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## Portfolio Optimization Theory

### 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.

• 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 optimization 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.

### Portfolio Problem Specification

To specify a portfolio optimization problem, you need the following:

• Proxy for portfolio return (μ)

• Proxy for portfolio risk (Σ)

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

Financial Toolbox™ has three objects to solve specific types of portfolio optimization problems:

• The Portfolio object (Portfolio) supports mean-variance portfolio optimization (see Markowitz [46], [47] at (Portfolio Optimization). This 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.

• The PortfolioCVaR object (PortfolioCVaR) implements what is known as conditional value-at-risk portfolio optimization (see Rockafellar and Uryasev [48], [49] at Portfolio Optimization), which is generally referred to as CVaR portfolio optimization. CVaR portfolio optimization works with the same return proxies and portfolio sets as mean-variance portfolio optimization but uses conditional value-at-risk of portfolio returns as the risk proxy.

• The PortfolioMAD object (PortfolioMAD) implements what is known as mean-absolute deviation portfolio optimizatio (see Konno and Yamazaki [50] Portfolio Optimization), which is generally referred to as MAD portfolio optimization. MAD portfolio optimization works with the same return proxies and portfolio sets as mean-variance portfolio optimization but uses mean-absolute deviation portfolio returns as the risk proxy.

### 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 forms 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,...,yS 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. Though 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 for x0

• BuyCost for b

• SellCost for s

### Risk Proxy

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

#### Variance

The variance of portfolio returns for a portfolio is

where C is the 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).

#### Conditional Value-at-Risk

The conditional value-at-risk for a portfolio , which is also known as expected shortfall, is defined as

where:

α is the probability level such that 0 < α < 1.

f(x,y) is the loss function for a portfolio x and asset return y.

p(y) is the probability density function for asset return y.

VaRα is the value-at-risk of portfolio x at probability level α.

The value-at-risk is defined as

An alternative formulation for CVaR has the form:

The choice for the probability level α is typically 0.9 or 0.95. Choosing α implies that the value-at-risk VaRα(x) for portfolio x is the portfolio return such that the probability of portfolio returns falling below this level is (1α). Given VaRα(x) for a portfolio x, the conditional value-at-risk of the portfolio is the expected loss of portfolio returns above the value-at-risk return.

 Note:   Value-at-risk is a positive value for losses so that the probability level α indicates the probability that portfolio returns are below the negative of the value-at-risk.

The risk proxy for CVaR portfolio optimization is CVaRα(x) for a given portfolio and . Note that the value-at-risk, or VaR, for a given probability level is estimated whenever CVaR is estimated.

In addition, keep in mind that VaR and CVaR are sample estimators for VaR and CVaR based on the given scenarios. Better scenario samples yield more reliable estimates of VaR and CVaR.

For more information, see Rockafellar and Uryasev [48], [49] at Portfolio Optimization.

#### Mean Absolute-Deviation

The mean-absolute-deviation (MAD) for a portfolio is defined as

where:

ys are asset returns with scenarios s = 1,...S (S collection of n vectors).

f(x,y) is the loss function for a portfolio x and asset return y.

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

such that

For more information, see Konno and Yamazaki [50] at Portfolio Optimization.

### Portfolio Set for 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 setting up your portfolio set, 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 these 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).

AI is the linear inequality constraint matrix (nI-by-n matrix).

bI is the linear inequality constraint vector (nI vector).

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

Portfolio object properties to specify linear inequality constraints are:

• AInequality for AI

• bInequality for bI

• NumAssets for n

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).

AE is the linear equality constraint matrix (nE-by-n matrix).

bE is the linear equality constraint vector (nE vector).

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

Portfolio object properties to specify linear equality constraints are:

• AEquality for AE

• bEquality for bE

• NumAssets for n

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).

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

uB 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 lB

• UpperBound for uB

• NumAssets for n

The default is to ignore these constraints.

Note that the default portfolio optimization problem (see Default Portfolio Problem) has lB = 0 with uB 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).

1 is the vector of ones (n vector).

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

uS 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 lS

• UpperBudget for uS

• NumAssets for n

The default is to ignore this constraint.

The default portfolio optimization problem (see Default Portfolio Problem) has lS = uS = 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 specifies how far portfolios can go into cash. For example, if lS = 0 and uS = 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 enforce "membership" among groups of assets. The constraints take the form

where:

x is the portfolio (n vector).

lG is the lower-bound group constraint (nG vector).

uG is the upper-bound group constraint (nG vector).

G is the matrix of group membership indexes (nG-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 lG

• UpperGroup for uG

• NumAssets for n

The default is to ignore these constraints.

#### Group Ratio Constraints

Group ratio constraints are specialized linear constraints that enforce relationships among groups of assets. The constraints take the form

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

x is the portfolio (n vector).

lR is the vector of lower-bound group ratio constraints (nR vector).

uR is the vector matrix of upper-bound group ratio constraints (nR vector).

GA is the matrix of base group membership indexes (nR-by-n matrix).

GB is the matrix of comparison group membership indexes (nR-by-n matrix).

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

Each row of GA and GB identifies 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 GA

• GroupB for GB

• LowerRatio for lR

• UpperRatio for uR

• NumAssets for n

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 computes portfolio turnover as the average of purchases and sales. Average turnover constraints take the form

where:

x is the portfolio (n vector).

1 is the vector of ones (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

• NumAssets for n

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 forms

where:

x is the portfolio (n vector)

1 is the vector of ones (n vector).

x0 is the Initial portfolio (n vector).

τB is the upper bound for turnover constraint on purchases (scalar).

τS is the upper bound for turnover constraint on sales (scalar).

To specify one-way turnover constraints, use the following properties in the Portfolio or PortfolioCVaR object:

• BuyTurnover for τB

• SellTurnover for τS

• InitPort for x0

The default is to ignore this constraint.

 Note:   The average turnover constraint (see Average Turnover Constraints) with τ is not a combination of the one-way turnover constraints with τ = τB = τS.

### 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.

• For mean-variance portfolio optimization, it is sufficient to set up the default problem. After setting up the problem, data in the form of a mean and covariance of asset returns are then used to solve portfolio optimization problems.

• For conditional value-at-risk portfolio optimization, the default problem requires the additional specification of a probability level that must be set explicitly. Generally, "typical" values for this level are 0.90 or 0.95. After setting up the problem, data in the form of scenarios of asset returns are then used to solve portfolio optimization problems.

• For MAD portfolio optimization, it is sufficient to set up the default problem. After setting up the problem, data in the form of scenarios of asset returns are then used to solve portfolio optimization problems.

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