Portfolio Optimization Theory

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Portfolio Optimization Problems Portfolio Problem Specification Return Proxy Risk Proxy Portfolio Set for Mean-Variance Portfolio Optimization Default Portfolio Problem

Portfolio Optimization Problems

Portfolio optimization problems involve identifying portfolios that satisfy three criteria: minimize a proxy for risk, match or exceed a proxy for return, and satisfy basic feasibility requirements.

Portfolios are points from a feasible set of assets that constitute an asset universe. A portfolio specifies either holdings or weights in each individual asset in the asset universe. The convention is to specify portfolios in terms of weights, although the portfolio object tools work with holdings as well.

The set of feasible portfolios is necessarily a nonempty, closed, and bounded set. The proxy for risk is a function that characterizes either the variability or losses associated with portfolio choices. The proxy for return is a function that characterizes either the gross or net benefits associated with portfolio choices. The terms "risk" and "risk proxy" and "return" and "return proxy" are interchangeable. The fundamental insight of Markowitz (see Portfolio Optimization ) is that the goal of the portfolio choice problem is to seek minimum risk for a given level of return and to seek maximum return for a given level of risk. Portfolios satisfying these criteria are efficient portfolios and the graph of the risks and returns of these portfolios forms a curve called the efficient frontier.

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Portfolio Problem Specification

To specify a portfolio optimization problem, you need:

• Proxy for portfolio return (μ)

• Proxy for portfolio risk (Σ)

• Set of feasible portfolios (X), called a portfolio set.

Financial Toolbox software supports a portfolio object for mean-variance portfolio optimization. The portfolio object has either gross or net portfolio returns as the return proxy, the variance of portfolio returns as the risk proxy, and a portfolio set that is any combination of the specified constraints to form a portfolio set.

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Return Proxy

The proxy for portfolio return is a function on a portfolio set that characterizes the rewards associated with portfolio choices. In most cases, the proxy for portfolio return has two general forms, gross and net portfolio returns. Both portfolio return models separate the risk-free rate r₀ so that the portfolio contains only risky assets.

Regardless of the underlying distribution of asset returns, a collection of S asset returns y₁, ... ,y_S has a mean of asset returns

and (sample) covariance of asset returns

These moments (or alternative estimators that characterize these moments) are used directly in mean-variance portfolio optimization to form proxies for portfolio risk and return.

Gross Portfolio Returns

The gross portfolio return for a portfolio is

where:

r₀ is the risk-free rate (scalar).

m is the mean of asset returns (n vector).

If the portfolio weights sum to 1, the risk-free rate is irrelevant. The properties in the portfolio object to specify gross portfolio returns are:

• RiskFreeRate for r₀

• AssetMean for m

Net Portfolio Returns

The net portfolio return for a portfolio is

where:

r₀ is the risk-free rate (scalar).

m is the mean of asset returns (n vector).

b is the proportional cost to purchase assets (n vector).

s is the proportional cost to sell assets (n vector).

You can incorporate fixed transaction costs in this model also, although in this case, it is necessary to incorporate prices into such costs. The properties in the portfolio object to specify net portfolio returns are:

• RiskFreeRate for r₀

• AssetMean for m

• InitPort forx₀

• BuyCost for b

• SellCost for s.

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Risk Proxy

The proxy for portfolio risk is a function on a portfolio set that characterizes the risks associated with portfolio choices.

Variance of Portfolio Returns

The variance of portfolio returns for a portfolio is:

where C is covariance of asset returns (n-by-n positive-semidefinite matrix).

The property in the portfolio object to specify the variance of portfolio returns is AssetCovar for C.

Although the risk proxy in mean-variance portfolio optimization is the variance of portfolio returns, the square root, which is the standard deviation of portfolio returns, is often reported and displayed. Moreover, this quantity is often called the "risk" of the portfolio. For details, see Markowitz (Portfolio Optimization).

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Portfolio Set for Mean-Variance Portfolio Optimization

The final element for a complete specification of a portfolio optimization problem is the set of feasible portfolios, which is called a portfolio set. A portfolio set is specified by construction as the intersection of sets formed by a collection of constraints on portfolio weights. A portfolio set necessarily and sufficiently must be a nonempty, closed, and bounded set.

When you set up your portfolio set, you need to ensure that the portfolio set satisfies these conditions. The most basic or "default" portfolio set requires portfolio weights to be nonnegative (using the lower-bound constraint) and to sum to 1 ( using the budget constraint). The most general portfolio set handled by the portfolio optimization tools can have any of the following constraints:

• Linear inequality constraints

• Linear equality constraints

• Bound constraints

• Budget constraints

• Group constraints

• Group ratio constraints

• Average turnover constraints

• One-way turnover constraints

Linear Inequality Constraints

Linear inequality constraints are general linear constraints that model relationships among portfolio weights that satisfy a system of inequalities. Linear inequality constraints take the form

where:

x is the portfolio (n vector).

A_I is the linear inequality constraint matrix (n_I-by-n matrix).

b_I is the linear inequality constraint vector (n_I vector).

n is the number of assets in the universe and n_I is the number of constraints.

Portfolio object properties to specify linear inequality constraints are:

• AInequality for A_I

• bInequality for b_I

The default is to ignore these constraints.

Linear Equality Constraints

Linear equality constraints are general linear constraints that model relationships among portfolio weights that satisfy a system of equalities. Linear equality constraints take the form

where:

x is the portfolio (n vector).

