portalloc

Optimal capital allocation to efficient frontier portfolios

Syntax

[RiskyRisk,RiskyReturn,RiskyWts,RiskyFraction,OverallRisk,OverallReturn] = portalloc(PortRisk,PortReturn,PortWts,RisklessRate)

[RiskyRisk,RiskyReturn,RiskyWts,RiskyFraction,OverallRisk,OverallReturn] = portalloc(___,BorrowRate,RiskAversion)

portalloc(PortRisk,PortReturn,PortWts,RisklessRate,BorrowRate,RiskAversion)

Description

[RiskyRisk,RiskyReturn,RiskyWts,RiskyFraction,OverallRisk,OverallReturn] = portalloc(PortRisk,PortReturn,PortWts,RisklessRate) calculates the optimal risky portfolio and the optimal allocation of funds between that risky portfolio of NASSETS and the risk-free asset.

Note

An alternative for portfolio optimization is to use the Portfolio object for mean-variance portfolio optimization. This object supports 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. For information on the workflow when using Portfolio objects, see Portfolio Object Workflow.

[RiskyRisk,RiskyReturn,RiskyWts,RiskyFraction,OverallRisk,OverallReturn] = portalloc(___,BorrowRate,RiskAversion) specifies options using one or more optional arguments in addition to the input arguments in the previous syntax.

portalloc(PortRisk,PortReturn,PortWts,RisklessRate,BorrowRate,RiskAversion) when invoked without any output arguments, a graph of the optimal capital allocation decision is displayed.

Examples

Compute the Optimal Risky Portfolio

This example shows how to compute the optimal risky portfolio by generating the efficient frontier from the asset data and then finding the optimal risky portfolio and allocate capital. The risk-free investment return is 8%, and the borrowing rate is 12%.

ExpReturn = [0.1 0.2 0.15]; 

ExpCovariance = [0.005   -0.010    0.004 
                -0.010    0.040   -0.002 
                 0.004   -0.002    0.023];
[PortRisk, PortReturn, PortWts] = portopt(ExpReturn,... 
ExpCovariance);

RisklessRate  = 0.08;
BorrowRate    = 0.12;
RiskAversion  = 3;

[RiskyRisk, RiskyReturn, RiskyWts, RiskyFraction, ... 
OverallRisk, OverallReturn] = portalloc(PortRisk, PortReturn,... 
PortWts, RisklessRate, BorrowRate, RiskAversion)

RiskyRisk = 
0.1283

RiskyReturn = 
0.1788

RiskyWts = 1×3

    0.0265    0.6023    0.3712

RiskyFraction = 
1.1898

OverallRisk = 
0.1527

OverallReturn = 
0.1899

Input Arguments

PortRisk — Standard deviation of each risky asset efficient frontier portfolio

vector

Standard deviation of each risky asset efficient frontier portfolio, specified as an NPORTS-by-1 vector.

Data Types: double

PortReturn — Expected return of each risky asset efficient frontier portfolio

vector

Expected return of each risky asset efficient frontier portfolio, specified an NPORTS-by-1 vector.

Data Types: double

PortWts — Weights allocated to each asset

numeric

Weights allocated to each asset, specified as an NPORTS by number of assets (NASSETS) matrix of weights allocated to each asset. Each row represents an efficient frontier portfolio of risky assets. Total of all weights in a portfolio is 1.

Data Types: double

RisklessRate — Risk-free lending rate

scalar decimal

Risk-free lending rate, specified as a scalar decimal.

Data Types: double

BorrowRate — Borrowing rate

NaN (default) | scalar decimal

(Optional) Borrowing rate, specified as a scalar decimal. If borrowing is not desired, or not an option, set the BorrowRate to NaN (which is the default value).

Data Types: double

RiskAversion — Coefficient of investor's degree of risk aversion

3 (default) | scalar numeric

(Optional) Coefficient of investor's degree of risk aversion, specified as a scalar numeric. Higher numbers indicate greater risk aversion. Typical coefficients range from 2.0 through 4.0. The default value of RiskAversion is 3.

Note

Consider that a less risk-averse investor would be expected to accept much greater risk and, consequently, a more risk-averse investor would accept less risk for a given level of return. Therefore, making the RiskAversion argument higher reflects the risk-return tradeoff in the data.

Data Types: double

Output Arguments

RiskyRisk — Standard deviation of the optimal risky portfolio

numeric

Standard deviation of the optimal risky portfolio, returned as a scalar.

RiskyReturn — Expected return of the optimal risky portfolio

numeric

Expected return of the optimal risky portfolio, returned as a scalar.

RiskyWts — Weights allocated to the optimal risky portfolio

vector

Weights allocated to the optimal risky portfolio, returned a 1-by-NASSETS vector. The total of all weights in the portfolio is 1.

RiskyFraction — Fraction of the complete portfolio allocated to the risky portfolio

numeric

Fraction of the complete portfolio (that is, the overall portfolio including risky and risk-free assets) allocated to the risky portfolio, returned as a scalar.

OverallRisk — Standard deviation of the optimal overall portfolio

numeric

Standard deviation of the optimal overall portfolio, returned as a scalar.

OverallReturn — Expected rate of return of the optimal overall portfolio

numeric

Expected rate of return of the optimal overall portfolio, returned as a scalar.

More About

Efficient Frontier

The efficient frontier is a key concept in modern portfolio theory (MPT) that represents a set of optimal portfolios offering the highest expected return for a given level of risk, or the lowest risk for a given level of expected return.

The efficient frontier illustrates the tradeoff between risk (often measured as the standard deviation of portfolio returns) and expected return. Investors aim to construct portfolios that lie on this frontier to maximize their returns for a given level of risk. Portfolios that lie on the efficient frontier are considered optimal because they provide the best possible expected return for their level of risk. Portfolios that lie below the frontier are suboptimal, as they do not provide sufficient return for the amount of risk taken. The efficient frontier emphasizes the importance of diversification. By combining different assets with varying returns and risks, investors can reduce the overall risk of the portfolio without sacrificing expected returns.

References

[1] Bodie, Z., Kane, A., and A. Marcus. Investments. McGraw-Hill Education, 2013.

Version History

Introduced before R2006a