holdings Always Mean More Diversification

Ten highly correlated positions may provide less diversification than two positions with genuinely different risk exposures.

Count alone is not the right measure of diversification.

Common Error: Treating Expected Return as Guaranteed Return

Expected return is a probability-weighted average.

The realized portfolio outcome can be above or below that value.

Common Error: Optimizing Estimated Inputs Too Aggressively

Small errors in estimated returns or correlations can produce large changes in calculated optimal weights.

Constraints, regularization, scenario analysis, and robust optimization can help reduce sensitivity to estimation error.

A Reliable Procedure for Basic Portfolio Calculations

List the alternatives.
Specify the amount or weight assigned to each.
Check that the weights satisfy the allocation constraints.
Compute weighted expected return.
Identify component variances and covariances when risk matters.
Compute portfolio variance.
Take the square root if portfolio standard deviation is needed.
Interpret the result together with concentration and downside exposure.

A Reliable Procedure for Allocation Decisions

Define the resource being allocated.
Define the objective.
List all constraints.
Estimate expected benefits and relevant risks.
Model dependencies among alternatives.
Compare feasible allocations.
Identify efficient tradeoffs.
Test sensitivity to uncertain assumptions.
Include transaction, switching, or implementation costs.
Choose an allocation consistent with the decision maker's objectives and risk tolerance.

Practice 1: Portfolio Weights

A total of $50,000 is allocated as:

$20,000 to A;
$15,000 to B;
$15,000 to C.

Find the three portfolio weights.

Practice 2: Weighted Return

A portfolio places $70\%$ in A and $30\%$ in B.

If A returns $12\%$ and B returns $4\%$, find the portfolio return.

Practice 3: Expected Portfolio Return

A portfolio has

\[ w_A=0.4,\qquad w_B=0.6, \]

with expected returns

\[ E[R_A]=9\%,\qquad E[R_B]=5\%. \]

Find expected portfolio return.

Practice 4: Two-Asset Variance with Zero Correlation

Suppose

\[ w_A=w_B=0.5, \] \[ \sigma_A=0.10,\qquad \sigma_B=0.06, \]

and

\[ \rho_{AB}=0. \]

Find portfolio variance.

Practice 5: Portfolio Standard Deviation

Using Practice 4, find portfolio standard deviation.

Practice 6: Perfect Positive Correlation

If two assets have

\[ \rho=1, \]

does combining them provide the strongest or weakest diversification benefit among correlation values from $-1$ to $1$?

Practice 7: Budget Constraint

A $120,000 budget is split between projects A and B.

If A receives $45,000, how much must B receive if the full budget is allocated?

Practice 8: Capacity Constraint

A project may receive at most $30\%$ of a portfolio.

Is a weight of

\[ 0.35 \]

feasible?

Practice 9: Expected Failure Cost Allocation

Allocation A has expected benefit $50$ and expected failure loss $8$.

Allocation B has expected benefit $47$ and expected failure loss $2$.

If net expected value is benefit minus expected loss, which allocation is preferred?

Practice 10: Diversification Concept

Portfolio X contains five investments that all respond almost identically to the same economic shock.

Portfolio Y contains three investments whose outcomes are much less correlated.

Which portfolio may provide greater diversification despite containing fewer positions?

Practice Answers

Practice 1

\[ w_A=\frac{20{,}000}{50{,}000}=0.40, \] \[ w_B=\frac{15{,}000}{50{,}000}=0.30, \] \[ w_C=\frac{15{,}000}{50{,}000}=0.30. \]

Therefore,

\[ \boxed{(0.40,0.30,0.30)}. \]

Practice 2

\[ R_P = 0.70(12\%)+0.30(4\%) \] \[ =8.4\%+1.2\% \] \[ \boxed{9.6\%}. \]

Practice 3

\[ E[R_P] = 0.4(9\%)+0.6(5\%) \] \[ =3.6\%+3.0\% \] \[ \boxed{6.6\%}. \]

Practice 4

\[ \sigma_P^2 = (0.5)^2(0.10)^2 + (0.5)^2(0.06)^2. \] \[ =0.25(0.01)+0.25(0.0036). \] \[ =0.0025+0.0009. \] \[ \boxed{0.0034}. \]

Practice 5

\[ \sigma_P=\sqrt{0.0034} \] \[ \boxed{\approx0.0583}. \]

This is approximately

\[ \boxed{5.83\%}. \]

Practice 6

Perfect positive correlation provides the

\[ \boxed{\text{weakest diversification benefit}} \]

among correlations from $-1$ to $1$.

Practice 7

\[ 120{,}000-45{,}000 \] \[ =\boxed{75{,}000}. \]

Practice 8

No. Since

\[ 0.35>0.30, \]

the weight violates the capacity constraint.

Practice 9

Allocation A:

\[ 50-8=42. \]

Allocation B:

\[ 47-2=45. \]

Therefore,

\[ \boxed{\text{Allocation B}} \]

has the larger net expected value.

Practice 10

Portfolio Y may provide greater diversification because diversification depends on dependence structure, not simply the number of positions.

What to Remember

Portfolio and allocation mathematics studies how multiple choices combine into one overall decision.

Allocation weights usually sum to one.
Portfolio outcomes are weighted combinations of component outcomes.
Expected portfolio return is a weighted average of expected returns.
Portfolio variance depends on individual variances and pairwise covariances.
Correlation determines how much diversification can reduce variability.
Diversification is about distinct risk exposures, not merely the number of holdings.
Real allocation problems include budget, capacity, and policy constraints.
Efficient portfolios improve one objective only by worsening another.
Robust allocation matters when model inputs are uncertain.
Optimization should include implementation costs and the decision maker's actual objectives.

The most important principle is: allocation quality depends not only on the expected performance of each choice, but also on how the choices interact when combined.

Where We Go Next

Portfolio analysis shows how uncertainty, dependence, constraints, and preferences shape resource allocation. The final lesson in this section addresses how the resulting risks and probabilities should be communicated to other people.

The next lesson, Ethical Communication of Risk, develops principles for presenting probabilities, uncertainty, forecasts, losses, and model limitations accurately and responsibly.