Interval Cross-Efficiency in Portfolio Optimization

• Interval Cross-Efficiency (ICE) is a methodology that extends DEA by evaluating portfolios over an interval of risk-free rates to account for parameter uncertainty.
• It aggregates cross-efficiency scores across all admissible risk-free rate values, reducing the impact of estimation errors in portfolio selection.
• The ICE framework offers closed-form solutions under short-selling and iterative techniques otherwise, enhancing robustness in Sharpe ratio optimization.

Interval Cross-Efficiency (ICE) is a methodology originally motivated by extensions of Data Envelopment Analysis (DEA) to settings with parameter uncertainty, most notably in uncertain or interval-valued risk-free rates for financial portfolios. ICE systematically evaluates and ranks decision-making units (DMUs) or candidate solutions by aggregating cross-efficiency scores across all admissible parameter values in a specified interval, rather than relying on an arbitrary fixed value. Recent advances have demonstrated the robustness and closed-form computability of ICE in portfolio optimization, where it yields portfolios less sensitive to specification errors or uncertainty in risk-free rates (Monge et al., 2016).

1. Conceptual Foundations of Cross-Efficiency and ICE

Classical cross-efficiency, developed in the context of DEA, evaluates each DMU’s efficiency not only according to its self-selected optimal weights but also by the optimal weights of all other DMUs. The interval cross-efficiency (ICE) approach generalizes this concept to account for uncertainty in evaluation parameters. In the context of portfolio selection, each DMU corresponds to a tangent portfolio parameterized by a candidate risk-free rate, $r_f \in [r_{\min}, r_{\max}]$.

For each DMU (portfolio corresponding to a specific $r_f$), a self-efficiency score is 1, while cross-efficiencies measure comparative performance under alternative parameterizations. ICE aggregates these cross-efficiencies, integrating over all possible $r_f$ values, thereby yielding an average cross-efficiency for each candidate solution.

2. ICE in Portfolio Optimization with Risk-Free Rate Uncertainty

The principal application of ICE in modern research is in robust Sharpe ratio portfolio construction when the risk-free rate is only known to lie within an interval $[r_{\min}, r_{\max}]$, as formalized in "Sharpe portfolio using a cross-efficiency evaluation" (Monge et al., 2016). The classical Maximum-Sharpe-Ratio (MSR) portfolio, for fixed $r_f$, solves: $$\max_{w^T1=1} \frac{w^T(\mu - r_f1)}{\sqrt{w^T \Sigma w}}$$ where $\mu$ is the vector of expected returns, $\Sigma$ is the covariance matrix, and $1$ is the unit vector.

When $r_f$ is interval-valued, one could pose a minimax problem, but ICE considers the full continuum of tangent portfolios $r_f$0 and evaluates each via cross-efficiency relative to all others across the interval.

3. Mathematical Structure of Interval Cross-Efficiency

Given the family of tangent portfolios and their risk-return pairs,

$r_f$1

the cross-efficiency of DMU $r_f$2 evaluated under the optimal weights of DMU $r_f$3 is: $r_f$4 where $r_f$5 denotes the Sharpe ratio of portfolio $r_f$6 using risk-free rate $r_f$7. Averaging over the interval, the cross-efficiency score is: $r_f$8 The ICE portfolio is the candidate $r_f$9 that maximizes $r_f$0.

4. Closed-Form ICE Portfolio Solution and Hyperbolic Geometry

When short selling is allowed ($r_f$1 invertible), each MSR portfolio admits the closed-form expression: $r_f$2 Writing $r_f$3, $r_f$4, $r_f$5, the global minimum variance (GMV) portfolio has $r_f$6, $r_f$7. Each tangent portfolio satisfies: $r_f$8 The ICE-optimal $r_f$9 is determined by maximizing $[r_{\min}, r_{\max}]$0 and is given by: $[r_{\min}, r_{\max}]$1 where $[r_{\min}, r_{\max}]$2, $[r_{\min}, r_{\max}]$3. The final weight vector is

$[r_{\min}, r_{\max}]$4

Closed-form evaluations for $[r_{\min}, r_{\max}]$5 and $[r_{\min}, r_{\max}]$6 involving logarithmic functions of the hyperbola’s asymptotes are presented in (Monge et al., 2016).

5. Algorithmic Implementation

In the absence of short-selling, ICE computation involves iterative quadratic programming. In the short-selling-allowed regime, the closed-form solution described above is employed.

The computational workflow consists of:

• Generating a grid of $[r_{\min}, r_{\max}]$7 risk-free rates $[r_{\min}, r_{\max}]$8 over $[r_{\min}, r_{\max}]$9,
• Solving for the tangency portfolio $r_f$0 at each $r_f$1,
• Computing each portfolio’s average cross-efficiency $r_f$2 across the interval,
• Selecting the portfolio achieving maximal $r_f$3.

The explicit pseudocode is as follows (Monge et al., 2016): $r_f$8

With short-sales allowed, use the closed-form $r_f$4.

6. Robustness and Interpretation

Portfolios constructed using ICE are robust to misspecification of $r_f$5. Rather than choosing a portfolio optimized for an arbitrary or potentially misestimated value of $r_f$6, the ICE approach performs a peer-evaluation across all admissible portfolios, relative to all values in $r_f$7. This model-averaging mitigates estimation risk by favoring portfolios that maintain strong relative Sharpe performance uniformly across the interval, rather than specializing for a single risk-free rate. This methodology generalizes beyond financial applications wherever DEA-based efficiency evaluations under parameter uncertainty are required (Monge et al., 2016).