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 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 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 implements what is known as mean-absolute deviation portfolio optimization (see Konno and Yamazaki [50] at Portfolio Optimization), which is 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 $μ : X \to R$ on a portfolio set $X \subset R^{n}$ that characterizes the rewards associated with portfolio choices. Usually, 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 $x \in X$ 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

$m = \frac{1}{S} \sum_{s = 1}^{S} y_{s},$

and (sample) covariance of asset returns

$C = \frac{1}{S - 1} \sum_{s = 1}^{S} (y_{s} - m) {(y_{s} - m)}^{T} .$

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 $x \in X$ is

$μ (x) = r_{0} + {(m - r_{0} 1)}^{T} x,$

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 $x \in X$ is

$μ (x) = r_{0} + {(m - r_{0} 1)}^{T} x - b^{T} \max {0, x - x_{0}} - s^{T} \max {0, x_{0} - x},$

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 $σ : X \to R$ on a portfolio set $X \subset R^{n}$ that characterizes the risks associated with portfolio choices.

Variance

The variance of portfolio returns for a portfolio $x \in X$ is

$V a r i a n c e (x) = x^{T} C x$

where C is the covariance of asset returns (n-by-n positive-semidefinite matrix). Covariance is a measure of the degree to which returns on two assets move in tandem. A positive covariance means that asset returns move together; a negative covariance means they vary inversely.

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 $x \in X$ , which is also known as expected shortfall, is defined as

$C V a R_{α} (x) = \frac{1}{1 - α} \int_{f (x, y) \geq V a R_{α} (x)} f (x, y) p (y) d y,$

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.

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

The value-at-risk is defined as

$V a R_{α} (x) = \min {γ : \Pr [f (x, Y) \leq γ] \geq α} .$

An alternative formulation for CVaR has the form:

$C V a R_{α} (x) = V a R_{α} (x) + \frac{1}{1 - α} \int_{R^{n}} \max {0, (f (x, y) - V a R_{α} (x))} p (y) d y$

The choice for the probability level α is typically 0.9 or 0.95. Choosing α implies that the value-at-risk VaR_α(x) for portfolio x is the portfolio return such that the probability of portfolio returns falling below this level is (1 –α). Given VaR_α(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.

To describe the probability distribution of returns, the PortfolioCVaR object takes a finite sample of return scenarios y_s, with s = 1,...,S. Each y_s is an n vector that contains the returns for each of the n assets under the scenario s. This sample of S scenarios is stored as a scenario matrix of size S-by-n. Then, the risk proxy for CVaR portfolio optimization, for a given portfolio $x \in X$ and $α \in (0, 1)$ , is computed as

$C V a R_{α} (x) = V a R_{α} (x) + \frac{1}{(1 - α) S} \sum_{s = 1}^{S} \max {0, - y_{s}^{T} x - V a R_{α} (x)}$

The value-at-risk, VaR_α(x), is estimated whenever the CVaR is estimated. The loss function is $f (x, y_{s}) = - y_{s}^{T} x$ , which is the portfolio loss under scenario s.

Under this definition, 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], and Cornuejols and Tütüncü, [51], at Portfolio Optimization.

Mean Absolute-Deviation

The mean-absolute deviation (MAD) for a portfolio $x \in X$ is defined as

$M A D (x) = \frac{1}{S} \sum_{s = 1}^{S} | {(y_{s} - m)}^{T} x |$

where:

y_s 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

$m = \frac{1}{S} \sum_{s = 1}^{S} y_{s}$

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