A_E is the linear equality constraint matrix (n_E-by-n matrix).

b_I is the linear equality constraint vector (n_E vector).

n is the number of assets in the universe and n_E is the number of constraints.

Portfolio object properties to specify linear equality constraints are:

• AEquality for A_E

• bEquality for b_E

The default is to ignore these constraints.

Bound Constraints

Bound constraints are specialized linear constraints that confine portfolio weights to fall either above or below specific bounds. Since every portfolio set must be bounded, it is often a good practice, albeit not necessary, to set explicit bounds for the portfolio problem. To obtain explicit bounds for a given portfolio set, use the method estimateBounds. Bound constraints take the form

where:

x is the portfolio (n vector).

l_B is the lower-bound constraint (n vector).

u_B is the upper-bound constraint (n vector).

n is the number of assets in the universe.

Portfolio object properties to specify bound constraints are:

• LowerBound for l_B

• UpperBound for u_B

The default is to ignore these constraints.

Note, the default portfolio optimization problem (see Default Portfolio Problem) has l_B = 0 with u_B set implicitly through a budget constraint.

Budget Constraints

Budget constraints are specialized linear constraints that confine the sum of portfolio weights to fall either above or below specific bounds. The constraints take the form

where:

x is the portfolio (n vector).

l_S is the lower-bound budget constraint (scalar).

u_S is the upper-bound budget constraint (scalar).

n is the number of assets in the universe.

Portfolio object properties to specify budget constraints are:

• LowerBudget for l_S

• UpperBudget for u_S

The default is to ignore this constraint.

The default portfolio optimization problem (see Default Portfolio Problem) has l_S = u_S= 1, which means that the portfolio weights sum to 1. If the portfolio optimization problem includes possible movements in and out of cash, the budget constraint is used to specify how far portfolios can go into cash. For example, if l_S = 0 and u_S = 1, then the portfolio can have 0% to 100% invested in cash. If cash is to be a portfolio choice, set RiskFreeRate (r₀) to a suitable value (see Return Proxy and Working with a Riskless Asset).

Group Constraints

Group constraints are specialized linear constraints that provide a useful way to enforce "membership" among groups of assets. The constraints take the form

where:

x is the portfolio (n vector).

l_G is the lower-bound group constraint (n_G vector).

u_G is the upper-bound group constraint (n_G vector).

G is the matrix of group membership indexes (n_G-by-n matrix).

Each row of G identifies which assets belong to a group associated with that row. Each row contains either 0s or 1s with 1 indicating that an asset is part of the group or 0 indicating that the asset is not part of the group.

Portfolio object properties to specify group constraints are:

• GroupMatrix for G

• LowerGroup for l_G

• UpperGroup for u_G

The default is to ignore these constraints.

Group Ratio Constraints

Group ratio constraints are specialized linear constraints that provide a useful way to enforce relationships among groups of assets. The constraints take the form

for i = 1, ... , n_R where:

x is the portfolio (n vector).

l_R is the vector of lower-bound group ratio constraints (n_R vector).

u_R is the vector matrix of upper-bound group ratio constraints (n_R vector).

G_A is the matrix of base group membership indexes (n_R-by-n matrix).

G_B is the matrix of comparison group membership indexes (n_R-by-n matrix).

n is the number of assets in the universe and n_R is the number of constraints. Each row of G_A and G_B identify which assets belong to a base and comparison group associated with that row. Each row contains either 0s or 1s with 1 indicating that an asset is part of the group or 0 indicating that the asset is not part of the group.

Portfolio object properties to specify group ratio constraints are:

• GroupA for G_A

• GroupB for G_B

• LowerRatio for l_R

• UpperRatio for u_R

The default is to ignore these constraints.

Average Turnover Constraints

Turnover constraint is a linear absolute value constraint that ensures estimated optimal portfolios differ from an initial portfolio by no more than a specified amount. Although portfolio turnover is defined in many ways, the turnover constraints implemented in Financial Toolbox software computes portfolio turnover as the average of purchases and sales. Average turnover constraints takes the form

where:

x is the portfolio (n vector).

x₀ is the initial portfolio (n vector).

τ is the upper-bound for turnover (scalar).

n is the number of assets in the universe.

Portfolio object properties to specify the average turnover constraint are:

• Turnover for τ

• InitPort for x₀

The default is to ignore this constraint.

One-Way Turnover Constraints

One-way turnover constraints ensure that estimated optimal portfolios differ from an initial portfolio by no more than specified amounts according to whether the differences are purchases or sales. The constraints take the form

with

• x — The portfolio (n vector)

• x0 — Initial portfolio (n vector)

• τ_B — Upper-bound for turnover constraint on purchases (scalar)

• τ_S — Upper-bound for turnover constraint on sales (scalar)

where n is the number of assets in the universe.

To specify one-way turnover constraints, use the following properties in the portfolio object: BuyTurnover for τ_B, SellTurnover for τ_S, and InitPort for x0.

Note   The average turnover constraint with τ is not a combination of the one-way turnover constraints with τ = τB = τS.

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Default Portfolio Problem

The default portfolio optimization problem has a risk and return proxy associated with a given problem, and a portfolio set that specifies portfolio weights to be nonnegative and to sum to 1. The lower bound combined with the budget constraint is sufficient to ensure that the portfolio set is nonempty, closed, and bounded. The default portfolio optimization problem characterizes a long-only investor who is fully invested in a collection of assets.

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Portfolio Optimization Tools

Portfolio Object

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